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Proceedings of the
IEEE Visualization '94 Conference

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Contents
Preface......................................................................................................................................x
Reviewers................................................................................................................................xi
Conference Committee...................................................................................................... xiii
Program Committee ...........................................................................................................xiv
Honorary Chair Address
Interactive Visualization via 3D User Interfaces......................................................................2
A. van Dam
Keynote Panel
Introduction: Visualization in the Information Highway .........................................................4
N. Gershon
Information Workspaces for Large Scale Cognition..................................................................5
S.K. Card
A Visualization System on Every Desk — Keeping it Simple...................................................6
S.F. Roth
The Future of Graphic User Interfaces.....................................................................................7
B. Shneiderman
Capstone Address
The Crucial Difference between Human and Machine Vision: Focal Attention.....................10
B. Julesz
Volume Visualization Systems
PAPERS
Integrated Control of Distributed Volume Visualization Through
the World-Wide-Web................................................................................................................13
C.S. Ang, D.C. Martin, and M.D. Doyle
A Distributed, Parallel, Interactive Volume Rendering Package ...........................................21
J.S. Rowlan, G.E. Lent, N. Gokhale, and S. Bradshaw
VolVis: A Diversified Volume Visualization System ...............................................................31
R. Avila, T. He, L. Hong, A. Kaufman, H. Pfister,
C. Silva, L. Sobierajski, and S. Wang
Applications
Implicit Modeling of Swept Surfaces and Volumes.................................................................40
W.J. Schroeder, W.E. Lorensen, and S. Linthicum
Visualizing Polycrystalline Orientation Microstructures
with Spherical Color Maps ......................................................................................................46
B. Yamrom, J.A. Sutliff, and A.P. Woodfield
Introducing Alpha Shapes for the Analysis of Path Integral Monte Carlo Results................52
P.J. Moran and M. Wagner
v

Surfaces
Piecewise-Linear Surface Approximation from Noisy Scattered Samples..............................61
M. Margaliot and C. Gotsman
Triangulation and Display of Rational Parametric Surfaces..................................................69
C.L. Bajaj and A. Royappa
Isosurface Generation by Using Extrema Graphs ..................................................................77
T. Itoh and K. Koyamada
Visualization Techniques
Wavelet-Based Volume Morphing ...........................................................................................85
T. He, S. Wang, and A. Kaufman
Progressive Transmission of Scientific Data Using Biorthogonal
Wavelet Transform ..................................................................................................................93
H. Tao and R.J. Moorhead
An Evaluation of Reconstruction Filters for Volume Rendering ..........................................100
S.R. Marschner and R.J. Lobb
Visualizing Flow with Quaternion Frames ...........................................................................108
A.J. Hanson and H. Ma
Flow Features and Topology
Feature Detection from Vector Quantities in a Numerically Simulated
Hypersonic Flow Field in Combination with Experimental Flow Visualization ..................117
H.-G. Pagendarm and B. Walter
3D Visualization of Unsteady 2D Airplane Wake Vortices ...................................................124
K.-L. Ma and Z.C. Zheng
Vortex Tubes in Turbulent Flows: Identification, Representation, Reconstruction.............132
D.C. Banks and B.A. Singer
The Topology of Second-Order Tensor Fields........................................................................140
T. Delmarcelle and L. Hesselink
Visualizing Geometry and Algorithms
GASP - A System for Visualizing Geometric Algorithms......................................................149
A. Tal and D. Dobkin
Virtual Reality Performance for Virtual Geometry...............................................................156
R.A. Cross and A.J. Hanson
A Library for Visualizing Combinatorial Structures.............................................................164
M.A. Najork and M.H. Brown
Strata-Various: Multi-Layer Visualization of Dynamics in Software System Behavior ......172
D. Kimelman, B. Rosenberg, and T. Roth
Volume Visualization Techniques
Differential Volume Rendering: A Fast Volume Visualization Technique
for Flow Animation ................................................................................................................180
H.-W. Shen and C. R. Johnson
Fast Surface Rendering from Raster Data by Voxel Traversal
Using Chessboard Distance...................................................................................................188
vi

M. Šrámek
Parallel Performance Measures for Volume Ray Casting.....................................................196
C.T. Silva and A.E. Kaufman
User Interfaces and Techniques
Spiders: A New User Interface for Rotation and Visualization
of N-Dimensional Point Sets..................................................................................................205
K.L. Duffin and W.A. Barrett
Restorer: A Visualization Technique for Handling Missing Data.........................................212
R. Twiddy, J. Cavallo, and S.M. Shiri
User Modeling for Adaptive Visualization Systems..............................................................217
G.O. Domik and B. Gutkauf
Flow Visualization Techniques
Streamball Techniques for Flow Vizualization .....................................................................225
M. Brill, H. Hagen, H.-C. Rodrian, W. Djatschin, and S.V. Klimenko
Volume Rendering Methods for Computational Fluid Dynamics Visualization...................232
D.S. Ebert, R. Yagel, J. Scott, and Y. Kurzion
Visualizing Flow over Curvilinear Grid Surfaces Using Line Integral Convolution ...........240
L.K. Forssell
Visualizing 3D Velocity Fields Near Contour Surfaces.........................................................248
N. Max, R. Crawfis, and C. Grant
Flow Visualization Systems
UFAT — A Particle Tracer for Time-Dependent Flow Fields...............................................257
D.A. Lane
The Design and Implementation of the Cortex Visualization System..................................265
D. Banerjee, C. Morley, and W. Smith
An Annotation System for 3D Fluid Flow Visualization.......................................................273
M.M. Loughlin and J.F. Hughes
Surface Extraction
Discretized Marching Cubes..................................................................................................281
C. Montani, R. Scateni, and R. Scopigno
Approximation of Isosurface in the Marching Cube: Ambiguity Problem ............................288
S.V. Matveyev
Nonpolygonal Isosurface Rendering for Large Volume Datasets .........................................293
J.W. Durkin and J.F. Hughes
Visualization Systems
Mix&Match: A Construction Kit for Visualization................................................................302
A. Pang and N. Alper
A Lattice Model for Data Display..........................................................................................310
W.L. Hibbard, C.R. Dyer, and B.E. Paul
An Object Oriented Design for the Visualization of Multi-Variable Data Objects...............318
vii

J.M. Favre and J. Hahn
XmdvTool: Integrating Multiple Methods for Visualizing Multivariate Data......................326
M.O. Ward
CASE STUDIES
Magnetohydrodynamics and Mathematics
Tokamak Plasma Turbulence Visualization..........................................................................337
S.E. Parker and R. Samtaney
Visualizing Magnetohydrodynamic Turbulence and Vortex Streets ........................................*
A. Roberts
Visualization and Data Analysis in Space and Atmospheric Science ...................................341
A. Mankofsky, E.P. Szuszczewicz, P. Blanchard, C. Goodrich,
D. McNabb, R. Kulkarni, and D. Kamins
Visualization for Boundary Value Problems .........................................................................345
G. Domokos and R. Paffenroth
Environment
Severe Rainfall Events in Northwestern Peru: Visualization of
Scattered Meteorological Data:..............................................................................................350
L.A. Treinish
Visualization of Mesoscale Flow Features in Ocean Basins .................................................355
A. Johannsen and R. Moorehead
Integrating Spatial Data Display with Virtual Reconstruction............................................359
P. Peterson, B. Hayden, and F.D. Fracchia
Medical Applications
Observing a Volume Rendered Fetus within a Pregnant Patient ........................................364
A. State, D.T. Chen, C. Tector, A. Brandt, H. Chen,
R. Ohbuchi, M. Bajura, and H. Fuchs
Visualization of 3D Ultrasonic Data......................................................................................369
G. Sakas, L.-A. Schreyer, and M. Grimm
New Techniques in the Design of Healthcare Facilities .......................................................374
T. Alameldin and M. Shepley
Fire and Brimstone
Visualization of an Electric Power Transmission System.....................................................379
P.M. Mahadev and R.D. Christie
Volume Rendering of Pool Fire Data.....................................................................................382
H.E. Rushmeier, A. Hamins, and M.-Y. Choi
_____________________________________________________
* Paper not received in time for publication
Visualization of Volcanic Ash Clouds ....................................................................................386
M. Roth and R. Guritz
viii

PANELS
Challenges and Opportunities in Visualization for NASA’s EOS Mission
to Planet Earth ......................................................................................................................392
Chair: M. Botts
Panelists: J.D. Dykstra, L.S. Elson, S.J. Goodman, and M. Lee
Visualization in Medicine: VIRTUAL Reality or ACTUAL Reality?.....................................396
Co-Chairs: C. Roux and J.-L. Coatrieux
Panelists: J.-L. Dillenseger, E.K. Fishman, M. Loew,
H.-P. Meinzer, and J.D. Pearlman
Visualization and Geographic Information Systems Integration: What Are the
Needs and the Requirements, If Any ?? ................................................................................400
Chair: T.M. Rhyne
Panelists: W. Ivey, L. Knapp, P. Kochevar, and T. Mace
Visualization of Multivariate (Multidimensional) Data and Relations.................................404
Chair: A. Inselberg
Panelists: H. Hinterberger, T. Mihalisin, and G. Grinstein
Visualizing Data: Is Virtual Reality the Key?.......................................................................410
Chair: L.M. Stone
Panelists: T. Erickson, B.B. Bederson, P. Rothman, and R. Muzzy
Validation, Verification and Evaluation................................................................................414
Chair: S. Uselton
Panelists: G. Dorn, C. Farhat, M. Vannier, K. Esbensen, and A. Globus
Color Plates .......................................................................................................CP-1 to CP-46
Author Index...................................................................................................................CP-47
ix

Integrated Control of Distributed Volume Visualization
Through the World-Wide-Web
The World-Wide-Web (WWW) has created a new paradigm for
online information retrieval by providing immediate and ubiquitous
access to digital information of any type from data repositories
located throughout the world. The web’s development enables not
only effective access for the generic user, but also more efficient and
timely information exchange among scientists and researchers. We
have extended the capabilities of the web to include access to threedimensional
volume data sets with integrated control of a distributed
client-server volume visualization system. This paper provides
a brief background on the World-Wide-Web, an overview of
the extensions necessary to support these new data types and a
description of an implementation of this approach in a WWWcompliant
distributed visualization system.
1. Introduction
Advanced scanning devices, such as magnetic resonance
imaging (MRI) and computer tomography (CT), have been
widely used in the fields of medicine, quality assurance and
meteorology [Pommert, Zandt, Hibbard]. The need to visualize
resulting data has given rise to a wide variety of volume
visualization techniques and computer graphics research groups
have implemented a number of systems to provide volume
visualization (e.g. AVS, ApE, Sunvision Voxel and 3D
Viewnix)[Gerleg, Mercurio, VandeWettering]. Previously
these systems have depended upon specialized graphics hardware
for rendering and significant local secondary storage for
the data. The expense of these requirements has limited the
ability of researchers to exchange findings. To overcome the
barrier of cost, and to provide additional means for researchers
to exchange and examine three-dimensional volume data, we
have implemented a distributed volume visualization tool for
general purpose hardware, we have further integrated that
visualization service with the distributed hypermedia [Flanders,
Broering, Kiong, Robison, Story] system provided by the
World-Wide-Web [Nickerson].
Our distributed volume visualization tool, VIS, utilizes a
Cheong S. Ang, M.S.
David C. Martin, M.S.
Michael D. Doyle, Ph.D.
University of California, San Francisco
Library and Center for Knowledge Management
San Francisco, California 94143-0840
pool of general purpose workstations to generate three dimensional
representations of volume data. The VIS tool provides
integrated load-balancing across any number of heterogeneous
UNIX workstations (e.g. SGI, Sun, DEC, etc…) [Giertsen]
taking advantage of the unused cycles that are generally available
in academic and research environments. In addition, VIS
supports specialized graphics hardware (e.g. the RealityEngine
from Silicon Graphics), when available, for real-time visualization.
Distributing information that includes volume data requires
the integration of visualization with a document delivery
mechanism. We have integrated VIS and volume data into
the WWW, taking advantage of the client-server architecture
of WWW and its ability to access hypertext documents stored
anywhere on the Internet [Obraszka, Nickersen]. We have
enhanced the capabilities of the most popular WWW client,
Mosaic [Andreessen] from the National Center for
Supercomputer Applications (NCSA), to support volume data
and have defined an inter-client protocol for communication
between VIS and Mosaic for volume visualization. It should be
noted that other types of interactive applications could be
"embedded" within HTML documents as well. Our approach
can be generalized to allow the implementation of object
linking and embedding over the Internet, similar to the features
the OLE 2.0 provides users of Microsoft Windows on an
individual machine.
1.1 The World-Wide-Web
The World-Wide-Web is a combination of a transfer
protocol for hyper-text documents (HTTP) and a hyper-text
mark-up language (HTML) [Nickersen]. The basic functionality
of HTTP allows a client application to request a wide
variety of data objects from a server. Objects are identified by
a universal resource locator (URL)[Obraczka] that contains
information sufficient to both locate and query a remote server.
HTML documents are defined by a document type definition

(DTD) of the Standard Generalized Mark-up Language
(SGML). These documents are returned to WWW clients and
are presented to the user. Users are able to interact with the
document presentation, following hyper-links that lead to
other HTML documents or data objects. The client application
may also directly support other Internet services, such as
FTP, Gopher, and WAIS, [Andreessen] or may utilize gateways
that convert HTTP protocol requests and return HTML
documents. In all interactions, however, the user is presented
with a common resulting data format (HTML) and all links are
accessible via URL’s.
1.2 Mosaic
THE INTERNET
SESSION
MANAGER
VISUALIZATION SERVERS DATA CENTER
HIGH SPEED LOCAL NETWORK (1 GBPS)
Figure 1: VIS client/server model.
The National Center for Supercomputer Applications
(NCSA) has developed one of the most functional and popular
World-Wide-Web clients: Mosaic. This client is available via
public FTP for the most popular computer interfaces (Motif,
Windows and Macintosh). Mosaic interprets a majority of the
HTML DTD elements and presents the encoded information
with page formatting, type-face specification, image display,
fill-in forms, and graphical widgets. In addition, Mosaic
provides inherent access to FTP, Gopher, WAIS and other
network services [Andreessen].
1.3 VIS
VIS is a simple but complete volume visualizer. VIS
provides arbitrary three-dimensional transformation (e.g. rotation
and scaling), specification of six axial clipping planes
(n.b. a cuboid), one arbitrary clipping plane, and control of
opacity and intensity. VIS interactively transforms the cuboid,
and texture-maps the volume data onto the transformed geometry.
It supports distributed volume rendering [Argrio, Drebin,
Kaufman] with run-time selection of computation servers, and
isosurface generation (marching cubes)[Lorenson, Levoy] with
software Gouraud shading for surface-based model extraction
and rendering. It reads NCSA Hierarchical Data Format
(HDF) volume data files, and has a graphical interface utility
to import volume data stored in other formats.
2. VIS: A Distributed Volume Visualization Tool
VIS is a highly modular distributed visualization tool,
following the principles of client/server architecture (figure 1),
and consisting of three cooperating processes: VIS, Panel, and
VRServer(s). The VIS module handles the tasks of transformation,
texture-mapping, isosurface extraction, Gouraud shading,
and manages load distribution in volume rendering. VIS
produces images that are drawn either to its own top-level
window (when running stand-alone) or to a shared window
system buffer (when running as a cooperative process). The
Panel module provides a graphical user-interface for all VIS
functionality and communicates state changes to VIS. The
VRServer processes execute on a heterogenous pool of general
purpose workstations and perform volume rendering at the
request of the VIS process . The three modules are integrated
as shown in figure 3 when cooperating with another process. A
simple output window is displayed when no cooperating
process is specified.
2.1 Distributed Volume Rendering
Volume rendering algorithms require a significant amount
of computational resources. However, these algorithms are
excellent candidates for parallelization. VIS distributes the
volume rendering among workstations with a “greedy” algorithm
that allocates larger portions of the work to faster
machines [Bloomer]. VIS segments the task of volume rendering
based on scan-lines, with segments sized to balance computational
effort versus network transmission time. Each of the

user-selected computation servers fetches a segment for rendering
via remote procedure calls (RPC), returns results and fetch
another segment. The servers effectively compete for segments,
with faster servers processing more segments per unit
time, ensuring relatively equal load balancing across the pool.
Analysis of this distribution algorithm [Giertsen, 93] shows
that the performance improvement is a function of both the
number of segments and the number of computational servers,
with the optimal number of sections increasing directly with
the number of available servers. Test results indictate that
performance improvement flattens out between 10 to 20
segments distributed across an available pool of four servers.
Although this algorithm may not be perfect, it achieves acceptable
results.
2.2 Cooperative Visualizaton
The VIS client, together with its volume rendering servers,
may be lauched by another application collectively as a
visualization server. The two requirements of cooperation are
a shared window system buffer for the rendered image and
support for a limited number of inter-process messages. VIS
and the initiating application communicate via the ToolTalk
service, passing messages specifying the data object to visualize
as well as options for visualization, and maintaining state
regarding image display. The VIS Panel application appears as
a new top-level window and allows the user control of the
visualization tool.
3. Visualization with Mosaic
We have enhanced the Mosaic WWW browser to support
both a three-dimensional data object and communication with
VIS as a coopezrating application (figure 2). HTTP servers
respond to requests from clients, e.g. Mosaic, by transferring
hypertext documents to the client. Those documents may
contain text and images as intrinsic elements and may also
contain external links to any arbitrary data object (e.g. audio,
video, etc…). Mosaic may also communicate with other
Internet servers, e.g FTP, either directly – translating request
results into HTML on demand – or via a gateway that provides
translation services. As a WWW client, Mosaic communicates
with the server(s) of interest in response to user actions (e.g.
selecting a hyperlink), initiating a connection and requesting
the document specified by the URL. The server delivers the file
specified in the URL, which may be a HTML document or a
variety of multimedia data files (for example, images, audio
files, and MPEG movies) and Mosaic uses the predefined
SGML DTD for HTML to parse and present the information.
Data types not directly supported by Mosaic are displayed via
user-specifiable external applications and we have extended
that paradigm to both include three-dimensional volume data,
as well as to integrate the external applications more completely
with Mosaic.
3.1 Mosaic 3D image support
We have extended the HTML DTD to support threedimensional
data via the introduction of a new SGML element:
EMBED. This element provides information to the
presentation system (i.e. Mosaic) about the content that is
referenced in the document. The EMBED element is defined
in the HTML DTD as shown in Example 1, which is translated
as “SGML document instance element tag EMBED containing
no content; four required attributes: TYPE, the type of the
external application, in the MIME-type format; HREF, the
location/URL of the datafile; WIDTH, the window width
and, HEIGHT, the window height. The TYPE attribute give
this specification the flexibility to accomodate different types
of external applications. In a HTML document, a 3D image
element would be represented as shown in Example 2, which
may be interpreted as “create a drawing-area window of width
400 pixels, height 400 pixels, and use the application associated
to hdf/volume MIME content-type to visualize the data
Embryo.hdf located at the HTTP server site
www.library.ucsf.edu”.
3.2 Interface with Mosaic
The VIS/Mosaic software system consists of three elements:
VIS, Mosaic, and Panel. Currently, the VIS application
communicates with Mosaic via ToolTalk, but the
system will work with any interclient communication protocol.
When Mosaic interprets the HTML tag EMBED, it
creates a drawing area widget in the document page presentation
and requests a shared buffer or pixmap from the windowing
system to receive visualization results. In addition, Mosaic
launches the Panel process, specifying the location of the data
object to render and identifying the shared image buffer. The
Panel process begins execution by first verifying its operating
parameters, then launching the VIS process. The Panel process
also presents the user with the control elements for data
manipulation and manages the communication between the
whole VIS application and Mosaic.
The VIS process, on the other hand, serves as a rendering
engine. It executes the visualization commands from the Panel
process, integrates the image data segments from various
VRServers, and presents the complete array of image data to
the Panel.
Thus the scenario following a user’s action on the Panel
will be (1) Panel issues visualization commands to the VIS
rendering engine, (2) VIS sends rendering requests to

Figure 2: VIS embeded within Mosaic for interactive visualization in a HTML document.
VRServer(s), then gathers the resulting image segments, (3)
Panel fetches the returned image data, then writes it to the
pixmap, (4) Panel notifies Mosaic upon completion, and (5)
Mosaic bit-blots the pixmap contents into its corresponding
DrawingArea widget. The interprocess communication issue
will be addressed in more details under section 3.3. The
configuration of this software system is depicted in figure 3.
3.3 Interclient communication
We recognized the minimum set of communication protocols
between Mosaic and a particular Panel process:
(a) Messages from Mosaic to a Panel process include the
following:
(i) ExitNotify - requesting the Panel to terminate
itself.when Mosaic exits.
(ii) MapNotify - requesting the Panel to map itself to
the screen when the HTML document containing the
DrawingArea corresponding to the above panel is visible.
(iii) UnmapNotify - requesting the Panel to unmap/
iconify itself when the HTML page containing the DrawingArea
corresponding to the above Panel is cached.
(b) Messages from a Panel process to Mosaic may be one
of the following:
(i) RefreshNotify - informing Mosaic of an update in
the shared pixmap, and requesting Mosaic to update the
correspinding DrawingArea.
(ii) PanelStartNotify - informing Mosaic the Panel is
started successfully, and ready to receive messages.
(iii) PanelExitNotify - informing Mosaic the Panel is
exiting, and Mosaic should not send any more messages to the
Panel.
We have packaged the above protocols and all the required
messaging functions into a library. Modification of an existing
external application merely involves registration of the external
application’s messaging window (the window to receive Mosaic’s
messages), installation of callback functions corresponding to

the messages from Mosaic, and addition of message-sending
routine invocations. The protocol is summarized in Table 1.
Messages Descriptions
ExitNotify Mosaic exiting
MapNotify DrawingArea visible
UnmapNotify DrawingArea cache
RefreshNotify DrawingArea update
PanelStartNotify Panel starting
PanelExitNotify Panel exiting
Table 1: Mosaic/VIS IPC communication.
4. Results
The results of the above implementation are very encouraging.
The Mosaic/VIS sucessfully allows users to visualize
HDF volume datasets from various HTTP server sites. Fig 2
shows a snapshot of the WWW visualizer. Distributing the
volume rendering loads results in a remarkable speedup in
image computations. Our performance ananlysis with a
homogeneous pool of Sun SPARCstation 2’s on a relatively
calm network produced reasonable results (Figures 4a, 4b, and
4c. Three trials per plot). The time-versus-number-ofworkstations
curve decreases as more servers participate, and
plateaus when the number of SPARCstations is 11 in the case
of 256x256 image (9 for 192x192 image, and 7 for 128x128
image). The speed increases at the plateaus are very significant:
about 10 times for the 256x256 image, 8 times for the 192x192
image, and 5 times for the 128x128 image. The outcomes
suggest that performance improvement is a function of the
number of volume rendering servers. Furthermore, the optimal
number of workstations and the speed increase are larger
when the image size is bigger. This is in complete agreement
with Giertsen’s analysis. We have also successfully tested the
software system in an environment consisting of heterogenous
workstations: a SGI Indigo2 R4400/150MHz, two SGI Indy
R4000PC/100MHz, a DEC Alpha 3000/500 with a 133MHz
Alpha processor, two Sun SparcStations 10, and two Sun
SparcStations 2, which were located arbitrarily on an Ethernet
network. To our knowledge this is the first demonstration of
the embedding of interactive control of a client/server visualization
application within a multimedia document in a distributed
hypermedia environment, such as the World Wide Web.
5. Ongoing/Future work
We have begun working on several extensions and improvements
on the above software system:
Example 1: SGML definition for EMBED element.
TYPE=”hdf/volume”
WIDTH=400
HEIGHT=400>
Example 2: EMBED element usage.
5.1 MPEG Data Compression
The data transferred between the visualization servers and
the clients consists of the exact byte streams computed by the
servers, packaged in the XDR machine independent format.
One way to reduce network transferring time would be to
compress the data before delivery. We propose to use the
MPEG compression technique, which will not only perform
redundancy reduction, but also a quality-adjustable entropy
reduction. Furthermore, the MPEG algorithm performs
MOSAIC
INTER-CLIENT
COMMUNICATION
VISUALIZATION
COMMANDS
RENDERING
REQUESTS
PANEL
IMAGE
DATA
VRSERVER(S)
Figure 3: Communication among Mosaic, VIS
and distributed rendering servers.
VIS
IMAGE
SEGMENTS

interframe, beside intraframe, compression. Consequently,
only the compressed difference between the current and the
last frames is shipped to the client.
5.2 Generalized External-Application-to-Mosaic-Document-Page
Display Interface
The protocols specified in Table 1 are simple, and general
enough to allow most image-producing programs be modified
to display in the Mosaic document page. We have successfully
incorporated an in-house CAD model rendering program into
Mosaic. Our next undertakings will be to extend the protein
1
2
3
4
5
Number of Servers
128x128
192x192
256x256
6
7
8
9
10
11
12
database (PDB) displaying program, and the xv 2D image
processing program, to create a Mosaic PDB visualization
server, and a Mosaic 2D image processing server.
5.3 Multiple Users
With multiple users, the VIS/Mosaic distributed visualization
system will need to manage the server resources, since
multiple users utilizing the same computational servers will
slow the servers down significantly. The proposed solution is
depicted in Fig 5. The server resource manager will allocate
servers per VIS client request only if those servers are not
Figure 4: Volume rendering performance for 128 2 , 192 2 , and 256 2 data sets .
13
14
128x128
192x192
256x256
12.00
10.00
8.00
6.00
4.00
2.00
0.00
20.00
18.00
16.00
14.00
Volume Size
Time (sec)

overloaded. Otherwise, negotiation between the resource
manager and the VIS client will be necessary, and, perhaps the
resource manager will allocate less busy alternatives to the
client.
5.4 Load Distributing Algorithm
Since the load distributing algorithm in the current VIS
implementation is not the most optimal load distribution
solution, we expect to see some improvement in the future
implementation, which will be using sender-initiated algorithms,
described in [Shivaratri].
NETWORK
BOUNDARY
ENCODED
IMAGE DATA
NETWORK
BOUNDARY
6. Conclusions
NETWORK
BOUNDARY
SERVER
RESOURCE
MANAGER
REQUEST
FOR SERVICE
VISUALIZATION
COMMANDS
CLIENT
VISUALIZATION
PROCESSES
VISUALIZATION
SERVER
PROCESS
POOL
Figure 5: Server Resource Management
PROCESS
ALLOCATION
Our system takes the technology of networked multimedia
system (especially the World Wide Web) a step further by
proving the possibility of adding new interactive data types to
both the WWW servers and clients. The addition of the 3D
volume data object in the form of an HDF file to the WWW
has been welcomed by many medical researchers, for it is now
possible for them to view volume datasets without a high-cost
workstation. Furthermore, these visualizations can be accessed
via the WWW, through hypertext and hypergraphics links
within an HTML page. Future implementations of this
approach using other types of embedded applications will
allow the creation of a new paradigm for the online distribution
of multimedia information via the Internet.
7. References
Argiro, V. “Seeing in Volume”, Pixel, July/August 1990, 35-
39.
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Please reference the following QuickTime movie located in the MOV
directory:
CHEONG.MOV
Copyright © 1994 by Cheong S. Ang, M.S.
QuickTime is a trademark of Apple Computer, Inc.

VolVis: A Diversified Volume Visualization System
Ricardo Avila ‡ ,Taosong He * ,Lichan Hong * ,Arie Kaufman * ,
Hanspeter Pfister * ,Claudio Silva * ,Lisa Sobierajski * ,Sidney Wang *
‡
Howard Hughes Medical Institute
*
Department of Computer Science
State University of New York at Stony Brook State University of New York at Stony Brook
Stony Brook, NY 11794-5230 Stony Brook, NY 11794-4400
Abstract
VolVis is a diversified, easy to use, extensible, high
performance, and portable volume visualization system for
scientists and engineers as well as for visualization
developers and researchers. VolVis accepts as input 3D
scalar volumetric data as well as 3D volume-sampled and
classical geometric models. Interaction with the data is
controlled by a variety of 3D input devices in an input
device-independent environment. VolVis output includes
navigation preview, static images, and animation
sequences. A variety of volume rendering algorithms are
supported, ranging from fast rough approximations, to
compression-domain rendering, toaccurate volumetric ray
tracing and radiosity, and irregular grid rendering.
1. Introduction
The visualization of volumetric data has aided many
scientific disciplines ranging from geophysics to the
biomedical sciences. The diversity of these fields coupled
with a growing reliance on visualization has spawned the
creation of a number of specialized visualization systems.
These systems are usually limited by machine and data
dependencies and are typically not flexible or extensible.
Afew visualization systems have attempted to overcome
these dependencies (e.g., AVS, SGI Explorer, Khoros) by
taking a data-flow approach. However, the added
computational costs associated with data-flow systems
results in poor performance. In addition, these systems
require that the scientist or engineer invest a large amount
of time understanding the capabilities of each of the
computational modules and how toeffectively link them
together.
VolVis is a volume visualization system that unites
numerous visualization methods within a comprehensive
visualization system, providing a flexible tool for the
scientist and engineer as well as the visualization
developer and researcher. The VolVis system has been
designed to meet the following key objectives:
Diversity: VolVis supplies a wide range of
functionality with numerous methods provided within each
functional component. For example, VolVis provides
various projection methods including ray casting, ray
tracing, radiosity, Marching Cubes, and splatting.
Ease of use: The VolVis user interface is organized
into functional components, providing an easy to use
visualization system. One advantage of this approach over
data-flow systems is that the user does not have to learn
how tolink numerous modules in order to perform a task.
Extensibility: The structure of the VolVis system is
designed to allow avisualization programmer to easily add
new representations and algorithms. For this purpose, an
extensible and hierarchical abstract model was developed
[1] which contains definitions for all objects in the system.
Portability: The VolVis system, written in C, is
highly portable, running on most Unix workstations
supporting X/Motif. The system has been tested on Silicon
Graphics, Sun, Hewlett-Packard, Digital Equipment
Corporation, and IBM workstations and PCs.
Freely available: The high cost of most
visualization systems and difficulties in obtaining their
source code often lead researchers to write their own tools
for specific visualization tasks. VolVis is freely available as
source code.
2. System Overview
Figure 1 shows the VolVis pipeline, indicating some
paths that input data could take through the VolVis system
in order to produce visualization output. Two of the basic
input data classes of VolVis are volumetric data and 3D
geometric data. The input data is processed by the
Modeling and Filtering components of the system to
produce either a 3D volume model or a 3D geometric
surface model of the data. For example, geometric data can
be converted into a volume model by the Modeling
component of the system, as described in Section 3, to
allow for volumetric graphic operations. A geometric
surface model can be created from a volume model by the
process of surface extraction.

Key:
3D Scalar Field
Volumetric
Model
Input Data
Internal Data
Output
Visualization
Action
Modeling
&
Filtering
Surface
Extraction
Measurement
Manipulation
Environment
Rendering
Image Animation
Figure 1:The VolVis pipeline.
3D Geometric
Objects
Geometric
Surface Model
Virtual Input
Device
Input Device
Abstraction
Physical Input
Device
Navigation
Preview
The Measurement component can be used to obtain
quantitative information from the data models. Surface
area, volume, histogram and distance information can be
extracted from volumes using one of several methods.
Isosurface volume and surface area measurements can be
taken either on an entire volume or on a surface-tracked
section. Additionally, surface areas and volumes can be
computed using either a simple non-interpolated voxel
counting method or a Marching Cubes [8] based
measurement method. For geometric surface models,
surface area, volume, and distance measurements can be
performed.
Most of the interaction in VolVis occurs within the
Manipulation component of the system. This part of the
system allows the user to modify object parameters such as
color, texture, and segmentation, and viewing parameters
such as image size and field of view. Within the
Navigation section of the Manipulation component, the
user can interactively modify the position and orientation
of the volumes, the light sources, and the view. This is
closely connected to the Animation section of the
Manipulation component, which allows the user to specify
animation sequences either interactively or with a set of
transformations to be applied to objects in the scene. The
Manipulation component is described in Section 4.
The Rendering component encompasses several
different rendering algorithms, including geometry-based
techniques such as Marching Cubes, global illumination
methods such as ray tracing and radiosity, and direct
volume rendering algorithms such as splatting. The
Rendering component is described in Section 5.
The Input Device component of the system maps
physical input device data into a device independent
representation that is used by various algorithms requiring
user interaction. As a result, the VolVis system is input
device independent, as described in Section 6.
3. Modeling
Aprimary responsibility of the Modeling component
is the voxelization of geometric data into volumetric model
representations. Voxelizing a continuous model into a
volume raster of voxels requires a geometrical sampling
process which determines the values to be assigned to
voxels of the volume raster. To reduce object space
aliasing, we adopt a volume sampling technique [14] that
estimates the density contribution of the geometric objects
to the voxels. The density of a voxel is determined by a
filter weight function which is proportional to the distance
between the center of the voxel and the geometric
primitive. Inour implementation, precomputed tables of
densities for a predefined set of geometric primitives are
used to assign the density value of each voxel. For each
voxel visited by the voxelization algorithm, the distance to
the predefined primitive is used as an index into the tables.
Figure 2:Avolumetric ray traced image of a volumesampled
geometric wine bottle and glasses.

Since the voxelized geometric objects are
represented as volume rasters of density values, we can
essentially treat them as sampled or simulated volume data
sets, such as 3D medical imaging data sets, and employ
one of many volume rendering techniques for image
generation. One advantage of this approach is that volume
rendering carries the smoothness of the volume-sampled
objects from object space over into image space. Hence,
the silhouette of the objects, reflections, and shadows are
smooth. Furthermore, by not performing any geometric
ray-object intersections or geometric surface normal
calculations, a large amount of rendering time is saved. In
addition, CSG operations between two volume-sampled
geometric models are accomplished at the voxel level
during voxelization, thereby reducing the original problem
of evaluating a CSG tree of such operations down to a
Boolean operation between pairs of voxels. Figure 2
shows a ray traced image of a wine bottle and glasses that
were modeled by CSG operations on volume-sampled
geometric objects. The upper right window inFigure 3
shows a ray traced image of a nut and bolt that were also
modeled by CSG operations.
Figure 3:Anexample VolVis session. The nut and bolt
are volume-sampled geometric models.
4. Manipulation
The Manipulation component of VolVis consists of
three sections: the Object Control section, the Navigation
section, and the Animation section. The Navigation and
Animation sections are also referred to as the Navigator
and Animator, respectively. Both the Navigator and
Animator produce output visualization, shown in Figure 1
as Navigation Preview and Animation, respectively.
The Object Control section of the system is
extensive, allowing the user to manipulate parameters of
the objects in the scene. This includes modifications to the
color, texture, and shading parameters of each volume, as
well as more complex operations such as positioning of cut
planes and data segmentation. The color and position of
all light sources can be interactively manipulated by the
user. Also, viewing parameters, such as the final image
size, and global parameters, such as ambient lighting and
the background color, can be modified.
The Navigator allows the user to interactively
manipulate objects within the system. The user can
translate, scale and rotate all volumes and light sources, as
well as the view itself. The Navigator can also be used to
interactively manipulate the view inamanner similar to a
flight simulator. Toprovide interactive navigation speed, a
fast rendering algorithm was developed which involves
projecting reduced resolution representations of all objects
in the scene. This task is relatively simple for geometric
objects, where calculating, storing, and projecting a
polygonal approximation requires little overhead.
However, when considering a volumetric isosurface the
cost of an additional representation increases considerably.
Asimple and memory efficient method available within
the Navigator creates a reduced resolution representation
of an isosurface by uniformly subdividing the volume into
boxes and projecting the outer faces of all the boxes that
contain a portion of the isosurface. These subvolumes
serve adual purpose in that they are also used by the
PARC (Polygon Assisted Ray Casting) acceleration
method [1] during ray casting and ray tracing.
Although the PARC subvolume representation can
be stored as a compact list of subvolume indices, the
resulting images are boxy and uninformative for many data
sets. To overcome this problem, another method is
provided which utilizes a reduced resolution Marching
Cubes representation of an isosurface. In order to reduce
the amount of data required for this representation, edge
intersections used to compute triangle vertices are
restricted to one of four possible locations. This results in
much smoother images which are typically more
informative than the uniform subdivision method. The
Navigator also supports the other VolVis rendering
techniques that are described in Section 5, although
interactive projection rates with these methods can be
achieved only on high-end workstations.
The Animator also allows the user to specify
transformations to be applied to objects within the scene,
but asopposed to the Navigator which is used to apply a
single transformation at a time, the Animator can be used
to specify a sequence of transformations to produce an
animation. The user can preview the animation using one
of the fast rendering techniques within the Navigator. The
user can then select a more accurate and time consuming
rendering technique, such as volumetric ray tracing, to

create a high quality animation. In addition to simple
rotation, translation and scaling animations, the Navigator
can be used to interactively specify a ‘‘flight path’’, which
can then be passed to the Animator, and rendered to create
an animation.
An example session of the VolVis system is shown in
Figure 3. The long window onthe left is the main VolVis
interface window, with buttons for each of the major
components of the system. The current scene is displayed
in the Navigator window onthe left, and in the Rendering
image window onthe right. A low resolution Marching
Cubes technique was used in the Navigator, while a ray
casting technique using the PARC acceleration method
was employed during rendering.
5. Rendering
Rendering is one of the most important and
extensive components of the VolVis system. For the user,
speed and accuracy are both important, yet often
conflicting aspects of the rendering process. For this
reason, a variety of rendering techniques have been
implemented within the VolVis system, ranging from the
fast, rough approximation of the final image, to the
comparatively slow, accurate rendering within a global
illumination model. Also, each rendering algorithm itself
supports several levels of accuracy, giving the user an even
greater amount of control. In this section, a few ofthe
rendering techniques developed for the VolVis system are
discussed.
Tw o of the VolVis rendering techniques, volumetric
ray tracing, and volumetric radiosity, are built upon global
illumination models. Standard volume rendering
techniques, which are also supported by VolVis, typically
employ only a local illumination model for shading, and
therefore produce images without global effects. Including
aglobal illumination model within a visualization system
has several advantages. First, global effects can often be
desirable in scientific applications. For example, by
placing mirrors in the scene, a single image can show
several views of an object in a natural, intuitive manner
leading to a better understanding of the 3D nature of the
scene. Also, complex surfaces are often easier to render
when represented volumetrically than when represented by
high-order functions or geometric primitives, as described
in Section 3. Volumetric ray tracing is described in
Section 5.1 and volumetric radiosity is discussed in
Section 5.2.
In order to reduce the large storage and transmission
overhead as well as the volume rendering time for
volumetric data sets, a data compression technique is
incorporated into the VolVis system. This technique allows
volume rendering to be directly performed on the
compressed data and is described in Section 5.3.
Although many scanning devices create data sets
that are inherently rectilinear, this restriction poses
problems for fields in which an irregular data
representation is necessary. These fields include
computational fluid dynamics, finite element analysis, and
meteorology. Therefore, support was added for irregularly
gridded data formats in the VolVis system, as discussed in
Section 5.4.
5.1. Volumetric Ray Tracing
The volumetric ray tracer provided within the VolVis
system is intended to produce accurate, informative images
[11]. In classical ray tracing, the rendering algorithm is
designed to generate images that are accurate according to
the laws of optics. In VolVis, the ray tracer must handle
classical geometric objects as well as volumetric data, and
strict adherence to the laws of optics is not always
desirable. For example, a scientist may wish to view the
maximum value along the segment of a ray passing
through a volume, instead of the optically-correct
composited value. Figure 4 illustrates the importance of
including global effects in a maximum-value projection of
a hippocampal pyramidal neuron data set which was
obtained using a laser-scanning confocal microscope.
Since maximum-value projections do not give depth
information, a floor is placed below the cell, and a light
source above the cell. This results in a shadow ofthe cell
on the floor, adding back the depth information lost by the
maximum-value projection.
In order to incorporate both geometric and
volumetric objects into one scene, the classical ray tracing
intensity equation, which is evaluated only at surface
locations, must be extended to include volumetric effects.
The intensity of light, Iλ(x, →
ω ), for a given wav elength λ,
arriving at a position x, from the direction →
ω , can be
computed by:
I λ(x, →
ω ) = I vλ(x, x′) + τ λ(x, x′)I sλ(x′, →
ω )
where x′ is the first surface intersection point encountered
along the ray →
ω originating at x. Isλ(x′, →
ω )isthe intensity
of light at this surface location, and can be computed with
a standard ray tracing illumination equation [15].
Ivλ(x, x′) isthe volumetric contribution to the intensity
along the ray from x to x′, and τ λ(x, x′) isthe attenuation
of Isλ(x′, →
ω )byany intervening volumes. These values are
determined using volume rendering techniques, based on a
transport theory model of light propagation [7]. The basic
idea is similar to classical ray tracing, in that rays are cast
from the eye into the scene, and surface shading is
performed on the closest surface intersection point. The
(1)

difference is that shading must be performed for all
volumetric data that are encountered along the ray while
traveling to the closest surface intersection point.
Figure 4:Avolumetric ray traced image of a cell using
amaximum-value projection.
For photo-realistic rendering, the user typically
wants to include all of the shading effects that can be
calculated within a given time limit. However,
visualization users may find it necessary to view
volumetric data with no shading effects, such as when
using a maximum-value projection. In VolVis, the user has
control over the illumination equations for both volumetric
and geometric objects, and can specify, for each object in
the scene, which shading effects should be computed. For
example, in Figure 4 no shading effects were included for
the maximum-value projection of the cell, while all parts
of the illumination equation were considered when shading
the geometric polygon. In another example, the user may
place a mirror behind a volumetric object in a scene in
order to capture two views in one image, but may not want
the volumetric object to cast a shadow onthe mirror, as
shown in Figure 5. The head was obtained using magnetic
resonance imaging, with the brain segmented from the
same data set. The mirror is a volume-sampled polygon
that was created using the modeling technique described in
Section 3.
5.2. Volumetric Radiosity
The ray tracing algorithm described in the previous
section can be used to capture specular interactions
between objects in a scene. In reality, most scenes are
dominated by diffuse interactions, which are not accounted
for in the standard ray tracing illumination model. For this
reason, VolVis also contains a radiosity algorithm for
volumetric data. Volumetric radiosity includes the
classical surface ‘‘patch’’ element as well as a ‘‘voxel’’
element. As opposed to previous methods that use
participating media to augment geometric scenes [10], this
method is intended to render scenes that may solely consist
of volumetric data. Each patch or voxel element can emit,
absorb, scatter, and transmit light. Both isotropic and
diffuse emission and scattering of light are allowed, where
‘‘isotropic’’ implies directional independence, and
‘‘diffuse’’ implies Lambertian reflection (i.e., dependent on
normal or gradient). Light entering an element that is not
absorbed or scattered by the element is transmitted
unchanged.
Figure 5:Avolumetric ray traced image of a human head.
In order to cope with the high number of voxel
interactions required, a hierarchical technique similar to
[5] is used. An iterative algorithm [2] is then used to shoot
voxel radiosities, where several factors govern the highest
level inthe hierarchy atwhich two voxels can interact.
These factors include the distance between the two voxels,
the radiosity of the shooting voxel, and the reflectance and
scattering coefficients of the voxel receiving the radiosity.
This hierarchical technique can reduce the number of
interactions required to converge onasolution by more
than four orders of magnitude.
After the view-independent radiosities have been
calculated, a view-dependent image is generated using a
ray casting technique, where the final pixel value is
determined by compositing radiosity values along the ray.
Figure 6 shows a scene containing a volumetric sphere,
polygon, and light source. The light source isotropically
emits light, and both the sphere and the polygon diffusely
reflect light. The light source is above the sphere and
directly illuminates the top half of the sphere. The bottom
half of the sphere is indirectly illuminated by light
diffusely reflected from the red polygon.

5.3. Compression Domain Volume Rendering
Another rendering method incorporated in VolVis is
adata compression technique for volume rendering. Our
volume compression technique is a 3D generalization of
the JPEG still image compression algorithm [13] , with
one important exception: the transform is a discrete Fourier
transform rather than a discrete cosine transform. The
original 3D data is subdivided into M×M×M subcubes,
where each subcube is Fourier transformed to the
frequency domain through a 3D discrete Fourier transform.
Each of the 3D Fourier coefficients in each subcube is then
quantized, and the resulting 3D quantized frequency
coefficients are organized as a linear sequence through a
3D zig-zag order. The resulting sequence of Fourier
transform coefficients is then fed into an entropy encoder
that consists of run-length coding and Huffman coding.
Figure 6: A volumetric radiosity projection of a
voxelized sphere and polygon.
To render in the compressed domain, we use a new
class of volume rendering algorithms [3, 9, 12] that are
based on the Fourier projection slice theorem. It states that
aprojection of the 3D data volume from a certain direction
can be obtained by extracting a 2D slice perpendicular to
the view direction out of the 3D Fourier spectrum and then
applying an inverse Fourier transform. In our approach we
apply the Fourier projection slice theorem to each subcube
in the Fourier domain, which results in a set of 2D planes
in the spatial domain called subimages that are composited
using spatial compositing to get the final projection of the
original 3D data set.
Using our compression-domain rendering approach,
we were able to achieve high compression ratios while
maintaining image quality. Figure 7 shows a CT scan of a
lobster that was rendered in the compressed domain.
We are currently investigating the adaptation of
subcube sizes to various spatial or frequency domain
criteria, such as subcube AC coefficient energy, which is a
measure of subcube activity, sample density, and
coefficient distribution.
Figure 7: Compression domain volume rendering of a
lobster.
5.4. Irregular Grid Rendering
An intuitive way to visualize irregularly gridded data
sets is to resample the data into a regular grid format.
Unfortunately, it is quite difficult to find a resampling
method that preserves details yet does not require a large
amount of memory. Consequently, wechose to extend the
traditional volume rendering algorithms to process the
irregularly gridded data directly. For example, we have
extended the ray tracing algorithms in VolVis to visualize
data represented in a spherical coordinate system, with
grids that are unevenly spaced in r, evenly spaced in θ ,and
unevenly spaced in φ. When rendering, we could cast rays
into the scene, uniformly stepping and compositing along
each ray. A problem with uniform stepping is that it
inevitably misses detailed information. To avoid this
problem, we traverse the ray cell by cell in the volume, in a
method similar to Garrity [4].
6. Input Devices
The Input Device component of the VolVis system
allows the user to control a variety of input devices in an
input device independent environment. For example, to
control the Navigator, the user can utilize a variety of
physical input devices such as a keyboard, a mouse, a
Spaceball, and a DataGlove. To achieve this, we have
developed the device unified interface (DUI) [6], which is
a generalized and easily expandable protocol for
communication between applications and input devices.

The key idea of the DUI is to convert raw data
received from different input sources into unified format
parameters of a ‘‘virtual input device’’. Depending on the
requirements of the application, the parameters may
include a number of 3D positions and orientations as well
as abstract actions. The abstract actions include direct and
simple actions like mouse or Spaceball button clicks , and
complex dynamic actions like two hand gestures or ‘‘snapdragging’’.
The conversion from the real device operations
to abstract actions is performed by the selected simulation
methods which are incorporated into the DUI.
The most important advantage of employing the
virtual input device paradigm is input device
independence. In the DUI, each application is interactively
assigned a virtual input device, whose configuration is also
interactively decided. Modification of either the input
device component or the application does not affect other
parts of the system. The simulation methods used to
convert different kinds of raw information into the unified
format are often difficult to design. For example, the
recognition of dynamic gestures of a DataGlove is fairly
difficult. By using the DUI, new simulation methods can
be easily incorporated and tested with no adverse effect on
the application or the other parts of the Input Device
component.
However, in order to fully utilize the capability of
different devices, a virtual input device should not totally
hide the device dependent information since different
devices are suitable for different applications. For
example, it is harder to control the Navigator with the
Spaceball than with the DataGlove, since the six degrees of
freedom provided by the Spaceball are not entirely
independent as they are in the DataGlove. Inthe DUI, a
device information-base is associated with each virtual
input device. All of the device dependent information
related to a virtual device is classified and stored in an
abstract form, which is then queried by an application
when necessary [6].
We are currently working on the expansion of the
DUI into a general-purpose interaction model. The model
is created based on lightweight threads and is designed to
handle simultaneous high bandwidth, multimodal, and
complex input from multiple users, even through the
network. A general abstract action and input device
description language is also being studied.
7. Implementation
Tw o major concerns during the implementation of
VolVis have been to ensure that the system could be
expanded to include new functionality and techniques, and
that the system would be relatively easy to port to new
platforms. Therefore, the development of the VolVis
system required the creation of a comprehensive, flexible,
and extensible abstract model [1]. The model is organized
hierarchically, beginning with low-level building blocks
which are then used to construct higher-level structures.
For example, low-level objects such as vectors and points
can be combined to create a coordinate system, while at
the highest level the World structure contains the state of
ev ery object in the system. The World structure includes
Lights, Volumes, Views, global cut planes, and global
shading parameters. Each Volume structure includes color
and texture information, local shading parameters, local
cut planes, and data which may be either geometric
descriptions, or rectilinear or irregularly gridded data.
The abstract model is flexible in that a structure can
assume one of many representations. For instance, a
segmentation structure can consist of either a threshold or
opacity and color transfer functions. A natural
consequence of flexibility is expandability. Since the
objects in the abstract model already provide for numerous
representations, the addition of a new segmentation type,
shading type, or even data type is fairly simple.
The VolVis system requires only Unix and X/Motif
to run. Unfortunately, only simple two-dimensional
graphics operations are supported in X. Therefore, all
viewing transformations, shading, and hidden surface
removal must be done in software. This greatly reduces the
rendering speed for the geometry-based projection routines
used in the Navigation section, and therefore also reduces
the overall interactivity of the system. Since many Unix
workstations now include graphics hardware, interactivity
can be maintained by utilizing the graphics language of the
workstation. To avoid rewriting large sections of the code,
we have dev eloped a library of basic graphics functions
that are used throughout the VolVis code. This simplifies
the process of porting the system to a new workstation that
has a different graphics language, since only the graphics
function library must be rewritten.
8. Conclusions
The VolVis system for volume visualization has been
used for many tasks in diverse applications and situations.
First, VolVis has been used to test new algorithms for
rendering, modeling, animation generation, and computerhuman
interaction. Due to the flexible nature of the
abstract model, testing new ideas within the system is
much easier and less time consuming than writing a new
application for each new algorithm. VolVis has also been
used by scientists and researchers in many different areas.
For example, neurobiologists have used VolVis to navigate
through the complex dendritic paths of nerve cells, which

is extremely useful since the function of nerve cells is
closely tied to their structure (see Figure 4).
VolVis is a rapidly growing system, with new plans
for future development continually being considered.
Since the system is currently being used by many research
labs and visualization developers, feedback from these
sources is used to make future versions of the system
easier to use and extend. To increase portability, auserinterface
library, similar to the graphics function library
described in the previous section, is being developed to
allow VolVis to be easily ported to new windowing
systems.
9. Acknowledgements
VolVis development has been supported by the
National Science Foundation under grant CCR-9205047,
Department of Energy under the PICS grant, Howard
Hughes Medical Institute, and the Center for
Biotechnology. Data for Figure 4 is courtesy of Howard
Hughes Medical Institute, Stony Brook, NY. Data for
Figure 5 is courtesy of Siemens, Princeton, NJ. Data for
Figure 6 is courtesy of AVS, Waltham, MA.
For information on how toobtain the VolVis system,
send e-mail to volvis@cs.sunysb.edu.
References
1. R.S. Avila, L.M. Sobierajski, and A.E. Kaufman,
“Tow ards a Comprehensive Volume Visualization
System,” Visualization ’92 Proceedings, pp. 13-20
(October 1992).
2. M. F. Cohen, S. E. Chen, J. R. Wallace, and D. P.
Greenberg, “A Progressive Refinement Approach to
Fast Radiosity Image Generation,” Computer
Graphics (Proc. SIGGRAPH) 22(4) pp. 75-84 (July
1988).
3. S. Dunne, S. Napel, and B. Rutt, “Fast Reprojection
of Volume Data,” Proceedings of the 1st Conference
on Visualization in Biomedical Computing, pp.
11-18 (1990).
4. M. Garrity, “Raytracing Irregular Volume Data,” San
Diego Workshop on Volume Visualization, Computer
Graphics 24(5) pp. 35-40 (December 1990).
5. P. Hanrahan, D. Salzman, and L. Aupperle, “A Rapid
Hierarchical Radiosity Algorithm,” Computer
Graphics (Proc. SIGGRAPH) 25(4) pp. 197-206
(July 1991).
6. T. Heand A. Kaufman, “Virtual Input Devices for
3D Systems,” Visualization ’93 Proceedings, pp.
142-148 IEEE Computer Society Press, (October
1993).
7. W. Krueger, “The Application of Transport Theory
to Visualization of 3D Scalar Data Fields,”
Computers in Physics, pp. 397-406 (July/August
1991).
8. W. E.Lorensen and H. E. Cline, “Marching Cubes:
A High Resolution 3D Surface Construction
Algorithm,” Computer Graphics (Proc. SIGGRAPH)
21(4) pp. 163-169 (July 1987).
9. T. Malzbender, “Fourier Volume Rendering,” ACM
Tr ansactions on Graphics 12(3) pp. 233-250 (July
1993).
10. H. E. Rushmeier and K. E. Torrance, “The Zonal
Method For Calculating Light Intensities in the
Presence of a Participating Medium,” Computer
Graphics (Proc. SIGGRAPH) 21(4) pp. 293-306
(July 1987).
11. L.M. Sobierajski and A.E. Kaufman, “Volumetric
Ray Tracing,” 1994 Symposium on Volume
Visualization, ACM Press, (October 1994).
12. T. Totsuka and M. Levo y, “Frequency Domain
Volume Rendering,” Computer Graphics (Proc.
SIGGRAPH), pp. 271-278 (1993).
13. G. K. Wallace, “The JPEG Still Picture Compression
Standard,” Communications of the ACM 34(4) pp.
30-44 (April 1991).
14. S.W. Wang and A.E. Kaufman, “Volume Sampled
Voxelization of Geometric Primitives,” Visualization
’93 Proceedings, pp. 78-84 IEEE Computer Society
Press, (October 1993).
15. T. Whitted, “An Improved Illumination Model for
Shaded Display,” Communications of the ACM
23(6) pp. 343-349 (June 1980).

Please reference the following QuickTime movies located in the MOV
directory:
MULTI_RO.MOV
HEAD_CUT.MOV
MAX_ROTA.MOV
Copyright © 1994 by the Research Foundation of the State University of
New York at Stony Brook
QuickTime is a trademark of Apple Computer, Inc.

Abstract
Swept surfaces and volumes are generated by moving
a geometric model through space. Swept surfaces
and volumes are important in many computer-aided
design applications including geometric modeling,
numerical cutter path generation, and spatial path
planning. In this paper we describe a numerical
algorithm to generate swept surfaces and volumes
using implicit modeling techniques. The algorithm is
applicable to any geometric representation for which
a distance function can be computed. The algorithm
also treats degenerate trajectories such as self-intersection
and surface singularity. We show applications
of this algorithm to maintainability design and
robot path planning.
Keywords: Computational geometry, object modeling,
geometric modeling, volume modeling, implicit
modeling, sweeping.
1.0 Introduction
A swept volume is the space occupied by a geometric
model as it travels along an arbitrary trajectory. A
swept surface is the boundary of the volume. Swept
surfaces and volumes play important roles in many
computer-aided design applications including geometric
modeling, numerical control cutter path generation,
and spatial path planning.
In geometric modeling swept curves and surfaces
are used to represent extrusion surfaces and surfaces
of revolution [1][2]. More complex geometry can be
generated by using higher order sweep trajectories
and generating surfaces [3]. In robot motion planning,
the swept volume can be used to evaluate safe
paths [4][5] or establish the footprint of a robot in a
Implicit Modeling of
Swept Surfaces and Volumes
William J. Schroeder
William E. Lorensen
GE Corporate Research & Development
Steve Linthicum
GE Aircraft Engines
workcell. Numerical control path planning uses
swept volumes to show the removal of material by a
tool [6].
Swept surfaces and volumes can also be applied to
resolve maintainability issues that arise during the
design of complex mechanical systems. The
designer needs to be able to create and visualize the
accessibility and removability of individual components
of the system. Typical questions related to
maintainability include:
• Can a mechanic remove the spark plugs?
• Is there room for an improved power supply?
• What is the impact on the maintenance of a system
if new components are included?
In maintenance design, the swept surface of the
part to be removed is called the removal envelope.
The removal envelope is the surface that a component
generates as it moves along a safe and feasible
removal trajectory. A safe trajectory is one that a
component can follow without touching other components
of the system. A feasible trajectory is one
that can be performed by a human. Lozano-Perez
[7] calls the calculation of safe trajectories the Findpath
problem. Although this trajectory may be automatically
calculated, restrictions on the trajectory’s
degrees of freedom [8], or the geometry of the
obstacles [9] prohibit practical application to realworld
mechanical systems. The technique presented
here assumes that a safe and feasible trajectory is
already available. For our application, we rely on
computer-assisted trajectory generation using commercial
computer-aided design [10] or robot simulation
software.
In the next section we review related work on
swept volumes and implicit modeling. Then we
describe the algorithm in detail along with a discus-

sion of error and time complexity. Examples from
maintainability and robot motion simulation illustrate
the effectiveness of the algorithm in application.
2.0 Background
Literature from two fields: spatial path planning and
implicit modeling contribute to swept volume generation.
2.1 Path Planning
Weld and Leu [11] present a topological treatment of
swept volumes. They show that the representation of
a swept volume in Rn generated from a n-dimensional
object is reduced to developing a geometric
representation for the swept volume from its (n-1)boundary.
However, they point out that the boundary
and interior of the swept volume are not necessarily
the union of the boundary and interior of the (n-1)boundary.
That is, there may exist points on the
boundary of the swept volume that are interior points
of the (n-1)-boundary of the swept volume. Likewise,
boundary points of the object may contribute to
the interior of the swept volume. This unfortunate
property of swept volumes limits conventional precise
geometric modeling to restricted cases. Martin
and Stephenson [12] recognize the importance of
implicit surface models for envelope representation
but seek to provide a closed solution. They present a
theoretical basis for computing swept volumes, but
note that complicated sweeps may take an unrealistic
amount of computer time.
Wang and Wang [6] present a numerical solution
that uses a 3D z-buffer to compute a family of critical
curves from a moving solid. They restrict the
generating geometry to a convex set, an appropriate
restriction for their numerical control application.
Other numerical techniques include sweeping
octrees[5]. Swept octrees produce very general
results at the cost of the aliasing effects due to octree
modeling.
2.2 Implicit Modeling
An implicit model specifies a scalar field value at
each point in space. A variety of field functions are
available depending on the application. Soft objects
created for computer animation [13] use a field that
is unity at a modeling point and drops to zero at a
specified distance from the point. The variation of
the fields is typically a cubic polynomial. Usually
these fields are represented on a regular sampling
(i.e., volume) as described by Bloomenthal [15].
Surface models are extracted using standard iso-surface
techniques such as marching cubes [16]. If the
field values are distance functions to the closest point
on the model, offset surfaces can be created by
choosing a non-zero iso-surface value [14].
3.0 Algorithm
The goal of the algorithm can be described as follows.
Given a geometric model and a trajectory
defined by a sequence of continuous transformations,
generate the swept volume, the volume occupied as
the model travels along the trajectory, and the swept
surface, the boundary of the swept volume.
The swept surface algorithm is implemented in
three basic steps (Figure 1).
1. Generate an implicit model from the original geometric
model. We use implicit techniques that
assign a distance value from each voxel to the
surface of the object. The geometric model may
be of any representational form as long as a distance
function can be computed for any point.
Common representations include polygonal
meshes, parametric surfaces, constructive solid
models, or implicit functions.
2. Sweep the implicit model through the workspace.
The workspace is a volume constructed to strictly
bound the model as it travels along the sweep trajectory.
The sweeping is performed by repeatedly
sampling the implicit model with the workspace
volume as it is transformed along the sweep trajectory.
The end result is a minimum distance
value at each voxel in the workspace volume.
3. Generate the swept surface using the iso-surface
extraction algorithm marching cubes. The value d
of the iso-surface is a distance measure. If d = 0 ,
the surface is the swept surface. For d ≠ 0 , a fam-
ily of surfaces offset from the swept volume is
created.
A detailed examination of each of these steps follows.
3.1 Generating the Implicit Model
This step converts the geometric model into a sampled
distance function in a 3D volume of dimension
( n1, n2, n3) . We refer to this representation as the
implicit model VI. Our approach is similar to Bloomenthal [15]. Any
n-dimensional object in R (we assume here )
can be described by an implicit function
where the function f() defines the distance of point
n
n = 3
f( p)
=
0

a) Generate implicit model
b) Sweep implicit model through
workspace volume
c) Generate swept surface via
iso-surface extraction
Figure 1. Overview of swept surface
generation.
p R to the surface of the object. To develop the
implicit model we compute the function f() in a 3D
volume of dimension . For geometric
models having a closed boundary (i.e., having an
inside and an outside), f() can generate a signed distance;
that is, negative distance values are inside the
model and positive values are outside of the model.
Geometric representations often consist of combinations
of geometric primitives such as polygons or
splines. In these cases f() must be computed as the
minimum distance value as follows. Given the n
primitives defining the n distance
values at point p, choose the
minimum value.,
3 ∈
( n1, n2, n3) G = ( g1, g2, …, gn) ( d1, d2, …, dn) d = f( p)
= Min ( d1 , d2 , …, dn )
For an implicit model, the minimum is equivalent to
the union of the scalar field [15].
The implicit model can be generated for any geometric
representation for which the distance function
f() can be computed. One common representation is
the polygonal mesh. Then f(p) is the minimum of the
distances from p to each polygon (Figure 2). For
more complex geometry whose distance function
may be expensive or too complex to compute, the
model can be sampled at many points, and then the
points can be used to generate the distance function.
Figures 6 and 7 illustrate the generation of an
d = Min(d 1 ,d 2 )
d 2
d 1
distance to plane
distance to edge
distance to vertex
Figure 2. Computing distance function f() for
polygonal mesh.
implicit model from a polygonal mesh. Figure 6
shows the original mesh consisting of 5,576 polygons.
The implicit model is sampled on a volume V I
at a resolution of 100 3 . An offset surface of the
implicit model is generated using a surface value d =
0.25 in Figure 7. In this example a non-negative distance
function has been used to compute the implicit
model. As a result, both an outer and inner surface
are generated. Any closed surface will necessarily
generate two or more surfaces for some values of d if
f() is non-negative.
3.2 Computing the Workspace Volume
The workspace volume Vw is generated by sweeping
the implicit model along the sweep trajectory, and
sampling the transformed implicit model. Vw must
be sized so that the object is strictly bounded
throughout the entire sweep. One simple technique
to size Vw is to sweep the bounding box of the geometric
model along the sweep trajectory and then
compute a global bounding box.
The sweep trajectory ST is generally specified as a
series of transformations ST = { t1, t2, …, tm} . Arbitrary
transformations are possible, but most applications
define transformations consisting of rigid body
translations and rotations. (The use of non-uniform
scaling and shearing creates interesting effects.)
The implicit model travels along the sweep trajectory
in a series of steps, the size of the step dictated
by the allowable error (see Discussion). Since these
transformations { t1, t2, …, tm} may be widely separated,
interpolation is often required for an intermediate
transformation t'.
We recover positions and
orientations from the provided transformations, and
interpolate these recovered values.
The sampling of the implicit model is depicted in
Figure 3. The values in Vw are initialized to a large
positive value. Then for each sampling step, each
point in Vw is inverse transformed into the coordi

ST
a) Inverse transform V W
nate system of V I. The location of the point within V I
is found and then its distance value is computed
using tri-linear interpolation. As in the implicit
model, the value of the point in V w is assigned the
minimum distance. The result is the minimum distance
value at each point in V w seen throughout the
sweep trajectory.
An alternative workspace volume generation
scheme calculates the distance function f() directly
from the transformed geometric model, eliminating
the need for V I entirely. Although this can be efficient
when f() is inexpensive to compute, we have
found that in our applications the performance of the
algorithm is unsatisfactory. Sampling the implicit
model into the transformed workspace volume
improves the performance of the algorithm significantly,
since the sampling is independent of the number
of geometric primitives in the original model,
and the cost of tri-linear interpolation is typically
much smaller than the cost of evaluating f().
3.3 Extracting Swept Surfaces
The last step in the algorithm generates the swept
surface from the workspace volume. Since Vw represents
a 3D sampling of distance function, the iso-surface
algorithm marching cubes is used to extract the
swept surface. Choosing d = 0 generates the surface
of the swept volume, while d ≠ 0 generates offset
surfaces. In many applications choosing d > 0 is
desirable to generate surfaces offset from the swept
volume by a certain tolerance.
As with any iso-surface extraction algorithm, the
surfaces often consist of large numbers of triangles.
Hence we often apply a decimation algorithm [17] to
reduce the number of triangles representing the surface.
4.0 Discussion
b) Sample V I
Figure 3. Sampling the implicit volume.
4.1 Degenerate Trajectories
Degenerate trajectories, as described by Martin
and Stephenson [12], are self-intersecting trajecto-
Dimensions
DxDxD
L
Figure 4. Sampling error in 3D volume.
L
L
ries or trajectories that result in surface singularities.
Self-intersection occurs when the generating geometry
intersects itself as it travels along the sweep trajectory.
Surface singularities occur when nonvolume
filling geometry is created, for example
when a plane travels perpendicular to its normal.
The algorithm presented here correctly treats
degenerate trajectories. Conventional analytical
techniques require sophisticated mathematical techniques
that do not yet have general solutions. Our
algorithm uses simple set operations that are immune
to both types of degeneracies.
4.2 Error Analysis
Due to sampling errors and differences in representation,
the generated swept surface is an approximation
to the actual surface. Sampling errors are introduced
at three points: 1) when creating VI from the geometric
model, 2) when sampling VI into the workspace
volume Vw, and 3) when incrementing the transformation
along the sweep trajectory ST. Representation
errors are due to the fact that the swept surface is a
polygonal mesh, while the original representation
can be combinations higher-order surfaces or even
implicit functions. Here we address sampling errors
only, since representational errors depend on the particular
geometric representation, which is beyond the
scope of this paper.
The sampling error bounds of the distance function
in a 3D volume is the distance from the corner of a
cube formed from eight neighboring voxels to its
center (Figure 4). Without loss of generality we can
assume that the volume size is length L, and that the
volume dimensions are n1 ,n2 ,n3 = D. It follows that
the cube length is then L/D. The maximum sampling
error e at a voxel is then
3
e ------
2
The error introduced by stepping along ST depends
upon the nature of the transformation. Translations
give rise to uniform error terms, but rotation gener-
L
≤
⎛
⎝
---
⎞
D ⎠
e
L/D

d i
t i
Transform
Figure 5. Approximating stepping error.
p
ti + 1
ates variable error depending upon distance from
center of rotation.The stepping error can be estimated
by linearizing the transformation and assuming
that the maximum positional change of any point
is Δx (Figure 5.). If the actual distance value to some
point p in Vw is given as d , and the distance functions
are di and di + 1 at the points ti and ti + 1 along
ST, then the maximum error occurs at Δx ⁄ 2 and is
given by e = di– d = di + 1 – d.
Since
2
d d
⎛Δx i -----
⎞
⎝ , the error is bounded by
2 ⎠
2
= –
The value of Δx can be estimated by transforming
the bounding box of the model and computing the
displacement of the extreme points.
These error terms can be combined to provide an
estimate to the total error:
The terms L I , L W , are the lengths and D I , D W are the
dimensions of the implicit and working volumes,
respectively. These terms arise when creating V I
from the geometric model and when sampling V I
into the workspace volume V w .
4.3 Complexity Analysis
The bulk of computation occurs when sweeping the
implicit model through the workspace volume. Generating
the implicit model and the final iso-surface
are one-time events at the beginning and end of the
algorithm.
Assuming that f() can be computed in constant
time for any point, the implicit model can be gener-
3
ated in time O〈 DI〉 . If the computation of f() is proportional
to the number of geometric primitives np in
3
the model, the time complexity is O〈 npD 〉 . The
3
I
complexity of sweeping is O〈 nsD 〉
where ns is the
W
d
Δx
di + 1
e di d
⎛
i ⎝
-----
⎞
2 ⎠
2 Δx
= – – ≤
⎛
⎝
-----
⎞
2 ⎠
e tot
≈
2 Δx
3
------
2
L ⎛ I
⎝
----- + -------
⎞ Δx
⎠
+ -----
2
D I
L W
D W
number of steps in the sweep. The final step, iso-surface
extraction, is a function of the size of the work-
3
ing volume O〈 DW〉 .
4.4 Multiple Surface Generation
More that one connected swept surface may be created
for certain combinations of geometry and isosurface
value d. As described previously if f( p)
≥ 0
for all p, both an inner an outer surface may be generated.
If the geometric model is non-convex, then
multiple inside and outside surfaces may be created.
There will be at least one connected outer surface,
however, that will bound all other surfaces. In some
applications such as spatial planning and maintainability
design this result is acceptable. Other applications
require a single surface.
One remedy that works in many cases is to compute
a signed distance function f() where values less
than zero occur inside the object. This approach will
eliminate any inner surfaces when d> 0.
Generally
this requires some form of inside/outside test which
may be expensive for some geometric representations.
Another approach to extract the single bounding
surface is to use a modified form of ray casting in
combination with a surface connectivity algorithm.
5.0 Results
Our initial application for swept surfaces was for
maintainability design. In this application the swept
surface is called a removal envelope. The removal
envelope graphically describes to other designers the
inviolable space required for part access and
removal. We have since used the swept surfaces to
visualize robot motion. Three specific examples follow.
The first two examples use a trajectory planned
using the McDonnell Douglas Human Modeling
System.
5.1 Fuel/Oil Heat Exchanger
The part shown in Figure 6 is a fuel/oil heat
exchanger used in an aircraft engine. Maintenance
requirements dictate that it must be able to be
replaced within 30 minutes while the aircraft is at the
gate. The implicit model of the heat exchanger is
shown in Figure 7. Figure 8 shows the resulting
swept surface. Both the implicit model and the workspace
volume were generated at a resolution of 100 3 .
The sweep trajectory contained 270 steps. The swept
surface, generated with an offset of d=0.2 inch, consists
of 15,108 triangles.
Figure 9 shows the removal envelope positioned
within surrounding parts. Of particular interest is the

piping that has been rerouted around the removal
envelope. The figure demonstrates the small clearances
typical in complex mechanical environments.
Because of tolerances in these systems, we generally
choose to create offset swept surfaces. Currently we
use no quantitative method to choose the offset value
and rely on design experience.
5.2 Fuel Nozzle
Figure 10 shows another application of swept surfaces
to maintainability design. A proposed design
change to the aircraft engine pre-cooler (shown in
turquoise) impacted the removal of the number 3
fuel nozzle. The removal path consists of many rotations
and is self-intersecting at many points.
The fuel nozzle was modeled using 6,100 polygons.
The implicit model was generated at a volume
resolution 50 3 and the workspace volume at 100 3 . A
total of 600 steps was taken along the sweep trajectory.
The offset swept surface of 16,572 triangles
was generated with a value d=0.2 inch.
5.3 Robot Motion
Swept surfaces are effective tools for visualizing
complex robot motion. In Figure 11 the swept surfaces
of the end effector, hand, wrist, and arm of the
robot are shown simultaneously. Each robot part was
sampled at 50 3 . The workspace volume was sampled
at 100 3 . The total number of triangles representing
the four surfaces was 681,800 using a distance value
d=1 inch (over a workspace volume size LW = 140
inches.)
6.0 Conclusion
We have demonstrated an algorithm that generates
swept surfaces and volumes from any geometric representation
for which a distance function can be
computed. This includes such common forms as
parametric surfaces, polygonal meshes, implicit representations,
and constructive solid models. The
algorithm treats complex trajectories including selfintersection
and surface singularities. We have successfully
applied the algorithm to a variety of complex
applications including maintainability design
and robot spatial planning.
Acknowledgments
Steve Rice of McDonnell Douglas assisted in the
extraction of removal paths from the McDonnell
Douglas Human Modeling System.
References
[1] D. F. Rogers and J. A. Adams. Mathematical Elements
for Computer Graphics. McGraw-Hill Publishing
Co., New York. 1990.
[2] M. E. Mortensen. Geometric Modeling. John Wiley
& Sons, New York, 1985.
[3] S. Coquillart. A control-point-based sweeping technique.
IEEE Computer Graphics and Applications
7(11):36-45, November 1987.
[4] T. Lozano-Perez. Spatial Planning: A configuration
space approach. IEEE Trans. on Computers C-
32(2):108-120, February 1983.
[5] M. Celenk and W. Sun. 3D visual robot guidance in
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[6] W. P. Wang and K. K. Wang. Geometric modeling
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197, March/April 1983.
[10] McDonnell Douglas Human Modeling System Reference
Manual. Report MDC 93K0281. McDonnell
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October 1990.
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three-dimensional objects. Computer-Aided Design
22(4):223-234, May 1990.
[13] B Wyvill, C. McPheeters, G. Wyville. Animating
soft objects. The Visual Computer 2:235-242.
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Computer Aided Geometric Design, 5(4):341-355,
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26(2):65-70, July 1992.

Figure 6. Original model of fuel/oil heat exchanger. Figure 7. Outer and inner offset surface generated
from implicit model.
Figure 8. Swept surface of heat exchanger.
Figure 10. Fuel nozzle removal envelope in context
with other parts.
Figure 9. Removal envelope (wireframe) in context
with other parts.
Figure 11. Swept surfaces of robot end effector
(brown), hand (blue), wrist (yellow), and
arm (orange).

Visualizing Polycrystalline Orientation Microstructures with Spherical Color Maps
Boris Yamrom * , John A. Sutliff * , Andrew P. Woodfield †
* GE Corporate Research and Development, 1 River Road, Schenectady, New York, 12345
† GE Aircraft Engine, One Neumann Way, Cincinnati, Ohio, 45215
Abstract
Spherical color maps can be an effective tool in the
microstructure visualization of polycrystals. Electron
backscatter diffraction pattern analysis provides large
arrays of the orientation data that can be visualized easily
using the technique described in this paper. A combination
of this technique with the traditional black and white
scanning electron microscopy imaging will enable scientists
to better understand the correlation between material
properties and their polycrystalline structure.
I. Introduction
Material scientists possess a large arsenal of tools for
the exploration of physical and chemical properties of
materials. Many of these tools directly address human
abilities to process visual information. Others generate
data sets that can be analyzed employing special-purpose
computer programs designed to extract meaningful and
useful information. Metallurgists and geologists often
study polycrystalline materials, that is materials composed
of many small grains of crystals. Physical properties
of these materials (strength, elasticity, etc.) may be
strongly influenced by the orientation of these grains, also
called crystallographic textures. These textures are the
subject of numerous investigations [9].
Typical scanning electron microscope images can
show boundaries between crystal grains and can reveal
compositional differences between grains of different
components called phases. In most cases, however, they
will not show particular orientation of individual grains
and, therefore, will not tell if several grains define a
domain of similar orientation.
To address the problem of imaging orientation-based
features in microstructures, a relatively new method of
electron backscatter diffraction pattern analysis(EBSP) is
used [10]. The diffraction pattern analysis by itself is not
new, but special enhancements (automation) in the diffraction
pattern collection combined with the computeraided
processing enables generation of up to 1800 sample
orientations per hour [10](in 1988 it was possible to
obtain only 120 measurements per hour, [2]).
Orientation of the crystals in the sample is usually
defined with Euler angles α, β, and γ, that represent three
consecutive rotations of the coordinate system associated
with the crystal to align it with the sample coordinate system
[9]. The first rotation is about the crystal’s z' axis (we
denote the sample coordinate system axes with x, y, z, and
the crystal coordinate axes with x', y' , and z' ), the second
rotation is about the crystal’s x' axis, and the third rota-
tion is again about the z' axis. All rotations are made
counterclockwise. Euler angles α, β, and γ satisfy the following
constraints:
0 ≤ α < 2π,
0 ≤ β < π, 0 ≤ γ < 2π .
There are variations in the Euler angles definition:
some authors use the y' axis instead of the x' axis for the
second rotation, angles boundaries may be shifted too, but
these are of no concern to us here. The data collected is
represented by the file of records
α β γ x y,
where α, β, and γ are Euler angles, and x, y represent two
Cartesian coordinates of a measured point relative to
some sample reference point. In the case of materials containing
multiple components the above record can be
expanded to contain the type of phase at a particular sample
location. Sample points coordinates x, y may belong
to a regular grid or be random.
A few visualization techniques exist for orientation
data. The most popular technique is the pole figure. A particular
direction is selected in a crystal coordinate system
( z' axis, for example) and for all points ( x, y)
in a sample,
the selected direction is mapped to a unit sphere
(where a ray of this direction from the origin intersects the

sphere) and then projected onto the xy plane with one of
the two widely used projection techniques: stereographic
or equal-area projection [9]. Thus orientation distribution
of a sample is represented as a set of points in the disk. We
get more information about the material if we plot pole
figures for all crystal major axes x', y' , and z' . Another
way to visualize orientation distribution is to display a set
of points representing Euler angle triplets in the Euler
angle volume of 3D space.
Though both techniques are widely used, their limitations
are well recognized. One of the limitations relates to
the fact that these are statistical visualizations, that is we
plot points in the function value domain (α, β, γ) and lose
all information about spatial distribution of orientations
(in x, y domain). Other limitation relates to distortion
introduced by the curved character of Euler angle space.
Recently some authors addressed the limitation of
Euler space mapping[6]. Other researchers recognize that
orientation, being multivariate, can be represented by
color or black and white texture [7]. However, all attempts
so far were not satisfactory. Either these mappings require
stereo displays [6], or the mappings were non-consistent
and not easy to use. General dissatisfaction with the existing
color maps for visualizing orientations is expressed in
the following quotation taken from [10]:
“However, there are two problems which can
arise from color visualizations of microstructure.
First, inherent to any color representation is the
danger of physiological responses to particular
colors, biasing the interpretation. Second, for
most parametrizations of orientation space, it is
difficult to find a color mapping which insures
that grains of similar orientation are assigned
similar colors.”
We try to address both of these concerns in our paper.
The first concern is addressed by using multiple color
maps instead of one, thus avoiding bias caused by specific
physiological responses to a particular color or group of
colors. The second concern is the major stimulus to our
investigation. We present a set of maps that is free of these
problems and that can be successfully used for producing
orientation images of polycrystalline microstructures. As
was indicated in [1] no single color map of orientation
space can be fully satisfactory, but their combination may
be. We limit the scope of this investigation by an issue of
color coding leaving out the geometry coding and other
deciphering and interpreting techniques [8]. Our experiments
show that those techniques may be very useful for
small arrays of data, but are substantially outperformed by
color coding in a case of very large data sets.
First, we observe that in many practical situations the
scientist is interested in the x'y' plane orientation of the
crystal only and is not interested in the rotation of the
crystal about the axis normal to its x'y' plane. This is typical
when we are interested in the strength of material and
we know that a crystal can be more easily split along the
x'y' plane than along any other plane. Other planes of
interest can be analyzed similarly simply by changing the
initial frame of reference. Orientation of the plane is
uniquely specified by the direction of its normal vector,
i.e., z' axis for the x'y' plane. To simplify the exposition,
we assume here that the crystal axes are orthogonal, but
final results still hold for arbitrary crystals.
The direction of the normal vector along the z' axis
can be uniquely represented by a point on a unit sphere.
Moreover, we should identify as equal the opposite points
on a sphere, since they correspond to the same orientation
of the x'y' plane relative to the sample coordinate system.
Therefore, the limited orientation space we are dealing
with is represented by a unit sphere in 3D space with
opposite points being identical. This is a so called two
dimensional projective plane. The arbitrary orientation,
however would require a unit sphere in 4D space (with the
same property of identity of the opposite points), or projective
space of three dimensions.
Now we can formulate the problem to be solved in
exact terms. We would like to create coloring of a sphere
that satisfies the following conditions:
(a) every point on a sphere is assigned a color;
(b) opposite points get the same color;
(c) different points get different colors;
(d) similar (close) points get similar colors.
Since we can represent color in RGB system with
three parameters [5], color gamut is a 3D topological
region. If we treat conditions (a)-(d) literally, we would
need an imbedding of projective plane (topologically
equivalent to the Klein bottle) into 3D space that we know
is impossible [3]. Luckily, we do not treat conditions (a)-
(d) to the letter. In reality, even if we could create a map
satisfying (a)-(d), it would not be very useful. If we allow
for every point on a sphere to have a unique color and use
this color map to color the orientation corresponding to
each sample point x, y , we get a picture that will contain
too much information to comprehend. The segmentation
of the image that is easily done with pre-attentive vision in
the case of limited number of colors becomes more difficult
or impossible to do when the amount of colors grows
dramatically. Therefore, we change the condition (a) to be
( a' ) the sphere is subdivided into a finite number of
regions, possibly of similar area, each region is
assigned a color;
and we require that (b)-(d) is applied to these regions
instead of points. We call these new conditions ( a' )-( d'
).

In the next section we construct maps that satisfy
these conditions. Then we discuss how the mapping of
orientation data into color space is implemented. At the
end we summarize and point out new directions of
research.
2. Spherical Color Maps
Since the ideal subdivision of a sphere into subregions
should possess many symmetries, it is natural that
we turn our attention to the subdivision induced by regular
polytopes inscribed into a sphere. Among all regular polytopes
the icosahedron has the maximum number of faces -
20. Each face of an icosahedron is a regular triangle and
all points opposite to ones in a face represent one of the
other faces of the icosahedron. We have the same property
satisfied by cube, octahedron, and dodecahedron. Thus all
regular polytopes except tetrahedron can be used for orientation
color maps (compare with[4]). We present here a
detailed description of the design of color maps based on
icosahedron and on octahedron geometries.
Coloring a cube satisfying ( a' )-( d' ) is easy, one just
has to assign primary colors red, green, and blue to pairs
of opposite faces. With the icosahedron it is not so easy
and the existence of appropriate coloring may be viewed
as some kind of magic. Indeed, let us consider one triangle
face of the icosahedron and look at its adjacent faces.
.
Figure 1. Three adjacent faces of an icosahedron
Let the central triangle in Figure 1 be colored in red,
then we choose adjacent faces colors to contain equal red
components but mixed with other primaries. One face will
become yellow and another will get magenta. Since there
is no fourth primary color, we mix red with white to color
the third adjacent triangle. If we assign two other primary
colors to other faces and repeat this construction, then,
with the realization that each face color will be repeated
twice in the icosahedron, we get two of each colors on the
surface of icosahedron: red, green, blue, yellow, magenta,
cyan, light red, light green, and light blue. All together
there are 18 faces. Just add two whites and we color all the
faces of the icosahedron. Question: can this be done in a
consistent way that is exemplified in Figure 1? Yes, this is
indeed possible and demonstrated in Figure 2, where an
icosahedron is shown from two opposite points of view so
all its faces can be observed unobstructed.
Figure 2. Two opposite views of an icosahedron
This coloring is characterized by other symmetries:
every combined color, yellow for example, is surrounded
by two primary colors it is combined with, red and green,
and also by a light blue. White color is surrounded by
three light primaries. All colors specifications can vary
slightly to accommodate for different display devices or
color printer devices.
The created color map satisfies ( a' )-( d' ) conditions
and contains ten colors. This color map is just an example
and a proof of existence by the demonstration. Other coloring
schemes can be designed for an icosahedron that
may be even more natural than this one.
If we want finer resolution of orientation in color
space, we need to subdivide the sphere into more regions.
We can build such subdivision by splitting each triangle
face of an icosahedron into four equal triangles, projecting
three new vertices (edge centers) onto the sphere, and connecting
them in a triangle mesh.
Instead, we start with an octahedron and by a similar
face subdivision we create color maps with 4, 16, 64 and
256 colors. To demonstrate a different approach to color
map design we use the HSV color system [5] and assign to
each triangle facet of the polyhedron the color corresponding
to its geometric center point. To find this color
we first check the sign of z components of the point coordinate
and, if it is negative, we negate all components of
the point. The resulting point in the northern hemisphere
we project onto the xy plane using stereographic or equalarea
projection [9]. Then we take the polar coordinate of
the projection point r, ϕ, ( 0 ≤ r ≤ 1,
– π ≤ ϕ < π)
and calculate
h, s, and v by the formulas (C language notation is
used here)

h = (ϕ > 0)? ϕ / π : (ϕ + π) / π,
s = r,
v = (ϕ > 0)? 1 : 0.7,
where h is hue, s is saturation, and v is value. The first
expression makes standard hue of the color wheel to be
repeated twice in one full turn around the origin. This is
done to assign equal hue to all points on a meridian. The
second formula makes color different for different points
on a sphere based on their latitude coordinate. Points
closer to the equator get more saturated colors and points
closer to the north pole get color close to white. The last
formula makes a distinction in coloring two symmetric
points, the points with projected coordinates (r, ϕ) and (r,
ϕ + π). These points will have the same hue and saturation
but can be distinguished by the value v (as previously,
variations in formulas are possible). The image of 64 colors
map equal-area projection of northern hemisphere is
presented in Figure 3.
Figure 3. The equal-area projection of 64 colors
map of the northern hemisphere
One unavoidable artifact of this color map is the contrast
boundary (more pronounced in the printed image
than on the color monitor) between the light and dark
quarters of the sphere (discontinuity in value along the
equator and zero meridian). The more colors we get in the
color map, the more difficult would be to avoid discontinuity
in mapping a projective plane to the color space.
With this comment in mind it immediately follows from
our construction that the designed color map satisfies con-
ditions ( a' )-( d' ). In cases when the sample orientation
data is limited in its range a simple color map of Figure 4
can be used. In this color map opposite points on the
boundary do not coincide and may have a different color.
Figure 4. A color map with 64 colors
3. Implementation and Examples
Figure 5 presents algorithm of processing orientation
data.
1. Select one of the color maps and the associated polyhedron
described in the previous section. Make the
face indices of polyhedron faces be equal to the color
indices in the color lookup table.
2. Create a rectangle subdivided into squares according
to all x,y sample points. (We assume that a regular
grid is used for simplicity).
3. For each sample point do the following:
3.1 Calculate a 3 x 3 orientation matrix based on the
Euler angles of the sample.
3.2 Apply the transformation matrix from the previous
step to the direction vector (0,0,1) (crystal’s z' axes).
3.3 Find the intersection (polyhedron face index)
between the ray from the origin along the transformed
z'
axes and the polyhedron centered at the origin.
3.4 Use the face index of the intersection point to
color the corresponding square in the rectangle
of step 2.
Figure 5. The conceptual scheme of the color
mapping orientation data
Figures 6-7 represent a sample of data taken with the
step size 1 micron on a 100 x 100 grid. A detail of the
image in Figure 6 is reproduced in Figures 8-9 using the
color map of Figure 4.

Figure 6. 100 x 100 sample of the orientation
data with the step size 1 micron colored with an
icosahedron color map
Figure 7. 100 x 100 sample of orientation data
with the step size 1 micron colored with the 64
colors map of Figure 3
Color can be used also to represent the misorientation
between the adjacent regions (grains) of homogeneous
orientations. This may be done semiautomatically by
selecting pairs of adjacent regions and by displaying the
misorientation data and color, but also can be automated
to display the grains adjacency graph with the links colored
according to a selected color map. Misorientation
between the two grains is measured by the smallest rotation
angle required to align one grain orientation with
another. The detailed description of the adjacency graph
and its coloring goes beyond the scope of this paper.
4. Summary and Conclusions
We have created a technique for visualizing the orientation
microstructure of polycrystalline materials. The
technique is based on a few key ideas. First, the general
orientation of the crystal grain is reduced to the orientation
of a fixed plane associated with the crystal coordinate
system. Second, the orientation of the plane in 3D space is
mapped to the projective plane realized in 3D space as a
unit sphere with identified opposite points. Third, the unit
sphere is triangulated in a way compatible with the topology
of projective plane (icosahedron, octahedron and their
subtriangulations suite well). Fourth, each triangle of the
triangulation is assigned a unique color with conditions
( a' )-( d' ). Thus each orientation of the crystal plane gets
its color code.
The color map based on the sphere subdivision
induced by the inscribed icosahedron contains 10 different
colors. For color maps with 64 and 256 colors solid angles
subtended by each region become much smaller. This
leads to less variations in color transition from region to
region. To satisfy the conditions of color separation and
color similarity, we used all three dimensions of the color
gamut. More research is required to find psychologically
the best uniform subdivision of the color space satisfying
conditions ( a' )-( d'
).
The small discontinuity in the HSV value parameter
along the zero meridian and the equator psychologically is
more visible than the changes in saturation and hue. One
way to preserve the separation of colors is to use colored
texture in two quarters of the sphere instead of reducing
the value in the HSV model. In cases when all orientations
vary in some solid angle less than π/2 in diameter the discontinuity
in the value parameter can be moved away by
changing the reference point in the northern hemisphere.
By default it is the north pole, but any other point can be
used.
We tried to create isotropic maps and for this reason
avoided a simplistic mapping of longitude and latitude
coordinates on a sphere into hue and saturation. In our
approach the proportion of a color in the image is defined
by the proportion of the corresponding orientation in the
raw data and is not distorted by the nonuniformity of color
representation that would be caused by the longitude/latitude
mapping. On the other hand, simple longitude/latitude
mapping could be much more efficient from the

computational point of view. For some applications anisotropic
maps may suite even better than the ones we have
created. Spherical color maps may be viewed as visual filters
and the search for particular features in the data may
pose specific requirements for the design of these filters.
Figure 8. 32 x 32 detail of orientation data in
Figure 6 colored with the 64 colors map of
Figure 4
Even without optimizations to find out ray-polyhedron
intersection, the processing time of the 10,000 sample
points on a SPARCstation 1+ was a matter of a few
minutes, which is negligibly small compared to the time
required to collect the data. Nevertheless, the application
of multiple color maps, the selection of subregions, and
the reprocessing of those regions interactively would
require some optimizations.
We only briefly mentioned the misorientation imaging
and presented some of the possible techniques but
there is much more to be said about it and we hope to do
so in a forthcoming publication. Finally, the comparison
of orientation patterns with the SEM images of the same
sample, we believe, will provide many new insights into
the structure of polycrystalline materials.
Figure 9. The same data as in Figure 8, each
square is oriented according to the z'
direction
References
[1] Alpern, B., Carter, L., Grayson, M., Pelkie, C., Orientation
Maps: Techniques for Visualizing Rotations, Proceedings
of Visualization 93, 1993
[2] Dingley, D.J., On-line microstructure determination
using backscatter Kikuchi diffraction in a scanning electron
microscope, ICOTOM 8, The Metallurgy Society,
1988.
[3] Encyclopaedia of Mathematics, vol. 5, p. 275, Kluwer
Academic Publishers, 1990.
[4] Fekete, G., Rendering and managing spherical data with
Sphere Quadtrees, Proceedings of Visualization 90, San
Francisco, 1990.
[5] Foley, D.F., van Dam, A., Feiner, S.K., Hughes, J.F.,
Computer Graphics, Addison-Wesley, 1990.
[6] Frank, F.C., Orientation Mapping, Metallurgical Transactions
A, Vol. 19A, March 1988, 403-408.
[7] Randle, V., Microtexture Determination and its Applications,
The Institute of Materials, 1992.
[8] Visualization of Multiparameter Images, Panel Session,
Proceedings of Visualization 90, San Francisco, 1990.
[9] Wenk, H.-R., ed., Preferred Orientation in Deformed
Metals and Rocks: An Introduction to Modern Texture
Analysis, Academic Press, 1985.
[10] Wright, S.I., Adams, B.L., Kunze, K., Application of new
automatic lattice orientation measurement technique to
polycrystalline aluminum, Material Science and Engineering,
A160 (1993) 229-240.

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Piecewise�Linear Surface Approximation From Noisy Scattered
Samples
Michael Margaliot Craig Gotsman
Dept. of Electrical Engineering Dept. of Computer Science
Technion � Israel Institute of Technology Technion � Israel Institute of Technology
Haifa 32000� Israel Haifa 32000� Israel
Abstract
We consider the problem of approximating a
smooth surface f�x� y�� based on n scattered samples
f�xi� yi� zi�n i�1g where the sample values fzig are con�
taminated with noise� zi � f�xi� yi� � �i. We present
an algorithm that generates a PLS �Piecewise Linear
Surface� f 0� de�ned on a triangulation of the sam�
ple locations V � f�xi� yi�n i�1g� approximating f well.
Constructing the PLS involves specifying both the tri�
angulation of V and the values of f 0 at the points of
V . We demonstrate that even when the sampling pro�
cess is not noisy� a better approximation for f is ob�
tained using our algorithm� compared to existing meth�
ods. This algorithm is useful for DTM �Digital Ter�
rain Map� manipulation by polygon�based graphics en�
gines for visualization applications.
1 Introduction
Let f�x� y� be a smooth surface. Assume we are
given n noisy samples f�xi� yi� zi�n i�1g of f at scat�
tered locations V � f�xi� yi� n i�1 g� such that zi �
f�xi� yi���i� where �i are independent identically dis�
tributed zero�mean Gaussian random variables with
known variance � 2 . We wish to construct a PLS
�Piecewise Linear Surface�� sometimes called a TIN
�Triangulated Iregular Network�� f 0 � de�ned on some
triangulation of V . The PLS f 0 should approximate
the original surface f�x� y� as closely as possible� i.e.
the distance jjf � f 0 jj should be minimal� for some
norm jj � jj.
This problem arises in the reconstruction of terrain
surfaces from random DTM�s �Digital Terrain Mod�
els� extracted by automatic methods� such as match�
ing stereo image pairs �see the many articles on this
subject in �1��. These modern methods obtain terrain
elevation samples wherever possible� usually at feature
points� resulting in a data set consisting of points at
essentially random locations in the plane. This con�
trasts with the traditional manual DTM extraction
procedures� where elevation samples are obtained on
a regular grid� or at signi�cantly correlated locations
in the plane. Both manual and automatic DTM ex�
traction methods are inaccurate� so noise is inevitably
introduced into the elevation samples.
We choose to reconstruct the terrain as a PLS�
namely� a collection of triangles� as these are stan�
dard geometric primitives in modern graphics engine
hardware. Terrain visualization with texture�mapped
aerial imagery are popular graphics applications in vi�
sual simulation environments �2�. Two issues must be
addressed�
� The PLS f 0 is a collection of triangles in 3D space.
The topology of the triangulation is identical to
the topology of the speci�c planar triangulation of
V used. As there are an exponential �in n� num�
ber of di�erent triangulations of V � each resulting
in a di�erent PLS f 0 � it is not obvious which is
the best for our solution.
� The values z 0 i � f0 �xi� yi� at the PLS vertices.
Since the data is noisy anyway� they do not necce�
sarily have to coincide with the sampled zi.
Many researchers have dealt with variants of the op�
timal surface triangulation problem in the non�noisy
case �� � 0�. In its most pure form� given an explicit
function f�x� y�� and a tolerance d� approximate f by
a PLS f 0 with a minimal number of triangles� such
that the lp distance between f and f 0� de�ned as
jjf � f 0 �Z 1 Z 1
jjp �
0 0 jf�x� y� � f0�x� y�j p �1�p
dxdy
�1�

is no larger than d. Nadler �9� studied the connection
between the the second derivatives of f at a point
and the optimal shape of a triangle in that vicinity
with respect to the l2 distance. D�Azevedo �4� used
coordinate transformations to generate optimal trian�
gulations with respect to the l1 distance.
In a similar vein� given a dense sample of f at m
locations f�xi� yi� � i � 1� ��� mg� it is sometimes re�
quired to dilute this sample to a sparse set V �some�
times called decimation� of n points� and approximate
f as a PLS f 0 on some triangulation of V . The lp dis�
tance is now measured relative to the original sample
set�
jjf � f 0 � mX 1
jjp � jf�xi� yi� � f
m
i�1
0�xi� yi�j p
�1�p
�2�
Schroeder at el. �15� present an algorithm for decimat�
ing a sample set and triangulating the result� while
preserving important geometric features. Quak and
Schumaker �12�� while not specifying how to decimate
the sample set� triangulate the n points to a PLS f 0
achieving a local minimum of the l2 distance jjf �f 0 jj2.
In both cases mentioned above� the objective is
clear� the PLS f 0 should approximate the explicitly
given f or its dense sample as closely as possible. Our
working point� for which the objective is not as clear�
is to triangulate an already sparse sample set of lo�
cations V � without removing or adding points. The
problem is ill�posed since� theoretically� we have no
reason to prefer one triangulation over another. How�
ever� we try to minimize some �smoothness� measure�
with the hope that since the �true� �but unknown� f
was probably smooth� a smoother PLS has a better
chance of approximating it well. Rippa �14� showed
that the standard Delaunay triangulation ��10�� Chap.
5� of V� which has many nice mathematical proper�
ties� yields suboptimal results in many cases and that
long thin triangles �which the Delaunay triangulation
avoids� are sometimes good for linear interpolation�
contrary to common belief. Dyn et al. �5� suggested
a method for the iterative improvement of an initial
triangulation� using an edge�swapping technique due
to Lawson �7�� minimizing a cost function measuring
the �roughness� of the PLS.
All the works mentioned above �except �12�� deal
only with the case of accurate �non�noisy� samples.
We extend the procedure of Dyn et. al to deal with
noisy samples. Additionally� even for the case of accu�
rate samples� we demonstrate that results superior to
theirs may be achieved by using our algorithm� not
constraining f 0 �xi� yi� � f�xi� yi� enables the algo�
rithm to reach a better local minimum of the cost
function than the one obtained when constraining the
PLS vertices. For a PLS f 0 � de�ne its �cost� C�f 0 � as
C�f 0 � � I�f 0 � � �R�f 0 � �3�
where I�f 0 � measures the inaccuracy of the �t of f 0 to
the sampled data� R�f 0 � measures the �roughness� of
f 0 � and � is a weighting factor. The best PLS is the f 0
minimizing C. This approach is standard for smooth�
ing splines �16�. For example� in the one�dimensional
case� g 2 C 2 �0� 1�� I is taken to be
I�g� � 1
n
nX
�g�xi� � yi�
i�1
2
and R is Z 1
R�g� �
0 g00�x�2 dx
which is an approximation for the average curvature
of the surface. In the case of a PLS� which is two�
dimensional and possesses noncontinuous derivatives�
an expression similar to I is still applicable� but a
discrete analog to R must be found. Once the cost
function is well de�ned� an e�cient procedure to de�
termine the PLS minimizing it must be described.
The rest of this paper is organized as follows� Sec�
tion 2 elaborates on the cost function C� Section 3 de�
scribes the optimization procedure minimizing C and
Section 4 deals with determining the scalar parameter
� present in C. The results of numerical experiments
are reported in Section 5. In Section 6 we summarize
and conclude.
2 The Cost Function
What type of PLS is considered �good� � In classic
approximation theory� a common answer is� a good
surface is one that passes close to the sampled data
and is smooth �implicitly we assume that the original
sampled function was smooth�. Towards this end� we
de�ne the following �cost� C�f 0 � of a PLS candidate
f 0 . Given the sample set f�xi� yi� zi� n i�1 g
where
C�f 0 � � I�f 0 � � �R�f 0 �
I�f 0 � �
nX
�zi � f
i�1
0�xi� yi��2 measures the in�delity of the PLS �t to the sam�
pled data� and R�f 0 � is a measure of the PLS rough�
ness� de�ned as follows� Let t1 � fp1� p2� p3g and

t2 � fp2� p3� p4g be two triangles of the PLS f 0 � with
common edge p2p3 �see Fig. 1�. Let ni be the vector
normal to ti� i � 1� 2. �and ABN�e� of edge e � p2p3
be the Angle Between the Normals n1 and n2. Then
R�f 0 � �
X
e2fedges of f 0 g
ABN�e� �4�
ABN�e� may be thought of as an estimate of the dis�
crete curvature of the surface f 0 at that edge. The
relative importance of the two components of the cost
function is controlled by the positive scalar parameter
�. For � � 0� the in�delity of the PLS f 0 to the sam�
pled data dominates� so any PLS with f 0 �xi� yi� � zi
minimizes C. For very large �� the roughness of the
surface dominates� so any constant�valued PLS mini�
mizes C.
p1
n1
p2
Figure 1� Two triangles of a PLS and their normals.
3 The Optimization Algorithm
p3
Our algorithm generates an initial PLS and itera�
tively improves it. The PLS f2 is an improvement of
PLS f1 if C�f2� � C�f1�. The input to the algorithm
is f�xi� yi� zi�n i�1g � the set of data samples� and � �
the smoothing parameter. The output of the algo�
rithm is T � a triangulation of V � f�xi� yi�n i�1g� and
f�z 0 i�ni�1g � the values of the PLS at the points of V .
An outline of the algorithm is �
Generate an initial PLS.
Repeat
�1� Improve the PLS values at points of V
�keeping the triangulation fixed�.
�2� Improve the triangulation of V
�keeping the PLS values at V fixed�.
Until �there is no change in the PLS�
The initial PLS f 0 is T � the Delaunay triangulation
of V � and z 0 i � zi� i � 1� ��� n. To improve the PLS
values at the set of vertices V � we systematically scan
the vertices and set z 0 i to be the z minimizing
h�z� � �zi � z� 2 � �
n2
X
e2Ei
p4
ABN�e� �5�
where Ei�fthe edges of f 0 incident on �xi� yi�g and
f 0 �xi� yi� � zi. Note that the second term of h�z�
implicitly depends on z. The z minimizing the one�
dimensional function h�z� may be found by standard
numerical optimization procedures. We used the sim�
ple golden section method ��11�� Chap. 10�. It is easy
to see that assigning this value to z 0 i improves the PLS.
To improve the triangulation of V � we use the same
LOP �Lawson Optimization Procedure� used in �5��
for every edge e which is a diagonal of a convex quadri�
lateral of T� replace e by the other diagonal of the
quadrilateral �replacing the two triangles by two oth�
ers� if this improves the resulting PLS �see Fig. 2�.
It is not obvious that this algorithm converges.
However� we have found in all our experiments that
this is indeed the case� requiring 6 iterations on the
average. We do not know whether the minimum
achieved is global.
p1
p2
p3
p4
Figure 2� Swapping edges in a convex quadrilateral.
4 Determining the �Optimal� �
The output of our algorithm obviously depends on
the value of � used. A large � causes the surface to be
smooth� while a small � forces the f 0 �xi� yi� to be close
to zi� strongly constraining the solution. An impor�
tant practical question is how to determine the opti�
mal value of � so that the resulting PLS f 0 will indeed
be a good approximation of the original �unknown�
surface f�x� y�. De�ne
e 2 ��� � 1
n
nX
p1
p2
p3
p4
�zi � z
i�1
0 i����2 � �6�
where z 0 i ��� are the values of the PLS f0 �xi� yi� pro�
duced by our algorithm when applied with parameter
�. If z 0 i were indeed the �true� values f�xi� yi�� then
e 2 ��� would be a good estimate of the sample error
variance� which we know to be � 2 . The optimal �
should then be
�opt � arg min H���� H��� � je 2 ��� � � 2 j � �7�

An approach similar to this was proposed by Reinsch
�13� in the context of C 2 smoothing splines. Finding
this �opt requires a search procedure that applies the
PLS construction algorithm for di�erent values of ��
calculates H��� and re�estimates � accordingly.
5 Experimental Results
We tested our algorithm on one set of real DTM
data and samples of two di�erent �synthetic� test
functions f � �0� 1�2 � IR. For the DTM� we started
with a set f�xi� yj� zi�j�g 100
i�j�1
of 100 � 100 data points
on a regular grid in the Dead Sea area �Fig. 3�a��. We
randomly selected 100 points from this set and added
to them Gaussian noise with � � 15m. These 100
samples� and a chosen �� served as input to our algo�
rithm� producing a PLS f 0 as output. To estimate the
PLS quality we calculated the l1 distance jjf � f 0jj1� as in �2�. The two test functions were taken from �5��
who adopted them from �6� and �8��
F1�x� � 3
4 exp
�
�9x � 2�
� 2 � �9y � 2�2� 4
� 3
4 exp
�
�9x � 1�
� 2 �
9y � 1
�
49 10
� 1
2 exp
�
�9x � 7�
� 2 � �9y � 3�2� 4
� 1
5 exp���9x � 4�2 � �9y � 7�2� where�
F8�x� � tanh��3g�x� y�� � 1
g�x� y� � 0�595576�y � 3�79762� 2 � x � 10
The function F1 is composed of two Gaussian peaks
and a sharp Gaussian dip �Fig. 3�c��. The func�
tion F8 simulates a sharp rise� whose contour lines
are the parabolas g�x� y� � const �Fig. 3�b��. Each of
the functions was sampled on 100 random points dis�
tributed uniformly in �0� 1� 2 and contaminated with
Gaussian noise of variance � 2 . These samples� and a
chosen �� served as input to our algorithm. Again� the
performance of the algorithm was measured by the l1
distance �1�. In practice� this integral was computed
numerically by Monte�Carlo integration. Note that
this can be computed only for synthetically generated
samples� such as ours� where f is available. To eval�
uate our results� and contrast them with those of �5��
for each input data set we constructed three PLS�s.
The �rst obtained using z 0
i � zi and T � the Delau�
nay triangulation �refered to as the Delaunay PLS�.
The second with z 0
i � zi and T � the triangulation
obtained by applying the LOP procedure on the De�
launay triangulation as suggested in �5� �refered to as
the LOP PLS�. The third is the �nal PLS obtained
using our algorithm �refered to as the Optimal PLS�.
5.1 Accurate Data
Although our algorithm was designed primarily for
noisy sample sets� we applied it also on relatively ac�
curate �very small �� samples of F1 and F8. With this
type of input� the LOP procedure on the Delaunay tri�
angulation seems to always improve the PLS� so it is
advantageous to swap the steps of our algorithm� such
that the �rst step improves the triangulation� and the
second the PLS values at the triangulation vertices.
Fig. 5 demonstrates that when � � 0� although the
LOP PLS reduces the distance by 48� relative to the
Delaunay PLS �as has been shown already in �5��� al�
lowing the sample values to move by applying our al�
gorithm with a small value of �� results in a PLS with
distance further improved by another 58� �relative to
the LOP PLS�. This is especially true for functions
with a clear prefered direction� such as F8� as the ex�
tra freedom allows the triangulation to align itself in
this prefered direction. For F1� as demonstrated by
Fig. 7� there is no signi�cant improvement.
5.2 Noisy Data
The main bene�t of our algorithm was in the case
of relatively noisy data �approximately 10� error in
the elevation values�. Fig. 4 shows our results on the
DTM data� and Figs. 8 and 6 show our results on F1
and F8. These are for the optimal values of �� found
experimentally.
The results of our procedure on the noisy DTM
data were not as good as those obtained for the noisy
synthetic data. This is probably due to the fact that
the original terrain surface is not very smooth� and
there are no signi�cant prefered directions.
For F1 and F8� the LOP PLS is not an improvement
over the Delaunay PLS �for F1 it is even worse�. In
contrast� the optimal PLS produced by our algorithm
reduces the distance relative to the Delaunay PLS by
15� and 23� respectively.
In all cases� the optimal value of � seems to be a
little smaller than that predicted by �7�� as was also
observed by Craven �3� in the case of C 2 smoothing
splines.

6 Summary and Conclusions
We have presented an algorithm generating a good
piecewise�linear approximation of a surface over some
triangulation from noisy samples. The algorithm pro�
duces the best results in one of the following cases�
1. The sampled function has a clear prefered direc�
tion �like F8�� It seems that the �exibility in the
PLS values at the triangulation vertices enables
the LOP to perform more edge swaps than when
the heights are constrained to �xed values. These
additional swaps improve the PLS. This is true
even when there is no noise.
2. The noise is signi�cant� The �rst LOP damages
the PLS quality because this procedure is very
sensitive to the sample values� which are very in�
accurate. Using our algorithm when �rst adjust�
ing the PLS heights reduces some of the noise�
enabling the LOP to perform better.
There are a few possible variations on our algorithm�
including the type of metric used to measure the dis�
tance� the traversal order of the PLS vertices during
the �rst step of the algorithm and the traversal order
of PLS convex quadrilaterals during the second step of
the algorithm. The exact threshold of � before which
the samples are considered relatively accurate� there�
fore bene�cial to reverse the order of the algorithm
steps� is not yet clear.
Acknowledgments
The second author wishes to thank Nira Dyn and
David Levin for helpful discussions on the subject of
the paper. The DTM data used in our experiments
was produced and kindly made available by John Hall
of the Israel Geological Survey.
References
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�3� P. Craven and G. Wahba. Smoothing noisy data
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�4� E.F. D�Azevedo. Optimal triangular mesh gen�
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�16� G. Wahba. Spline Models for Observational Data.
Society for Industrial and Applied Mathematics�
1990.

y
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Figure 3� Test cases� �a� DTM. �b� F8. �c� F1.
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Figure 4� PLS�s on a 100�point sample of DTM data �Fig. 3�a�� with � � 15m� �a���b� Delaunay PLS �l1 distance
� 13.3�. �c���d� LOP PLS �l1 distance � 14.0�. �e���f� Optimal PLS at � � 150 �l1 distance � 12.8�.
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Figure 5� PLS�s on a 100�point sample of F8 with � � 0� �a���b� Delaunay PLS �l1 distance � .050�. �c���d� LOP
PLS �l1 distance � .026�. �e���f� Optimal PLS at � � �005 �l1 distance � .011�.
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Figure 6� PLS�s on a 100�point sample of F8 with � � �1� �a���b� Delaunay PLS �l1 distance � .098�. �c���d�
LOP PLS �l1 distance � .097�. �e���f� Optimal PLS at � � �01 �l1 distance � .075�.
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Abstract
This paper presents a technique for performing
volume morphing between two volumetric datasets in the
wavelet domain. The idea is to decompose the volumetric
datasets into a set of frequency bands, apply smooth
interpolation to each band, and reconstruct to form the
morphed model. In addition, a technique for establishing a
suitable correspondence among object voxels is presented.
The combination of these two techniques results in a
smooth transition between the two datasets and produces
morphed volume with fewer high frequency distortions
than those obtained from spatial domain volume morphing.
1. Introduction and motivation
Recently, 3D metamorphosis, the process of
simulating the deformation of one 3D model to another,
has gained popularity in animation and shape design.
Previously published techniques [5, 6] deal mainly with
the metamorphosis between two polygonal-based models.
The general method of these algorithms is to displace the
vertices, edges, and faces of the first model over time to
coincide in position with the corresponding vertices, edges,
and faces of the second model. However, establishing the
suitable correspondence among surface elements is
complex. In addition, these algorithms generally impose
topological restrictions on the models in order to maintain
the face connectivity during morphing.
Motivated in part by the difficulties presented in
morphing surface-based 3D models and in part by the
desire to morph sampled/simulated datasets directly, the
volume graphics [4] approach represents the 3D models in
voxel space and performs volume morphing between the
two volumetric models. One of the main advantages of this
approach is that the topology restriction on the datasets is
eliminated, since there is no explict topology description of
the volume and the voxel correspondence can be directly
established between any two volumes. However, the
problem of finding the appropriate correspondence among
voxels still exists. One naive solution is simply to crossdissolve
between the two volumes over time. In other
words, to morph from model g(x, y, z) to f (x, y, z), a new
model k t(x, y, z) = (1 − t)g(x, y, z) + tf(x, y, z) isformed.
Wa velet-Based Volume Morphing
Taosong He, Sidney Wang, and Arie Kaufman
Department of Computer Science
State University of New York at Stony Brook
Stony Brook, NY 11794-4400
Although simple, this method is often ineffective for
creating the smooth transition from one model to another.
For example, given two concentric spheres with identical
iso-values but with different radii, an ideal morphing from
the larger sphere to the smaller one should be a sphere with
constant iso-value but a gradually shrinking radius. But
the naive technique described above would generate a
sudden shrinkage from the large sphere to the small one at
a certain time T when a surface rendering method is
employed. This is because when 0 ≤ t < T , the region
between the two spheres has the density value above the
iso-value, and at time t = T , the density values of the
region uniformly drop below the iso-value.
Another problem when performing volume
morphing is that the direct interpolation of the highfrequency
components in the models might cause
distortion and unsatisfactory results. Hughes [3] deals
with this problem by performing volume morphing in the
Fourier domain. Basically, his approach takes the first
volumetric model, gradually removes the high frequencies,
interpolates over to the low frequencies of the second
model, and then smoothly adds in the high frequencies of
the second model. Although effective in reducing high
frequency distortion, the technique does not solve the
problem of unsmooth transformation of iso-surfaces
because the Fourier transform does not localize in spatial
domain. In Hughes’ implementation, in order to have a
smooth transition during morphing, the voxel values of the
entire volume are modified according to the distance to the
nearest iso-surface. Hence, new datasets need to be created
solely for the morphing application.
In this paper, a technique for performing volume
morphing in the wav elet domain is introduced. Since
wavelet transform localizes in both frequency domain and
spatial domain, the problems of high frequency distortion
and unsmooth transformation can be alleviated
simultaneously. The idea is to decompose the volumes
into a set of frequency bands, apply smooth interpolation
between the volumes to each band, and then reconstruct
the morphed volume. Furthermore, the decomposition and
reconstruction processes are accomplished in a
multiresolution fashion so that high frequency distortion

can be adjusted to the desired level. By taking advantage
of the spatial information within each frequency band, we
can extract and correspond the object voxels of the first
model to the object voxels of the second model
intelligently. In the next section, the volume
correspondence problem is presented. In Section 3
wavelet theory is briefly introduced. Wav elet application to
volume morphing in described in Section 4.
2. The correspondence problem
Unlike polygonal-based modeling, where every
surface element of the mesh contributes to the modeling of
apart of an object, voxel-based modeling captures not
only the object itself but also the space surrounding the
object. Hence, in order to have a gradual deformation
from one object to another, itisessential to map only those
voxels which belong to parts of an object. In our
implementation, iso-values are used to distinguish these
voxels from empty space. The algorithm for solving the
correspondence problem is first described in 1D space,
followed by the extension of the algorithm into 3D space.
Given anobject A in a 1D raster and an object B in another
1D raster, the first step of the algorithm is to classify these
two rasters into segments of object and non-object.
Without loss of generality, let object A consist of m object
segments, object B consist of n object segments, and
m ≥ n (Figure 1). As in the case of surface-based
morphing, the two important quality criterion are
maintaining the correct topology and minimizing the shape
distortion during transformation. First, to satisfy the
topology criterion, each object segment in A can be
mapped to only one object segment in B. This restriction
is needed to ensure that the number of object segments
does not increase during morphing. Under the condition
that the first criterion is satisfied, object segments of A
should be distributed onto object segments of B as
"evenly" as possible to minimize the shape distortion.
.
T = 0
T = 0.5
T = 1
Object A’s segments
Object B’s segments
Figure 1:1DCorrespondence Problem
Formally, the 1D correspondence problem is stated as
follows. Given:
determine:
where
subject to:
and
A = ⎧ ⎨a1a 1′, a2a2′,..., ama m′
⎩
⎫ ⎬, ⎭
B = ⎧ ⎨b1b1′, b2b2′,..., bnb n′
⎩
⎫ ⎬, m ≥ n
⎭
P = {p 1, p 2,..., p n} and
W = {w 1, w 2,...,w n}
pi = ⎧ ⎫
⎨x j,...,x k⎬,
1 ≤ i ≤ n, 1 ≤ j ≤ k ≤ m,
⎩ ⎭
and x l ∈A for j ≤ l ≤ k
wi = Σ (al′−al) j ≤ l ≤ k
1. (x u ∈p i1 Λ x v ∈p i2 Λ i 1 < i 2) → (u < v)
for all u, v
Σ
2. min n
i=1
⎛
⎜ wi ⎜
⎜
(bi′−bi) ⎝
−
m
i=1
Σ(ai′−ai) n
i=1
Σ(b
⎞
⎟
⎟ .
i′−bi) ⎟
⎠
In Equation 2, the set P represents the partition of A‘s
object segments into n partitions, and the set W represents
the corresponding weight for each member of P. As
shown in Equation 4, the weight for each partition is the
total length of object segments within each partition.
Equation 5 guarantees that the partition is in consecutive
order from left to right. Equation 6 insures that object
segments of A are "evenly" distributed onto object
segments of B by minimizing the variance.
In our implementation, dynamic programming has
been used to solve the 1D correspondence problem. Once
the correspondences have been established for the object
segments in A and B, each object segment in B needs to be
partitioned to accommodate the corresponding object
segments from A. For example, if
2
(1)
(2)
(3)
(4)
(5)
(6)

.
Correspond_3D ( A, B )
Volume_Data *A, *B;
{
}
Z1 = Object_Segments ( A, Z_AXIS );
Z2 = Object_Segments ( B, Z_AXIS );
Correspond_1D ( Z1, Z2 );
for each non-empty scan plane u in A
{
}
u’ = the corresponding scan plane in B;
Y1 = Object_Segments ( u , Y_AXIS );
Y2 = Object_Segments ( u’ , Y_AXIS );
Correspond_1D ( Y1, Y2 );
for each non-empty scan line v in u
{
}
v’ = the corresponding scan line in u’ ;
X1 = Object_Segments ( v , X_AXIS );
X2 = Object_Segments ( v’ , X_AXIS );
Correspond_1D ( X1, X2 );
Figure 2:Pseudo-code for the 3D Correspondence
pi = ⎧ ⎨a ja j′,..., ak ak′ ⎩
⎫ ⎬
⎭
then segment b ib i′ of B needs to be partitioned into
k − j + 1sub-segments of lengths
where
⎧
⎨(a
j′−a j) r,...,(ak′−ak) r
⎩
⎫ ⎬
⎭
r = (bi′−bi) k
Σ(al′−al) l= j
.
The correspondence problem in 3D space is
essentially accomplished by applying the above 1D
(7)
(8)
(9)
correspondence algorithm, which we call Correspond_1D,
to each of the three principal axes in an nested fashion.
The 3D correspondence algorithm, described in the
pseudo-code presented in Figure 2, establishes a mapping
from object scan-planes of A to object scan-planes of B,
from object scan-lines of A to object scan-lines of B, and
finally from object voxels of A to object voxels of B. With
these correspondence relations, volume morphing is
achieved through the interpolations over time of the
corresponding scan-planes, scan-lines, and voxels.
3. Wav elet theory
Generally, the high frequency components in 3D
functions represented by volumetric data tend to generate
small wiggles on the iso-surfaces of the models [3]. If the
morphing algorithm described above is directly performed
in the spatial domain, these wiggles could cause distortions
on the iso-surfaces of the intermediate functions (see
Figure 3a). Recently, wav elet theory, which is rooted in
time-frequency analysis, has been widely used in a variety
of applications, such as shape description of volumetric
objects [8] and radiosity [2]. Since a wav elet transform has
local property in both spatial and frequency domain, it is
an ideal solution to the problem of high frequency
distortion during morphing. In this section wav elet theory
is briefly introduced, then in Section 4 the wav elet-based
morphing algorithm is presented.
Multiresolution signal analysis decomposes a
function into a smooth approximation of the original
function and a set of detailed information at different
resolutions [7]. Formally, let L 2 (R) denote all functions
with finite energy; the smooth approximation of a function
f ∈L 2 (R) atany resolution 2 i is a projection denoted as
A 2 i: L 2 (R) → V 2 i, V 2 i ∈L 2 (R), and the detail of f at any
higher resolution 2 j is a projection of f onto a subspace
O 2 j of L 2 (R) denoted as P 2 j: L 2 (R) → O 2 j, j ≥ i.
Consequently, the finest detailed information is contained
in P 2 j with the highest resolution. By choosing the
appropriate projection functions such that O 2 j are
orthogonal to both each other and V 2 i,wehav e V 0 = L 2 (R)
and L 2 (R) =+ 1 j = i O 2 j + V 2 i (when O 2 j is the orthogonal
complement of V 2 j, V 2 j+1 is written as V 2 j+1 = V 2 j + O 2 j ).
For the discrete functions, it can be proven that there exist
two families of functions:
ψ j, n = 2 − j /2 ψ (2 j t − n) n ∈ Z
φ j, n = 2 − j /2 φ(2 j t − n) n ∈ Z,
(10)
(11)
which constitute the orthonormal basis of V 2 j and O 2 j,
respectively. ψ j,n are called wavelets and φ j,n are the
corresponding scaling functions.

Using wav elets and scaling functions, the discrete
detail signal and discrete approximation at resolution 2 j
are respectively defined as:
(D 2 j f ) n = 2 − j /2 < f (u), ψ j ,n >
(A d 2 j f ) n = 2 − j /2 < f (u), φ j,n >
(12)
(13)
and the detailed information and smooth approximation
are:
P O2 j f =Σ∞ n =−∞ (D 2 j f ) n ψ (2 j t − n)
A 2 j f =Σ ∞ n =−∞ (A d 2 j f ) n φ(2 j t − n).
(14)
(15)
Instead of calculating the inner products in Equations 12
and 13, a pyramidal algorithm [7] is applied for the
decomposition of the function (Figure 4a), where
H = H(−n) and G(n) = G(−n). The impulse response of
the filters used is defined as:
H(n) =
G(n) =
(16)
(17)
By repeating the algorithm for −1 ≥ j ≥−M, both the
discrete detail signal and the discrete approximation at
resolution 2 j can be computed. Using the same pair of
filters, the original discrete samples can be computed by
the reverse pyramidal algorithm, as shown in Figure 4b.
.
2
. 2
2 j
d
A
2 j
d
A
D 2 j
+1
f
f
f
2
2
G
H
: 1 sample out of 2
: multiplication by 2
(a)
(b)
H
G
X
2
2
2
D 2 j
2 j
d
A
f
f
. 2
2 j
d
A
+1
: a 0 between 2 samples
: convolve with filter X
Figure 4:Wav elet Decomposition and Reconstruction
f
Wa velet theory can be easily expanded to any
dimension by constructing high dimension orthonormal
wavelets using the tensor product of several subspaces of
L 2 (R) [7, 8]. To decompose or reconstruct a 3D function,
the one dimensional pyramidal algorithm described in
Figure 4 is applied sequentially along the principal axes.
Since the convolution along each axis is separable, for a
volume of size n 3 ,the decomposition and reconstruction
can be implemented in O(n 3 )time, which is asymptotically
optimal. Figure 5shows that the smooth approximation of
a volume at resolution 2 j+1 decomposes into a smooth
approximation at resolution 2 j and the discrete detail
signals along seven orientations. Since the wav elets and
scaling functions are orthogonal, the multiresolution
representation
(A d 2 −M f , (D i 2 j f ) −M ≤ j ≤−1, 1 ≤ i ≤ 7)
(18)
has the same total number of samples as the original signal
A d 1 f .
4. Wav elets for morphing
Equation 10 and Equation 11 indicate that both the
wavelets and scaling functions are the translation and
dilation of a mother function ψ (t) or φ(t). It can thus be
proven that the wav elet decomposition has the property of
both spatial and frequency locality. The algorithm shown
in Figure 4a can be interpreted as the separation of the
detailed information, which corresponds to high pass
filtering, and the generation of a smooth function, which
corresponds to low pass filtering. In addition, a signal that
is nonzero only during a finite time span has a wav elet
transform whose nonzero elements are concentrated
around that time. For a 3D volume, this means that the
spatial information, which is essential for volume
morphing, is maintained in wav elet domain.
.
2 j
d
A
+1
f
4 5
6 7
D j f D j f
2 2
D j f D j f
2 2
2 j
d
A
1
D j
2
Figure 5:Decomposition of a Volume Discrete Approximation
by a 3D Wav elet Transform
f
f
D
3
2 j
f

The basic idea of wav elet-based morphing is to solve
the correspondence problem between the general shape of
the objects without the interference of high frequencies. To
achieve this, the two volumes f and g are first decomposed
into smooth approximation at resolution 2 −M and the
detailed information:
(A 2 −M f , (P O2 j f ) −M ≤ j ≤−1) and
(A 2 −M g, (P O2 j g) −M ≤ j ≤−1).
(19)
Next, the direct morphing algorithm, which is described in
Section 2, is applied on the smooth approximation
A 2 −M f and A 2 −M g to generate a smooth approximation
A 2 −M k of an intermediate model. Then, the same
correspondence relations found between A 2 −M f and A 2 −M g
are employed for the interpolation between the detailed
information of f and g to generate PO2 k. Finally, the
j
morphed model k is reconstructed from the smooth
approximation A 2 −M k and detailed information P O2 j k.
To establish the correspondence between the smooth
approximation A 2 −M f and A 2 −M g, anatural approach is to
first reconstruct the smooth approximation of the functions
from the discrete approximations A d 2−M f and Ad 2−M g at
resolution 2 −M back to the original resolution using the
expansion of Equation 15 in 3D. Then, the correspondence
problem can be solved at the original resolution. This
approach can generate good results, but is computationally
expensive because of the reconstruction process.
Another approach is to directly establish the
correspondence between the discrete approximation A d 2−M f
and A d 2−M g (see Figure 6). The reason why this method is
reasonable is that the filter H used in the decomposition
can be seen as a low pass filter. Consequently, A d 2−M f and
A d 2−M g can be interpreted as the representation of the
original functions at a lower resolution, and the
correspondence between A d 2−M f and Ad 2−M g presents the
relation between the general shape of the two objects. In
addition, given the correspondence, since the scaling
.
2 j
d
A
2 j
d
A
f
g D 2 j
D 2 j
C
f
g
I
2 H
. 2
2 H . 2
D j+1 f 2
I 2 G
I
D j+1 g 2
C : correspondence I
2
: interpolation
Figure 6:Wav elet Morphing Algorithm
functions ψ (2 j t − n) are all the translations of a single
function ψ (2 j t), interpolation before or after
reconstruction using Equation 15 would generate the same
intermediate model. Unlike the first approach, there is no
need to reconstruct the smooth approximation at the
original resolution. In addition, since the time of solving
the correspondence problem depends on the size of
volume, it is much cheaper to establish the correspondence
at a lower resolution.
As for the high frequency components, the discrete
detail signals are interpolated at the same resolution and
the same orientation using the same correspondence
relation found between the smooth approximation. Again,
the theoretical base of this approach is the spatial and
frequency locality of the wav elet transform. To establish
the correspondence between D i 2 j f and Di 2 j g when j > M,
we treat the subvolumes with size of (2 j−M ) 3 as unit
volume elements in D i 2 j f and Di 2 j g.
Once we have the multiresolution representation,
different interpolation schedules, similar to [3], are applied
at different resolutions. The effect is the blending of the
general shape of the two models, with the gradual removal
of the high frequencies of the first model and the gradual
appearance of the high frequencies of the second model.
This is demonstrated in Figure 3b and Figure 7b, where
schedules are designed so that finer details of the first
model disappear faster than the coarser details, while the
coarser details are blended in before the finer details for
the second model.
An advantage of wav elet multiresolution
representation is that the detailed information along seven
different orientations is separately saved in
D i 2 j f (1 ≤ i ≤ 7). Although not implemented yet, different
schedules can be designed for the detail signals at the same
resolution but with different orientations, since visual
sensitivity depends not only on the frequency components
but also on the orientation of the stimulus [7]. In
summary, bydesigning different schedules, high frequency
G
....

distortion can be adjusted to the desired level (ev en
magnified, if desired) and different morphing effects can
be achieved.
Another flexibility of the wav elet-based morphing is
the wide selection of wav elets that can be employed, since
there is an infinite number of wav elets with different
characteristics. In our implementation, the Battle-Lemarie
wavelet is used for its symmetry and exponential decay.
5. Results and conclusions
We hav e presented a technique for performing
volume morphing in the wav elet domain. The advantage
of our method over aFourier volume morphing [3] is that
our approach not only effectively reduces high frequency
distortion, but also establishes a suitable correspondence
between the two volumetric datasets without the data
modification process.
The wav elet-based morphing technique presented
can be applied to sampled, simulated and modeled
geometric datasets. Tw o sequences of morphing from a
CT scanned lobster to an MRI head are illustrated in
Figures 3a and 3b. In Figure 3a, volume morphing is
performed directly in the spatial domain, while in Figure
3b our technique of wav elet-based volume morphing is
applied to the datasets. The alleviation of high frequency
distortion is most apparent during the middle stages of the
animation in Figure 3b, where the morphing is performed
mainly on the general shape of the models. Similarly, the
comparison between spatial and wav elet-based morphing
for geometric datasets is shown in Figures 7a and 7b,
where in these two sequences a binary voxelized goblet is
morphed into a binary voxelized torus. It is shown that in
these two figures the correct topology is maintained during
morphing. Furthermore, in Figure 7b our technique of
wavelet-based morphing gradually alleviates the high
frequency distortion caused by the aliasing existed in the
original binary voxelized models. Binary voxelized models
were used here just to demonstrated the effectiveness of
the wav elet-based volume morphing. In practice, volumesampled
voxelized models [9] are used as original models,
resulting in even smoother wav elet-based morphing.
The multiresolution representation of the volume
can be explored for an adaptive morphing. We are
currently building a "previewer" so that the morphing can
be interactively performed at low resolutions. This kind of
"preview" tool is very useful for adjusting morphing
parameters in the interpolation schedules. By taking
advantage of the spatial locality property of wav elet
transform, subvolumes of the models can be selected for
morphing. Thus, similar to 2D feature-based morphing [1],
3D features of one volume can be extracted and mapped to
the desired feature of another. Weare currently developing
auser interface to accomplish this task. Other future work
includes the investigation of the wav elet selection for the
specific morphing effect, and the design of interpolation
schedules for the information along different orientations.
6. Acknowledgments
This work has been supported in part by the National
Science Foundation under grant number CCR-9205047
and the Department of Energy under the PICS grant. The
MRI head data for Figure 3 is courtesy of Siemens,
Princeton, NJ, and the CT lobster data is courtesy of AVS,
Waltham, MA. The authors would like to thank Steve
Skiena for his suggestions on the correspondence problem.
7. References
1. Beier, T. and Neely, S., ‘‘Feature-based Image
Metamorphosis’’, Computer Graphics, 26, 2 (July 1992),
35-42.
2. Gortler, S.J., Schroder, P., Cohen, M. F. and Hanrahan, P.,
‘‘Wa velet Radiosity’’, SIGGRAPH 93, August 1993,
221-230.
3. Hughes, J. F., ‘‘Scheduled Fourier Volume Morphing’’,
Computer Graphics, 26, 2(July 1992), 43-46.
4. Kaufman, A., Cohen, D. and Yagel, R., ‘‘Volume
Graphics’’, IEEE Computer, 26, 7(July 1993), 51-64.
5. Kaul, A. and Rossignac, J., ‘‘Solid-Interpolating
Deformations: Construction and Animations of PIPs’’,
Proceeding of EUROGRAPHICS’91, September 1991,
493-505.
6. Kent, J., Parent, R. and Carlson, W., ‘‘Establishing
Correspondence by Topological Merging: A New
Approach to 3-D Shape Transformation’’, Proceedings of
Graphics Interface’91, June 1991, 271-278.
7. Mallat, S. G., ‘‘A Theory for Multiresolution Signal
Decomposition: The Wav elet Representation’’, IEEE
Tr ansaction on Patten Analysis and Machine Intelligence,
11, 7(July 1989), 674-693.
8. Muraki, S., ‘‘Volume Data and Wav elet Transform’’, IEEE
Computer Graphics and Applications, 13, 4(July 1993),
50-56.
9. Wang, S. W. and Kaufman, A. E., ‘‘Volume Sampled
Voxelization of Geometric Primitives’’, Proceedings
Visualization ’93, San Jose, CA, October 1993, 78-84.

Figure 3a: Spatial Domain Volume Morphing from a CT Lobster to an MRI Head.

Figure 3b: Wav elet Domain Volume Morphing from a CT Lobster to an MRI Head.

Figure 7a: Spatial Domain Volume Morphing from a Binary Voxelized Goblet to a Bi nary
Voxelized Torus.

Figure 7b: Wav elet Domain Volume Morphing from a Binary Voxelized Goblet to a Bi nary
Voxelized Torus.

Please reference the following QuickTime movies located in the MOV
directory:
HE.MOV
HE_2.MOV
Copyright © 1994 by the Research Foundation of the State University of
New York at Stony Brook
QuickTime is a trademark of Apple Computer, Inc.

Progressive Transmission of Scientific Data Using
Biorthogonal Wavelet Transform
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NSF Engineering Research Center for Computational Field Simulation
P.O. Box 6176, Mississippi State University, Mississippi State, MS 39762
Abstract
An important issue in scientific visualization systems
is the management of data sets. Most data sets in scientific
visualization, whether created by measurement or simulation,
are usually voluminous. The goal of data management
is to reduce the storage space and the access time of these
data sets to speed up the visualization process. A new progressive
transmission scheme using spline biorthogonal
wavelet bases is proposed in this paper. By exploiting the
properties of this set of wavelet bases, a fast algorithm involving
only additions and subtractions is developed. Due
to the multiresolutional nature of the wavelet transform,
this scheme is compatible with hierarchical–structured rendering
algorithms. The formula for reconstructing the functional
values in a continuous volume space is given in a
simple polynomial form. Lossless compression is possible,
even when using floating–point numbers. This algorithm
has been applied to data from a global ocean model. The
lossless compression ratio is about 1.5:1. With a compression
ratio of 50:1, the reconstructed data is still of good
quality. Several other wavelet bases are compared with the
spline biorthogonal wavelet bases. Finally, the reconstructed
data is visualized using various algorithms and the
results are demonstrated.
1. Introduction
Progressive transmission is a good framework for data
management in scientific visualization when a large volume
of data is involved. The principle is to transmit the
least amount of data necessary to generate a usable approximation
of the original data. If the user requires further refinement,
more data is fetched into the memory and a higher–resolution
data set is reconstructed. See Fig. 1.
Progressive transmission algorithms based on the
DCT, tree–structured pyramids, bit plane techniques, etc.,
have been proposed [2]. Blandford [3] presented a scheme
based on a 8–tap discrete wavelet transform (WT) and concluded
it was very suitable for progressive transmission in
the case of non–uniform resolution. Muraki [6] applied the
truncated version of Battle–Lemarié wavelets to volume
data and proposed a fast superposition algorithm for reconstruction
of functional values in continuous space. However,
there are still some unsolved problems. Most transform–
based approaches slow down the system performance by
introducing a time–consuming decoding procedure. This
greatly limits their value in applications with strict speed
HH HL
LH
Original
Data
WT
HH HL
LH LL
Refinement Information
Refine–
ment
Control
super–
position
IWT
Rendering
algorithms
Reconstruct–
ed Data
Figure 1: Block diagram of progressive transmisson using
the wavelet transform.
requirements. In most visualization techniques such as isosurface
rendering and fluid topology extraction, the functional
values in the continuous space must be calculated
conveniently. By using an infinitely supported function as
the superposition basis, a complex approximation algorithm
has to be utilized. Another drawback is that since
most low bit–rate algorithms use transforms such as the
DCT or WT, lossless data compression is difficult because
the denominators of the coefficients often are not in the
form of 2 n .

In this paper, a scheme based on the biorthogonal
spline wavelet transform is proposed. The lengths of all filters
are less than 6. Experimental results indicate that they
perform better in most cases than a family of optimized
6–tap wavelet bases where the same filter is used in both the
decomposition and the reconstruction sides. Because all the
filter coefficients are dyadic rationals, multiplication and
division operations can be simplified to addition and subtraction
for floating–point numbers or to shifting for integer
numbers. This makes the transform much faster. This property
also makes lossless coding possible. In the case of the
biorthogonal wavelet, symmetry is more easily achieved,
resulting in a better handling of the data boundary. Because
the transform basis functions are compactly supported and
can be explicitly formulated in a polynomial form, the reconstruction
of function values from wavelet transform coefficients
is much easier.
2. Method
Basically, the scheme illustrated in Fig. 1 contains four
steps. The wavelet transform and entropy coding, the entropy
decoding and inverse wavelet transform, the refinement
strategy and the rendering algorithm. Since sophisticated
methods for entropy coding and refinement control exist
[2][3][4], we will focused on the wavelet transform and the
reconstruction of functional values in continuous space.
2.1 Biorthogonal Wavelet Transform
Because of its multiresolutional nature, the wavelet
transform is very suitable for hierarchical manipulations
such as progressive data transmission and tree–structured
graphics rendering. A simple WT decomposition and reconstruction
scheme [4] using quadrature mirror filters
(QMF) is shown in Figure 2.
Suppose that c j�1 is the original input at a given resolution
level j � 1. In the decomposition stage, data are
convoluted with a pair of filters h and g. h is often a low–
pass filter while g is often a high–pass filter. As a result, the
c j can be viewed as a coarser approximation of c j�1 . By
downsampling the filtered data, the total number of samples
in level j is the same as that of level j � 1. This procedure
can be repeated several times. After reconstruction,
the final result c ~ j�1 is identical to c j�1 . g is given by:
g n � (� 1) n h �n�1
There are many families of wavelet bases with reasonable
decay both in the time and the frequency domain. To
guarantee perfect reconstruction (PR), the filter can not be
truncated. This implies that the wavelet basis must be compactly
supported. Also, to more easily deal with the data
(1)
boundary, symmetric bases are preferred. However, it is
well known from wavelet theory that symmetry and perfect
reconstruction are incompatible if we use the transform in
Fig. 2. This difficulty can be overcome by using the biorthogonal
wavelet transform [1], which is diagrammed in
Fig. 3.
h
2�
2�
c j
c
+
j�1 ~ j�1
c
g
d j
put one zero between 2� keep one sample out
each sample of two
Figure 2: Block diagram of the basic wavelet decomposition
and reconstruction scheme. Only one h is needed.
c j�1
2�
2�
h convolve with h(n) g convolve with g(n)
h convolve with h(–n) g convolve with g(–n)
2�
h
g
2�
2�
c j
d j
Figure 3: Block diagram of the biorthogonal wavelet de–
composition and reconstruction scheme. and h are
different.
~
h
2�
2�
h ~
g ~
h
g
+ c~ j�1
In this implementation, h and h ~
are different but are
both symmetric . Also, g and g ~ satisfy:
g ~
n � (� 1)n h �n�1
gn � (� 1) n h ~
�n�1 (2)
If H(�) and H ~
(�) are the Fourier–transforms of h and h ~
, according
to [1], a sufficient condition on H and H ~
to make
them PR filters is:
� �
l�1
�l � 1 � p
p �· sin(��2)
p�0
2p � sin(��2) 2l H(�)H
R(�)�
~
(�) � cos(��2) 2l ·
where R(�) is an odd polynomial in cos(�), and
2l � k � k ~
, which means the length of h and h ~
(3)

should be both even or both odd. If R � 0 and if
H ~
(�) � cos(��2) k~
e j ���2 , where � � 0 if k ~
is even and
� � 1 if k ~
is odd, h and h ~
are called spline filters since the
related scaling function � ~
is a B–spline function. Table I
gives some example bases of this family. H ~
(z) and H(z) are
z–transforms of h and h ~
. Notice that the denominators of
the coefficients are all in the form 2 n . This property makes
them good candidates for our application. There is still
some freedom left to choose bases from this family depending
on implementation requirements. One basic principle
is if a set of longer bases is used, the frequency separation
property of h and g will be better , but at the expenses of
more computational complexity and a lower lossless compression
ratio�
k ~
H ~
TABLE I
(z) k H(z)
1 (1 � z) 1
1
(1 � z)
2
3 � z�2 z�1
� �
1
�
z z2 z3
� �
16 16 2 2 16 16
2
3
1
2 (z�1 � 2 � z) 2 � z�2
4
3z �4
128
8
� 3z�3
64
� z�1
4
� z�2
8
� z2
8
1
4 (z�1 � 3 � 1 � z�1
4
3z � z 2 )
3
3z �3
64
� 9z�2
64
� 7z2
64
� 3
4
� 19z�1
64
�
z z2
�
4 8
3z3 3z4
� �
64 128
� 3
4
� 7z�1
64
�
3z z2
�
4 4
9z3 3z4
� �
64 64
�
45
�
19z
64 64
�
45
�
45z
64 64
Our application involves rendering the data from eighteen
75 MB data files simultaneously. For efficient visualization,
the inverse wavelet transform must be very fast. A
system using the wavelet bases in Table I with k ~
� 2 and
k � 2 has been implemented. Fig. 4 shows the scale function
� ~
(x) and the wavelet function � ~ (x).
2.2 Fast Wavelet Transform
The efficient wavelet transform method introduced by
Mallat [4] borrowed the QMF scheme from subband coding
theory (Fig. 2). In this algorithm, the major computational
burden is caused by the convolution operation which in-
volves a lot of floating–point multiplication. Because h, h ~
,
g and g ~ are all symmetric in our scheme, the number of
multiplications can be reduced by approximately 50%. After
investigating the IEEE standard–floating point data format
(Fig. 5), it was found that for a floating–point number,
multiplication by 2 n can be simplified to the addition of n
to the exponent, while for an integer number only a shift
1.0
0.0
–0.5
2.0
1.0
0.0
–1.0
–1.0 0.0 1.0
–1.0 0.0 1.0
� ~
(x)
� ~ (x)
Figure 4: Scale function and wavelet function of
biorthogonal WT in Table I when k and .
~
�
� 2
~ � (x)
~ (x)
k � 2
operation is needed. This fact was exploited to develop a
multiplication–free WT algorithm.
8–BIT
EXP.
23–BIT
FRACTION
SIGN OF FRACTION
2.0
SINGLE REAL
Figure 5: Single precision (32 bits) floating–point data for–
mat defined by IEEE 754 standard.
2.3 Superposition
To combine a progressive transmission scheme with a
visualization algorithm such as isosurface generation or
ray casting, the functional values in the continuous volume
must be approximated efficiently from the transform coef-
ficients. In a general 1–D case, if the WT with k ~
� 2 and
k � 2 is used, the function value at x can be reconstructed
using the following formula:
f ^ j��
(x) � �
j���
� ~
K, j
i�K
(x) � �
j��
i�1
j���
� ~
i, j (x)
where � ~
K, j (x) � 2�K � � ~
( 2�K � j )
� ~
i, j (x) � 2�i � � ~ ( 2�i � j )
K � levels of the WT (4)
When � ~
(x) is a set of compactly supported piecewise B–

spline functions, f(x) can be written in a closed polynomial
form with order k ~
� 1. As shown in Fig. 4, the � ~
(x) and
� ~ (x) in our algorithm are of order 1. The process of reconstructing
the function values in a continuous space from the
WT coefficients is illustrated in Fig. 6. In each interval [i,
i+1) , the reconstructed functions are linear functions. Thus
(4) can be simplified to:
f ^ (x) � (1 � q ) · f ^ (l) � q · f ^ (l�1)
where x � [l, l � 1)
q � x�l
This suggests that the exact function value at any real x can
be computed through linear interpolation of the function
values at the two discrete neighbors which surround x. In
this case, the resulted f ~
(x) is C 0 continuous. When the order
of � ~
(x) and � ~ (x) is larger than 1, we can still write (4)
in a closed polynomial form with order of k ~
(x) will
� 1. f ~
be C k~ �2 continuous. If we simply take the tensor product
of the one–dimension bases as the bases for the multidimension
WT, the same conclusion can be derived.
2.4 Lossless Compression is Possible
IEEE single–precision floating–point numbers have
23+1 bits of precision. Multiplication by 2 n will require no
extra bit to retain the same precision. However, addition
may result in some extra bits. As an example, consider the
WT in our scheme. For h, the convolution formula is:
cn � � 1
8 f n�2 � 1
4 f n�1 � 3
4 fn � 1
4 f n�1 � 1
8 f n�2
Where f i is the discrete input function value. Suppose the
maximum and minimum exponents of all five f i values are
Pmax and P min. Then the maximum number of extra bits we
need to record the precise c n is:
(5)
(6)
be � Pmax � P min � 3 (7)
Similarly, for h ~
, the maximum number of extra bits needed
is no more than:
b ~
e � Pmax � P min � 2 (8)
This is the worst case. Usually b e and b ~
e are less than the
value given here. To increase the lossless compression ratio,
the fewest extra bits for each coefficient in the transform
domain is desired. Recording the information about
extra bits using an oct–tree has produced good results.
...
...
–4
–4
. . . . . .
–4
C 1 �1 �~
1,�1
–2
d 1 �2 �~
1,�2
–2
–2
d 1
�1 �~
1,�1
C 1 0 �~
1, 0
C 1 1 �~
1, 1
0 2 4
d 1 0 �~
1, 0
d 1 1 �~
1, 1
0 2 4
f ^
(x)
0 2 4
...
+
=
...
. . . . . .
Figure 6: Reconstruction (superposition) of function values
in continuous space using WT coefficients in our
scheme. Notice the f is linear function in the interval of
[i, i+1).
^
(x)
3. Results
We applied the scheme described in this paper to ocean
model data [14]. The model output was available on a
337 � 468 � 6 grid 120 times per year. Layer thickness
and current data in floating–point format — variables from
the model — were used in our experiments. Each of the
six time–varying layers were considered separately, the top
layer being of greatest interest.
First, we compared the spline biorthogonal wavelet
transform with a family of 6–tap WTs where all hi are functions
of two parameters — � and � — according to the formulae
�� ���� Several 1–D signal sequences with 120 samples
were randomly chosen from the ocean model data set.
For each test signal, we searched exhaustively in the � � �
x
x
x

h �2 � ��1 � cos � � sin �)· (1 � cos � � sin �) � 2 cos � sin �]�4
h �1 � ��1 � cos � � sin �)· (1 � cos � � sin �) � 2 cos � sin �]�4
h 0 � [1 � cos(� � �) � sin(� � �)]�2
h 1 � [1 � cos(� � �) � sin(� � �)]�2
h 2 � 1 � h �2 � h 0 h 3 � 1 � h �1 � h 1
where � � � �,� � � (9)
parameter space for the minimum MSE and compared it
with the MSE generated by the spline biorthogonal wavelet
compression algorithm. Table II gives the results of some
typical signals in our application. From these data, we concluded
that although the lengths of the spline biorthogonal
WT filters are less than 6, they perform the same or better
than the optimized 6–tap WTs in a scheme where both forward
and inverse transforms use the same filters.
TABLE II
Test Signal 6–Tap WT Biorth. Spline WT
� �
k MSE
~
120 samples Optimal Optimal MSE � 2, k � 2,
A –1.0210 0.4712 0.0241 0.0143
B 1.4137 0.8639 0.0805 0.0537
C 1.4922 1.0210 0.0806 0.0513
D –1.0210 0.3141 0.0099 0.0048
E 1.0995 0.3141 0.1521 0.0912
F –1.4137 –0.8639 0.2364 0.0831
G 1.1780 0.4712 0.1947 0.0786
To apply the biorthogonal WT to our data set, the
boundary conditions had to be considered. In each frame
(timestep) of ocean model data, valid function values were
only available within the ocean area (Fig. 7a). This area was
constant over time. A 2–D WT requires the ocean area be
approximated using square blocks. The blocksize used was
2 n � 2 n , where n was larger than the level of the WT in
each data frame. The approximated ocean area had to cover
the entire original ocean area (Fig. 7b). For grid points in
the approximated ocean area but not in the original area, interpolation
and extrapolation techniques were used to yield
function values. This introduced some discontinuity at the
data boundary. In our implementation, second order Lagrange
interpolation was used. Since all our WT bases are
symmetric, the function values outside the boundary were
achieved simply by reflection.
As shown in Figure 8, after applying the transform
twice in each spatial dimension in a frame and once in the
temporal dimension, only 1/32 of the transform coefficients
were transmitted to the decoder. The reconstructed
data had good quality and met the requirements of further
scientific visualization processes. By using the refinement
control strategy proposed by Blandford [3], only a small
amount of data was needed for a better reconstruction.
In Fig. 9, layer thickness, which is a scalar field, compressed
by our algorithm at the rate of 50:1 is shown. For
y
t
x
2, v dF c 2
F
1, v dF 2, d dF d 2,h
F
1, v dB 1, d dF 1, h dF 464
1, d dB 1, h dB 120
336
Figure 8: Wavelet transform coefficients of 3–D data of
ocean model.
a 2–D vector field, the two components are encoded independently.
Streamlines are generated both for the original
field and the field reconstructed using only 1/32 of WT coefficients,
the results are shown in Fig. 10.
Applying a topology extraction algorithm [10] to this
vector field, the global and the stable topology structures
remained relatively unchanged, even at a 16:1 compression
ratio. With progressive refinement, more and more local
and unstable features can be extracted (Fig. 11). This can
be explained as the low–pass effects of the WT.
The decoding speed, an important factor in interactive
visualization system, is about 10 frames/sec on an SGI Indigo
2 machine. This satisfies the need of fast volume rendering
in our application.
4. Conclusions and Future works
A new scheme of progressive transmission using a
spline biorthogonal wavelet transform is proposed in this
paper. This family of wavelet base functions is symmetric,
compactly supported, and the QMF coefficients are dyadic
rationals. These attractive features make this scheme advantageous
in several aspects. The transform itself is fast
and high compression ratios can be achieved. The reconstructed
data is of good quality and can be refined with a
small amount of additional data. Data boundaries can be
tackled gracefully, The reconstruction of function values in
continuous space from the WT coefficients is a simple
polynomial. This interesting property is especially useful
for scientific visualization applications.
Future work includes more investigation of the WT’s
effects on the topology of the flow field, combining this algorithm
with some other hierarchical scientific visualiza-

tion techniques, and constructing better wavelet bases.
5. Acknowledgments
We wish to thank Scott Nations for his many helpful
comments on an early version of this paper. This work has
been supported in part by ARPA and the Strategic Environmental
Research and Development Program (SERDP).
6. References
[1] I. Daubechies, Ten Lectures on Wavelets, CBMS–NSF Regional
Conference Series in Applied Mathematics, no. 61, Society for Industrial
and Applied Mathematics, Philadelphia, PA, 1992.
[2] Kou–Hu Tzou, “Progressive Image Transmission: A Review
and Comparison of Techniques,” Optical Engineering, 26(7), pp.
581–589, July 1987.
[3] R. P. Blandford, “Wavelet Encoding and Variable Resolution
Progressive Transmission,” NASA Space & Earth Science Data
Compression Workshop Proceedings, pp. 25–34, April 1993.
[4] S. G. Mallat, “A theory for multiresolution signal decomposition:
the wavelet representation,” IEEE Trans. Pattern Anal. Machine
Intell., vol. 11, no. 7, July 1989.
[5] M. A. Cody, “The Fast Wavelet Transform,” Dr. Dobb’s Journal,
vol. 17, no. 4, April 1992.
[6] S. Muraki, “Volume Data and Wavelet Transforms,” IEEE Computer
Graphics & Applications, pp. 50–56, July 1993.
[7] P. Ning and L. Hesselink, “Fast Volume Rendering of Compressed
Date,” Visualization ’93 Proceedings, pp. 11–18, 1993.
[8] M. Antonini, M. Barlaud, P. Mathieu and I. Daubechies, “Image
Coding Using Wavelet Transform,” IEEE Trans. on Image Processing,
vol. 1, no. 2, pp. 205–220, April 1992.
[9] R. H. Abraham and C. D. Shaw, Dynamics: The Geometry of Behavior,
part 1–4, Arie Press, Santa Cruz, CA, 1984.
[10] J. L. Helman and L. Hesselink, “Representation and Display of
Vector Field Topology in Fluid Flow Data Sets,” IEEE Computer,
pp. 27–36, Aug. 1989.
[11] O. Rioul and M. Vetterli, “Wavelets and Signal Processing,”
IEEE SP Magazine, pp 14–38, Oct. 1991.
[12] A. H. Tewfik, D. Sinha and P. Jorgensen, “On the Optimal
Choice of a Wavelet for Signal Representation,” IEEE Trans. on
Info. Theory, vol. 38, no. 2, pp 747–765, March 1992.
[13] J. N. Bradley and C. M. Brislawn, “Applications of Wavelet–
Based Compression to Multidimensional Earth Science Data,”
NASA Space & Earth Science Data Compression Workshop Proceedings,
pp. 13–24, April 1993.
[14] H. E. Hurlburt, A. J. Wallcraft, Z. Sirkes and E. J. Metzger,
“Modeling of the Global and Pacific Oceans,” Oceanography, vol.
5, no. 1, pp. 9–18, 1992.
[15] M. Rabbani and P. Jones, Digital Image Compression Techniques,
SPIE, vol. TT7, 1991.
Pacific Ocean
(a)
Approximated Ocean Area
(b)
North America
Land
Figure 7: Land and ocean areas in each data frame. (a)
Ocean area in original data (white area). (b) Approximation
of the ocean area using 8�8 blocks. This approximated
version entirely covers the original ocean area.

Step 19
Figure 9a. Layer thickness for the NE Pacific Ocean-
Original thickness data.
Figure 10a. Velocity data for the NE Pacific Ocean-
Original data.
Step 19
Figure 9b. Layer thickness for the NE Pacific Ocean-
Reconstructed data (compression ratio = 50:1).
Figure 10b. Velocity data for the NE Pacific Ocean-
Reconstructed data (compression ratio = 50:1).
Figure 11. Vector field topology extraction on reconstructed data. (a) Using original data. (b) Data reconstructed
using 1/4 WT coefficients. (c) Data reconstructed using 1/16 WT coefficients.

An Evaluation of Reconstruction Filters for Volume Rendering
Stephen R. Marschner and Richard J. Lobb †
Program of Computer Graphics
Cornell University, Ithaca NY 14853
srm@graphics.cornell.edu richard@cs.auckland.ac.nz
Abstract
To render images from a three-dimensional array of
sample values, it is necessary to interpolate between the
samples. This paper is concerned with interpolation methods
that are equivalent to convolving the samples with a reconstruction
filter; this covers all commonly used schemes,
including trilinear and cubic interpolation.
We first outline the formal basis of interpolation in
three-dimensional signal processing theory. We then propose
numerical metrics that can be used to measure filter
characteristics that are relevant to the appearance of images
generated using that filter. We apply those metrics
to several previously used filters and relate the results to
isosurface images of the interpolations. We show that the
choice of interpolation scheme can have a dramatic effect
on image quality, and we discuss the cost/benefit tradeoff
inherent in choosing a filter.
1 Introduction
Volume data, such as that from a CT or MRI scanner, is
generally in the form of a large array of numbers. In order
to render an image of a volume’s contents, we need to construct
from the data a function that assigns a value to every
point in the volume, so that we can perform rendering operations
such as simulating light propagation or extracting
isosurfaces. This paper is concerned with the methods of
constructingsuch a function. We restrict our attention to the
case of regular sampling, in which samples are taken on a
rectangular lattice. Furthermore, our analysis is in terms of
uniform regular sampling, in which we have equal spacing
along all axes, since the non-uniform case can be included
by a simple scaling.
Given a discrete set of samples, the process of obtaining
a density function that is defined throughout the volume
is called interpolation or reconstruction; we use these terms
interchangeably. Trilinear interpolation is widely used, for
example in the isosurface extraction algorithms of Wyvill,
et. al. [19], and Lorensen, et. al. [13, 4], and in many raytracing
schemes (e.g., that of Levoy [12]). Cubic filters
have also received attention: Levoy uses cubic B-splines
for volume resampling, and Wilhelms and Van Gelder mention
Catmull-Rom splines for isosurface topology disam-
† On leave from the Department of Computer Science, University of
Auckland, Auckland, New Zealand until August 1994.
biguation [18]. The “splatting” methods of Westover [17]
and Laur & Hanrahan [11] assume a truncated Gaussian filter
for interpolation, although the interpolation operation is
actually merged with illumination and projection into a single
fast but approximate compositing process. Carlbom [3]
discusses the design of discrete “optimal” filters based on
weighted Chebyshev minimization. All of these schemes
fall within the standard signal processing framework of reconstruction
by convolution with a filter, which is the model
we use to analyze them.
The process of interpolation is often seen as a minor
aside to the main rendering problem, but we believe it
is of fundamental importance and worthy of closer attention.
One needs to be aware of the limitations of interpolation,
and hence of the images produced, which are usually
claimed to represent the original density function prior to
sampling. Sampling and interpolation are also basic to volume
resampling, and the cost of using more sophisticated
interpolation schemes may well be outweighed by the potential
benefits of storing and using fewer samples.
2 Reconstruction Theory
2.1 Review of Fourier analysis
We will review Fourier analysis and sampling theory
in two dimensions to make diagrams feasible; the generalization
to three dimensions is straightforward. Some initial
familiarity is assumed; introductions to this subject can be
found in [6] and [7].
Fourier analysis allows us to write a complex-valued
function g : R 2 � C as a sum of “plane waves” of the form
exp�i�ωxx � ωyy��. For a periodic function, this can actually
be done with a discrete sum (a Fourier series), but for
arbitrary g we need an integral:
g�x�y� � 1
Z
2π R2 ˆg�ωx�ωy�e i�ωxx�ωyy�
dωx dωy
The formula to get ˆg from g is quite similar:
ˆg�ωx�ωy� �
Z
R 2 g�x�y�e�i�ωxx�ωyy� dxdy
One intuitive interpretation of these formulae is that
ˆg�ωx�ωy� measures the correlation over all �x�y� between
g and a complex sinusoid of frequency �ωx�ωy�, and that

g k gk
gˆ kˆ gˆ kˆ = gˆ k
Figure 1: Two-dimensional sampling in the space domain (top) and the frequency domain (bottom).
g�x�y� sums up the values at �x�y� of sinusoids of all possible
frequencies �ωx�ωy�, weighted by ˆg. We call ˆg the
Fourier transform of g, and j ˆgj the spectrum of g. Since
the Fourier transform is invertible, g and ˆg are two representations
of the same function; we refer to g as the space
domain representation, or just the signal, and ˆg as the frequency
domain representation. Of particular importance is
that the Fourier transform of a product of two functions is
the convolution of their individual Fourier transforms, and
vice versa: c gh � ˆg � ˆh; d g � h � ˆgˆh.
2.2 Basic sampling theory
We represent a point sample as a scaled Dirac impulse
function. With this definition, sampling a signal is equivalent
to multiplyingit by a grid of impulses, one at each sample
point, as illustrated in the top half of Figure 1.
The Fourier transform of a two-dimensional impulse
grid with frequency fx in x and fy in y is itself a grid of impulses
with period fx in x and fy in y. If we call the impulse
grid k�x�y� and the signal g�x�y�, then the Fourier transform
of the sampled signal, b gk, is ˆg� ˆk. Since ˆk is an impulse grid,
convolving ˆg with ˆk amounts to duplicating ˆg at every point
of ˆk, producing the spectrum shown at bottom right in Figure
1. We call the copy of ˆg centered at zero the primary
spectrum, and the other copies alias spectra.
If ˆg is zero outside a small enough region that the alias
spectra do not overlap the primary spectrum, then we can
recover ˆg by multiplying b gk by a function ˆh which is one
inside that region and zero elsewhere. Such a multiplication
is equivalent to convolving the sampled data gk with h,
the inverse transform of ˆh. This convolution with h allows
us to reconstruct the original signal g by removing, or filtering
out, the alias spectra, so we callh a reconstruction filter.
The goal of reconstruction, then, is to extract, or pass, the
primary spectrum, and to suppress, or stop, the alias spectra.
Since the primary spectrum comprises the low frequencies,
the reconstruction filter is a low-pass filter.
It is clear from Figure 1 that the simplest region to
which we could limit ˆg is the region of frequencies that are
less than half the sampling frequency along each axis. We
call this limiting frequency the Nyquist frequency, denoted
fN, and the region the Nyquist region, denoted RN. We define
an ideal reconstruction filter to have a Fourier transform
that has the value one in the Nyquist region and zero
outside it. 1
2.3 Volume reconstruction
Extending the above to handle the three-dimensional
signals encountered in volume rendering is straightforward:
the sampling grid becomes a three-dimensional lattice, and
the Nyquist region a cube. See [5] for a discussion of signal
processing in arbitrary dimensions.
Given this new Nyquist region, the ideal convolution
filter is the inverse transform of a cube, which is the product
of three sinc functions:
hI�x�y�z� � �2 fN� 3 sinc�2 fNx� sinc�2 fNy� sinc�2 fNz��
Thus, in principle, a volume signal can be exactly reconstructed
from its samples by convolving with hI, provided
that the signal was suitably band-limited 2 before it
was sampled.
In practice, we can not implement hI, since it has infinite
extent in the space domain, and we are faced with
choosing an imperfect filter. This will inevitably introduce
some artifacts into the reconstructed function.
3 Practical reconstruction issues
The image processing field, which makes extensive use
of reconstruction filters for image resampling (e.g., [10,
16, 15]), provides a good starting point for analyzing volume
reconstruction filters. In particularly, Mitchell and
1 Other definitions of an ideal filter are possible—for example, a filter
h such that ˆh is one inside a circle of radius fN.
2 A signal is band-limited if its spectrum is zero outside some bounded
region in frequency space, usually a cube centered on the origin.

Ideal reconstruction filter
– fs
Primary spectrum
ˆg
Reconstructed spectrum
Aliasing
0 N f
– f
freq.
N
f s
Alias spectrum
Netravali [15] identified postaliasing, blur, anisotropy, and
ringing as defects arising from imperfect image reconstruction.
3.1 Postaliasing
Postaliasing arises when energy from the alias spectra
“leaks through” into the reconstruction, due to the reconstruction
filter being significantly non-zero at frequencies
above fN. The term postaliasing is used to distinguish the
problem from prealiasing, which occurs when the signal is
insufficiently band-limited before sampling, so that energy
from the alias spectra “spills over” into the region of the primary
spectrum. In both cases, frequency components of the
original signal appear in the reconstructed signal at different
frequencies (called aliases). The important distinction
between the two types of aliasing is illustrated in Figure 2.
Sample frequency ripple is a form of postaliasing that
arises when the filter’s spectrum is significantly non-zero at
lattice points in the frequency domain. The zero-frequency,
or “DC”, component of the alias spectra, which is very
strong even for signals with little density variation, then appears
in the interpolated volume as an oscillationat the sample
frequency. Near-sample-frequency ripple, which occurs
when filters are non-zero in the immediate vicinity of frequency
domain lattice points, can also be significant.
3.2 Smoothing (“blur”)
This term refers to the removal of rapid variations in a
signal by spatial averaging. Some degree of smoothing is
normal during reconstruction, since practical filters usually
start to cut off well before fN. In image processing, excessive
smoothing results in a blurred image. In volume rendering,
it results in loss of fine density structure. Theoretically,
smoothing is a filter defect, but in practice noisy volume
data may benefit from some smoothing, since most of
its fine structure is spurious. Also, smoothing is often necessary
to combat Gibbs phenomenon (see below).
3.3 Ringing (overshoot)
Low-pass filtering of step discontinuities results in oscillations,
or ringing, just before and after the discontinuity;
this is the Gibbs phenomenon (see for example [14]). Severe
ringing is not necessary for band-limitedness: Figure 3
shows two band-limited approximations to a square wave,
Imperfect reconstruction filter
– fs
Primary spectrum
Figure 2: Prealiasing (left) and postaliasing (right).
ˆg
Reconstructed spectrum
Aliasing
0 N f s f
– f
freq.
N
Alias spectrum
one generated with an ideal low pass filter and the other
with a filter that cuts off more gradually but with the same
ultimate cut-off frequency. Perceptually, the latter seems
preferable. 3
When a signal is being sampled, we have seen that it
must be band-limited if we are to reconstruct it correctly.
Natural signals are not generally band-limited, and so must
be low-pass filtered before they are sampled (or, equivalently,
the sampling operation must include some form of
local averaging). The usual assumption is that an ideal lowpass
filter, cutting off at the Nyquist frequency, is optimal.
However, we have just seen that such a filter causes ringing
around any discontinuities, regardless of any subsequent
sampling and reconstruction. If we then reconstruct the
sampled signal with an ideal reconstruction filter, we will
end up with exactly the filtered signal we sampled, which
has ringing at the discontinuities. To avoid such problems,
either the sampling filter or the reconstruction filter should
have a gradual cutoff if the signal to be sampled contains
discontinuities.
sharp cutoff gradual cutoff
Figure 3: Band-limited square waves.
3.4 Anisotropy
If the reconstruction filter is not spherically symmetric,
the amount of smoothing, postaliasing, and ringing will
vary according to the orientation of features in the volume
with respect to the filter. Anisotropy manifests itself as an
asymmetry in smoothing or postaliasing artifacts; in the absence
of those, anisotropy can not occur. We therefore regard
anisotropy as a secondary effect, and do not measure
it separately.
3 But the former is the optimal band-limited approximation under the
L 2 norm, which demonstrates the dangers of assuming that the L 2 norm is
always appropriate.

3.5 Cost
The remaining critical issue in filter design is that of
cost. Any practical filter takes a weighted sum of only a limited
number of samples to compute the reconstruction at a
particular point; that is, it is zero outside some finite region,
called the region of support. If a filter’s region of support
is contained within a cube of side length 2r, we call r the
radius of the filter. In this paper, we consider a range of filters
of different radii. It is important to realize that larger
filters are generally much more expensive: a trilinear interpolation
involves a weighted sum of eight samples, while a
tricubic filter involves 64. In general, the number of samples
involved increases as the cube of the filter radius.
The effect of filter radius on the run time of a volume
rendering program depends on the algorithm. Run times for
simple ray tracing algorithms tend to increase with the cost
of each density calculation, i. e., as the cube of the filter radius.
Run times for splattingalgorithms, which precompute
the two-dimensional “footprint” of a filter, tend to increase
as the square of the filter radius. Lastly, when resampling an
image or volume on a new lattice that is parallel to the old
lattice, separable filters (see Section 4.1) allow linear time
complexity with respect to filter radius, using a multi-pass
algorithm that filters once along each axis direction.
4 Filters to be Analyzed
The filters we wish to analyze fall into two categories,
separable and spherically symmetric. However, a subclass
of separable filters, the pass-band optimal filters, is defined
in a different way from all other filters, and is discussed separately
below.
In the defining equations that follow, we use the notation
�P� to be one if P is true and 0 otherwise. All but the first
two of the filters below need to be normalized by a constant
so that their integral over R 3 is equal to one.
4.1 Separable filters
Separable filters can be written
Included in this category are:
h�x�y�z� � hs�x� hs�y� hs�z��
� The trilinear filter. Trilinear interpolationis equivalent to
convolution by a separable filter:
hs�x� � �jxj � 1� �1 � jxj�
� A two-parameter family of cubic filters, with parameters
B and C, studied in two dimensions in [15]:
1
6
8
������
hs�B�C��x� �
������
�12 � 9B � 6C�jxj 3 �
��18 � 12B � 6C�jxj2 � �6 � 2B�
if jxj �1,
��B � 6C�jxj3 � �6B � 30C�jxj2 �
��12B � 48C�jxj � �8B � 24C�
if 1 �jxj �2,
0 otherwise.
This family includes the well-known B-spline (B � 1,
C � 0) and Catmull-Rom spline (B � 0, C � 0�5). We
confine our attention to filters in the range �B�C� � �0�0�
to �1�1�.
� The (truncated) Gaussian filter, which is often used in
splatting algorithms for volume rendering:
hs�xm�σ��x� � �jxj � xm�e �x2 �2σ 2
� The cosine bell filter, which has been widely used as a
window (see below) in one-dimensional signal processing
[1], but can also be used as a reconstruction filter in
its own right:
hs�xm��x� � �jxj � xm� �1 � cos�πx�xm��
� Windowed sinc filters. These filters approximate the
ideal sinc filter by a filter with finite support. Simply
truncating the sinc at some distance leads to problems
with ringing and postaliasing. Instead, the sinc is multiplied
by a window function that drops smoothly to zero.
This family approximates a sinc filter arbitrarily closely
as the radius of the window is increased. We consider
only one window, namely a cosine bell that reaches zero
after two cycles of the sinc function. The defining equation
of the windowed sinc is:
hs�xm��x� � �jxj � xm��1 � cos�πx�xm�� sinc�4x�xm�
4.2 Spherically symmetric filters
The value of a spherically symmetric filter depends
only on the distance from the origin. Such filters can be
written
p
h�x�y�z� � hr� x2 � y2 � z2�� The two such filters we investigate are:
� a rotated version of the cosine bell. This is simply a filter
whose hr is the same as the separable version’shs above.
� a spherically symmetric equivalent of the separable windowed
sinc, which we call a windowed3-sinc. The 3-sinc
(which is not the same as the separable sinc defined earlier)
is the inverse Fourier transform of a function that is
one inside a unit sphere and zero outside. For this filter,
hr�rm��r� � �α � 1��1 � cos�πα���sinα � αcosα��α 3
where α � r�rm.
4.3 Pass-band optimal discrete filters
These filters, described by Hsu and Marzetta [8] and
recommended for use in volume rendering by Carlbom [3],
are separable. Hence, the following discussion relates to
one-dimensional interpolation;the three-dimensionalfilters
are the products of three one-dimensional filters.
All previous filters are defined by continuous functions;
for any given interpolation position, a filter is centered
on the point of interest, and its values at sample points

provide the weights to apply to the data points. That set
of weights can be regarded as a discrete filter that resamples
the input data at new sample points displaced by some
fixed offset from the original sampling points. Carlbom defines
an optimal interpolation filter as a set of such discrete
filters, each individually optimized to minimize smoothing.
For each interpolation offset, a weighted Chebyshev
minimization program [9] is used to obtain a discrete filter
whose Fourier transform has (approximately) a minimum
weighted departure from ideal up to some frequency
fm � fN.
By computing a sequence of these fixed-length optimal
discrete filters for offsets in the range 0 - 1, and interpolating
between adjacent members, we can construct an underlying
continuous filter. Figure 4 shows two such underlying
filters. 4 The design method handles only odd-length discrete
filters, and thus the underlying filters are asymmetric,
unlike all other filters we study.
A problem with this approach to filter design is that
postaliasing is ignored, giving filters that are (in a sense)
optimal in the pass band at the expense of relatively poor
performance in the stop band.
5-point
filter
1
-3 2
9-point
filter
1
-5 -3 2 4
Figure 4: Two pass-band-optimal filters
5 Metrics for filter quality
5.1 Definitions
One of our goals in this research was to obtain some
quantitativemeasures of filter quality. As already indicated,
choosing a filter requires trading off benefits and defects according
to the the nature of the signal, how it was sampled,
how much noise is present, how costly a filter we can tolerate,
and what rendering algorithm is being used. For this
reason, a single number describing the quality of a filter—
for example, the L 2 norm of the difference between a particular
filter and the ideal filter—is not an appropriate goal.
Accordingly, we define separate metrics for the most important
filter qualities: smoothing, postaliasing, and overshoot
(ringing).
Formally, we define our smoothing metric, S, of a filter
h, to be
S�h� � 1 � 1
Z
jˆhj
jRNj RN 2 dV�
where RN is the Nyquist region, jRNj is the frequency-space
volume of RN, and dV is an infinitesimal volume element in
4 The discrete filters were computed using the program [9], modified as
described in [8] and [3], and with fm values of 0.3 and 0.4 for the 5-point
and 9-point filters respectively, as in [3].
RN. We define our postaliasing metric, P, to be
P �h� � 1
Z
jˆhj
jRNj
2 dV�
where RN is the complement of RN.
The smoothing and postaliasing metrics measure the
difference between a particular filter and our ideal filter inside
and outside the Nyquist region respectively; the difference
is measured in terms of energy. (The filter energy in
a region is the integral of the square of the filter over that
region.)
Our overshoot metric, O�h�, measures how much overshoot
occurs if a filter h is used to band-limit the unit step
function, ρs. More formally, O�h� � max�ρs �h��1, where
ρs is 1 if x � 0 and 0 otherwise.
5.2 Computation
The smoothing and postaliasing metrics are based on
the three-dimensional Fourier transforms of the filters. All
except the pass-band optimal filters are even functions, for
which the transform simplifies to the cosine transform [14]:
ˆh�ωx�ωy�ωz� �
Z
R 3 h�x�y�z�cos�ωxx�cos�ωyy�cos�ωzz� dxdydz�
For the separable filters, the transform further simplifies to
the product of three one-dimensional transforms.
For spherically symmetric filters, the three-
dimensional integral can be simplified [2] to
ˆhr�ωr� � 4π
ωr
Z ∞
0
R N
rhr�r�sin�ωrr� dr�
The smoothing metric is obtained directly from its definition
by numerical integration. The postaliasing metric is
computed from the smoothing metric and the total filter energy.
We can compute the total energy in the space domain,
where the filter has finite support, since Parseval’s theorem
[14] shows that the result is the same in both space and frequency
domains.
Metrics for the pass-band optimal filters were computed
from the underlying continuous filters illustrated in
Figure 4.
6 Filter Testing
6.1 The test volume
Although numerous datasets are publicly available, we
are unaware of any that are correctly sampled from some
exactly known signal. This makes it difficult to evaluate reconstruction
techniques, since the ultimate measure of the
quality of a reconstruction is how closely it approximates
the original signal. Accordingly, we use a test signal
where
ρ�x�y�z� �
�1 � sin�πz�2� � α�1 � ρr�
2�1 � α�
ρr�r� � cos�2π fM cos� πr
2 ���
p
x2 � y2�� �

We sampled this signal on a 40 by 40 by 40 lattice in
the range �1 � x�y�z � 1, with fM � 6 and α � 0�25. The
function has a slow sinusoidal variation in the z direction
and a perpendicular frequency-modulated radial variation.
With the given parameters, it can be shown that the onedimensional
radial signal has 99.8% of its energy below a
frequency of 10, and our analysis suggests that the spectrum
of the volume as a whole is similarly band-limited. This
makes it acceptable to point sample the function over the
range �1 � x�y�z � 1 at 20 samples per unit distance. Note,
however, that a significant amount of the function’s energy
lies near the Nyquist frequency, making this signal a very
demanding filter test—all our filters show some perceptible
postaliasing and smoothing.
Figure 5 shows a ray-traced image of the test volume’s
isosurface ρ�x�y�z� � 0�5.
Figure 5: The unsampled test signal.
6.2 Test image rendering
To demonstrate the behaviour of the various filters, we
display isosurfaces of reconstructed test volumes. It is important
that we show the exact shape of the isosurface, including
small irregularities that can be seen only with detailed
shading. This means we need a gradient that corresponds
exactly to the reconstructed density function. The
usual schemes for rendering isosurfaces (e. g., Lorensen and
Cline [13]) approximate the gradients using central differences
at sample points and then interpolate those gradients;
the resulting estimate does not track small-scale changes in
the isosurface orientation.
Since our reconstructed density function is the convolution
of the samples with the reconstruction filter, the density
gradient is the convolution of the samples with the gradient
of the filter. For any differentiable filter h, we can thus
obtain an exact formula for the gradient of the reconstructed
function, which can be evaluated at any point in the volume.
For rendering, we use a ray tracer that displays isosurfaces
of arbitrary functions by using a root-finding algorithm
to locate the first crossing of the isosurface level along
each ray.
Postaliasing
0.1
0.08
0.06
0.04
0.02
0
pass-band
optimal
filters
9-pt
windowed sincs
r=3.79
(0,0.5)
= Catmull-Rom
7-pt
(0,1)
��� ��� ��
5-pt
r=4.28
r=4.78
(0,0)
Trilinear
(1,0)
= B-Spline
0.2 0.4 0.6 0.8 1
Smoothing
Figure 6: Smoothing and postaliasing metrics.
7 Results
7.1 Smoothing and Postaliasing
Figure 6 shows the smoothing and postaliasing metrics
for the trilinear filter, the family of cubic filters, a range of
windowed sincs, and three pass-band optimal filters. The
metrics for our ideal filter would be (0,0), although, as discussed
in Section 3.3, some smoothing is usually required,
if only to combat overshoot.
Cubic Filters. This family is shown in the figure as a
10 by 10 mesh. The mapping from B-C space to smoothingpostaliasing
space is not one-to-one: the �1�1� corner of
the mesh is “folded” over. The B-spline smoothes the most
heavily, but has low postaliasing, while the Catmull-Rom
spline produces much less smoothing but has poor postaliasing
properties. The images in Figures 9(a) and 9(b) support
these measurements: the B-spline smoothes out the
large variations in the signal—the waves get shallower with
increasing frequency—and the Catmull-Rom preserves the
depth of the waves at the cost of aliasing, which shows up
as scalloped crests.
According to our metrics, the filters along the fold
should be best. However, Figure 9(c) shows the test volume
reconstructed using one such filter (B � 0�5, C � 0�85). We
can see that, while the overall geometry is reproduced quite
faithfully, the surface has a dimpled texture, due to nearsample-frequency
ripple. The ripples, although of low amplitude,
are of high frequency, and so produce large local
variations in gradient, and therefore in shading. It is perhaps
a limitation of our postaliasing metric that it weights
leakage at all frequencies equally.
Our experience corroborates the space-domain convergence
analysis of Mitchell and Netravali [15], which suggests
that filters along the line 2C � B � 1 (which includes
Catmull-Rom and B-splines as extreme cases) are among
the best: we find that these filters have negligible nearsample-frequency
ripple. But we see no reason in general
to prefer any particular filter along that line a priori, since
we must always settle for a tradeoff between smoothing and
postaliasing.

Trilinear filter. The trilinear filter is plotted in Figure
6. It can be seen that its metrics are the same as for
a cubic of approximately B � 0�26, C � 0�1. Images for
these two filters are shown in Figure 9(d) and 9(e). They
look similar, except that the trilinear filter introduces gradient
discontinuities, which our metrics do not measure.
Windowed sinc filter. The metrics for our particular
cosine-windowed sinc are shown in Figure 6 for a range of
radii. It can be seen from the figure that these filters are
in a sense superior to the entire family of cubics, since for
any cubic filter there are windowed sincs with both better
postaliasing and better smoothing. However, because of
their size, they are much more expensive to use than the cubics.
Also, because sample-frequency ripple is so offensive,
only the labelled points are of interest, since they are the
only ones for which the filter’s spectrum has zeroes at the
nearest lattice points (see Section 3.1).
The results for a radius of 4.78 are shown in Figure 9(f).
The wave structure is free of both scalloping and excessive
smoothing. (As in all the images, we must ignore the pronounced
effects of filtering the discontinuous outer edge of
the volume.) However, the filter’s anisotropy causes significant
variations in the heights of the circular crests: the filter
smoothes more in directions near the coordinate axes than
along diagonals.
The results for a radius of 4.28 are similar, but with
slightly more postaliasing. Both these filters are roughly
two orders of magnitude more expensive (in an O�r 3 � algorithm)
than trilinear interpolation.
Pass-band optimal filters The metrics for three different
pass-band optimal filters are shown in Figure 6. As expected,
their excellent pass band performance (low smoothing)
is achieved at the expense of relatively poor postaliasing.
The 5-point optimal filter is wider than the cubic filters
(twice the cost, in an O�r 3 � algorithm) and the 9-point filter
is comparable in cost with the windowed sinc of radius
4.78. Also, the pass-band optimal filters are more difficult
to calculate and manipulate generally (e.g., to obtain gradients)
than other filters, so we do not recommend them for
general-purpose reconstruction. Their primary use is probably
for image and volume resampling at a fixed offset when
minimal smoothing is the goal.
Other filters. Table 1 shows the smoothing and postaliasing
metrics for some representative separable Gaussian
and separable cosine bell filters. From the metrics, and from
several test images, we conclude that the cubics generally
perform better for similar cost and are more flexible. One
exception is the cosine bell of radius 1.5, which, as the metrics
in Table 1 suggest, produces images similar to a Bspline,
but at a lower cost. The filter does introduce slight
local gradient variations, but in many rendering contexts
these would not be apparent.
We investigated a range of spherically symmetric filters.
However, since the zeroes of their spectra fall on
Filter Radius Smooth. Postalias.
Cosine bell 1.0 0.67 0.096
Cosine bell 1.5 0.88 0.002
Cosine bell 2.0 0.95 0.00008
Gauss. σ � 0�50 2.5 0.81 0.014
Gauss. σ � 0�60 2.0 0.90 0.002
Gauss. σ � 0�75 2.5 0.95 0.0001
Table 1: Miscellaneous separable filter metrics.
spherical surfaces, rather than axis-aligned planes, it proved
impossible to adequately reject sample-frequency ripple
with filters of any reasonable cost. For very high-quality
reconstruction, the isotropy of spherically symmetric windowed
sinc filters could be a significant advantage.
overshoot %
8
0
1
B
Figure 7: Overshoot metrics for cubic filters.
7.2 Overshoot
Our overshoot metrics for the cubic filters are graphed
in Figure 7. As discussed in Section 3.3, overshoot is primarily
of concern with volumes containing inadequately
smoothed discontinuities;filters with high overshootshould
be avoided in such cases. Figure 8 illustrates the effects of
reconstructing a point-sampled cube with a B-spline, which
has no overshoot, and with the cubic filter most prone to
overshoot, B � 0, C � 1.
8 Conclusions
Interpolation underpins all volume rendering algorithms
working from sampled signals. We have considered
the family of interpolationschemes that can be expressed as
convolution of a sample lattice with a filter.
The artifacts resulting from imperfect reconstruction
fall into three main categories: smoothing, postaliasing,
Figure 8: A point-sampled cube reconstructed with a
B-spline (left) and with the cubic (0,1).
0
0
C
1

and overshoot. Since reconstruction is necessarily imperfect,
choosing a filter must involve tradeoffs between these
three artifacts.
We have defined metrics to quantify the characteristics
of a filter in terms of these artifacts. In general, the metrics
correlate well with the observed behavior of the filters, although
the postaliasing metric does not adequately address
the troublesome problem of ripple in the reconstructed signal
at or near the sample frequency.
Trilinear interpolation is certainly the cheapest option,
and will likely remain the method of choice for time-critical
applications. Where higher quality reconstruction is required,
especially in the presence of rapidly varying signals,
the family of cubics is recommended. Cubics offer
considerable flexibility in the tradeoff between smoothing
and postaliasing. For applications in which near-samplefrequency
ripple could be a problem, we recommend cubics
for which B � 1�2C; otherwise, filters along the “fold” line
in Figure 6 are preferred.
For the most demanding reconstruction problems, windowed
sincs can provide arbitrarily good reconstruction.
Their large radii make them extremely expensive in O�r 3 �
algorithms, such as ray-tracing, but they could certainly be
used in an O�r� resampling algorithm. The radius should be
chosen so that the Fourier transform is zero at the sampling
frequency, in order to eliminate sample-frequency ripple.
Spherically symmetric filters tend to produce samplefrequency
ripple, and do not seem to offer any significant
advantages over separable filters for most applications.
8.1 Future work
Although the metrics presented in this paper provide a
useful guideline, we believe they can be improved. In particular,
the postaliasing metric could be made more sensitive
to the frequencies that produce the most objectionable
artifacts.
Given the better reconstruction techniques outlined in
this paper, it should be possible to represent volume data
with a sparser sampling lattice. Using a precise definition
of the reconstructed signal gives us a framework in which
to evaluate errors introduced by such subsampling or other
forms of data compression.
9 Acknowledgements
This work was supported by the NSF/ARPA Science
and Technology Center for Computer Graphics and Scientific
Visualization(ASC-8920219). We gratefully acknowledge
the generous equipment grants from Hewlett-Packard
Corporation, on whose workstations the images in this paper
were generated. We especially wish to thank James
Durkinfor providingsupport and motivationthroughoutthe
research.
References
[1] K. G. Beauchamp. Signal Processing. George & Allen Unwin
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(a) B-spline (b) Catmull-Rom
(c) Cubic (B � 0�5, C � 0�85) (d) Trilinear
(e) Cubic (B � 0�26, C � 0�1) (f) Windowed sinc (r � 4�8)
Figure 9: Isosurface images of the test signal reconstructed using various filters.

Please reference the following QuickTime movies located in the MOV
directory:
CIRCLE.MOV..(Macintosh only)
LINE.MOV (Macintosh only)
Copyright © 1994 Cornell University Program of Computer Graphics
These two animations illustrate the effects of using a range of cubic filters to
reconstruct the sampled test signal described in the paper. They show images
generated using filters along two paths through the (B, C) parameter space of
the cubics. The “line” movie is a sequence of reconstructions using filters
along the line segment joining (B, C) = (1, 0), the B-spline filter, with (B, C)
= (0, 0.5), the Catmull-Rom filter. This movie makes the most sense played
in a “loop back and forth” mode. The second movie, the “circle” movie, is a
sequence moving around a circle inscribed in the square with corners at (0,
0), (1, 0), (1, 1), and (0, 1). This one makes more sense played in a “loop
from beginning to end” mode. The movies show that there is a continuous
range of cubic filters, with varying properties, to choose from.
QuickTime is a trademark of Apple Computer, Inc.

Visualizing Flow with Quaternion Frames
Abstract
Flow �elds� geodesics� and deformed volumes are
natural sources of families of space curves that can
be characterized by intrinsic geometric properties such
as curvature� torsion� and Frenet frames. By express�
ing a curve�s moving Frenet coordinate frame as an
equivalent unit quaternion� we reduce the number of
components that must be displayed from nine with six
constraints to four with one constraint. We can then
assign a color to each curve point by dotting its quater�
nion frame with a 4D light vector� or we can plot
the frame values separately as a curve in the three�
sphere. As examples� we examine twisted volumes used
in topology to construct knots and tangles� a spherical
volume deformation known as the Dirac string trick�
and streamlines of 3D vector �ow �elds.
1 Introduction
We propose new tools for the visualization of a class
of volume data that includes streamlines derived from
3D vector �eld data as well as deformed volumes. The
common feature of such data that we exploit is the
existence of a family of static or time�varying space
curves that do not intersect� but which may exhibit
extremely complex geometry� typically� portions of
neighboring curves are similar in some regions of the
volume� but become inhomogeneous when interesting
phenomena are taking place. Our approach is based
on the observation that families of curves in three�
space possess several intrinsic but locally computable
geometric properties� these include the curvature� the
torsion� and a local coordinate frame� the Frenet frame
�also called the Frenet�Serret frame�� de�ned by the
tangent� normal and binormal at each point of each
curve.
While it is awkward to represent the Frenet frame
itself visually in high�density data because it consists
of three 3D vectors� or nine components� it has only
three independent degrees of freedom� it is well known
that these three degrees of freedom can conveniently
be represented by an equivalent unit quaternion that
Andrew J. Hanson and Hui Ma
Department of Computer Science
Indiana University
Bloomington� IN 47405
corresponds� in turn� to a point in the three�sphere
�see� e.g.� �15��. We exploit the quaternion representa�
tion of rotations to reexpress the Frenet frame of a 3D
space curve as an elegant unit four�vector �eld over
the curve� the resulting quaternion Frenet frame can
be represented as a curve by itself� or can be used to
assign a color to each curve point using an interactive
4D lighting model.
2 Frenet Frames
The Frenet frame is uniquely de�ned for �almost�
every point on a 3D space curve �see� e.g.� �5� 6�� and
also �1� 8��. If �x�s� is any smooth space curve� its
tangent� binormal� and normal vectors at a point on
the curve are given by
�T�s� �
�B�s� �
�x 0 �s�
k�x 0�s�k �x 0�s� � �x 00�s� k�x 0�s� � �x 00�s�k �N�s� � �B�s� � �T�s� �
�1�
The Frenet frame obeys the following di�erential equa�
tion in the parameter s�
2
4
3
�T 0�s� �N 0�s� �B 0 5 � v�s�
�s�
2
4
0 ��s� 0
���s� 0 ��s�
0 ���s� 0
3 2
5 4
�T�s�
�N�s�
�B�s�
�2�
where v�s� � k�x 0 �s�k is the scalar magnitude of the
curve derivative� ��s� is the curvature� and ��s� is the
torsion. All these quantities can in principle be calcu�
lated in terms of the parameterized or numerical local
values of �x�s� and its �rst three derivatives as follows�
��s� � k�x0 �s� � �x 00 �s�k
k�x 0 �s��k 3
��s� � ��x0 �s� � �x 00 �s�� � �x 000 �s�
k�x 0 �s� � �x 00 �s�k 2
�
3
5
�3�
If we are given a non�vanishing curvature and a torsion
as smooth functions of s� we can theoretically integrate

the system of equations to �nd the unique numerical
values of the corresponding space curve �up to a rigid
motion�.
3 Theory of Quaternion Frames
A quaternion frame is a unit�length four�vector
q � �q0� q1� q2� q3� � �q0� �q� that corresponds to ex�
actly one 3D coordinate frame and is characterized by
the following properties�
� Unit Norm. The components of a unit quater�
nion obey the constraint�
�q0� 2 � �q1� 2 � �q2� 2 � �q3� 2 � 1 �4�
and therefore lie on S 3 � the three�sphere.
� Multiplication rule. The quaternion product
of two quaternions q and p� which we write as
q � p� takes the form
2 3
�q � p�0 6
6
�q � p� 7
1 7
4 �q � p� 5
2
�q � p�3 �
2
6
6
4
q0p0 � q1p1 � q2p2 � q3p3
q0p1 � p0q1 � q2p3 � q3p2
q0p2 � p0q2 � q3p1 � q1p3
q0p3 � p0q3 � q1p2 � q2p1
3
7
7
5 �
This rule is isomorphic to multiplication in the
group SU�2�� the double covering of the ordinary
3D rotation group SO�3�.
� Mapping to 3D rotations. Every possible 3D
rotation R �a 3�3 orthogonal matrix� can be con�
structed from either of two related quaternions�
q � �q0� q1� q2� q3� or �q � ��q0� �q1� �q2� �q3��
using the quadratic relationship�
R �
2
4
Q�� � �� D � �123� D � �312�
D � �123� Q�� � �� D � �231�
D � �312� D � �231� Q�� � ��
3
5 �
�5�
where Q����� � q 2 0 �q2 1 �q2 2 �q2 3 and D � �ijk� �
2q iq j � 2q0q k.
� Quaternion Frenet Frame. All 3D coordinate
frames can be expressed in the form of quater�
nions using Eq. �5�� if we assume the columns
of Eq. �5� are the vectors � �T� �N� �B�� respectively�
one can show �see �8�� that jq 0 �s�j takes the form
2
6
6
4
q 0
0
q 0 1
q 0 2
q 0 3
3
7 v
7 � 5 2
2
6
6
4
0 �� 0 ��
� 0 � 0
0 �� 0 �
� 0 �� 0
3
7
7
5 �
2
q0
3
6
6
4
q1
q2
7
7
5 �
q3
�6�
As a check� we verify that the matrices
A �
B �
C �
2
4
2
4
2
4
q0 q1 �q2 �q3
q3 q2 q1 q0
�q2 q3 �q0 q1
�q3 q2 q1 �q0
q0 �q1 q2 �q3
q1 q0 q3 q2
q2 q3 q0 q1
�q1 �q0 q3 q2
q0 �q1 �q2 q3
give rise directly to the usual quantities� jAj�jq 0 j �
�T 0 � v� �N� jBj � jqj � �N 0 � �v� �T � v� �B� jCj �
jdqj � �B 0 � �v� �N� where we have applied Eq. �6�
to get the �nal terms.
Just as the Frenet equations may be integrated ex�
plicitly to generate a unique moving frame with its
space curve for non�vanishing ��s� �6�� we could in
principle integrate the quaternion equations� Eq. �6��
to get the needed information� Equation �6�� having
only four components and automatically enforcing the
constraint Eq. �4�� is much simpler for numerical anal�
ysis than Eq. �2�.
4 Assigning Smooth Quaternion
Frames
The Frenet frame equations are pathological� for
example� when the curve is perfectly straight for some
distance or when the curvature vanishes momentarily.
Thus� real numerical data for space curves will fre�
quently exhibit behaviors that make the assignment
of a smooth Frenet frame di�cult or impossible. In
addition� since any given 3 � 3 orthogonal matrix cor�
responds to two quaternions that di�er in sign� meth�
ods of deriving a quaternion from a Frenet frame are
intrinsically ambiguous. Therefore� we prescribe the
following procedure for assigning smooth quaternion
Frenet frames to points on a space curve�
� Assign an initial orientation to the beginning of
each curve. Appropriate choices may depend on
context.
� Given a sequence of points for a sampled space
curve� compute the numerical derivatives at a
given point and use those to compute the Frenet
frame according to Eq. �2�. If any critical quanti�
ties vanish� keep the frame of the previous point.
� Check the dot product of the previous binormal
�B�s� with the current value� if it is much less than
one� choose a correction procedure to handle this
3
5
3
5
3
5

singular point. Among the correction procedures
we have considered are �1� simply jump discontin�
uously to the next frame to indicate the presence
of a point with very small curvature� �2� create an
interpolating set of points and perform a geodesic
interpolation �15�� or �3� deform the curve slightly
before and after the singular point to �ease in�
with a gradual rotation of the frame. Creating a
jump in the frame assignment is the easiest� and
is an intuitively reasonable choice� since the ge�
ometry is changing dramatically at such points.
� Apply a suitable algorithm such as that of Shoe�
make �15� to compute a candidate for the quater�
nion corresponding to the Frenet frame.
� If the 3 � 3 Frenet frame is smoothly changing�
make one last check on the 4D dot product of the
quaternion frame with its own previous value� if
there is a sign change� choose the opposite sign to
keep the quaternion smoothly changing �this will
have no e�ect on the corresponding 3 � 3 Frenet
frame�. If this dot product is near zero instead
of �1� you have detected a radical change in the
Frenet frame which should have been noticed in
the previous tests.
� If the space curves of the data are too coarsely
sampled to give the desired smoothness in the
quaternion frames� but are still close enough
to give consistent qualitative behavior� one may
choose to smooth out the intervening frames using
the desired level of recursive slerping �15� to get
smoothly splined intermediate quaternion frames.
In Figure 1� we plot an example of a torus knot�
a smooth space curve with everywhere nonzero cur�
vature� together with its associated Frenet frames� its
quaternion frame values� and the path of its quater�
nion frame �eld projected from four�space. Figure 2
plots the same information� but this time for a curve
with a discontinuous frame that �ips too quickly at a
zero�curvature point. This space curve has two planar
parts drawn as though on separate pages of a partly�
open book and meeting smoothly on the �crack� be�
tween pages. We see the obvious jump in the Frenet
and quaternion frame graphs at the meeting point� if
the two curves are joined by a long straight line� the
Frenet frame is ambiguous and may optionally be de�
clared unde�ned in this segment.
Alternative Frames. We have chosen to deal di�
rectly with the more familiar properties and anoma�
lies of the standard Frenet frame in the treatment pre�
sented here� however� it is worth noting that alterna�
tive methods such as the parallel transport frame for�
mulation proposed by Bishop �1� can be used to avoid
frame discontinuities. The quaternion frame method
extends also to parallel transport frames �8�.
5 Visualization Methods
Once we have calculated the quaternion Frenet
frame� the curvature� and the torsion for a point on the
curve� we have a family of tensor and scalar quantities
that we may exploit to expose the intrinsic proper�
ties of a single curve. Furthermore� and probably of
greater interest� we also have the ability to make visual
comparisons of the similarity and di�erences among
families of neighboring space curves.
The Frenet frame �eld of a set of streamlines is po�
tentially a rich source of detailed information about
the data. However� the Frenet frame is unsuitable
for direct superposition on dense data due to the
high clutter resulting when its three orthogonal 3�
vectors are displayed� direct use of the frame is only
practical at very sparse intervals� which prevents the
viewer from grasping important structural details and
changes at a glance. The 4�vector quaternion frame
is potentially a more straightforward and �exible ba�
sis for frame visualizations� below� we discuss several
alternative approaches to the exploitation of quater�
nion frames for data consisting of families of smooth
curves.
5.1 Direct Three�Sphere Plot of Quater�
nion Frame Fields
We now make a crucial observation� For each 3D
space curve� the quaternion frame de�nes a completely
new 4D space curve lying on the unit three�sphere em�
bedded in 4D Euclidean space.
These curves can have entirely di�erent geometry
from the original space curve� since distinct points on
the curve correspond to distinct orientations. Families
of space curves with exactly the same shape will map
to the same quaternion curve� while curves that fall
away from their neighbors will stand out distinctly in
the three�sphere plot. Regions of vanishing curvature
will show up as discontinuous gaps in the otherwise
continuous quaternion frame �eld curves.
Figures 1d and 2d present elementary examples of
the three�sphere plot. More complex examples are in�
troduced in Section 7 and shown in Figures 3b� 4b�
and 5b. The quaternion frame curves displayed in
these plots are 2D projections of two overlaid 3D solid
balls corresponding to the �front� and �back� hemi�
spheres of S 3 . The 3�sphere is projected from 4D to
3D along the 0�th axis� so the �front� ball has points
with 0 � q0 � �1� and the �back� ball has points

�a� �b�
�c� �d�
Figure 1� �a� Projected image of a 3D �3�5� torus knot. �b� Selected Frenet frame components displayed along
the knot. �c� The corresponding smooth quaternion frame components. �d� The path of the quaternion frame
components in the three�sphere projected from four�space. Gray scales indicate the 0�th component of the curve�s
four�vector frame �upper left graph in �c��.
with �1 � q0 � 0. The q0 values of the frame at each
point are displayed as shades of gray in the curves of
Figures 1 and 2.
5.2 Probing Quaternion Frames with 4D
Light
As an alternative display method� we now adapt
techniques we have explored in other contexts for deal�
ing directly with 4D objects �see �11�� �12�� and �10��.
In our previous work on 4D geometry and lighting� the
critical element has always been the observation that
4D light can be used as a probe of geometric struc�
ture provided we can �nd a way �such as thickening
curves or surfaces until they become true 3�manifolds�
to de�ne a unique 4D normal vector that has a well�
de�ned scalar product with the 4D light� when that
objective is achieved� we can interactively employ a
moving 4D light and a generalization of the standard
illumination equations to produce images that selec�
tively expose new structural details.
Once we have generated our �possibly piecewise�
smooth quaternion frames for each curve� we see that
the quaternion frame q�s� is precisely what is required
to apply 4D lighting and thus instantly identify those
sections of the curves with similar orientation proper�
ties by their matching re�ectances. The lighting equa�
tion we adopt is simply
I�s� � I a � I d�L � q�s�� � �7�
where L is a 4D unit vector representing a 4D light
direction� and we have included both an ambient and

�a� �b�
�c� �d�
Figure 2� �a� Projected image of a pathological curve segment. �b� Selected Frenet frame components� showing a
sudden change of the normal. �c� The quaternion frame components. �d� The discontinuous path of the quaternion
frame components in the three�sphere. Gray scales indicate the 0�th component of the curve�s four�vector frame
�upper left graph in �c��.
a di�use term. The 4D dot product may be treated in
several ways when it becomes negative� if the partic�
ular application actually has a reason to distinguish
two equivalent quaternion frames because of their po�
sition in a time sequence� one may distinguish positive
and negative dot products by assigning di�erent color
ranges �this method is used in Figures 3�5� or by set�
ting the di�use value to zero when the dot product is
negative� if this distinction is not signi�cant� the ab�
solute value may be used. We have found it useful to
use a variety of color maps in place of a strict gray
scale of intensities. A moment�s re�ection reveals that
this amounts to assigning shells of color to the three�
sphere and tagging the quaternion vector �eld lines in
the three�sphere plot with the values of the color shells
through which they pass� rotating the light is equiv�
alent to rotating the 4D orientation of the color map
distributed across the three�sphere. Yet another vari�
ation would assign each point of the three�sphere to a
unique color in the 3D hue�saturation�intensity space�
making 4D rotation of the �light� irrelevant except for
permuting the emphasized orientations. It is also pos�
sible to add a heuristic specular term to Eq. �7�� e.g.�
by considering the projection direction of the three�
sphere plot to be a 4D �view vector.�
Figures 3a� 4a� and 5a show streamline data sets
rendered by computing a pseudocolor index at each
point from the 4D lighting formula Eq. �7�.
5.3 Additional Geometric Cues
Gray �6� 3� has advocated the use of curvature and
torsion�based color mapping to emphasize the geomet�
ric properties of single curves such as the torus knot.

Since this information is trivial to obtain simultane�
ously with the Frenet frame� we also o�er the alter�
native of encoding the curvature and torsion as scalar
�elds on a volumetric space populated either sparsely
or densely with streamlines� examples are shown in
Figures 3c�d� 4c�d and 5c�d.
Another extension one might adopt is based on the
idea that� from Eq. �4�� the curvature ��s� is propor�
tional to the norm of a 3�vector� we can thus construct
the four�component object
�K�s� �
�
��s�� �x0 �s� � �x 00 �s�
j�x 0 �s�j 3
�
�8�
and then use the normalized �four�vector� �K�s� �
�K�s��k �K�s�k as the quantity to be probed with 4D
light. Since we normally start the interactive system
with the light shining directly in the direction of the
component assigned to ��s� in Eq. �8�� we can begin
with an image coded strictly by the strength of the
curvature� and move the 4D light out into the other di�
mensions to reveal the components of the vector whose
magnitude is the curvature.
Finally� we remark that stream surfaces �14� can in
principle also be mapped to the three�sphere� e.g.� by
choosing an interpolation of the quaternion frames of
the streamline curves from which the stream surface
is derived.
6 Interactive Interfaces
4D Light Orientation Control. Direct manipula�
tion of 3D orientation using a 2D mouse is typically
handled using a rolling ball �7� or virtual sphere �2�
method to give the user a feeling of physical control.
Here we brie�y outline the extension of this philosophy
to 4D orientation control �see �4� 9��
To control a single 3D lighting vector using a 2D
mouse� we note that the unit vector in 3D has only
two degrees of freedom� so that picking a point within
a unit circle determines the direction uniquely up to
the sign of its view�direction component. By choos�
ing some convention for distinguishing vectors with
positive or negative view�direction components �e.g.�
solid or dashed line display�� we can make the direction
unique. To control the light vector using the rolling
ball� begin with the vector pointing straight out of the
screen� and move the mouse diagonally in the desired
direction to tilt the vector to its new orientation� ro�
tating past 90 degrees moves the vector so its view
component is into the screen.
The analogous control system for 4D lighting is
based on a similar observation� since the 4D nor�
mal vector has only 3 independent degrees of freedom�
choosing an interior point in a solid sphere determines
the vector uniquely up to the sign of its component
in the unseen 4th dimension �the �4D view�direction
component��. The rest of the control proceeds analo�
gously.
Three�sphere projection control. A natural
characteristic of the mapping of the quaternion frames
to the three�sphere is that a particular projection must
be chosen from 4D to 3D when actually displaying
the quaternion data. In order to expose all possible
relevant structures� the user interface must allow the
viewer to freely manipulate the 4D projection parame�
ters. This control is easily and inexpensively provided
using the interface just described for controlling the
4D light orientation. Our �MeshView� 4D viewing
utility supports real�time interaction with such struc�
tures.
7 Examples
We present three di�erent examples of streamline or
�ow data. Each data set is rendered four ways� �1� as
a Euclidean space picture� pseudocolored by 4D light�
�2� as a four�vector quaternion frame �eld plotted in
the three�sphere� �3� in Euclidean space color�coded
by curvature� �4� in Euclidean space color�coded by
torsion.
� Figure 3. An AVS�generated streamline data
set� the �ow is obstructed somewhere in the cen�
ter� causing sudden jumps of the streamlines in
certain regions.
� Figure 4. The �Dirac string trick� deformation
�13� of a spherical solid consisting of concentric
spheres in Euclidean space.
� Figure 5. A complicated set of streamlines de�
rived from twisting a solid elastic Euclidean space
as part of the process of tying a topological knot.
In principle� any of the images for which we have
shown only streamline curves can also be displayed
with stream surfaces or interpolated solid volume ren�
dering� we have a module for solid rendering that dis�
plays data such as that in Figure 5 as slices of the solid
object.
8 Conclusion
In this paper� we introduced a visualization method
for distinguishing critical features of streamline�like
volume data by assigning to each streamline a quater�
nion frame �eld derived from its moving Frenet frame�
curvature and torsion �elds may be incorporated as
well. The quaternion frame is a four�vector �eld that

is a piecewise smoothly varying map from each orig�
inal space curve to a new curve in the three�sphere
embedded in four�dimensional Euclidean space. Fur�
thermore� this four�vector can be interpreted as a four�
dimensional normal direction at each curve point� thus
allowing us to build on our previous work exploiting
4D illumination� by assigning a shade to each curve
point that depends on the scalar product of the quater�
nion frame with a 4D light� we can explore a wide va�
riety of local data features by interactively varying the
light direction.
Acknowledgments
This work was supported in part by NSF grant IRI�
91�06389. We thank Brian Kaplan for his assistance
with the vector �eld data set� and John Hart� George
Francis� and Lou Kau�man for the Dirac string trick
data description. Bruce Solomon brought reference �1�
to our attention.
References
�1� Bishop� R. L. There is more than one way
to frame a curve. Amer. Math. Monthly 82� 3
�March 1975�� 246�251.
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Figure 3: Vector field streamlines. (a) With 4D light. (b) Quaternion
field path. (c) Curvature. (d) Torsion.
Figure 4: Dirac Strings. (a) With 4D light. (b) Quaternion field path.
(c) Curvature. (d) Torsion.
Figure 5: Knot volume deformation. (a) With 4D light. (b) Quaternion
field path. (c) Curvature. (d) Torsion.

Feature Detection from Vector Quantities in a Numerically Simulated Hypersonic
Flow Field in Combination with Experimental Flow Visualization
Abstract
In computational fluid dynamics visualization is a frequently
used tool for data evaluation, understanding of flow characteristics,
and qualitative comparison to flow visualizations
originating from experiments. Building on an existing visualization
software system, that allows for a careful selection
of state-of-the-art visualization techniques and some extensions,
it became possible to present various features of the
data in a single image. The visualizations show vortex position
and rotation as well as skin-friction lines, experimental
oil-flow traces, and shock-wave positions. By adding experimental
flow visualization, a comparison between numerical
simulation and wind-tunnel flow becomes possible up to a
high level of detail. Since some of the underlying algorithms
are not yet described in detail in the visualization literature,
some experiences gained from the implementation are illustrated.
1 Introduction
Numerical flow simulation creates demanding data of high
complexity, which requires a well selected combination of
visualization techniques to visualize the dominant properties
of the data and the significant structures needed to gain
insight into the flow. By carefully reducing the number of
visual elements required for the representation of a given
feature in the data, one may free space in the image for the
visualization of other features present at the same time.
Eventually, even data from different sources, such as an
image from an experiment and visual elements from numerically
simulated flows, may be visualized in a single image.
It turns out that interesting discoveries become possible only
due to this combined visualization. Complex visualization
that provides a collective presentation of different features
offers better insight into the interaction of physical phenomena.
The major benefit for aerodynamicists, however, was
only achieved due to the successful combination of vortex
visualization, shock visualization, skin-friction lines, and
oil-flow patterns.
Hans-Georg Pagendarm
Birgit Walter
DLR, German Aerospace Research Establishment
D37073 Göttingen, Germany
- 1 -
2 The flow problem
Fig. 1 Geometry of the flat plate, wedge, and fin
configuration placed in a hypersonic flow field
(schematic view)
The visualization of phenomena in flows will be demonstrated
using data kindly supplied by T. Gerhold [1],[2]. The
flow simulation results from a Navier-Stokes solution for a
blunt-fin / wedge configuration (see: Fig. 1) in a hypersonic
flow at a Mach number of 5. The flow shows a horseshoe
type of vortex. The simulation also resolves a smaller secondary
vortex that rotates in the opposite direction. These
vortices produce characteristic traces on the wall, which
have been visualized in the numerical data as well as in a
wind-tunnel flow for direct comparison. A second significant
phenomenon in this flow field is a pattern of shock
waves. This flow is studied to learn more about the interaction
of shock waves and complex three-dimensional boundary
layers or viscous flows in general. Even though the
geometry appears to be simple an extremely complex threedimensional
flow field is generated.

3 Visualization of wall friction
As in this case experimental fluid mechanics of hypersonic
flows is frequently done in blow-down wind tunnels. These
facilities allow only a short-duration of constant-flow condition.
Therefore the data collected within a single experiment
are extremely limited. Scalar quantities such as pressure are
often measured at a small number of locations on the wall.
Usually there are no measurements in the three-dimensional
flow field. The use of oil-flow visualization allows global
acquisition of near-wall-velocity directional information.
Thus oil-flow patterns are a powerful representation of flow
behaviour.
The technique employs a dye dispersed in a special oil.
This oil is sprayed on the solid walls of the model in the
wind tunnel. Due to viscous action of the flow velocity close
to the wall, the oil moves slowly in the local flow direction.
When the oil evaporates it leaves behind a trace of dye,
which marks the local flow direction.
In order to compare numerical flow simulations with
such experiments, one has to visualize the near- wall flow
field. The direction of a flow velocity field is frequently visualized
using streamlines. In simulations of viscous flows
the velocity of the flow at a solid wall is zero by definition.
This prevents the calculation of streamlines directly on the
walls. One would like to find the limiting streamline at locations
where the velocity goes to zero while the direction of
the velocity vector is determined by the direction of the
velocity near the wall. Limiting streamlines on a simpler fin/
flat-plate configuration were presented already by Hung and
Buning [3]. Helman and Hesselink [4] point out some of the
difficulties of such an approach using an example from [4].
At that time those limiting streamlines were probably generated
by performing a particle tracing in the flow regime
close to the wall (in the case of [4] and [5] by using additional
constraints on the velocity component normal to the
wall.. Today skinfriction lines are commonly used instead of
near-wall particle traces. Visualization of skin-friction lines
has been used for a long time for comparison with oil-flow
patterns. (e.g. see [6] and [7] ). Therefore, the main interest
here is concerned with the particular implementation within
a visualization system.
All images in this paper where created using the visualization
system HIGHEND [8]. This system is easily extendible
and the data processing needed for visualization of skinfriction
lines was quickly performed using existing tools
after adding small pieces of code.
The expression “wall-friction lines” results from using
the wall shear vector for visualization purposes. The wall
shear vector τw is the derivative normal to the wall of the
velocity vector v . In general it is nonzero and points in
the direction of the near-wall velocity vectors when projected
normal to the wall. The wall shear τ w may be calcu-
- 2 -
lated from the gradient of the velocity vector v after
introducing a wall coordinate system where
dinate vector normal to the wall.
n is the coor-
τ w
= ∇v – ( ∇v ⋅ n)
The gradient ∇v
may be calculated using a Gaussian integral
formulation
∇v
⎛ 1
-- v dA⎞
⎝ V∫
⎠
This allows for an easy implementation[9],[10] that is insensitive
to a pathological or degenerated grid cell in the
numerical data set. To calculate the vorticity, a staggered
grid is generated by creating new vertices in the centre of
each grid cell (see Fig. 2). This is done by calculating the
coordinates of the centre vertex as the arithmetical mean of
the coordinates of the surrounding vertices.
i-1,j-1,k+1
=
lim
V → 0
i-1,j+1,k+1 i+1,j+1,k+1
i-1,j-1,k-1 i+1,j-1,k-1
Fig. 2 A staggered grid around the grid point i,j,k is
used. The integration is performed on the
surface of the inner volume. For simplicity, this
figure shows a cartesian grid where cells are
cubes. However the approach may be
generalized towards curvilinear or even
unstructured grids.
After the wall shear τ w has been calculated on all solid
boundaries represented in the data, a standard streamline
integration algorithm using a second-order Runge-Kutta
scheme is used to integrate the friction lines from the shear
vector field. The skin-friction lines provide global information
about the near-wall flow field (Fig. 3) Experienced aerodynamicists
will be able to imagine even qualitatively the
A
i+1,j+1,k-1

overall three-dimensional flow pattern in the flow field from
these wall patterns.
In particular, skin-friction lines show the location of separation
and reattachment of the flow at the wall. As mentioned,
earlier experiments were performed in a wind tunnel to provide
measurements and visualization to match the numerical
flow simulation. Oil-flow patterns were captured as a photograph.
Due to the limited access to the wind tunnel during
the experiments, the photograph shows a perspective view
of the fin taken from a side of the wind tunnel.
Fig. 3 Skin-friction lines on walls of the the blunt fin
and wedge.
Unfortunately, some of the details of the positions and the
equipment used when recording the experiments were not
available anymore. Therefore, for direct comparison of significant
lines and oil-flow pattern, the perspective had to be
reconstructed from the edges of the model visible in the
image.
Fig. 4 Skin-friction lines projected perspectively on
the oil-flow pattern
- 3 -
After careful interactive matching of the viewing conditions
which was required in this case, the skin-friction lines could
be projected into the image showing the oil-flow pattern in
the wind tunnel experiment.
The resulting combined image (Fig. 4) increases confidence
in the overall correct numerical simulation. Some
local discrepancies could be explained by a local lack of resolution
in the numerical simulation. However, the experiment
clearly shows a second weak separation line in the
lower side wall of the fin, which does not show in the
numerically generated skin-friction lines (Fig. 5). The
experiment suggests that there is a vortex at some distance
from the wall, which is the reason for this pattern.
4 Vortex visualization
s 1
s2 ss22 Fig. 5 The oil-flow pattern shows a second weak
separation trace s 2 , which is not visible in the
numerical data
A vortex would be expected to be present in the numerical
solution as well. However, due to a lack of resolution for
very small vortices it might be too weak to influence the
skin-friction lines visibly.
The visualization of vortices typically relies on particle
traces. Calculation of particle traces within a flow field is a
well known visualization technique that similary to flow visualization
by injection of dye into real flows. The construction
of trajectory lines is generally applicable to all kinds of
vector data. Presumably it is because of the similarity to
experimental flow visualization techniques that the visualization
of such lines is frequently mentioned in the literature
of computational fluid dynamics (see among others
[11],[12],[13],[4]). Some authors study the accuracy problem
in detail (see: [14],[15],[16]). Today, most visualization
systems make use of a second-order Runge-Kutta integra-

tion scheme. However, some authors report alternative
methods [17], [18]. Volpe [13] suggested the display of
stream ribbons for a more understandable visualization of
flow fields. He demonstrated how to obtain the impression
of a ribbon floating in the flow field by calculating a large
number of adjacent streamlines. Those lines must be placed
very close to each other to keep the gaps invisible in the
image. Another way of creating stream ribbons is by constructing
polygons between two neighbouring lines. A simple
method to create such polygons by the use of a marching
triangulation algorithm is illustrated in [19]. A similar
approach is reported by [20]. Both approaches fail if the
flow field is diverging because the two limiting streamlines
depart from each other. Eventually they will be far enough
from each other that physically unfeasible effects occur in
the visualization. Such effects could be the crossing of
stream ribbons or ribbons passing through solid flow boundaries.
To produce valid visualizations in such circumstances,
it is necessary to calculate the stream ribbon from a single
trajectory line. Again, this approach has been used in the
past. In particular the famous NCSA video on a ‘Severe
Thunderstorm’ shows swirling ribbons. Therefore, the following
description concentrates on implementation issues.
Calculating stream ribbons from more than one streamline
often fails because the numerical values of two near by
velocity vectors differ significantly. This allows for a growing
distance between two adjacent lines. A straightforward
approach to avoid this considers one of the two lines as a
primary line that marks the location of the ribbon and uses
the second streamline calculation only for obtaining the orientation
of the line in space while keeping the two lines at a
constant distance. Such an approach may be derived from an
algorithm suggested for tracking of surface particles by van
Wijk [20]. However, in fluid dynamics information on vortical
behaviour is commonly obtained by applying the rotation
or curl operator of the velocity vector field v (e.g. see
[21]).
rot v
=
∂vz ∂vy ------- – -------
∂y ∂z
∂vx ∂vz ------- – -------
∂z ∂x
∂vy ∂vx ------- – -------
∂x ∂y
The resulting vector represents the axis of rotation as well as
the amount of rotation at a given point in the vector field
v .The angular velocity, or vorticity ω , is simply one half
rot v .
- 4 -
ω ( x, y, z, t)
1
-- rot v
2
Since flow simulation data frequently are stored on curvilinear
or even unstructured grids, it may require some effort to
evaluate the rotation or curl because the transformation of
the coordinates has to be taken into account. However, one
may again take advantage of replacing the rotation or curl
by a Gaussian integral:
This approach allows an implementation very similar to the
one suggested earlier for the evaluation of the gradient of a
scalar field [9]. Once the vorticity ω is calculated everywhere
in the vector field, the vorticity vector may be projected
onto the local flow direction to obtain the angular
velocity of rotation in the local streamwise direction, which
allows the detrmination of the stream-ribbon swirl.
Note that ωribbon is a scalar quantity, since the direction is
defined by the local direction of the velocity vector or the
streamline. This quantity is a measure of rigid body rotation
in the flow field. It is not identical to another scalar quantity
often used to find vortical motion in flow fields, the helicity,
which is simply the cosine of the angle between the vorticity
vector and the velocity vector.
4.1 Construction of a stream ribbon
First, a streamline or particle path is calculated using an
established particle tracing algorithm. This program produces
a line as a series of coordinates. The system was
extended in such a way that for each point along the line the
magnitudes of the local velocity vector v and the local
streamwise vorticity are stored as well. These
values are obtained within the grid cell by trilinear interpolation.
Knowledge of the local velocity of the flow along a
streamline allows the calculation of the elapsed time
between two points on the line at a distance s.
=
rot v = ∇× v = – lim
ω ribbon
t ( s)
V → 0
⎛ 1
-- v × dA⎞
⎝ V∫
⎠
ω ⋅ V
= ωstreamwise = -----------
V
ω streamwise
=
s
∫
s 0
A
v
---------------------- ds
s ( x, y, z)

At the starting point of a streamline, the initial direction and
the width of the stream ribbon is defined. Marching from
point to point along the streamline the rotation α may be
integrated by multiplying the known angular velocity of the
flow around the streamline and the elapsed time
.
s
∫
α ( s)
= ω ( s)
dt
s 0
Simple Eulerian integration was found to be accurate
enough for visualization purposes after comparing the
results with an alternative method [19] for a number of
cases.
The ribbon is constructed of a series of polygons.
These follow the streamline and are oriented according to
the local angle integrated from the angular velocity. In general,
ribbons which extend symmetrically to both sides of
the defining streamline are more suitable for visualization
purposes.
α(s)
·
·
i+1
·
i
·
b
i-1
Fig. 6 Construction of a stream ribbon along the path
of a streamline while rotating it around the line
with the given angle α(s)
The normal of the edge i is rotated in a plane normal to the
streamline. The rotation angle relative to the edge i-1 is
given by α(s). The width b of the ribbon is kept constant.
This marching procedure constructs a stream ribbon of constant
width (see Fig. 6). The initial angle of the first polygon
on a line may be chosen arbitrarily. Often, this angle will be
identical with a physical coordinate axis. It is recommended
to choose the same initial angle for all ribbons within one
image.
Analogous to the method pointed out in [21], the orientation
of the ribbon could be deducted using a Frenet frame
with similar results. The Frenet frame needs special treatment
to resolve ambiguities for straight streamlines. The
method presented here is straight forward to implement and
was found accurate enough in various test cases. Obviously,
the ribbon will not change orientation in an irrotational field.
In a field of solid body rotation with axial translation super-
s
- 5 -
imposed, the ribbon shows the correct swirl whether it is
placed on the vortex axis or away from it. The width of the
ribbon may be chosen arbitrarily as well. Therefore, it is
possible to adjust the width to the scale of the ribbon within
the image. Small vortical structures may require a comparatively
large ribbon to make the swirl visible. This is a major
advantage compared to the technique of constructing a ribbon
between two streamlines, because in that case both lines
are restricted to stay close to the centre of the vortex else the
method would fail. The technique presented here however is
not restricted.
4.2 Vortex pattern near the blunt fin
The flow shows a horseshoe type of vortex. The simulation
also resolves a smaller secondary vortex that rotates in
opposite direction (Fig. 7). In particular, in the vincinity of
the stagnation region ahead of the fin the vortices are accelerated
when they pass the fin. This is advantageous for the
visualization since a streamline release in this region is
sucked into the vortex and remains very close to the central
core for a long time.
Fig. 7 Ribbons represent vortices in the flow field. The
image allows the examination of threedimensional
phenomena with respect to their
traces on the solid walls
Therefore, streamlines were carefully selected to hit the vortex
close to the plane of symmetry then follow the centre of
the vortex past the fin. The ribbons were calculated for two
streamlines, one for each of the vortices.
To find streamlines that stay within the vortex core, the
starting point for the streamline must be placed accurately in
the centre of the vortex. This may be done interactively.
While in the two-dimensional or surface-flow case significant
starting points may be found by means of topology
analysis [4], this is not necessarily sufficient in the threedimensional
case. However visualization of restricted

streamlines in a plane through the flow domain provide a
good starting point for interactive refinement.The full path
of the line consists of a part integrated forward through the
converging vortex core while the part of the line that
approaches the fin from the inflow boundary was integrated
backwards to meet the vortex exactly. Careful selection of
streamlines and calculation of ribbons allows representing
the vortex pattern with effective and easy to perceive visual
objects that do not clutter the image
In the case discussed here, the horseshoe vortex passes
the fin with increasing distance while the secondary vortex
stays very close to the side wall of the fin. When visualized
in combination with the oil-flow pattern obtained in the win
tunnel experiment, the position of this secondary vortex
explains nicely the weak trace of a separating flow half way
up the fin. Fig. 8 shows the vortex core close to the fin,
slightly below the lower separation trace. The vortex must
be below this trace to topologically explain the pattern on
the fin.
Fig. 8 Three-dimensional vortex cores from the
numerical simulation visualized in combination
with oil-flow traces from wind tunnel
experiments
As suspected earlier, this vortex may be represented a bit too
weak, due to spatial resolution in the simulation. This could
explain why the skin friction lines as they are shown in Fig.
3 and Fig. 4 do not show this separation.
5 Vortex and shock wave interaction
The location of the vortices are typically very much determined
by their interaction with the shock waves present in
this three-dimensional flow field. In the past, most visualizations
of shock phenomena relied on pseudo colours or isolines
on cuts through the data space. These methods were
- 6 -
already used by Hung and Buning [3] to visualize shock
locations in a comparable flow field. Their paper as well as
the related work of Hung and Kordulla [23] provide visualization
of scalar data such as Mach number or pressure in
various two-dimensional slices of the flow field for presentaion
of the complex flow field. In the plane of symmetry
they provide an early example of a combined visualization
showing “two-dimensional streamlines” together with isocontours.
To visualize the three-dimensional shape of shock waves, a
shock detection and visualization technique was implemented
using the HIGHEND visualization system mentioned
earlier [9].
Fig. 9 Three-dimensional shock waves from the
numerical simulation visualized in combination
with oil-flow traces from wind tunnel
experiments
This method constructs a surface that matches the maximum
of the local density change everywhere in the flow field. The
gradient of the scalar quantity density is projected onto the
local flow direction vector to find the location of maximal
compression in the streamwise direction. By connecting
these locations, a surface is formed, which may be visualized
as a transparent surface. In the case discussed here, various
shock waves are present in the flow field.
The blunt fin creates a dominant detached bow shock.
The sharp kink in the bottom wall formed by the wedge
forms an oblique ascending shock wave, which starts at the
location of the kink and is steeper than the wedge. Both
shock waves intersect. As a result of the intersection of the
fin with the bottom wall and the horseshoe type vortex, there
is a third part of the shock pattern that encloses the vortex.
More details still await further analysis. However, these
prominent phenomena already provide insight into the interaction
of different flow properties.

Because of viscous action near the walls, shocks become
weaker there and will not be traced. Their position and
shape in the vicinity of the fin clearly indicate the interaction
between the shock wave and the separation traces in the oilflow
image (Fig. 9).
The shapes of the shock waves are bent by interaction with
vortices, as may be seen when visualizing the vortex cores
in combination with the shock waves (Fig. 10).
Fig. 10 Three-dimensional shock waves from the
numerical simulation visualized in combination
with vortex cores seen from the flow exit plane
6 Conclusion
Visualization has been technology driven for a long time,
pseudocolour playing a dominant role. A large variety of
visualization methods is known today. However, only the
integrated application of carefully selected techniques provide
better insight even into complex data with different
phenomena being simultaneously present. Each technique
applied in this example provides a compact, meaningful representation
of a selected feature in the data without occupying
too much space in the image. The combined
visualization of different flow features provides insight into
the interaction of phenomena. Further improvements were
obtained by including the combined visualization of data
from different sources such as numerical simulations and
wind tunnel experiments. Due to a perspective mapping and
blending technique, comparison between data from different
sources was not restricted to image-level comparison by
placing images side by side. Presentation of both data within
a single image allows for quantitative evaluation, for example
of displacement of features.
- 7 -
7 References
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Flow Past a Blunt Fin/Wedge Configuration”, AIAA-93-5026, 5th
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11, Nov. 1984

3D Visualization of
Unsteady 2D Airplane Wake Vortices
Kwan�Liu Ma
ICASE� Mail Stop 132C
NASA Langley Research Center
Hampton� Virginia
Abstract
Air �owing around the wing tips of an airplane forms
horizontal tornado�like vortices that can be dangerous
to following aircraft. The dynamics of such vortices�
including ground and atmospheric e�ects� can be pre�
dicted by numerical simulation� allowing the safety and
capacity of airports to be improved. In this paper� we
introduce three�dimensional techniques for visualizing
time�dependent� two�dimensional wake vortex compu�
tations� and the hazard strength of such vortices near
the ground. We describe a vortex core tracing algo�
rithm and a local tiling method to visualize the vortex
evolution. The tiling method converts time�dependent�
two�dimensional vortex cores into three�dimensional
vortex tubes. Finally� a novel approach is used tocal�
culate the induced rolling moment on the following air�
plane at each grid point within a region near the vortex
tubes and thus allows three�dimensional visualization of
the hazard strength of the vortices.
1 Introduction
Aircraft wakes represent potential hazards near air�
ports. This hazard is so severe that it can control
the required separation between aircraft� thus limit air�
port capacity. When multiple runways are used� cross�
winds and density strati�cation can alter the trajec�
tories and lifetimes of these wake vortices� producing
hazardous �ight conditions which can persist during
subsequent �ight operations. Thus improved forecast�
ing techniques could result in safer airport operation
and higher passenger throughput.
In laboratories� experimental �ow visualization
techniques have been used to allow better understand�
ing of vortex interactions and of merging character�
istics in multiple vortex wakes. For example� smoke
and laser light sheet have been used to obtain both
qualitative and quantitative information on the evolu�
tion of vortices. Up�to�date experimental methods re�
main the primary source of design information. On the
Z. C. Zheng
Department of Aerospace Engineering
Old Dominion University
Norfolk� Virginia
other hand� because of the advances in computational
technology� numerical predictions of wakevortices have
become feasible and are beginning to produce results
consistent with experiments and physical observations.
However� numerical predictions of wake vortex trajec�
tories near the ground are still di�cult. This is be�
cause the �ow is unsteady and is characterized by at
least one pair of moving viscous�trailing�line vortices�
which respond to an essentially inviscid background�
but are coupled to an unsteady� viscous ground�plane�
boundary�layer region. The boundary�layer region can
include separated �ows during portions of the vortex
wake history and thus create secondary vortices.
During the development of more sophisticated nu�
merical models� appropriate visualization methods are
needed to monitor and verify the results from numeri�
cal simulations. Compared to the di�culties of experi�
mental �ow visualization� three�dimensional computer
visualization is both �exible and repeatable. In this pa�
per� we describe the visualization techniques that we
have developed� along with development ofnumerical
prediction models.
Visualization of wake vortices can be divided into
three steps� locating vortex cores� direct visualization
of the vortices� and visualization of quantities asso�
ciated with vortices� such asvelocity� rolling moment�
etc. The identi�cation of a vortex core has been treated
di�erently for di�erent types of �ows. A rigorous�
widely�accepted de�nition of a vortex for unsteady� vis�
cous �ows is needed �7�. Direct visualization of vortic�
ity �elds is often misleading in recognizing the struc�
ture of vortex cores.
Singer and Banks used both the vorticity and pres�
sure �eld to trace the skeleton line of vortices in three�
dimensional transitional �ow �8�. Pressure �eld is used
to correct the numerical errors that might beintro�
duced during the integration of a vorticity line. A
skeleton line passes the center of vortices. Pressure
�eld is also used to determine the boundary of the vor�
tex tube. Yates and Chapman examined both theo�
retically and computationally two de�nitions of vortex

cores for steady �ow� a minimum in the streamline
curvature and a maximum in the normalized helicity
�10�. However� the minimum streamline curvature does
not give a viscous vortex core edge.
A somewhat vague de�nition of a vortex core is a
vorticity�concentrated area characterizing the viscous
aspect of the vortex. We are interested in the details of
the vortex core such as its size and shape that change
in time. Furthermore� for our time�dependent two�
dimensional data� at each time step� there is only one
primary vortex and the center of vortex is well de�ned�
where the maximum vorticity occurs. Our goal is to
identify the boundary of the vortex core.
We describe a tracing algorithm for identifying the
vortex core at each time step using both the vorticity
and velocity �elds. Visualization of these vortex cores
is done by tiling them into a tube�like surface� using
a local surface construction algorithm. The surface is
composed of Gouraud shaded triangle strips and can
be e�ciently displayed on a graphics workstation. It
is colored by di�erent scalar �elds to examine di�erent
aspects of the vortex history. Two�dimensional tech�
niques such as slicing and superimposed xy�plots are
also used in conjunction with visualization of the vor�
tex tube to present more information simultaneously.
The techniques we have developed and the visualiza�
tion results we have obtained help verify and under�
stand the predicted �ow �eld.
An important quantity that may indicate vortex
hazards is the induced rolling moment on the following
airplane� which can be calculated using the unsteady
two�dimensional predicted �ow �elds� combined with
lifting line theory �4�. A novel approach presented here
is to calculate the induced rolling moment on the fol�
lowing airplane at each grid point within a region near
the vortex tube. The resulting data allow us to com�
prehend the hazard strength in three dimensions near
the vortices with techniques such as direct volume ren�
dering. Direct visualization of hazard zone in three
dimensions can assist the design of aircrafts as well as
the �ight control at an airport.
2 Numerical Model
The di�culties related to numerical prediction of wake
vortices near the ground have been discussed by Zheng
and Ash �12�. We show brie�y here the computational
scheme for laminar cases. The coordinate system for
this problem is shown in Figure 1. Because the problem
is symmetric �without crosswind e�ects� in terms of
the y�axis� only the �rst�quadrant of the �ow �eld is
computed. Treating the problem as an unsteady� two�
dimensional �ow� the velocity�vorticity formulation in
Cartesian coordinates can be written as follows�
��
�t
� ���U�
�x
���V �
�
�y � ���2 �
�x2 � �2� � �1�
�y2 x
��������������������������������������
��������������������������������������
��������������������������������������
��������������������������������������
GROUND
��������������������������������������
Figure 1� Coordinate System.
� �U
�y
where � � �V
is the vorticity. The stream func�
�x
tion was governed by the Poisson equation�
with
U � ��
�y
y
r 2 � � ��� �2�
��
and V � � � �3�
�x
The vorticity transport equation is integrated by using
an alternating direction implicit �ADI� scheme with
upwind �ux�splitting. The Poisson equation governing
the stream function is computed using Swarztrauber
and Sweet�s fast Poisson solver �9�.
Computations were started when the two vortices
were located at X0 � 1�Y0 � 2 in non�dimensional
units� where one unit is one half�span of the initial vor�
tex pair. The initial vortex cores were assumed to have
radii of 0.2� and symmetry was employed to reduce
the size of the computational domain. A non�uniform
grid� constructed by mapping a uniformly incremented
exponential distribution into physical space� was em�
ployed� which was a 150�300 net. The grid points were
packed in this manner to permit the vortex and bound�
ary layer regions to be adequately resolved. A typical
laminar case� used in the following discussion� requires
four megaword memory and four�hour CPU time on a
Cray YMP to march 180 non�dimensional time units
with �t � 0.01.
3 Vortex Cores
The vortex core history is an important feature of wake
vortex behavior. It is a way ofchecking the computa�
tional schemes used to calculate vortex problems and
to characterize the vortex decay rate. In single vor�
tex cases� as shown in Figure 2� the vortex core radius
is de�ned as the distance from the vortex center to
the point of maximum tangential velocity. In the cur�
rent problem� a pair of vortices interact� and at the

core boundary
V tangential
0
r core
Figure 2� A Single Vortex Core.
same time� interact with the ground boundary. The
above de�nition of vortex core radius becomes vague
and hard to apply in this case. The viscous interac�
tion between the vortices and the ground causes de�
formation of vortex cores� and thus they are no longer
circular. In addition� the complicated motion of the
vortex in both vertical and lateral directions� and the
use of Cartesian coordinates� makes the determination
of the tangential velocity very di�cult and somewhat
misleading.
We are investigating unsteady� two�dimensional
laminar�vortex �ows. The time dimension can be con�
sidered as the vortex axial dimension with constant
speed motion in that direction. The unsteadiness� how�
ever� makes visualization of three�dimensional vortex
cores di�cult. First� the vorticity values at the vortex
core edge is di�erent at each time step �see Figure 3��
so that isovalue contours can not be used. Second�
the same vorticity values may be generated near the
ground boundary and thus even within a single time
frame� isovalue contours may not reveal true vortex
core shape. Finally� since the vortex core changes size
and shape� the number of points included in the core
changes.
While it is natural to visualize the history of vortex
cores by connecting them into a vortex tube� general�
purpose visualization packages like FAST and Tecplot
cannot correctly construct the desirable vortex tubes.
Some preprocessing is needed to extract the tube� af�
ter which we can perhaps make use of general�purpose
visualization software. In the following sections� we
describe a vortex core tracing algorithm and a surface
construction method� which together form such a pre�
processing step.
3.1 Vortex Core Tracing
The vortex core tracing algorithm developed here is
based on the vorticity �eld� which is obtained di�
r
7.9362
Vorticity
6
4
2
0
�2
�4
�5.2167
threshold value
6 30 90
Time
120 180
Figure 3� Vorticity Values at the boundary of each
Vortex Core.
rectly with the vorticity�streamfunction formulation
described in the previous section. The tangential ve�
locity isnow used only at the altitude of the vortex
center �de�ned as the maximum vorticity point in the
region of the vortex� to determine vorticity levels at
the outer edge of vortex core. The tangential veloc�
ity at this altitude is simply the vertical component of
the velocity �eld� which can be calculated using Equa�
tion 3� once the streamfunctions have been obtained.
The left and right vortex core edges have thus been de�
�ned� where the maximum tangential velocity occurs.
The average of the vorticity values at these two edge
points is then used to establish the upper and lower
boundaries of the core around the vortex at each lat�
eral positions between the two edge points� marching
from left to right.
Figure 3 shows typical threshold values derived from
one of our test data set for locating the boundary of the
vortex core at each time frame. While the maximum
vorticity is 7.93 and the minimum vorticity is �5.21
�near the ground�� the threshold values vary between
0.052 and 1.792. Therefore� the use of a single isovalue
would be inappropriate. This is also why we cannot
use general�purpose visualization software.
The tracing algorithm is called for each time frame
�every 6 time units� saved for �ow visualization� and
therefore a time history of the vortex core can be
shown. Figure 4 summarizes the tracing algorithm in
C. Figure 5 shows a typical contour from data obtained
at time � 180 when the simulation ends. The points
near the left and right edges are sparse due to the left�
to�right traversal along the rectangular computational�
grid that we used. The �ner grid employed in y�
direction is designed to resolve the ground boundary

for �t�0� t�nof�time�frames� t��� �
FindCenterOfVortexCore��center�x� �center�y��
FindLeftRightEdge��left�edge�x� �right�edge�x��
�� find the boundary of the vortex core ��
threshold��vorticity�left�edge�x� center�y��
vorticity�right�edge�x�center�y���2�
for �i�left�edge�x�1� i�right�edge�x� i��� �
�� find the upper boundary point ��
for �j�center�y� j�ydim� j���
if �vorticity�i�j���threshold�
� AddUpperPoints�i�j�� break� �
�� find the lower boundary point ��
for �j�center�y� j�0� j���
if �vorticity�i�j���threshold�
� AddLowerPoints�i�j�� break� �
�
�
Figure 4� Vortex Core Tracing Algorithm.
layer beneath the vortex. Redistributing points by us�
ing� for example� a cubic spline� would produce a bet�
ter point set for the subsequent tiling task� probably
resulting in a smoother surface. Nevertheless� this ex�
tra e�ort does not provide the computational scientist
with additional information. Figure 5 also shows that
at time � 180� the vortex core has grown in size and
has been deformed� from a circle to an ellipse.
3.2 From Vortex Cores to a Vortex Tube
After the vortex core at each time frame is located� the
next task is to tile consecutive vortex cores into a vor�
tex tube. We can treat these consecutive vortex cores
as a set of planar contours similar to those generated in
medical imaging techniques such as computer tomog�
raphy. The tiling problem for successive contours has
been studied thoroughly and a summary of previous
work can be found in the survey paper by Meyers et
al. �6� on the subject of surfaces from contours.
The e�ciency of the surface construction process
and the quality of surfaces generated is related to the
properties of the contours. Our contours �vortex cores�
have the following features�
1. Contours are parallel.
2. Contours are usually well aligned.
3. Contours are well shaped � no severe concavity.
4. Contours have no holes.
5. Each contour has a well de�ned center.
6. The spacing between consecutive contours need
not be exact.
7. Each contour may have a di�erent number of
points.
Figure 5� Points Forming a Vortex Core.
Feature 2 indicates that the contour �vortex core�
changes its shape and position gradually in time. As to
Feature 6� unlike� for example� in medical applications
where the distance between consecutive contours needs
to be precise to make correct interpretation� the spac�
ing can be less restricted. Depending on the speed of
the airplane� the actual spacing can be very large with
respect to the x and y dimensions. The total time units
taken in the simulation may be equal to miles in the
spatial dimension. So the spacing is usually selected
for the ease of displaying and viewing with respect to
the x and y dimensions.
Although Features 1�6 greatly simplify the tiling
problem� Feature 7 introduces some di�culties. How�
ever� two existing metrics are reported in the literature
for connecting consecutive contours. The �rst was in�
troduced by Cook et al. �2�� and is based on match�
ing directions of points from each contour�s center of
mass. The second metric was introduced by Chris�
tiansen and Sederberg �1� and is based on minimiz�
ing the length of diagonal between consecutive con�
tours. Both of these metrics are linear�time heuristic
�greedy� methods that make the best choice locally�
with no guarantee of �nding a globally optimal result.
In contrast� Fuchs et al. �3� proposes a globally opti�
mal solution� which is computationally more expensive.
In practice� linear�time heuristic methods are usually
�good enough� and a lot faster when the contours have
a lot of points.
We havechosen the method proposed by Cooketal.
and the result has been satisfactory. Figure 6 shows a
typical mesh generated. The sparsity of points near the
left and right edges of the vortex tube results in less

Figure 6� Mesh of a Vortex Tube.
desirable triangle combinations. However� this does
not distract from the visualization of the history of the
vortices.
3.3 Visualization Results
We applied the above techniques to two�dimensional
simulation results containing 180 time units. The vi�
sualization domain was a 140�280 non�uniform grid.
The vortex tube generated was displayed on a Silicon
Graphics Indigo 2. The three�dimensional graphics was
implemented in GL and the user�interface was imple�
mented in Motif.
Plate 1 displays two color�contour slices through the
vortex tube. The right�most slice shows the vorticity
�eld at the �rst time frame� which touches the head
of the tube. Color is mapped to vorticity and the
colormap used is shown underneath. The most nega�
tive vorticity value is mapped to blue� zero vorticity is
mapped to white� and the most positive vorticity value
is mapped to purple. Colors in between are linearly
blended. In both Plate 1 and 2� the color of the vortex
tube is mapped to vorticity values� �. Consequently�
the head of the tube� in the right side of the image�
is more red and purple indicating high vorticity val�
ues. The xy�plot superimposed on each color�contour
slice shows the vertical velocity values along the lateral
line passing through the center of the vortex core. The
slice taken at a later time frame in Plate 1 shows that
secondary negative vorticity has been induced by the
primary vortex� after which separation occurs near the
ground.
In Plate 2� for the contour slice at the head of the
vortex tube� the colors are mapped to kinetic energy
and the colormap used is shown underneath. Low ki�
netic energy values are mapped to white�blue and high
to red�purple. Values in between are linearly mapped.
Again� the xy�plot shows the vertical velocity. Asyou
can see� the maximum kinetic energy region does not
coincide with the vortex core �unlike in the single vor�
tex case�. The tube becomes bigger as vorticity de�
creases with increasing time.
In Plate 3� a volume rendered image of the vorticity
�eld is shown. Volume rendering o�ers direct visu�
alization of complex structures and three�dimensional
relationships. The image was generated by treating the
stack of contours as a volume� and mapping vorticity
values to color and opacity. By carefully selecting the
color and opacity mapping� the volume rendered image
also captures a vortex tube of similar shape. Unfortu�
nately� this tube is less precise and is not identical to
the one shown in Plate 1. The problem� which has
been described in the previous section� is that a single
mapping cannot capture the actual vortex tube. The
true edge of the tube� where the maximum tangen�
tial velocity occurs� cannot be determined with direct
volume visualization. The tangential velocity plot in
Plate 1 shows that edge.
In Plate 4� we compare the vortex tube extracted
previously by using the algorithm described in Sec�
tion 3.1 with tubes extracted by using only an isovalue
of vorticity. The image shows as meshes the vortex
tube superimposed with an isosurface of vorticity. An
isovalue of 0.1 was used. As expected� the isovalue
tubes do not coincide with the vortex core tube.
Further� compare the slice taken at a later time
frame in Plate 1 with the volume rendered image.
The color contours display the vorticity �eld around
the vortex core as well as near the ground plane. As
mentioned before� the vorticity �eld generated on the
ground may smear the displayed vortex core in the
three�dimensional image. This phenomenon is shown
in the volume rendered image too. The red vortex tube
�positive vorticity values� is covered by a blue sheet
�negative vorticity values� from the bottom.
The vortex tube in Plate 1 gives an intuitive� three�
dimensional display of the decay ofvortex. This cer�
tainly resembles results obtained from experimental
�ow visualization and �eld observation �vapor trails�.
Additionally� computer visualization may allow re�
searchers to consider various atmosphere conditions
such as turbulence� strati�cation and cross wind. The
three�dimensional vortex tube would give a vivid pic�
ture of the behavior of the vortex under those di�erent
conditions.
4 Vortex Hazard � Rolling Moment
In order to assess the vortex hazard� some measure
of hazard strength is required. Since the computa�
tional domain is an unbounded quadrant� overall or

global measures of circulation or velocity levels are of
little value. The approach zone method used in �11�
by Zheng and Ash could be meaningful in some sense�
which calculated the circulation and kinetic energy in a
bounded zone. However� the approach zone was de�ned
somewhat arbitrarily� and the circulation and kinetic
energy were not a direct measure of the vortex haz�
ard. Therefore� a more systematic method has been
developed by Zheng� Ash and Greene �13�. The rolling
moment induced on following aircraft by the wake vor�
tex �ow �eld generated from leading aircraft is used as
a measure of the hazard.
In the rolling moment method� the down�wash ve�
locity �eld� derived from Equation 3� is the computa�
tional result obtained from the numerical scheme in
Section 2. Following aircraft are modeled as rectangu�
lar wings. In the current model� the in�uence of the fol�
lowing aircraft on the upstream �ow �eld is neglected�
as is the thickness of their wings. Then� employing
Prandtl�s lifting line theory �4�� the induced rolling
moment coe�cient can be obtained using Fourier se�
ries expansions. The rolling moment data used in the
following discussion required 1�200 CPU seconds on a
Cray YMP.
4.1 Visualization Results
The example shown here is a case with the follow�
ing airplane having the same wing span as that of the
generating airplane. Since the �ow �eld is symmetric
�without crosswind e�ects�� only half of the domain is
displayed� as in the previous section. In the past� we
calculated and displayed� with xy�plot� the history of
rolling moment point by point. In order to visualize
vortex hazard of the whole �ow �eld� we use direct
volume rendering of rolling moment coe�cients� as de�
picted in Plate 5. Only the region near the vortex
tube is rendered. The colormap used is the same as
the one shown in Plate 1. The rolling moment coef�
�cients visualized here� ranging from �0.189 to 0.054�
are mapped to color and opacity. Negative values are
mapped to green�blue� white represents no rolling mo�
ment� and positive values are mapped to yellow�red�
purple. Higher absolute values are mapped to higher
opacity. As a result� regions of lower rolling moment
are more transparent and thus the region of hazard
stands out.
The right image in Plate 5 shows a view from the
top �positive y�. Further exploration of the data can
be made by using the interactive volume visualization
techniques described in �5�. It is seen that there are
regions of positive rolling moment at both the left and
right sides of the region of negative rolling moment.
However� the left side is very small because it was an�
nulled by the negative moment from the other side of
the plane.
It should be noted that the value at each point rep�
resents the rolling moment coe�cient a following air�
plane experiences when the center of the wing span
reaches that point in space. The following airplane
is assumed to have the same size as the leading air�
plane. From our results� it is found that when the
following airplane is near the region of the vortex core�
the rolling moment is large �negative sign�� and there�
fore the hazard is more signi�cant. In fact� the loca�
tion with largest negative rolling moment is the center
of vortex. When it moves laterally away from the cen�
ter� the rolling moment becomes smaller. If it moves
laterally further to a position where only one side of
the wing is in�uenced by the vortex� the rolling mo�
ment sign is changed� which isphysically correct. Fig�
ure 7 shows such sign�change graphically. When the
following airplane is in di�erent positions relative to
the center of the vortex� it experiences di�erent di�
rection of rolling motion. In Region 2� the following
airplane moves into the region around the vortex core
and counterclockwise rolling motion is induced on it.
In Region 1� the port side of the following airplane ex�
periences the rolling motion caused by the up�wash ve�
locity of the vortex� and thus has a clockwise roll� while
in Region 3 a clockwise rolling motion caused by the
down�wash velocity exerts on the following airplane.
It is observed that the maximum positive value is
much smaller than the largest negative value� so one
can conclude that the most hazardous region is the
region around the vortex core. On the other hand�
the kinetic energy contours in Plate 2 has lower values
around the vortex center and higher values at both
the left and right sides of the vortex core. Therefore�
kinetic energy cannot be used to judge the hazardous
regions.
Plate 6 displays two isosurfaces �blue and green�
of rolling moment superimposed with the vortex tube
�red� extracted previously. The isovalues chosen are
0.01 �green� and �0.01 �blue�. As an example� the value
of rolling moment coe�cient �0.01 can be considered as
ahypothetical hazard threshold for the following air�
plane. That is� the region inside the blue tube can be
treated as where the induced rolling moment on the
follower is beyond its controllable capacity and there�
fore the follower should avoid �ying into this region. In
addition� by superimposing tubes� the resulting visu�
alization allows us to discern the relationship between
the size of the vortex core and the hazard. It is easily
seen that when the core size increases� the hazardous
region decreases� which explains the importance of pre�
dicting correct core sizes in numerical simulations.
5 Conclusions
The study of the evolution of vortices behind a wing
and its potential hazards is of considerable interest in
aeronautic design and terminal control. In addition to
experimental studies� numerical models have also been
designed which might be used in practice for �ight
control at an airport. In this research� visualization

y
region 1
region 3 region 2
x
region 1
region 2 region 3
Figure 7� Rolling Moment.
techniques have been developed which allow the na�
ture of numerically predicted wake vortices to be seen
and analyzed. A vortex core tracing algorithm and
a tiling method have been implemented which enable
researchers to interactively examine the structure of
vortex cores. Rolling moment� a quantity that may be
used to measure the degree of hazard� is calculated
throughout the domain of interest and visualized in
three�dimensions using direct volume rendering.
The comprehensive images generated by using these
techniques suggest more intuitive ways of visualizing
wake vortices and their corresponding hazard strength.
These techniques� implemented with an interactive
user interface� can be used not only for researchers to
tune their numerical models� but also for �ight control
at airports. Consequently� our future work will include
developing interactive visualization of vortex hazards
and integrating the visualization techniques into the
numerical simulation for real�time monitoring.
Acknowledgements
This work was supported by the National Aeronautics
and Space Administration under contract NAS1�19480
and research grant NAG1�1437. The authors would
like to thank Robert Ash� George Greene� Bart Singer
and John Van Rosendale for their helpful comments.
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ference �Octorber 1991�� pp. 31.1�31.30. Wash�
ington� D.C.
�12� Zheng� Z. C.� and Ash� R. L. Prediction of
Turbulent Wake Vortex Motion. In Proceedings of
the Fluids Engineering Conference �1993�� L. Kral
and T. Zang� Eds.� American Society of Mechani�
cal Engineers� pp. 195�207. Transitional and Tur�
bulent Compressible Flows� FED�Vol. 151.
�13� Zheng� Z. C.� Ash� R. L.� and Greene� G. C.
A Study of the In�uence of Cross Flow on the Be�
havior of Aircraft WakeVortices Near the Ground.
In the 19th Congress of the International Council
of the Aeronautical Sciences� Anaheim� California.

Plate 1: Vortex tube and vorticity cut planes.
Plate 3: Direct volume rendering of vorticity field.
Plate 5: Direct volume rendering of rolling moment.
Plate 2:Vortex tube and kinetic energy cut plane.
Plate 4: Vorticity (blue).
Plate 6: Rolling moment (blue, green).

Abstract
A new algorithm for identifying vortices in complex flows
is presented. The scheme uses both the vorticity and pressure
fields. A skeleton line along the center of a vortex is
produced by a two-step predictor-corrector scheme. The
technique uses the vector field to move in the direction of
the skeleton line and the scalar field to correct the location
in the plane perpendicular to the skeleton line. With
an economical description of the vortex tube’s cross-section,
the skeleton compresses the representation of the
flow by a factor of 4000 or more. We show how the reconstructed
geometry of vortex tubes can be enhanced to
help visualize helical motion.
1 Introduction
Vortex Tubes in Turbulent Flows:
Identification, Representation, Reconstruction
Vortices are considered the most important structures
that control the dynamics of flow fields. Large-scale vortices
are responsible for hurricanes and tornadoes.
Medium-scale vortices affect the handling characteristics
of an airplane. Small-scale vortices are the fundamental
building blocks of the structure of turbulent flow. One
would like, therefore, to visualize a flow by locating all of
its vortices and displaying them.
This paper presents a novel predictor-corrector technique
for locating vortex structures in three-dimensional
flow data. The technique is effective at locating vortices
even in turbulent flow data. As an additional benefit, the
technique provides a terse, one-dimensional representation
of vortex tubes which offers significant compression
of the flow data. Such compression is important if one
wishes to visualize unsteady (i.e., time-varying) flows
interactively.
*Institute for Computer Applications in Science and Engineering, MS
132C, NASA Langley Research Center, Hampton, VA 23681
(banks@icase.edu). Work supported under contract NAS1-19480.
**High Technology Corporation, MS 156, NASA Langley Research
Center, Hampton, VA 23681 (b.a.singer@larc.nasa.gov). Work supported
by the Theoretical Flow Physics Branch at NASA Langley
Research Center under contract NAS1-20059.
David C. Banks* and Bart A. Singer**
Section 2 presents a survey of the efforts by various
other researchers to define mathematical characteristics
satisfied by vortices. Section 3 presents our predictor-corrector
scheme for identifying vortices and discusses some
of the programming considerations that are necessary to
make the scheme efficient. Section 4 describes how we
calculate the cross-sections of the vortex tube and how
we represent them. In section 5 we show how the vortex
skeletons, together with an efficient representation of the
cross-sections, offer a substantial amount of data compression
to represent features of a flow. We then describe
the process of reconstructing the vortex tubes from the
compressed format and show an enhanced reconstruction
that helps visualize the motion of the fluid along the vortex
tube.
2 Survey of Identification Schemes
The term “vortex” connotes a similar concept in the
minds of most fluid dynamicists: a helical pattern of flow
in a localized region. There are mathematical definitions
for “vorticity” and “helicity,” but vortical flow is not
completely characterized by them. For example, a shear
flow exhibits vorticity at every point even though there is
no vortical motion. A precise definition for a vortex is
difficult to obtain — a fact supported by the variety of
efforts outlined below.
Spiral Moving With the Core
Robinson [1] suggests the following working definition
for a vortex.
A vortex exists when instantaneous streamlines mapped
onto a plane normal to the vortex core exhibit a roughly
circular or spiral pattern, when viewed from a reference
frame moving with the center of the vortex core.
Robinson [2] and Robinson, Kline, and Spalart [3] use the
above rigorous definition to confirm that a particular
structure is, in fact, a vortex. Unfortunately, this definition
requires a knowledge of the vortex core before one
can determine whether something is a vortex.
Low Pressure
Robinson and his colleagues find that elongated lowpressure
regions in incompressible turbulent flows almost

always indicate vortex cores. Isosurfaces of low pressure
are usually effective at capturing the shape of an individual
vortex (Figure 1a). Pressure surfaces become indistinct
where vortices merge, however, and a high-quality
image can easily require thousands of triangles to create
the surface. The need to compress the representation
becomes acute when visualizing time-varying data.
Vorticity Lines
Vorticity is a vector quantity that is proportional to
the angular velocity of a fluid particle. It is defined as
ω = ∇ × u
where u is the velocity at a given point. Vorticity lines are
integral curves of vorticity (Figure 3). Moin and Kim [4]
[5] use vorticity lines to visualize vortical structures in
turbulent channel flow. The resulting curves are
extremely sensitive to the choice of initial location x 0 for
the integration. As Moin and Kim point out [4],
If we choose x 0 arbitrarily, the resulting vortex line is
likely to wander over the whole flow field like a badly
tangled fishing line, and it would be very difficult to
identify the organized structures (if any) through which
the line may have passed.
They illustrate the potential tangle in Figure 2 of [5].
To avoid such a confusing jumble, they carefully select
the initial points. However, Robinson [2] shows that even
experienced researchers can be surprisingly misled by
ordinary vorticity lines.
Cylinder With Maximum Vorticity
Villasenor and Vincent [6] present an algorithm for
locating elongated vortices in three-dimensional timedependent
flow fields. They start from a seed point and
compute the average length of all vorticity vectors contained
in a small-radius cylinder. They repeat this step for
a large number of cylinders that emanate from the seed
point. The cylinder with the maximum average becomes
a segment of the vortex tube. They use only the magnitudes
(not the directions) of vorticity; as a consequence
the algorithm can inadvertently capture structures that are
not vortices.
Vorticity and Vortex Stretching
Zabusky et al. [7] use vorticity |ω| and vortex
stretching |ω ⋅ ∇u| /|ω| in an effort to understand the
dynamics of a vortex reconnection process. They fit ellipsoids
to the regions of high vorticity. Vector field lines of
vorticity and of vortex stretching emanate from the ellipsoids.
In flows with solid boundaries or a mean straining
field, the regions with large vorticity magnitudes do not
necessarily correspond to vortices (Figure 1b); hence, the
ellipsoids do not always provide useful information.
(a)
(b)
(c)
(d)
(e)
Figure 1. Different schemes used to identify a vortex. From top:
(a) isosurface of constant pressure; (b) isosurfaces of constant
vorticity; (c) isosurfaces of complex-valued eigenvalues of the
velocity-gradient matrix; (d) isosurface of constant helicity; (e)
spiral-saddles (dark lines) compared with isopressure vortex
tube (pale surface). A reconstruction based on our method is
shown in Figure 8. Each image visualizes the same flow.
Velocity Gradient Tensor
Chong, Perry, and Cantwell [8] define a vortex core
as a region where the velocity-gradient tensor has complex
eigenvalues. In such a region, the rotation tensor
dominates over the rate-of-strain tensor. Soria and
Cantwell [9] use this approach to study vortical structures
in free-shear flows. At points of large vorticity, the eigen-

values of the velocity-gradient matrix are determined: a
complex eigenvalue suggests the presence of a vortex.
This method correctly identifies the large vortical
structures in the flow. However, the method also captures
many smaller structures without providing a way to link
the smaller vortical volumes with the larger coherent vortices
of which they might be a part (Figure 1c).
Curvature and Helicity
Yates and Chapman [10] carefully explore two definitions
of vortex cores. Unfortunately, the analyses and
conclusions for both definitions are appropriate only for
steady flows.
By one definition, the vortex core is the line defined
by the local maxima of normalized helicity (the dot product
of the normalized velocity and vorticity). Figure 1d
shows an isosurface of constant helicity. Notice that the
surface fails to capture the “head” on the upper-right side
of the hairpin vortex. This shows that the local maxima
fail to follow the core.
In the other definition, a vortex core is an integral
curve that has minimum curvature. If there is a critical
point on a vortex core, then that point must be a spiralsaddle.
The eigenvector belonging to the only real eigenvalue
of the spiral-saddle corresponds, locally, to an integral
curve entering or leaving the critical point. By
integrating this curve, the entire vortex core may be visualized
[11]. Figure 1e, however, shows that these curves
can miss the vortex completely. (The red spot is a critical
point; the horizontal integral curves are colored blue).
User-guided Search
Bernard, Thomas, and Handler [12] use a semi-automated
procedure to identify quasi-streamwise vortices.
Their method finds local centers of rotation in user-specified
regions in planes perpendicular to the streamwise
direction of a turbulent channel flow. Experienced users
can correctly find the critical vortices responsible for the
maintenance of the Reynolds stress. Their method captures
the vortices that are aligned with the streamwise
direction, but in free-shear layers and transitional boundary
layers, the significant spanwise vortices go undetected.
Because it depends heavily on user intervention, the
process is tedious and is dependent upon the individual
skill of the user.
3 The Predictor-corrector Method
The methods listed above all experience success in
finding vortices under certain flow conditions. But all of
them have problems capturing vortices in unsteady shear
flow, which is of interest to us because of its importance
in understanding the transition from laminar to turbulent
flow. We were led, therefore, to develop another tech-
nique which could tolerate the complexity of such a transitional
flow.
Our predictor-corrector method produces an ordered
set of points that approximates a vortex skeleton. Associated
with each point are quantities that describe the local
characteristics of the vortex. These quantities may
include the vorticity, the pressure, the shape of the crosssection,
or other quantities of interest. This method produces
lines that are similar to vorticity lines, but with an
important difference. Whereas vorticity is a mathematical
function of the instantaneous velocity field, a vortex is a
physical structure with coherence over a region of space.
In contrast to vorticity lines (which may wander away
from the vortex cores), our method is self-correcting: line
trajectories that diverge from the vortex core reconverge
to the center.
In this section we discuss the procedure used to find
an initial seed point on the vortex skeleton. We then
explain the predictor-corrector method used for growing
the vortex skeleton from the seed point. Finally, we
address how to terminate the vortex skeleton.
3.1 Finding a Seed Point
Vorticity lines begin and end only at domain boundaries,
but actual vortices have no such restriction. Therefore
we must examine the entire flow volume in order to
find seed points from which to initiate vortex skeletons.
We consider low pressure and a large magnitude of vorticity
to indicate that a vortex is present. Low pressure in
a vortex core provides a pressure gradient that offsets the
centripetal acceleration of a particle rotating about the
core. Large vorticity indicates that such rotation is probably
present.
In our implementation, the flow field (a three-dimensional
rectilinear grid) is scanned in planes perpendicular
to the streamwise direction. The scanning direction
affects the order in which vortices are located, but not the
overall features of the vortices. In each plane, the values
of the pressure and the vorticity magnitude are checked
against threshold values of these quantities. (Threshold
values can be chosen a priori, or they can be a predetermined
fraction of the extrema.) A seed point is a grid
point that satisfies the two threshold values.
We next refine the position of the seed point so that it
is not constrained to lie on the grid. The seed point moves
in the plane perpendicular to the vorticity vector until it
reaches the location of the local pressure minimum. From
this seed point we develop the vortex skeleton in two
parts: forward and backward.
3.2 Growing the Skeleton
Once a seed point has been selected, the skeleton of
the vortex core can be grown from the seed. This is where

(1) (2)
p i
ω i
Compute the vorticity at a
point on the vortex core.
(3) ωi+1 (4)
Compute the vorticity at the
predicted point.
p i+1
Figure 2. Four steps of the predictor-corrector algorithm.
p i+1
Step in the vorticity direction
to predict the next point.
we apply the two-stage predictor-corrector method. With
this technique, the next position of the vortex skeleton is
predicted by integrating along the vorticity vector. This
candidate location is corrected by adjusting the position
to the pressure minimum in the plane that is perpendicular
to the vorticity vector. The rationale is that rotation
about the vorticity vector is supported by low pressure at
its center: the vortex tube’s cross-section has its lowest
pressure at the center of the tube. Integral curves of vorticity
or of the pressure gradient are both unreliable at
capturing vortex skeletons. Remarkably, the combination
of the two provides a robust method of following the vortex
core. The continuous modification of the skeleton
point lessens the sensitivity to both the initial conditions
and the integration details.
The predictor-corrector algorithm is illustrated in the
schematic diagrams of Figure 2. The details for continuing
the calculation from one point to the next are indicated
by the captions. Steps 1-2 represent the predictor
stage of the algorithm. The corrector stage is summarized
by steps 3-4.
The effectiveness of the predictor-corrector scheme
is illustrated in Figure 3, in which data from the direct
numerical simulations of Singer and Joslin [13] are analyzed.
The transparent vortex tube (a portion of a hairpin
vortex) is constructed with data from the full predictorcorrector
method. Its core is indicated by the darker skeleton.
The lighter skeleton follows the uncorrected integral
curve of the vorticity. It is obtained by disabling the
corrector phase of the scheme. The vorticity line deviates
from the core, exits the vortex tube entirely, and wanders
within the flow field.
P
Correct to the pressure min in
the perpendicular plane.
Figure 3. Vorticity line (light) compared to predictor-corrector line
(dark). Note that the vorticity line exits from the vortex tube while
the predictor-corrector skeleton line follows the core
3.3 Terminating the Vortex Skeleton
Vorticity lines extend until they intersect a domain
boundary, but real vortices typically begin and end inside
the domain. Therefore, the algorithm must always be prepared
to terminate a given vortex skeleton. A simple and
successful condition for termination occurs when the vortex
cross-section (discussed in section 4) has zero area.
As Figures 3 and 7 show, the reconstructed vortex tubes
taper down to their endpoints (where the cross-section
vanishes).
3.4 Implementation Details
Although the general behavior of the predictor-corrector
algorithm is reliable and robust, optimal performance
of the technique requires careful attention to
implementation details. This section addresses issues that
are important to the successful use of this method. It is by
no means exhaustive; additional details are provided by
Singer and Banks [15].
Eliminating Redundant Seeds and Skeletons
Sampling every grid point produces an overabundance
of seed points, and hence a multitude of nearlycoincident
vortex skeletons (Figure 4). Each of these
skeletons lies at the core of the very same vortex tube;
one representative skeleton suffices. The redundancies
are eliminated when points inside a tube are excluded
from the pool of future seed points. We accomplish this
by flagging any cell (in the computational 3D grid) that is
found to lie in a vortex tube. Future seed points must not
be flagged.

Figure 4. Multiple realizations of the same vortex tube from different
seed points. Each seed point generates a slightly different
skeleton line, although all the skeletons remain close to the vortex
core.
Eliminating Spurious Feeders
A seed near the surface of the vortex tube can produce
a “feeder” vortex skeleton that spirals toward the
vortex center. Intuitively, most of these are seeds that
should have been flagged, but were missed because they
lie near the boundary of a vortex tube. Examples of these
feeders are illustrated in Figure 5. We eliminate feeders
by taking advantage of the fact that the predictor-corrector
method is convergent to the vortex core. A feeder
skeleton, begun on the surface of the tube, grows toward
the core; by contrast, a skeleton growing along the core
does not exit through the surface of the tube. To validate
a candidate seed p 0 , we integrate forward n steps to the
point p n and then backward again by n steps. If we return
very close to p 0 then it is a “true” seed point.
Numerical Considerations for Interpolation
Neither the predictor nor the corrector step is likely
to land precisely on a grid point; hence, we must interpolate
the pressure and vorticity at arbitrary locations in the
flow field. To reduce any bias from the interpolation, a
four-point Lagrange interpolation is used in each of the
three coordinate directions. The interpolation scheme
works quite well, although it is the most expensive step in
our implementation.
The interpolation scheme makes the predictor-corrector
method at least first-order accurate: skeleton points
are located to within the smallest grid dimension. This
ensures that, on data sets with well-resolved vorticity and
pressure, the method successfully locates vortex cores.
Figure 5. Feeders merge with a large-scale hairpin vortex. Three
points that satisfy the threshold criteria lie on the edge of vortex
tube. Their trajectories curve inward toward the core and then follow
the main skeleton line.
Corrector Step
The pressure-minimum correction scheme uses the
method of steepest descent to find the local pressure minimum
in the plane perpendicular to the vorticity vector.
The smallest grid-cell dimension is used as a local length
scale to march along the gradient direction.
The corrector phase can be iterated in order to converge
to the skeleton, but that convergence is not guaranteed.
We therefore limit the angle that the vorticity can
change during a repeated iteration of the corrector phase.
In the case of overshooting the limit, we simply quit the
corrector phase. We could choose a smaller step-size and
retry, but we have not found this to be necessary.
4 Finding the Cross-section
Since it is unclear how to precisely define what
points lie in a vortex, it is also unclear how to determine
the exact shape of a vortex tube’s cross-section. Determining
an appropriate measure of the vortex cross-section
has been one of the more difficult practical aspects of
this work.
A point on the vortex skeleton serves as a convenient
center for a polar coordinate system in the plane perpendicular
to the skeleton line. We have chosen therefore to
characterize the cross-section by a radius function. Note
that this scheme correctly captures star-shaped cross-sections.
Cross-sections with more elaborate shapes are truncated
to star shapes (with discontinuities in the radius
function). In practice this choice does not seem to be very
restrictive, as section 4.2 indicates.

In examining the cross-section plane there are two
important questions to address. First, what determines
whether a point in the plane belongs to the vortex tube?
Second, how should the shape of the tube’s cross-section
be represented? This section summarizes the strategies
that we found to be successful.
4.1 Criteria for Determining Membership
For isolated vortices, a threshold of pressure provides
an effective criterion to determine whether a point
belongs to a vortex. When two or more vortices interact,
their low-pressure regions merge and distort the radius
estimate of any single vortex. This difficulty is resolved if
the angle between the vorticity vector on the skeleton line
and the vorticity vector at any radial position is restricted.
Any angle greater than 90 degrees indicates that the fluid
at the radial position is rotating in the direction opposite
to that in the core. We have found that the 90-degree
restriction works well in combination with a low-pressure
criterion for the vortex edge.
For the actual computation of the radial distance, the
pressure and the vorticity are sampled along radial lines,
emanating from the skeleton, lying in the perpendicular
plane. We step along each radial line until a point is
reached that violates the vorticity or the pressure-threshold
criterion.
4.2 Representation of the Cross-section
If the radius of the cross-section were sampled at 1degree
increments, then 360 radial distances (and a reference
vector to define the 0-degree direction) would be
associated with each skeleton point. That is a great deal
of data to save for each point of a time-varying set of vortex
skeletons. We have found that average radius is sufficient
to describe the cross-section of an isolated vortex
tube.
When vortices begin to interact, the cross-section is
non-circular and so the average radius does not provide a
good description of it. A truncated Fourier series of the
radial distance provides a convenient compromise
between the average radius and a full set of all radial
locations. The series is easy to compute, easy to interpret,
and allows a large range of cross-sectional shapes. In our
work, we keep the constant term, the first and second sine
and cosine coefficients, and a reference vector. Most of
the cases that we have checked have a factor-of-10 drop
in the magnitude of the first and second coefficients, indicating
that the neglected terms are not significant. That
observation also validates our assumption that the crosssection
is well-represented by a continuous polar function.
Figure 6 illustrates a single cross section of a vortex
educed from direct numerical simulation data. The
Figure 6. Comparison of different ways to represent the crosssection
of a vortex tube. The shaded region is the finely-sampled
radius function. The thin line is an approximating circle. The thick
line is a 5-term Fourier representation.
shaded region is the interior of the vortex tube, sampled
at 1-degree intervals. The thin line is a circle, centered at
the skeleton, using the averaged radius of the vortex tube.
The thick line is the truncated Fourier series representation
of the vortex cross-section, providing a better
approximation than the circle.
5 Data Compression & Reconstruction
Our particular interest is to visualize the transition to
turbulence in a shear flow. We have performed a lengthy
simulation using Cray computers over the course of two
calendar years, using about 2000 Cray2 hours of processing
time. The numerical grid grows with the size of the
evolving flow structures, but a grid size of 461×161×275
in streamwise, wall-normal, and spanwise directions is
representative. Each grid point holds several numerical
quantities, including pressure and vorticity. Thus the storage
exceeds 650 MB (megabytes) of data per time step.
By using vortex skeletons we are able to compress
the data significantly and then reconstruct the vortex
tubes locally on a workstation.
5.1 Compression
An animation of the vortices, based on the original
computational volumetric data, would consume over
650 GB of data for 1000 time steps. At present this is a
prohibitive requirement of an interactive 3D animation
on a workstation. Storing all the individual polygons
offers some compression, but not enough to bring an animation
within reach.
In general, a vortex skeleton is adequately represented
by 30 to 200 samples. The complex scene in Figure
7 is represented by about 2000 skeleton points, each

Figure 7. Interacting vortices (from a numerical simulation)
within a complex flow are identified with the predictor-corrector
algorithm. The ball to the upper left represents the light source.
endowed with 72 bytes of data (representing position,
tangent, normal, binormal, cross-section, and velocity
magnitude). Thus a reduction from 650 MB to 144 KB is
achieved, representing more than a 4000-fold factor of
compression. This amount of compression offers the
promise of workstation-based interactive animations,
even for a 1000-frame simulation.
5.2 Reconstruction
The significant data-compression that vortex skeletons
provide does not come without cost. There is still the
matter of reconstructing polygonal tubes from the skeletons.
If the tubes have circular cross-sections, they are
generalized cylinders. Bloomenthal gives a clear exposition
of how to reconstruct a generalized cylinder from a
curve through its center [14]. The coordinate system of
the cross-section rotates from one skeleton point to the
next. The key issue is how to keep the rate of rotation
(about the skeleton’s tangent vector) small. Excessive
twist is visible in the polygons that comprise the tube:
they become long and thin and their interiors approach
the center of the tube.
In our implementation, we project the coordinate
bases from one cross-section onto the next cross-section.
This produces a new coordinate system that has not
twisted very much. This normal vector might be different
from the reference vector (which indicates the 0-degree
direction) for the Fourier representation of the cross-section.
To reconstruct the cross-section, we phase-shift the
angle in the Fourier series by the angular difference
between the normal and the reference vector. In general,
30 to 80 samples suffice to reconstruct a cross-section of
good quality.
Sometimes there is good reason for a “reconstruction”
that is not faithful to the original shape of the vortex
tube. A static image does not convey the spiraling motion
Figure 8. Enhanced reconstruction of a hairpin vortex tube. The
grooves follow integral curves of velocity, constrained to follow
the surface of the tube.
along the surface of the vortex tube. We experimented
with different methods of visualizing the velocities on the
tube itself. One helpful technique is to create a texture on
the surface, drawing curves to indicate the helical flow.
This visualization is enhanced dramatically when the
curves are displaced inward to produce grooves. Figure 8
demonstrates this technique on a single hairpin vortex.
The grooves follow integral curves of the surface-constrained
velocity vectors. In an informal survey of about a
dozen colleagues, we found that none could estimate the
amount of helical motion in a faithful reconstruction (as
in Figures 3 and 7) of a vortex tube. On the other hand,
the same subjects instantly identified the direction and
amount of rotation in the enhanced image of Figure 8.
There are two important issues in reconstruction that
we have not yet addressed in our implementation; both
relate to the representation and compression of the vortices
as well. First, we would like to minimize the number
of samples carried by a vortex skeleton. Where the vortex
skeleton has high curvature or where the cross-section
changes shape quickly, many samples are required to permit
an accurate reconstruction. But most vortex tubes
have long, straight portions with nearly-circular crosssections
of nearly-constant radius. This characteristic
should permit us to represent the vortex tube with fewer
samples along its skeleton.
The second issue concerns interpolation. In reviewing
the development of a vortical flow, a scientist may be
especially interested in narrowing the interval of animation
to only a few of the original time steps. It would be
helpful to generate in-between frames from the given
data. We could interpolate the original volumetric grids to
extract interpolated vortex skeletons, but that would
require a great deal of data communication. Interpolating
between the skeletal representations, on the other hand,
could be done in memory. Unfortunately, it is difficult to

interpolate between irregular branching structures like
the time-varying vortex skeletons. It would be helpful to
reconstruct the topology of the vortex tubes as they
appear, branch, merge, and disappear over time. These
issues remain as future work.
Conclusions
The innovative use of a two-step predictor-corrector
algorithm has been introduced to identify vortices in
flow-field data. Unlike other approaches, our method is
able to self-correct toward the vortex core. The principle
of using a vector field to predict the location of the next
point and a scalar field to correct this position distinguishes
this method from others. The theoretical justification
for the technique is that vortices are generally
characterized by large magnitudes of vorticity and low
pressures in their core. The presence of these two characteristics
in a cross-section defines the shape of the vortex
interior.
This paper discusses a number of novel approaches
that we have developed to deal with matters such as eliminating
redundant vortices, eliminating feeders, and representing
the cross-section of a vortex tube. Sample
extractions of vortices from various flow fields illustrate
the different aspects of the technique.
The vortex skeletons are an economical way to represent
flow data, offering a 4000-fold compression factor
even in a complex flow. This offers the possibility of storing
a multi-frame flow animation in a workstation’s
memory. The vortex tubes can be enhanced during reconstruction
in order to help visualize the dynamics of vortical
flow.
Acknowledgments
The images in Figure 1 were rendered on a Silicon
Graphics Indigo workstation using the FAST visualization
system. The images in Figures 3, 4, and 5 were rendered
on a Silicon Graphics Indigo 2 using the Explorer
visualization system. Figure 7 was produced using a visualization
system written by Michael Kelley. The image in
Figure 8 was rendered on an Intel Paragon using PGL
(Parallel Graphics Library), which was developed at
ICASE by Tom Crockett and Toby Orloff.
We thank Gordon Erlebacher for his helpful insights
regarding vortex identification schemes. We thank Greg
Turk and the reviewers for their suggested improvements
to this paper.
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The Topology of Symmetric� Second�Order Tensor Fields
Thierry Delmarcelle Lambertus Hesselink
Department of Applied Physics Department of Electrical Engineering
Stanford University Stanford University
Stanford� CA 94305�4090 Stanford� CA 94305�4035
Abstract
We study the topology of symmetric� second�order tensor
�elds. The goal is to represent their complex structure by
a simple set of carefully chosen points and lines analogous
to vector �eld topology. We extract topological skeletons
of the eigenvector �elds� and we track their evolution over
time. We study tensor topological transitions and correlate
tensor and vector data.
The basic constituents of tensor topology are the degen�
erate points� or points where eigenvalues are equal to each
other. Degenerate points play a similar role as critical points
in vector �elds. We identify two kinds of elementary degen�
erate points� which we call wedges and trisectors. They can
combine to form more familiar singularities�such as sad�
dles� nodes� centers� or foci. However� these are generally
unstable structures in tensor �elds.
Finally� we show a topological rule that puts a constraint
on the topology of tensor �elds de�ned across surfaces� ex�
tending to tensor �elds the Poincar�e�Hopf theorem for vector
�elds.
1 Introduction
Many physical phenomena are described in terms
of continuous vector and tensor data. In �uid �ows�
for example� velocity� vorticity� and temperature gradi�
ents are vector �elds. Stresses� viscous stresses� rate�
of�strain� and momentum �ux density are symmetric
tensor �elds.
Both vector and tensor �elds are multivariate� they
involve more than one piece of information at every
point of space. In fact� vector and symmetric tensor
�elds in N dimensional space embody as much infor�
mation as N and 1 2N�N � 1� independent scalar �elds�
respectively� Visualizing such data is a di�cult chal�
lenge� mainly because of the necessity of rendering the
underlying continuity while avoiding problems of visual
clutter. �See for example Reference �1� for a uni�ed
expos�e of vector and tensor visualization techniques.�
Representing vector �elds by their topology is pow�
erful at ful�lling this requirement. The topology is ob�
tained by locating critical points�i.e.� points where the
magnitude of the vector �eld vanishes�and by display�
ing the set of their connecting streamlines �2� 3�. From
this simple and austere depiction� an observer can infer
the structure of the whole vector �eld.
In this article we discuss topological representations
of 2�D symmetric� second�order tensor �elds �referred to
here simply as �tensor �elds��. That is� we investigate
data of the type
� �
T11�x� y� T12�x� y�
T�x� �
�1�
T12�x� y� T22�x� y�
T�x� is fully equivalent to two orthogonal eigenvectors
v i�x� � � i�x�e i�x� �2�
where i � 1� 2 �Figure 1�. � i�x� are the eigenvalues of
T�x� and e i�x� the unit eigenvectors. �The reader un�
acquainted with these concepts will �nd Reference �4�
especially useful.� The eigenvectors v i�x� represent all
λ 2
Figure 1� The two orthogonal eigenvectors v i repre�
sented as bidirectional arrows.
the amplitude information �� i�x�� and all the directional
information �e i�x�� represented in matrix notation by
the components T ij�x�. In a stress�tensor �eld� for ex�
ample� the vectors v i�x� describe the magnitude and
direction of the principal stresses. We represent v1 and
v2 in Figure 1 as bidirectional arrows because their sign
is not determined.
To obtain continuous representations of tensor �elds�
we integrate a series of curves along one of the eigenvec�
tors v1�x� or v2�x�. We refer to these curves as �tensor
v 2
λ 1
v 1

�eld lines� �5� or as �hyperstreamline trajectories� for
consistency with our earlier work �6�.
The topology of a tensor �eld T�x� is the topology
of its eigenvector �elds v i�x�. As with regular vector
�elds� we seek topological skeletons that provide sim�
ple depictions of the structure of the eigenvector �elds.
We obtain these skeletons by locating degenerate points
�Section 2� and integrating the set of their connecting
hyperstreamline trajectories �Section 3�. Due to their
sign indeterminacy� eigenvectors have a di�erent struc�
ture from regular� signed vector �elds. For example� we
show a tensor topological rule constraining the struc�
ture of tensor �elds de�ned across surfaces �Section 4�.
Finally� we discuss succinct extensions of our theory to
3�D and to unsymmetric tensor data �Section 5�.
2 Degenerate Points
We build a topological analysis of tensor �elds from
the concept of degenerate points� which play the role of
critical points in vector �elds.
Streamlines in vector �elds never cross each other
except at critical points and� as we show below� hy�
perstreamlines in tensor �elds meet each other only at
degenerate points. Similar to critical points� degenerate
points are the basic singularities underlying the topol�
ogy of tensor �elds. We de�ne them mathematically as
follows.
De�nition 1 �Degenerate point� A point x0 is a
degenerate point of the tensor �eld T�x� i� the two
eigenvalues of T�x� are equal to each other at x0�i.e.�
i� �1�x0� � �2�x0�.
Let us denote by � the common eigenvalue at the degen�
erate point x0. At x0� the tensor �eld is proportional
to the identity matrix�
� �
� 0
T�x0� �
0 �
which implies that T�x0�e � �e for every vector e. At
most points� there is only one eigenvector associated
with each eigenvalue but� at degenerate points� there
are an in�nity of such eigenvectors. So hyperstreamlines
cross each other at degenerate points.
Degenerate points satisfy the following conditions�1 �
T11�xo� � T22�xo� � 0
�3�
T12�xo� � 0
which we use to locate them. When the data are de�ned
on a discrete grid� we use bilinear interpolation of the
tensor components between vertices.
1 Valid in any coordinate system.
2.1 Index� sectors� and separatrices
In vector �elds there are various types of criti�
cal points�such as nodes� foci� centers� and saddle
points�that correspond to di�erent local patterns of
the neighboring streamlines. These patterns are char�
acterized by the vector gradients at the positions of the
critical points �2�.
Likewise in tensor �elds� di�erent types of degenerate
points occur that correspond to di�erent local patterns
of the neighboring hyperstreamlines. These patterns
are determined by the tensor gradients at the positions
of the degenerate points.
Consider the partial derivatives
a � 1 2 ��T11�T22�
�x
c � �T12
�x
b � 1 2 ��T11�T22�
�y
d � �T12
�y
�4�
evaluated at the degenerate point x0. In the vicinity of
x0� we can expand tensor components to �rst�order as
� T11�T22
2 � a�x � b�y
T12 � c�x � d�y
�5�
where ��x� �y� are small displacements from x0. An
important quantity for the characterization of degener�
ate points is
� � ad � bc �6�
The appeal of � arises from its being invariant under
rotation. That is� if you rotate the coordinate sys�
tem� both tensor components T ij and partial derivatives
fa� b� c� dg change� but � remains constant �7�.
To proceed further we de�ne the concept of an index
at a degenerate point. This extends from vector �elds to
tensor �elds the classical notion of an index at a critical
point �8�.
De�nition 2 �Tensor index� The index at the de�
generate point x0 of a tensor �eld is the number of
counter�clockwise revolutions made by the eigenvectors
when traveling once in a counter�clockwise direction
along a closed path encompassing x0. The path is cho�
sen close enough to x0 so that it does not encompass
any other degenerate points.
While indices at critical points of continuous vector
�elds must be integer quantities �1 for a node� �1 for
a saddle� etc.�� indices at degenerate points of continu�
ous tensor �elds are half�integers. This arises from the
sign ambiguity of the eigenvectors. In fact� we show in
Reference �7� that� if � 6� 0� the index I at x0 is given
by
I � 1
sign��� � �1
�7�
2 2

S 3
β 1
S 2
α 2
S 4
θ 2
β 2
α 1
Figure 2� Hyperbolic �� i� and parabolic �� j� sectors at
a degenerate point.
The index at the degenerate point x0 characterizes
the pattern of neighboring hyperstreamlines. When
traveling along a closed path encompassing x0� we en�
counter two types of angular sectors �Figure 2��
S 5
α 3
S 1
1 hyperbolic sectors � i� where trajectories sweep
past the degenerate point� and
2 parabolic sectors � j� where trajectories lead
away or towards the degenerate point. 2
For example� the singularity in Figure 2 has three hy�
perbolic and three parabolic sectors. By analogy with
vector �elds� we call �separatrices� the dividing hyper�
streamlines that separate one sector from the next� such
as s1 to s6 in Figure 2. Let � k be the angle between the
separatrix s k and the x�axis. We show in Reference �7�
that x k � tan � k must be a root of the cubic equation
dx 3 � �c � 2b�x 2 � �2a � d�x � c � 0 �8�
Thus� there are at maximum three separatrices �real
roots xk� and degenerate points have no more than three
sectors.
Consider a hypothetical singularity with np parabolic
and nh hyperbolic sectors �np � nh � 3�. The parabolic
sectors span angles �j �j � 1� � � � � np� and the hyper�
bolic sectors span angles �i �i � 1� � � � � nh�. The eigen�
vectors rotate an angle �j within a parabolic sector and
�i � � within a hyperbolic sector �Figure 2�. Thus�
during one counter�clockwise revolution around the de�
Pgenerate point� the eigenvectors rotate an angle 2�I �
np
j�1 �j� Pnh i�1��i���. Since Pnp j�1 �j� Pnh i�1 �i � 2��
2 The reader familiar with sectors at critical points in vector
�elds may remember the existence of another type of sector called
�elliptic� �8�. In the case of unsigned eigenvector �elds� elliptic
and parabolic sectors are indistinguishable and we group them in
a unique parabolic class.
θ 1
β 3
S 6
S 1
S2 δ < 0
I = −1/2
S3
S 2
δ > 0
I = 1/2
Figure 3� Trisector �� � 0� and wedge �� � 0� points.
� � ad � bc and I � index.
the index at the degenerate point is given by
I � 1 � n h
2
It follows from Equation 7 that the number of hyper�
bolic sectors at a degenerate point is
n h � 2 � sign���
2.2 Trisector and wedge points
When � � 0� nh � 3� the degenerate point has three
hyperbolic sectors and� since np � nh � 3� there is no
parabolic sector. The pattern of hyperstreamlines cor�
responds to the trisector point shown by the texture3 in
Figure 6. See Figure 3 for a schematic depiction. We
show in Reference �7� that each hyperbolic sector at a
trisector point is less than 180o wide.
When � � 0� nh � 1� the degenerate point has one
hyperbolic sector. The local pattern corresponds to the
wedge point represented in Figures 6 and 3. We show in
Reference �7� that the hyperbolic sector at a wedge point
is always wider than 180o . There are np � 2 parabolic
sectors. When np � 2� the two parabolic sectors are
contiguous and we combine them into a unique sector.
Hence the pattern in Figure 3 where np � 1. �If np � 0�
separatrices s1 and s2 are identical� the parabolic sector
reduces to a single line.�
To summarize� the most elementary singularities in
tensor �elds are trisector and wedge points. The invari�
ant � at the location of a degenerate point characterizes
the nature of this point. � � 0 corresponds to a trisector
point �I � �1 2� and � � 0 corresponds to a wedge point
�I � 1 2�. The crossing of the boundary � � 0 denotes a
topological transition which we study in the next sec�
tion. We defer until Section 3 a discussion of the global
implications of the patterns delineated in Figure 3.
3 We create the textures in this article and in the accompanying
video by a technique discussed in References �7� 9�.
S 1

Saddle
+ δ = =+
0
=
I = −1
+ =
Node
Center
δ = 0
I = 1
= +
+
=
Focus
δ = 0
I = 1
= +
Figure 4� Merging degenerate points. � � invariant given by Equation 6� I � index.
2.3 Merging degenerate points
Wedges and trisectors are stable structures in con�
tinuous tensor �elds� they can not be broken into more
elementary singularities with smaller index. In time�
dependent �ows� however� they move and can merge
with each other� creating combined singularities of
higher index.
A combination of degenerate points looks in the far
�eld as a singularity whose index is the sum of the in�
dices of its constituent parts. The following pattern�
for example� is made up of 4 wedges and 2 trisec�
tors. Its total index is 4 � 1 2 � 2 � 1 2 � 1 and the
structure looks indeed like a center �I � 1� in the far
�eld. Figure 4 shows how merging trisectors create
saddle points �I � �1 2 � 1 2 � �1� and how merging
wedges create nodes� centers� or foci �I � 1 2 � 1 2 � 1�.
Trisectors and wedges cancel each other by merging
�I � �1 2 � 1 2 � 0��i.e.� the singularity vanishes. Con�
versely� wedge�trisector pairs can be created from reg�
ular points. Pair creation is topologically consistent
since it conserves the local index. �We show examples
in Section 3.�
The merging of wedges and trisectors corresponds to
� � 0. A more quantitative study of the patterns in
Figure 4 is di�cult since it requires developing tensor
components at least to second order in Equations 5.
As opposed to critical points in vector �elds� degen�
erate points with integral indices are usually unstable.
They split into elementary wedges or trisectors soon af�
ter their creation by merging. They are nevertheless
important for the study of instantaneous topologies.
3 Tensor Field Topology
We build on the theory of degenerate points to ex�
tract the topology of tensor �elds and to study topo�
logical transitions.
The technique is similar to vector �eld topology with
degenerate points playing the role of critical points. We
represent each eigenvector �eld by a topological skeleton
obtained by locating degenerate points and integrating
the set of their connecting separatrices. We illustrate
these concepts by visualizing the topology of the stress
tensor in a 2�D periodic �ow past a cylinder.
Fluid elements undergo compressive stresses while
moving with the �ow. Stresses are described mathe�
matically by the stress tensor� which combines isotropic
pressure and anisotropic viscous stresses. Both eigen�
values of the stress tensor are negative� and the two
orthogonal eigenvectors� v1 and v2 �Equation 2�� are
along the least and the most compressive directions� re�
spectively. At a degenerate point� the viscous stresses
vanish and both eigenvalues are equal to the pressure�
degenerate points are points of pure pressure.
The texture in Figure 7 shows the �ow �velocity �eld�
at one representative time step. The �ow consists in the
periodic detachment of a separation bubble. Overlaid
are the degenerate points of the stress tensor.
2 Video Clip 1 � The moving texture shows the �ow
evolving over time. Color encodes velocity magnitude from
fast �red� to slow �blue�.
3.1 Tracking degenerate points
The instantaneous representation in Figure 7 con�
tains valuable information but we can learn more about
the spatiotemporal structure of the tensor �eld by
tracking the motion of degenerate points over time.
Figure 8 shows the trajectories followed by degener�
ate points in 3�D space. The third dimension is time�

increasing from front to back. The �gure represents one
period of the evolution of the �ow. Red dots are wedge
points and green dots are trisectors. C�events are cre�
ations of wedge�trisector pairs from regular �ow� and
M�events correspond to pair cancellation by merging.
In some instances� pair creations a�ect only the lo�
cal �ow� the two newly created points move together
and eventually disappear by merging. Two C�events�
however� are di�erent� the newly created points move
far away from each other� inducing a topological tran�
sition in the tensor �eld. These new wedge�trisector
pairs are created periodically in a location alternatively
above then below the cylinder symmetry axis. New
wedge points are quickly dragged into the wake about
the cylinder axis while new trisectors move downstream
away from the axis.
2 Video Clips 2 and 3 � We visualize the motion of
degenerate points of the stress�tensor �eld. The colored
background encodes the magnitude �2 of the most compres�
sive force� from very compressive �red� to mildly compres�
sive �orange� yellow� green� to little compressive �blue�. We
show wedge points as black dots and trisectors as white dots.
Video Clip 2 represents the overall structure of the motion
and Video Clip 3 focuses on the region closer to the body.
The pair�creation events are clearly tight to each region of
low compressive stresses �blue color�.
3.2 Correlating vector and tensor data
Tensor data are highly multivariate and rich in in�
formation content but they are complex and poorly un�
derstood. Vector data are simpler and more familiar to
scientists. It is useful to correlate visually tensor and
vector �elds� not only for our basic understanding of
tensor data but also for gleaning new physical insights
into vector �elds.
2 Video Clip 4 � The moving texture encodes the di�
rection of the velocity �eld. Color encodes the magnitude
of the most compressive eigenvalue �2. Overlaid are the de�
generate points of the stress tensor.
Figure 7 represents one frame from this clip. Tex�
ture and color indicate clearly that detachment bubbles
�saddle�center pairs of the velocity �eld� are regions of
low compressive stresses. Red and white dots are wedge
and trisector points of the stress tensor� respectively.
The motion of the degenerate points is interesting. The
wedge point A� which originated by pair creation� fol�
lows the detachment bubble in its motion downstream.
In fact� a new pair is created with each new bubble.
The oscillating pair B is closely associated to the recir�
culation regions close to the body surface. The wedge
C follows a stable orbit shaped as an 8. It rolls back
and forth between two consecutive bubbles without ever
venturing inside.
3.3 Topological skeletons
We obtain topological skeletons by detecting degen�
erate points and integrating the set of their connecting
separatrices.
Trisector points in tensor �elds play the topologi�
cal role of saddle points in vector �elds. As shown in
Figures 3 and 6� they de�ect adjacent trajectories in
any one of their three hyperbolic sectors toward topo�
logically distinct regions of the domain. Wedge points
possess both a hyperbolic and a parabolic sector. They
de�ect trajectories adjacent in their hyperbolic sector
and terminate trajectories impinging on their parabolic
sector.
Here follows an algorithm to extract the topology of a
tensor �eld. This simpli�ed version assumes that there
are no merged degenerate points with integral index�
1 locate degenerate points by searching in every grid
cell for solutions to Equations 3�
2 classify each degenerate point as a trisector �� � 0�
or a wedge �� � 0� by evaluating a� b� c� d using
Equations 4 and computing � as in Equation 6�
3 select an eigenvector �eld�
4 use Equation 8 to �nd the three separatrices
fs1� s2� s3g at each trisector point and the two sep�
aratrices fs1� s2g at each wedge point �Figure 3��
integrate hyperstreamlines along the separatrices�
terminate the trajectories wherever they leave the
domain or impinge on the parabolic sector of a
wedge point.
Figure 9 shows an example. The texture represents the
most compressive eigenvector of the stress tensor �v2�.
Color encodes as before the magnitude of the compres�
sive force ��2�� from most compressive �red� to least
compressive �blue�. We emphasize the structure of the
tensor �eld by superimposing the topological skeleton of
v2. The structure of these time�dependent data is very
complex and we simplify the topology �in Figure 9 as in
the remaining of this article� by computing only those
separatrices that originate from trisector points� leav�
ing on the side separatrices that emanate from wedge
points.
We can mentally infer the orientation of the eigen�
vector at any point in the plane from the topological
skeleton. Hyperstreamline trajectories curve so as to
follow the shape of the separatrices� bending around
wedge points.
With time� the repeated creation of new wedge�
trisector pairs induces periodic topological transitions�

W 2
W 3
W 1
T 1
T 2
W 4
W 5
W 2
W 3
T 3 W6
Figure 5� Two consecutive frames showing a topological transition of the stress�tensor �eld.
M ��M�
sphere 2
torus 0
2�holed torus �2
n�holed torus 2 � 2 n
Table 1� Euler characteristic of generic surfaces.
which we illustrate in Figure 5. The newly created pair
fT3� W6g changes the topological structure of the tensor
�eld.
As with vector �eld topology� the power of the rep�
resentation comes from its simplicity� a few points and
lines su�ce to reveal the directional information other�
wise buried within abundant multivariate data.
2 Video Clips 5 and 6 � The two clips show the evolu�
tion with time of the topological skeleton in Figure 9� with
and without the textured background. Black dots represent
wedge points and white dots are trisectors.
3.4 Trivariate data visualization
By using textures or topological skeletons� we render
tensor information only partially. Indeed� we see from
Equation 1 that 2D tensor data are truly trivariate. If
the goal is to correlate full tensor information within
a single display� one must visualize simultaneously two
eigenvalues and the orientation of the eigenvectors.
In Figure 10� we use texture� color� and elevation as
channels to encode eigenvector direction� longitudinal
eigenvalue� and transverse eigenvalue� respectively. 4 In
4 The vertical stretching creates an unwanted distortion of the
texture which can be compensated for by techniques such as those
described in Reference �10�.
W 1
T 1
T 2
W 4
W 5
addition to topological information the display reveals a
strong correlation between the two eigenvalues�a fact
that was previously overlooked in representations such
as Figures 5 and 9.
4 Tensor Topological Rule
When a tensor �eld is de�ned across a surface M�
the topology of M puts a constraint on the number and
nature of degenerate points� limiting considerably the
variety of possible tensor patterns. We investigate this
constraint in this section.
The topology of any surface M is unambiguously
characterized by a single number ��M� called the sur�
face�s Euler characteristic �8�. All orientable 5 home�
omorphic surfaces�i.e.� the set of orientable surfaces
that can be distorted to look identical by continuous
bending� stretching� or squashing� but without tear�
ing or gluing�have the same value of ��M�. For ex�
ample� a sphere and a cube are homeomorphic with
��M� � 2. A torus and a co�ee mug are homeomor�
phic with ��M� � 0. Table 1 lists ��M� for a few
generic surfaces.
A classical theorem of surface topology� known as the
Poincar�e�Hopf theorem �11�� stipulates that the sum of
the indices at the critical points of a vector �eld de�ned
across a surface M is equal to ��M�. Thus� if such
a vector �eld has N nodes� C centers� F foci� and S
saddles� the total index is N �C �F �S � ��M�. This
important result shows how the topology of the surface
M�i.e.� ��M��a�ects the structure of any vector �eld
5 See Reference �8� for a precise de�nition of surface orientabil�
ity. Most of the surfaces in every day life are orientable. Notable
exceptions include M�obius bands and Klein bottles.

de�ned across M�i.e.� N � C � F � S.
In order to extend the Poincar�e�Hopf theorem from
vector �elds to tensor �elds� we make the assumption
that the sum of the indices at the degenerate points of a
tensor �eld T�x� de�ned across the surface M depends
only on the topology of M and not on the particular ten�
sor �eld T�x�. The following topological rule results�
Tensor topological rule � Let T�x� be a tensor
�eld de�ned across an orientable surface M having Eu�
ler characteristic ��M�. If T�x� has only isolated de�
generate points consisting exclusively of W wedges� T
trisectors� N nodes� C centers� F foci� and S saddles�
then the sum of the indices at the degenerate points of
T�x� is equal to ��M�. Hence the topological rule�
1
�W � T � � N � C � F � S � ��M� �9�
2
We refer the reader to Reference �7� for a proof. As with
vector �elds� this rule establishes a connection between
the topology of the surface M�i.e.� ��M��and the
structure of any tensor �eld de�ned across M�i.e.� the
sum of indices.
Equation 9 restricts considerably the number of pos�
sible surface tensor patterns. For example� Figure 11
shows two complex tensor �elds�one de�ned across a
torus and another one across a sphere. A topological
analysis reveals N � C � F � S � 0� W � T � 18
for the torus� and N � C � 1� F � 0� W � T � 3 for
the sphere. Both sets of values satisfy Equation 9 with
��torus� � 0 and ��sphere� � 2� respectively.
5 Extensions and Conclusions
We can extend the theory of degenerate points to 3�D
symmetric tensor �elds� which have three real eigenval�
ues and three orthogonal eigenvectors. At a degener�
ate point where two eigenvalues are identical� locally
two�dimensional patterns such as wedges and trisec�
tors �Figure 3� occur in the plane orthogonal to the
third eigenvector. However� it remains to characterize
the fully three�dimensional patterns that exist in the
vicinity of degenerate points where three eigenvalues
are identical.
The results presented above are also useful for un�
symmetric tensor �elds. We show in Reference �6� that
it is always possible to extract a symmetric tensor com�
ponent from unsymmetric data. We can then apply
the topological analysis to the symmetric component
for unveiling� at least partially� the structure of the ten�
sor �eld.
In conclusion� visualization allowed us to elucidate
the structure of symmetric tensor �elds� demonstrating
the tremendous potential of the �eld for building new
knowledge beyond the usual goal of inspecting results
from experiments and computations.
Acknowledgements
We are most indebted to Dan Asimov from NASA Ames
for a useful discussion on topology� and to Mark Peercy
from Stanford University for his critical comments and some
of his software. The authors are supported by NASA un�
der contract NAG 2�911 which includes support from the
NASA Ames Numerical Aerodynamics Simulation Program
and the NASA Ames Fluid Dynamics Division� and also by
NSF under grant ECS9215145.
References
�1� T. Delmarcelle and L. Hesselink� �A uni�ed frame�
work for �ow visualization�� in Computer Visualization
�R. Gallagher� ed.�� ch. 5� CRC Press� 1994.
�2� J. L. Helman and L. Hesselink� �Visualization of vector
�eld topology in �uid �ows�� IEEE Computer Graphics
and Applications� vol. 11� no. 3� pp. 36�46� 1991.
�3� A. Globus� C. Levit� and T. Lasinski� �A tool for visu�
alizing the topology of three�dimensional vector �elds��
in Proc. IEEE Visualization �91� pp. 33�40� 1991.
�4� A. I. Borisenko and I. E. Tarapov� Vector and Tensor
Analysis with Applications. Dover Publications� New
York� 1979.
�5� R. R. Dickinson� �A uni�ed approach to the design
of visualization software for the analysis of �eld prob�
lems�� in Proc. SPIE� vol. 1083� pp. 173�180� SPIE�
Bellingham� WA.� 1989.
�6� T. Delmarcelle and L. Hesselink� �Visualizing second�
order tensor �elds with hyperstreamlines�� IEEE Com�
puter Graphics and Applications� vol. 13� no. 4� pp. 25�
33� 1993.
�7� T. Delmarcelle� The Visualization of Second�Order
Tensor Fields. PhD thesis� Stanford University� 1994.
To be published.
�8� P. A. Firby and C. F. Gardiner� Surface Topology. Ellis
Horwood series in Mathematics and its Applications�
John Willey � Sons� New York� 1982.
�9� B. Cabral and L. C. Leedom� �Imaging vector �elds
using line integral convolution�� Computer Graphics
�SIGGRAPH�93 Proc.�� pp. 263�272� 1993.
�10� J. Maillot� H. Yahia� and A. Verroust� �Interactive tex�
ture mapping�� Computer Graphics �SIGGRAPH�93
Proceedings�� vol. 27� pp. 27�34� 1993.
�11� J. W. Milnor� Topology from the Di�erentiable View�
point. The University Press of Virginia� Charlottesville�
1965.

Figure 6� Textures representing the two eigenvector �elds in the vicinity of a trisector point �top� and a wedge
point �bottom�. Color encodes the di�erence between the two eigenvalues.

Figure 7� A frame of Video Clip 4 showing the correlation between the velocity �eld �moving texture� and the
degenerate points of the stress tensor. Color encodes the most compressive stress. Red dots � wedges� white dots
� trisectors. �See the video tape accompanying the Visualization �94 proceedings.�

Figure 8� Spatiotemporal trajectories of degenerate points in the stress�tensor �eld. Time increases from front to
back. Red spheres � wedges� green spheres � trisectors. M and C indicate merging and creation of wedge�trisector
pairs� respectively. �See the video tape accompanying the Visualization �94 proceedings.�

Figure 9� A frame of Video Clip 5 showing the instantaneous topology of the most compressive eigenvector
v2. Color encodes �2. W � wedge� T � trisector. �See the video tape accompanying the Visualization �94
proceedings.�

Figure 10� Trivariate data visualization to fully represent the stress�tensor �eld.

Figure 11� Illustration of the tensor topological rule for a torus and a sphere.

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This paper describes a system, GASP, that facilitates
the visualization of geometric algorithms. The user need
not have any knowledge of computer graphics in order to
quickly generate a visualization. The system is also intended
to facilitate the task of implementing and debugging
geometric algorithms. The viewer is provided with
a comfortable user interface enhancing the exploration of
an algorithm’s functionality. We describe the underlying
concepts of the system as well as a variety of examples
which illustrate its use.
I s����������
The visualization of mathematical concepts goes back
to the early days of graphics hardware [19], [2], and continues
to the present [16], [14, 13], [17]. These videos use
graphics and motion to explain geometric ideas in three
dimensions and higher. They have been widely accepted
as the necessary companions to the traditional medium of
journal publication [27], [28]. Similar gains in exposition
are found in the algorithm animation work that has become
popular in recent years [1], [7], [4, 5], [23, 24], [6],
[21], [20]. The limiting force has been the difficulty of
generating the graphics for such animations.
We have chosen a restricted domain, that of computational
geometry, to build a system that greatly facilitates
the visualization of algorithms regardless of their complexity.
The visual nature of geometry makes it one of
the areas of computer science that can benefit greatly from
visualization. Even the simple task of imagining in the
mind a three-dimensional geometric construction can be
hard. In many cases the dynamics of the algorithm must
be understood to grasp the algorithm, and even a simple
animation can assist the geometer.
The main principle guiding our work is that algorithm
designers want to visualize their algorithms but are limited
by current tools. In particular, visualizations would
be less rare if the effort to create them was little. In
the past, visualizations have been produced by developing
sophisticated software for a particular situation but
there has been little movement towards more widely usable
systems. By limiting our domain, we are able to
create such a system that enables others to easily use it.
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Indeed, two colleagues have already published visualizations
built with our system [3], [9].
We describe in this paper our system, GASP, (Geometric
Animation System, Princeton). We present the basic
ideas that underlie the development and implementation
of our system, and we demonstrate its utility in the accompanying
video. Our system differs from its predecessors
(e.g., Balsa [7], Balsa-II [4, 5], Tango [23, 24] and Zeus
[6]) in several ways. In particular, the development of
GASP was driven by three major goals, which we feel
represent a radical departure from previous work.
GASP allows the very quick creation of three dimensional
algorithm visualizations. A typical animation
can be produced in a matter of days or even hours.
In particular, GASP allows the fast prototyping of
algorithm animations.
Even highly complex geometric algorithms can be
animated with ease. This is an important point, because
it is our view that complicated algorithms are
those that gain the most from visualization. To create
an animation, it is sufficient to write a few dozen
lines of code.
Providing a visual debugging facility for geometric
computing is one of the major goals of the GASP
project. Geometric algorithms can be very complex
and hard to implement. Typical geometric code is often
heavily pointer-based and thus standard debuggers
are notoriously inadequate for it. In addition,
running geometric code is plagued by problems of
robustness and degeneracies.
There are many ways in which the system can be used.
First, it can be used simply as an illustration tool for geometric
constructions. Second, stand-alone videotapes to
accompany talks and classes can be created by GASP.
Third, GASP can ease the task of debugging. Fourth,
GASP can significantly enhance the study of algorithms
by allowing students to interact and experiment with the
animations. Fifth, GASP enables users to create animations
to attach to their documents.
Computational geometers describe configurations of
geometric objects either through ASCII text as generated
by symbolic computing tools (e.g., Mathematica [29]) or
through hand drawn figures created with a graphics editor.
Our system offers an alternative to this by allowing
the geometer to feed ASCII data into a simple program
and get a three-dimensional dynamic (as well as static)
visualization of objects.

Often, the dynamics of the algorithm must be understood.
Animations can assist the geometer and be a powerful
adjunct to a technical paper. With GASP, generating
an animation requires no knowledge of computer graphics.
The interaction with the system is tuned to the user’s
area of expertise, i.e., geometry.
Until recently, most researchers have been reluctant to
implement, let alone visualize, their algorithms. In large
part, this has been due to the difficulty in using graphics
systems combined with that of implementing geometric
algorithms. This combination made it a major effort to
animate even the simplest geometric algorithm. Our system
can ease some of the unique hardships of coding and
debugging geometric algorithms. The inherent difficulty
in checking a geometric object (e.g., listing vertices, edges
and faces of a polyhedron) in a debugger, can be eliminated
once it becomes possible to view the object. In
practice, a simple feature such as being able to visualize
a geometric object right before a bug causes the program
to crash has been an invaluable debugging tool.
Visualization can have a great impact in education.
Watching and interacting with an algorithm can enhance
the understanding, give insight into geometry and explain
the intuition behind the algorithm. The environment in
which the animation runs is designed to be simple and
effective. The viewer is able to observe, interact and experiment
with the animation.
An important consideration in the design of GASP is
the support of enclosures of animations in online documents.
GASP movies can be converted into MPEG
movies which can be included in a Mosaic document.
A user is able to include animations in document. The
reader of the document is presented with an icon in the
document and clicking on the icon causes the animation
to play. This way, researchers are able to pass around
documents with animations replacing figures.
In the next section we describe the specification of the
system. We focus on the ways the system meets the needs
of both the geometer and the viewer. In Section 3 we describe,
through examples, how our system has been used
in various scenarios. This section is accompanied with
a videotape that demonstrates the various cases. Section
4 discusses some implementation issues. We conclude in
Section 5.
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The scenes of interest to us are built out of geometric
objects and displays of data structures. Typical geometric
objects are lines, points, polygons, spheres, cylinders and
polyhedra. Typical data structures include lists and trees
of various forms. The operations applied to these objects
depend upon their types. A standard animation in the
domain of computational geometry is built out of these
building blocks.
A geometer creating an animation cares about the structure
of the animation (which building blocks are included),
and the structure of each scene. The geometer is less
concerned, however, about how each building block animates,
about colors, about the speed of the animation etc.
To draw an analogy to L—T E X [15], the creator of the document
is concerned more with the text the paper includes
and less with the margins, spacings and fonts. Our system
allows the user to generate an animation with minimal effort
by specifying only the aspects of the animation about
which the user cares. There are times, however, when the
creator of the animation does want to change the viewing
aspects (e.g., colors) of the animation, just like the writer
of a document would like to change fonts. Our system
allows this also.
The end-user (the viewer) need not know how the animation
was produced. By analogy, the reader of a document
does not care how it was created. The viewer would
like to be able to play the animation at low or high speed,
to pause and alter the objects being considered and to run
the animation on an input of the user’s choosing among
other things. Our system provides an environment that
enables this.
In this section we discuss the design of the system
which comes to answer the above needs. We consider
two interfaces: the programmer interface and the viewer
interface. For the first of these, making the animation creation
a quick and easy task is the aim of our system. The
viewer interface is designed to be simple and effective,
and to allow the user to experiment with the animation.
Programmer Interface: To generate an animation, the
programmer needs to write C code which includes calls to
GASP’s functions. Making the snippets of C code which
generate the animation short and powerful is our goal.
To do this, we follow two principles: First, the programmer
does not need to have any knowledge of computer
graphics. Second, we distinguish between what is being
animated and how it is animated. The application specifies
what happens and need not be concerned with how to
make it happen on the screen. For example, the creation
of a polyhedron (The What) is different from the way it
is made to appear through the animation (The How). It
can be created by fading into the scene, by traveling into
its location etc. The code includes only manipulations of
objects and modifications of data structures. Style files
can be used by the programmer to change from default
aspects of the animation to other options. In other words,
the interface we provide allows the programmer to write
brief snippets of C code to define the structure of an animation
and ASCII style files to control any single viewing
of the animation.
The programmer interface contains three classes of
operations - geometric operations, operations on datastructures
and motion.
Geometric objects, such as polyhedra, spheres, cylinders,
lines and points, can be created, removed and modified.
The way each such operation is visualized depends
on the type of object. For instance, by default,
Create polyhedron fades in the given polyhedron and
Create point causes a point to blink. Removal of an
object is executed in a reverse fashion. Modification of
an object is constrained by its type. We can add faces to
a polyhedron by calling Add faces, but naturally there is
no equivalent operation for atomic objects such as spheres.
A second class of operations deals with data structures.
GASP has a knowledge of combinatorial objects
such as trees, and allows the user to visualize their manipulation.
For example, Create tree fades in a tree in
three dimensions level by level, starting from the root (as

shown in the accompanying videotape). Add subtree is
visualized similarly. Remove subtree fades out the appropriate
subtree, level by level, starting from the leaves.
A third class of operations involves the motion of objects
and the scene. GASP can rotate, translate, scale,
float (i.e., on a Bézier curve) and linearly-float an object,
a group of objects, or the whole scene. Typical
functions are Rotate object, Translate world,
Float object, and Linear float world. For the latter
two, GASP is given a number of positions, rotations
and scales and it moves the object smoothly (similarly,
linearly) through these specifications.
In addition, GASP supports an Undo function which
plays the animation backwards. Every primitive has a
way to reverse itself visually.
The snippets of C code containing these operations are
grouped into logical phases, called atomic units. Atomic
units allow the programmer to isolate phases of the algorithm.
The user encloses the operations which belong
to the same unit within an atomic unit phrase and GASP
executes their animation as a single unit. For example, if
adding a new face to a polyhedron, creating a new plane
and rotating a third object constitute one logical unit, these
operations are animated concurrently. The code to do it
is:
Begin_atomic("Example");
Add_faces(‘‘Poly’’, face_no, faces);
Create_plane(‘‘Plane’’, point1, point2,
point3, point4);
Rotate_object(‘‘ThirdObj’’);
End_atomic();
The parameter data for the various functions is part of
the algorithm being animated.
The programmer need not specify how the animation
appears. The user does not state in the code how each of
the above operations is visualized, what colors to assign
to objects in the scene, how long each atomic unit should
take, what line width to choose, what fonts to use for
the titles etc. Instead, GASP automatically generates the
animation, trying to create a visually pleasing one.
Each operation generates a piece of animation which
demonstrates the specific operation in a suitable way.
For instance, the operation Split Polyhedron which
removes vertices from the given polyhedron, is animated
by creating new polyhedra - a cone for every removed
vertex. The new cones travel away from the initial polyhedron,
creating black holes in it. Each cone travels in
the direction of the vector which is the difference between
the vertex (which created the cone) and the center of the
split polyhedron.
To get a uniform appearance for the overall animation,
objects of similar types are created in the same way, and
removed in the reverse way. An object which is created
by, say, fading in is removed by fading out.
Special attention is given to the issue of colors. “Color
is the most sophisticated and complex of the visible language
components” [18]. Most users, especially inexperienced
ones, do not know how to select colors. Choosing
colors becomes harder when a video of a computer animation
needs to be produced because colors come out
very different from the way they appear on the screen.
Therefore, pre-selected colors are useful. GASP maintains
palettes of pre-selected colors, and picks colors which are
appropriate for the device they are presented on.
Colors are assigned to objects (or other features such
as faces of a polyhedron) on the basis of their creation
time. All the objects created during a single logical phase
of the algorithm get the same color, which was not used
before. This way we group related elements and make it
clear to the observer what happened at which phase of the
algorithm.
If a programmer wants freedom to accommodate personal
taste, the parameters of the animation can be modified
by editing a “Style File”. The animation is still generated
automatically by the system but a different animation
will be generated if the style file is modified. The style
file affects the animation, not the implementation. This
allows the programmer to experiment with various animations
of an algorithm without ever modifying or compiling
the code.
Many parameters can be changed in the style file. They
include, among others, the way each primitive is visualized,
the color of the objects, the speed of each atomic
unit and the number of frames it takes, the kind of colors
for the animation (for videotapes or for the screen), the
width of lines, the radius of points, the font of text etc.
For instance, Remove object when the object is a tree,
fades out the tree level after level by default. However,
a parameter can be set in the style file, to fade out the
whole tree at once.
For example, the following is part of the style file for
an animation which will be discussed in the next section.
The style file determines the following aspects of the animation.
The background color is light gray. The colors to
be chosen by GASP are colors which fit the creation of a
video (rather than the screen). Each atomic unit spans 30
frames, that is, the operations within an atomic unit are
divided into 30 increments of change. If the scene needs
to be scaled, the objects will become 0.82 of their original
size. Rotation of the world is done 20 degrees around
the Y axis. The atomic unit pluck is executed over 100
frames, instead of over 30. The colors of the faces to be
added in the atomic unit add faces are green.
begin_global_style
background = light_gray;
color = VIDEO;
frames = 30;
scale_world = 0.82 0.82 0.82;
rotation_world = Y 20.0;
end_global_style
begin_unit_style pluck
frames = 100;
end_unit_style
begin_unit_style add_faces
color = green;
end_unit_style
Note that the syntax of the style file is eminently simple.
Viewer Interface: The GASP environment, illustrated
in Plate 1, consists of a Control Panel through which the
student controls the execution of the animation, several
windows where the algorithm runs, called the Algorithm

Windows, along with a Text Window which explains the
algorithm.
The Control Panel, which uses the VCR metaphor,
helps viewers to explore the animation at their own pace.
A viewer (typically a student or a programmer who debugs
the code) might want to stop the animation at various
points of its execution. Sometimes it is desirable to fastforward
through the easy parts and single-step through the
hard ones to facilitate understanding. The viewer may
want to “rewind” the algorithm in order to observe the
confusing parts of the algorithm multiple times. GASP’s
environment allows this. The panel allows running the algorithm
at varying speeds: fast(bb), slow(b), or unit by
unit (b �). The analogous `, ``, and � ` push buttons
run the algorithm “backwards”. The viewer can PAUSE
at any time to suspend the execution of the algorithm or
can EJECT the movie.
The viewer observes the algorithm in the Algorithm
Windows. Algorithm windows use Inventor’s Examiner
Viewer [25] and thus are decorated with thumbwheels and
push buttons. With the thumbwheels, the viewer can rotate
or scale the scene. The push buttons allow the user
to reset the camera to a ”home” position, or reposition it.
This allows the user to perceive a more complete structure
of the scene, to observe the objects “behind”, to view the
object of interest from a different angle, or to check the
relations of objects.
In addition, the user can get information by pressing
the push buttons in the Algorithm Window. It is possible
to list the atomic units, to list the objects in the
scene, print description of a chosen object (for example,
when a polyhedron is picked, its vertices and faces are
printed out), print the current transformation, and create
a postscript file of the screen.
A Text Window, supported by GASP, adds the ability
to accompany the animation running on the screen
with verbal explanations. Text can elucidate the events
and direct the student’s attention to specific details. Every
atomic unit is associated with a piece of text which
explains the events occurring during this unit. When
the current atomic unit changes, the text in the window
changes accordingly. Voice is also supported by GASP.
The viewer can listen to the explanations that appear in
the Text Window.
Q qeƒ€ �� e����
In this section we describe different scenarios for which
we produced animations to accompany geometric papers.
Excerpts from the animations are given in the accompanying
video. For each case we present the problem of study,
the goal in creating the animation and the animation itself.
Building and Using Polyhedral Hierarchies: This algorithm,
which is based on [10, 11], builds an advanced
data structure for a polyhedron and uses it for intersecting
a polyhedron and a plane. The main component of the
algorithm is a preprocessing method for convex polyhedra
in 3D which creates a linear-size data structure for the
polyhedron called its Hierarchical Representation. Using
hierarchical representations, polyhedra can be searched
(i.e., tested for intersection with planes) and merged (i.e.,
tested for pairwise intersection) in logarithmic fashion.
The basic geometric primitive used in constructing the
hierarchical representation is called the Pluck: Given a
polyhedron, €H, we build a polyhedron €I by removing
vertices in † (€H) 0 † (€I). The cones of faces attached
to the vertices are also removed. This leaves holes in the
polyhedron €H. These holes are retriangulated in a convex
fashion. Repetition of plucking on the polyhedron €I
creates a new polyhedron €P. The sequence €H €I €P ...
€� forms the hierarchical representation.
There were two goals for creating the animation. First,
we wanted to create a video that explains the data structure
and the algorithm for educational reasons. Second,
since the algorithm for detecting plane-polyhedral intersection
had not been implemented before, we wanted the
animation as an aid in debugging the implementation.
The animation explains how the hierarchy is constructed
and then how it is used. In the accompanying
video, however, we show only the construction process.
For the first of these we explain a single pluck and then
show how the hierarchy progresses from level to level.
First, we show a single pluck. The animation begins
by rotating the polyhedron to identify it to the user. Next
we highlight a vertex and lift its cone of faces by moving
them away from the polyhedron (Plate 2). Then, we add
the new triangulation to the hole created (Plate 3). Finally,
we remove the triangulation and reattach the cone. This
is done in our system by the following piece of C code,
which is up to the creator of the animation to write.
explain_pluck(int poly_vert_no,
float (*poly_vertices)[3],
int poly_face_no, long *poly_faces,
char *poly_names[],
int vert_no, int *vertices,
int face_no, long *faces)
{
/* create and rotate the polyhedron */
Begin_atomic("poly");
Create_polyhedron("P0", poly_vert_no,
poly_face_no, poly_vertices,
poly_faces);
Rotate_world();
End_atomic();
}
/* remove vertices and cones */
Begin_atomic("pluck");
Split_polyhedron(poly_names, ‘‘P0’’,
vert_no, vertices);
End_atomic();
/* add new faces */
Begin_atomic("add_faces");
Add_faces(poly_names[0], face_no, faces);
End_atomic();
/* undo plucking */
Undo(2);
Each of the operations described above is a single
GASP primitive. Create polyhedron fades in the
given polyhedron. Rotate world makes the scene spin.
Split polyhedron highlights the vertex and splits the
polyhedron as described above. Add faces fades in the

new faces. Undo removes the triangulation and brings the
cone back to the polyhedron.
Notice that the code does not include the graphics:
Coloring, fading, traveling, speed etc. are not mentioned
in the code. In the related style file, these operations are
controlled. This allows the user to experiment with the
animation without modifying and recompiling the code.
After explaining a single pluck, the next step is to
show the pluck of an independent set of vertices. This is
no more difficult than a single pluck and is achieved by
the following code.
animate_one_level_hierarchy(
char *atomic1_name, char *atomic2_name,
char *atomic3_name, char *poly_name,
int vert_no, int *vertices,
int face_no, long *faces,
char *new_polys_names[])
{
Begin_atomic(atomic1_name);
Split_polyhedron(new_polys_names,
poly_name, vert_no, vertices);
End_atomic();
}
Begin_atomic(atomic2_name);
Add_faces( new_polys_names[0], face_no,
faces);
Finish_early(0.5);
for (i = 1; i

along the sweepline is maintained in a lazy fashion, meaning
that the nodes of the tree representing the cross section
might correspond to segments stranded past the sweepline.
In the first pass of the animation, red line segments
fade into the scene. While they fade out, a green visibility
map fades in on top of them, to illustrate the correlation
between the segments and the map. Yellow points, representing
the “interesting” events of the algorithm, then
blink. At that point, the scene is cleared and the second
pass through the algorithm begins. The viewer can watch
as the sweep-line advances by rolling to its new position
(the gray line in Plate 6). The animation also demonstrates
how the map is built - new subsegments fade in
in blue, and then change their color to green to become
a part of the already-built visibility map. The third pass
adds more information about the process of constructing
the map by showing how the the red-black tree which is
maintained by the algorithm changes. The animation also
presents the “walks” on the map (marked in yellow in
Plate 6).
There are only eleven GASP’s calls necessary for the
creation of this animation and they are:
Begin atomic, End atomic, Rotate world,
Scale world, Create point, Create line,
Create Sweepline, Modify Sweepline,
Create tree, Add node to tree,
Remove object.
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GASP is written in C and runs under UNIX on a Silicon
Graphics Iris. It is built on top of Inventor [25] and
Motif/Xt [12].
GASP consists of two processes which communicate
with each other through messages, as shown in Figure
1. Process 1 includes the collection of procedures which
make up the programmer interface. Process 2 is responsible
for executing the animation and handling the viewer’s
input.
message
user’s process process
code 1
2
type
u
n
i
o
n
ID
Figure 1: GASP’s Architecture
x
y
z
Inventor
The user’s code initiates calls to procedures which belong
to Process 1. Process 1 prepares one or more messages
containing the type of the operation required and
the relevant information for that operation and sends it
to Process 2. Upon receiving the message, Process 2
updates its internal data structure or executes the animation,
and sends an acknowledgement to Process 1. The
acknowledgement includes internal IDs of the objects (if
necessary). Process 1, which is waiting for that message,
updates the hash table of objects and returns to the user’s
code.
This hand-shaking approach has a few advantages.
First, it enables the user to visualize the scene at the time
when the calls to the system’s functions occur and thus
facilitates debugging. Since rendering is done within an
event mainloop, it is otherwise difficult to return to the
application after each call. Second, compilation becomes
very quick since the ’heavy’ code is in the process the application
does not link to. Finally, the user’s code cannot
corrupt GASP’s code. This is an important point, because
one of the major goals of GASP is to ease debugging.
During debugging, it is always a problem to figure out
whose bug is it - the application’s or the system’s.
Process 2, which is responsible for the graphics, works
in an event mainloop. We use Inventor’s Timer-Sensor to
update the graphics. This sensor goes off at regular intervals.
Every time it goes off, Process 2 checks which
direction the animation is running. If it is running backwards,
it updates the animation according to the phase it
is in. If it is running forwards, it checks whether there is
still work to do updating the animation (if yes, it does it)
or it is at the point when further instructions from Process
1 are needed. In the latter case, it checks to see
whether there is a message sent by Process 1. It keeps
accepting messages, updating its internal data structure,
and confirming the acceptance of messages until it gets
an END ATOMIC message. At that point, Process 2 starts
executing all the commands specified for the atomic unit.
It informs the first process upon termination.
S g��������
This paper has described a system GASP for automatically
generating visualizations for geometric algorithms.
The main benefits of the GASP are:
We have defined a hierarchy of users. The programmer
need not have any knowledge of computer
graphics. The code includes only manipulations of
objects and modifications of data structures. GASP
makes heuristic guesses for the way the animation
should appear. The advanced programmer experiments
with the animation by editing an ASCII style
file, without ever modifying or compiling the code.
The end-user explores the algorithm and controls the
execution of the algorithm in an easy way. The
GASP environment is very useful for education and
debugging.
Ease of use has been a main consideration in developing
GASP. A typical animation can be generated
in a very short time. This is true even for highly
complex geometric algorithms.
We have shown several animations of geometric algorithms.
The system is now at the stage where other
people are starting to use it. Three [3] [9] [26] out of
the eight segments of animations which appeared in the

Third Annual Video Review of Computational Geometry
were created by GASP. Two of them were created by the
geometers who wrote the papers. They took less than a
week to produce.
In the future, we intend to extend GASP to support
four-dimensional space. This can be an invaluable tool
for research and education. We would like to experiment
with GASP in an actual classroom. We believe
that animations can be used as a central part of teaching
computational geometry, both for demonstrating algorithms,
and for accompanying programming assignments.
Finally, many intriguing possibilities exist in making an
electronic book out of GASP. A user will then be able to
sit on the network, capture an animation, and experiment
with the algorithm.
e��������������
We would like to thank Bernard Chazelle for many
useful comments.
‚��������
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pages 255–262, August 1993.
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and Bartlett, 1991.
[15] L. Lamport. A Document Preparation System L—T E X
User’s Guide and Reference Manual. Addison Wesley,
1986.
[16] D. Lerner and D. Asimov. The sudanese mobius
band (video). In SIGGRAPH Video Review, 1984.
[17] S. Levy, D. Maxell, and T. Munzner. Outside in
(video). In SIGGRAPH Video Review, 1994.
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Company, 1988.

Plate 1: GASP’s Environment
Plate 2: Removing the Cone of Faces
Plate 3: Retriangulating the Polyhedron
Plate 4: Objects that cannot be Taken Apart wi th
Two Hands Using Translation
Plate 4: Objects that cannot be Taken Apart wi th
Two Hands Using Translation
Plate 6: Building the Visibility Map

Virtual Reality Performance for Virtual Geometry
Abstract
We describe the theoretical and practical visualiza�
tion issues solved in the implementation of an interac�
tive real�time four�dimensional geometry interface for
the CAVE� an immersive virtual reality environment.
While our speci�c task is to produce a �virtual geom�
etry� experience by approximating physically correct
rendering of manifolds embedded in four dimensions�
the general principles exploited by our approach re�
�ect requirements common to many immersive virtual
reality applications� especially those involving volume
rendering. Among the issues we address are the clas�
si�cation of rendering tasks� the specialized hardware
support required to attain interactivity� speci�c tech�
niques required to render 4D objects� and interactive
methods appropriate for our 4D virtual world applica�
tion.
1 Introduction
In this paper we describe how we have combined
general requirements for a broad class of virtual reality
applications with the capabilities of special�purpose
graphics hardware to support an immersive virtual
reality application for viewing and manipulating four�
dimensional objects. We present general issues con�
cerning the application of virtual reality methods
to scienti�c visualization� discuss how the resulting
requirements are re�ected in fundamental rendering
tasks� and point out where hardware features have cru�
cial roles to play. The proving ground for our general
observations is the design and implementation of an
application for visualizing a 4D mathematical world
through interaction with 3D volume images. We intro�
duce a task independent rendering paradigm through
which� with proper hardware support� we can produce
complex realistic images at interactive speeds.
4D Visualization. There have been a variety of
systems devoted to the general problem of 4D visu�
alization� ranging from the classic work of Bancho�
�2� 1� to geometry viewers such as Geomview �17�� our
local �MeshView� 4D surface viewer� and specialized
high�performance interactive systems such as that of
Banks �3�. Our virtual�reality�oriented work builds on
these previous e�orts and adds new features of 4D ren�
dering �see� e.g.� �20� 4� 14�� that have only recently be�
come technically feasible to exploit interactively �11�.
Such systems are valuable tools for mathematical re�
search �16� as well as for volume and �ow��eld visu�
alization applications �13� 15�. One of our goals is to
Robert A. Cross and Andrew J. Hanson
Department of Computer Science
Indiana University
Bloomington� IN 47405
develop techniques applicable to a real�time demon�
stration of 4D and volume�based 3D rendering applica�
tions in the CAVE �6�. In its present con�guration� the
CAVE is a Silicon Graphics Onyx�4 RealityEngine 2
with multiple graphics channels driving projectors for
two wall displays and the �oor� with simulator support
for workstation code development.
2 Virtual Reality and Visualization of
Geometry
In this paper we emphasize the visualization of chal�
lenging classes of mathematical objects through 3D
volumetric rendering. This section outlines our view�
points on the general issues involved in creating a vir�
tual reality for such domains.
Mental models. Philosophers have long wrestled
with the question of the nature of reality� we consider
reality to be embodied in our personal mental models
that derive from experience with natural phenomena
and allow us to cope with the qualitative physics of
everyday life. Virtual reality� then� is achievable in
one of two ways� we may create simulated experiences
that involve the subject by exploiting existing mental
models and perceptual expectations� or we may at�
tempt� by simulating phenomena with no real�world
correspondence� to create new classes of mental mod�
els� thereby extending the human experience.
Necessary features. The basic features of the vir�
tual reality systems that concern us here are�
1. Immersion. The system must physically involve
the user by responding to viewer motions and ac�
tions� e.g.� using a head�tracker� �ying mouse�
wand� etc. Regardless of whether the display
medium is a simple through�the�window view or
a CAVE� this intensi�es the intuition�building ex�
perience.
2. Interaction. The virtual environment must re�
spond to the participant�s actions at a high frame
rate and provide smooth and accurate tracking
of the input devices to support realistic feedback
to the viewer. The user must be able to make
changes and observe immediate results in order
to draw intuitive conclusions about the structure
and behavior of the simulated environment.
3. Visual realism. Redundant realistic visual cues
are needed to involve the participant� so we

should strive to include e�ects such as perspec�
tive� attenuation with distance� specular and dif�
fuse shading� shadows� motion parallax� and oc�
clusion. Providing such cues creates a more sat�
isfying visual experience� in addition to provid�
ing qualitative intuitive information at a pre�
conscious level �7�.
Anticipating the future. One task of the vir�
tual reality developer is to avoid the pitfalls of short�
sightedness. In this respect� our philosophy is to ex�
tend our attention also to approaches that are not
feasible in terms of current performance� but could be
drastically accelerated in future hardware generations.
Indeed� the development of appropriate algorithms to
keep up with the rapid evolution of the hardware may
be viewed as one of the most serious challenges we
face� we may become imagination�limited long before
the limits of the hardware technology are reached.
3 Interactive Realism
The goals of interactivity and realism are contra�
dictory� we must apparently compromise between the
speed of scan�conversion approximations and physi�
cally accurate but time�consuming ray�tracing meth�
ods. Here we present the fundamentals of a rendering
semantics that has the potential to support an accept�
able compromise.
3.1 The Toolbox
The following image�level abstractions form a set
of fundamental tools in terms of which we can express
a remarkable number of complex geometric rendering
e�ects�
z�bu�er. The z�bu�er tests and optionally replaces
a geometric value normally representing an object�s
distance seen through the pixel� complex e�ects can
be achieved by selecting appropriate tests and replace�
ment rules.
Frame bu�er and accumulation bu�er mathe�
matics. Frame bu�ers support operations that act
selectively on images and include addition� multiplica�
tion� logical operations� convolution� and histograms.
The accumulation bu�er� which is separate from the
frame bu�er� is dedicated to image addition and scal�
ing and usually has more bits per color component
than the frame bu�er. Typical applications involve
averaging a number of images� as in Monte�Carlo in�
tegration.
Static and dynamic textures. A static texture
is a common surface or volume texture map� while a
dynamic texture is one that may vary between frames.
For instance� we might simulate a window as a plane
with the current outside view mapped onto it� as the
scene outside changes� so does the texture map. In
addition to storing surface color� texture maps may
also serve as lookup tables for re�ectance or shading
functions.
Automatic texture vertex generation. Given a
texture map containing a lookup table for a function�
we may not know in advance what portion of the tex�
ture map is required. Texture vertices can be gen�
erated automatically by supplying a �possibly projec�
tive� transformation function from the geometry to
the texture space. For example� given a volumetric
woodgrain texture� the system automatically chooses
the correct woodgrain to map onto a slice through the
virtual block. For a more complete discussion of tex�
ture mapping operations� see �8�.
3.2 First�order E�ects
We consider a �rst�order e�ect to be one that re�
quires a �xed� environment�independent number of
primitive operations per scene element. In this sec�
tion� we describe a set of useful �rst�order techniques
that can be used to approximate rendering e�ects.
Planar re�ection. The re�ections from a �at sur�
face can be de�ned by rendering the environment from
a virtual viewpoint placed on the opposite side of the
re�ecting surface� this image is then mapped onto the
surface� giving the impression of a mirror.
Non�planar re�ection. An environment map is
de�ned as the image of a perfectly re�ecting sphere lo�
cated in the environment when the viewer is in�nitely
far from the sphere. Given the view direction and
normal at each vertex of a polygon� we can use au�
tomatic texture vertex generation to choose texture
coordinates� this gives an approximation of in�nite fo�
cal length re�ection �8�.
Shading maps. Texture maps can also be used
as repositories for pre�computed re�ectance functions
�e.g.� di�use� Phong� Cook�Torrance� etc.�. This
method produces much better behavior than hardware
lighting �i.e.� Gouraud shading�� which de�nes colors
only at the vertices� and so cannot place specularities
inside a large polygon.
Physically based luminaires. A di�use emitter
can be approximated by placing a projection point be�
hind the planar emitter and using projective textures
to shine a pre�computed cosine distribution texture
map into the environment. The di�use light thus emit�
ted is multiplied by each surface�s di�use light color
coe�cient. Other distributions can be approximated
by projecting di�erent lookup tables. If we use shad�
ing maps to approximate the cosine term and distance
attenuation at target polygons� we can approximate
physical luminaires.
Shadows. Areas lit by a particular light can be de�
�ned as areas that that light �sees�� i.e.� for sharp�
edged shadows� we test whether a particular point can
describe an unobstructed path to the luminaire. Us�
ing the z�bu�er� we can de�ne a depth texture map
from the light�s point of view. By projecting this tex�
ture map into the environment and z�bu�ering from
the eye�s point of view� we can compare the distance�
to�the�light values of visible surfaces to the projected
nearest�to�the�light value. If the eye sees a surface
whose distance to the light is larger than the indicated
minimum� it must be in shadow� if not� it is lit. If this
function is applied over all pixels to de�ne a binary
black�white image� this can be multiplied by an image

containing a shadowless lit image to construct a �nal
image including appropriate shadows �18� 19�
3.3 Second�order E�ects
Second�order e�ects provide more sophisticated im�
age features� and involve iteration or multiple samples
to generate a single image. These methods are su��
ciently expensive� at present� to preclude their use in
most interactive applications. However� these meth�
ods greatly increase visual satisfaction and hardware
improvements will make them increasingly practical.
Multiple samples. The accumulation bu�er allows
an elegant implementation of Monte�Carlo methods
over entire images. Thus� dynamic images such as
shadow maps or re�ection images can be de�ned by
probabilistic sampling� producing smooth shadows or
blurred re�ections. Psychological research indicates
that smooth shadows� in particular� are important for
visual realism �7�.
Iterated di�use and specular re�ection. If we
approximate di�use emission from a single polygon
and produce an image of the incident light at another�
we can then emit some of this light back into the en�
vironment. When this process is iterated� it becomes
an approximation to radiosity and global illumination.
This method has the advantage of being much faster
than similar precomputations and has lower algorith�
mic complexity while maintaining important visual
features.
Participating media. Approximate volume images
may be produced by cutting multiple additive slices
through the viewed space. By projecting lighting dis�
tributions and shadows onto these slices� we can ap�
proximate the scattering of light as it passes through a
foggy environment� producing visible beams and sim�
ilar volume�rendering e�ects.
3.4 Hardware Support
At present� only the simplest of the above tech�
niques are viable in a software�only interactive system
with a complex environment. As our needs for real�
ism increase� so do our needs for graphics hardware
support. For example� if we require Phong shading�
but must implement it in software� the complexity
of environments with which we can interact will be
severely limited. However� given hardware texture�
mapping support� we can precompute a lookup table
to support an implementation of specular re�ectance
functions� the hardware can wrap this texture around
the objects� interpolating between computed points to
produce correct images of specular objects.
Our virtual geometry applications are designed to
take full advantage of the hardware support of the Sil�
icon Graphics Onyx RealityEngine 2 . Its support for
high�speed texture mapping� in particular� enables us
to map large portions of our graphics computations
directly onto hardware�supported primitives. For ex�
ample� dynamic texture mapping and automatic tex�
ture vertex generation allow us to interactively simu�
late bizarre physical illumination models such as 4D
light. E�ectively� we have transformed the mathemat�
ical rendering model into a form expressible in terms
of our hardware�supported high�speed toolbox. The
exploitation of such transformations can greatly en�
hance user comprehension through improved feedback
and perceived visual realism.
4 Visualization E�ects Design
The particular system that we have implemented
to explore virtual reality paradigms focuses on math�
ematical visualization in higher dimensions �11�. We
create the illusion that the participant is immersed in
the 3D� volumetric retina of a 4D cyclops interacting
with objects in a 4D world. Among the issues we must
address to achieve this are the following�
4D Depth. Perceiving 4D depth requires binocular
fusion of a pair of distinct 3D volume images� a 3D
human would e�ectively need 2 pairs of 3D eyes to
see these images at the same time �and could not fuse
them anyway�. Thus we need to obtain 4D depth cues
from other sources such as motion� occlusion� or depth
color codes. For example� occlusions of surfaces by
surfaces can be emphasized for visibility by painting or
cutting away the more distant of two surfaces around
an illusory intersection in a particular projection.
4D Motion Cues. Motion is an important factor
in our ability to perceive 3D structure monocularly�
either constant�angular�velocity rigid 3D rotation or
periodic rocking is an adequate substitute for a stereo
display. We use rigid 4D rotations to generate mo�
tion cues for our 4D monocular world that resemble
familiar 3D motion cues.
3D Depth. Typical objects that we project out of
4D produce volume images� though our rendered im�
ages of thickened surfaces are simpli�ed since we use a
thin�surface approximation to achieve acceptable ren�
dering performance. Thus seeing the 3D opaque ex�
terior of our objects is not enough � we want to see
inside the projected shape. This causes a problem� if
we make a surface transparent but featureless� like a
highly in�ated balloon� there are not enough distinct
features in the image to activate 3D stereo percep�
tion except along the outer edges of the object. Sim�
ilarly� for volumetric objects whose interior is made
of smooth internal �jellylike� solids� it is di�cult to
produce a strong impression of what may be a very
complex internal 3D structure.
Texture. For human binocular vision to perceive
a full 3D structure in one glance� smooth rendering
methods are often de�cient� they do not generate the
image gradients necessary for the edge matching pro�
cess used in stereo depth reconstruction. One tech�
nique to circumvent this problem is to spice up the
featureless jelly with surface or volumetric textures.
Such textures� which can be as simple as a set of grid
lines or a regular or random lattice of points� provide a
richer collection of image gradients to drive the stere�
ographic matching process.
No Slices. A common approach to representing 4D
objects is to slice them up and consider them as a
sequence of 3D objects� often presented as a time�
sequenced animation. We insist on holistic images for
our imaginary 4D retina� humans are not adept at per�
ceiving 3D objects from a time sequence of 2D slices�
so we do not expect that 3D slices of 4D objects will
be any easier.

Lighting. In everyday life� we are able to perceive
3D shapes in static photographs and drawings. We
make certain assumptions about the nature of the ob�
jects and lighting conditions� and apparently infer the
3D structure using what is known in computer vision
as a �shape from shading� algorithm. We see objects
whose structure is revealed by the intensity gradations
re�ected from the object and by its cast shadows. Dif�
fuse and specular highlights reveal additional informa�
tion about the directions of surface or volume patches
that is more speci�c than� for example� gradient or
isosurface information. 4D lighting permits a similar
holistic depiction of a 4D object� and the structure of
the lighting in the 3D projection contains many sub�
tle clues about the 4D structure and its orientation
relative to the 4D lights and camera.
Shadows. To enhance the scene perception experi�
ence� we can provide auxiliary cues such as 3D shad�
ows to supplement 3D stereo perception. One can also
generate 4D shadow volumes to help reveal hidden 4D
structure �12�.
Occlusion. We exploit occlusion information to in�
fer structure in 2D drawings representing the 3D
world� a typical mathematical application would be
the �crossing diagram� showing the unique 3D struc�
ture of a knotted loop of string using over�under cross�
ing markings on a 2D diagram alone. Similar phenom�
ena occur in 4D� non�intersecting surfaces in 4D may
appear to intersect in a curve when projected to 3D�
just as 3D lines may appear falsely to intersect when
projected to 2D� pieces of 4D volumes may completely
block out other 4D volumes in the 3D projection� just
as 3D surfaces block �occlude� one another in 2D imag�
ing. Single convex 3D and 4D objects have no occlu�
sions� and so can be easily rendered using back�face
culling. For 4D multiple�object scenes and non�convex
objects� we can provide occlusion handling� crossing
markings or depth�cued colors to emphasize the oc�
currence of 4D occlusion. This is not always possi�
ble to achieve interactively� since processing occlusions
may be a memory�intensive or combinatorially explo�
sive process.
Depth Coding. A number of techniques have been
proposed to provide a sense of depth in 4D graph�
ics �see� e.g.� �2��� these include pseudocolor coding of
depth� application of depth�dependent static or mov�
ing textures� and 4D�depth dependent opacity or blur�
ring.
Redundancy. Typical 3D terrain maps and graphs
have redundant coding of properties such as elevation.
Pseudocolor� isolevel contours� ruled surface markings�
and oblique views exhibiting occlusion� illumination
e�ects� and shadows may all be combined in a single
representation. 4D data representations also pro�t
from such redundancy� so we add multiple 4D cues
when possible.
In summary� the family of visual e�ects that we
wish to achieve involves a wide variety of issues and
representation technologies. The common thread is
this� we examine holistic perceptual processes such as
lighting and motion that serve us well in dealing with
our 3D world� and exploit the 4D analogs of those
processes to encode relevant information in the 3D
volume image perceived by our hypothetical 4D being.
5 Interactive Interface Design
Our philosophy of 4D interaction is based on sev�
eral fundamental assumptions about how human be�
ings learn about the 3D world. We are all familiar with
the fact that if we are driven around a strange town�
we are much less able to �nd our way later than if we
do the driving ourselves. Thus we seek 4D interaction
modes that emphasize the involvement of the user and
promote the feeling of direct manipulation� as though
4D objects were responding in some physical way to
the motions of our input devices. Successful strate�
gies should therefore signi�cantly reduce the required
user training time by exploiting analogs of familiar 3D
direct manipulation.
Restricting ourselves for now to single compact ob�
jects lying in the center of our perceived CAVE space�
we need several basic types of control� �1� the user
moving around the 3D projected object itself� �2� rigid
3D motions of the object� �3� rigid 4D rotations �and
perhaps translations� of the object� �4� 4D control of
the orientation of the light ray �or rays� used in the
shading and shadowing processes. The �rst two capa�
bilities are standard for almost all CAVE applications.
The two 4D rotation tasks� however� require the
following application�speci�c design considerations�
4D Orientation Control. Direct manipulation of
3D orientation using a 2D mouse is typically handled
using a rolling ball �9� or virtual sphere �5� method to
give the user a feeling of physical control. Figure 1a
shows the e�ect of horizontal and vertical 3D rolling
ball motions on a cube� supposing that the cube ini�
tially shows only one face perpendicular to the viewer�s
line of sight� moving the mouse in the positive x di�
rection exposes an oblique sliver of the left�hand face�
while motion in the y direction exposes the bottom
face� reversing directions exposes the opposite faces.
Long or repeated motions in the same direction bring
cycles of 4 faces towards the viewer in turn. In the
rolling ball method� circular mouse motions counter�
rotate the cube about the viewing axis� in the virtual
sphere method� the mouse acts as if glued to a glass
sphere� so that at a certain radius along the x�axis
from the center� a vertical mouse motion causes spin�
ning about the viewing axis.
The extension of this approach to 4D is outlined in
the appendix and described in more detail in �10�. Fig�
ure 1b is the 4D analog of Figure 1a� beginning with
a fully visible transparent �volume�rendered� cube�
which represents a single hyperface of a hypercube per�
pendicular to the 4D viewing vector� we move the 3D
mouse along the x�axis to expose a volumetric sliver
on the left� this is the oblique view of the left hyper�
face. Moving the 3D mouse along the y�axis exposes a
volumetric sliver on the bottom� moving the 3D mouse
along the z�axis exposes a volumetric sliver on the back
of the original volumetric cube. Reversing directions
brings up the opposite hyperfaces� long motions reveal
cycles of 4 hyperfaces� but now there are three cycles�
one each in the x� y� and z directions. How do we
get ordinary rotations� say in the x�y plane� Moving

y
�a�
x
Figure 1� Schematic diagram comparing �a� 2D mouse control of a 3D object� and �b� 3D �ying mouse control of
a 4D object.
the 3D mouse in small circles in any plane produces
counter�rotations of that plane� thus giving 3 more
degrees of freedom� exhausting the 6 degrees of orien�
tational freedom in 4D. The 4D virtual sphere action
follows by exact analogy to the 3D case.
4D Light Control. Figure 2a shows a schematic
diagram of a method for controlling the 3D lighting
vector using a 2D mouse� the unit vector in 3D has
only two degrees of freedom� so that picking a point
within a unit circle determines the direction uniquely
�up to the sign of its view�direction component�. With
a convention for distinguishing vectors with positive
or negative view�direction components �e.g.� solid or
gray�� we can uniquely choose and represent the 3D
direction. Control of the vector is straightforward us�
ing the rolling ball� the lighting vector initially points
straight out of the screen �up in the oblique view of
Figure 2b�� and moving the mouse in the desired di�
rection tilts the vector to its new orientation� whose
projection to the plane of Figure 2a is shown in the
gray ellipse in Figure 2b. Rotating past 90 degrees
moves the vector so its view�direction component is
into the screen.
The analogous control system for 4D lighting�
shown in Figure 2c� is based on a similar observation�
since the 4D normal vector has only 3 independent
degrees of freedom� choosing an interior point inside
a solid sphere determines the vector uniquely up to
the sign of its component in the unseen 4th dimension
�the �4D view�direction component��. The rest of the
control proceeds analogously. Since we cannot easily
interpret 4D oblique views� we do not attempt to draw
the 4D analog of Figure 2b.
z
y
�b�
6 Examples
A classic example of a non�trivial surface embed�
ded in 4D is a knotted sphere� and this is the central
demonstration we have implemented for the CAVE�
Figure 3 shows the spun trefoil� the 4D knotted sphere
closest in spirit to an ordinary 3D trefoil knot� while
Figure 4 shows the twist�spun trefoil� which� astonish�
ingly� can be shown to be unknotted in 4D.
The real�time display of these images is made possi�
ble by replacing the techniques of �14�� which required
up to half an hour per frame to render� with a dy�
namic texture map implementation of �11�� resulting in
update rates of up to 30 frames�second. The texture
mapping support of the RealityEngine permitted us to
represent the 4D lighting distributions on the surface
using a dynamic texture map� the resulting transpar�
ent volumetric image was rendered using frame bu�er
addition with multiplicative opacity.
Occlusion computations remain too expensive for
real time� and so we precompute the occlusions for
one particular viewpoint and �x them to the object�
this has the curious advantage that� when the object
is rotated in 4D� one can see the explicit separation of
the apparent self�intersections� and convince oneself
that the �side view� shows that no self�intersections
exist.
In Figure 5 we show a closeup of the 4D control
feedback display� which reads out the current 4D light
position and 4D orientation of the central knotted
sphere. Figure 6 is a true volume�rendered object� the
hypersphere� projected from 4D to show a 3D view of
its �northern hemisphere�� grid lines and a volumetric
speckle texture are added within the featureless vol�
ume of this object to give a clear stereographic image�
as noted in Section 4.
Finally� in Figure 7� we step back to show how
x

�a� �b� �c�
Figure 2� Schematic diagram comparing �a� selecting 2D point in disk to specify 3D light direction� shown
obliquely in �b�� and �c� 3D �ying mouse control of a 4D light direction by picking 3D point inside solid sphere.
our mathematical world appears in a rich virtual
workspace. To illustrate some of our other capabil�
ities for general virtual geometry� we show in Figure 8
how the knotted sphere appears in a room illuminated
by a light shining through foggy air. Note the satisfy�
ing e�ect of the 3D shadows cast in the room by the
mathematical objects.
7 Conclusions and Future Work
We have described a wide spectrum of issues in�
volved in the development of an ambitious virtual re�
ality system that attempts to immerse the viewer in a
technically correct four�dimensional world. The tech�
niques required for this system include the optimiza�
tion of rendering approximations through the use of
hardware graphics operations� as well as task�speci�c
approaches to enhancing user interaction. The opti�
mizations used in this system apply also to general
virtual reality performance problems.
While adapting our software to the CAVE� we faced
parallelization issues that did not arise on a single
screen. In addition to parallelizing the mathematics
�e.g.� multi�threading the projection of four dimen�
sions to three�� we dealt with other problems involving
shared memory and resources. For example� one must
avoid collisions on the graphics hardware� particularly
during geometry transformations.
In its present form� this project is approaching the
limits of the target graphics hardware. We now face
the standard bottleneck of textured polygon �ll rate�
for instance. Extending the features of the system
will require computations for which the RealityEngine
hardware has no particular advantage� for example�
the 4D occlusion calculation is still too computation�
ally expensive for adequate interactive performance�
as is depth�ordered transparency. However� the addi�
tion of resources such as hardware support for large
dynamic 3D textures would enable us to attack even
more challenging problems in virtual geometry.
Acknowledgments
This work was supported in part by NSF grant IRI�
91�06389. We thank George Francis� Chris Hartman�
and the NCSA CAVE personnel� as well as the mem�
bers of the Electronic Visualization Laboratory at the
University of Illinois�Chicago for their support.
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�1� Banchoff� T. F. Visualizing two�dimensional
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of two�dimensional surfaces in four�dimensional
space. In Computer Graphics �1992 Sympo�
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�5� Chen� M.� Mountford� S. J.� and Sellen�
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vol. 22� pp. 121�130. Proceedings of SIGGRAPH
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�6� Cruz�Neira� C.� Sandin� D. J.� and De�
Fanti� T. A. Surround�screen projection�based
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the CAVE. In Computer Graphics �SIGGRAPH
�93 Proceedings� �Aug. 1993�� J. T. Kajiya� Ed.�
vol. 27� pp. 135�142.
�7� Goldstein� E. B. Sensation and Perception.
Wadsworth Publishing Company� 1980.
�8� Haeberli� P.� and Segal� M. Texture map�
ping as a fundamental drawing primitive. In

Fourth EUROGRAPHICS Workshop on Render�
ing �June 1993�� M. Cohen� C. Puech� and F. Sil�
lion� Eds.� pp. 259�266.
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ics Gems III� D. Kirk� Ed. Academic Press� San
Diego� CA� 1992� pp. 51�60.
�10� Hanson� A. J. Rotations for n�dimensional
graphics. Tech. Rep. 406� Indiana University
Computer Science Department� 1994.
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Computer Society Press� pp. 196�203.
�12� Hanson� A. J.� and Heng� P. A. Visualizing
the fourth dimension using geometry and light.
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�13� Hanson� A. J.� and Heng� P. A. Four�
dimensional views of 3d scalar �elds. In Proceed�
ings of Visualization �92 �1992�� IEEE Computer
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the fourth dimension. Computer Graphics and
Applications 12� 4 �July 1992�� 54�62.
�15� Hanson� A. J.� and Ma� H. Visualization �ow
with quaternion frames. In Proceedings of Vi�
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�16� Hanson� A. J.� Munzner� T.� and Francis�
G. K. Interactive methods for visualizable geom�
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�17� Phillips� M.� Levy� S.� and Munzner� T. Ge�
omview� An interactive geometry viewer. No�
tices of the Amer. Math. Society 40� 8 �Octo�
ber 1993�� 985�988. Available by anonymous ftp
from geom.umn.edu� The Geometry Center� Min�
neapolis MN.
�18� Reeves� W. T.� Salesin� D. H.� and Cook�
R. L. Rendering antialiased shadows with depth
maps. In Computer Graphics �SIGGRAPH �87
Proceedings� �July 1987�� M. C. Stone� Ed.�
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�19� Segal� M.� Korobkin� C.� van Widenfelt�
R.� Foran� J.� and Haeberli� P. E. Fast shad�
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In Computer Graphics �SIGGRAPH �92 Proceed�
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pp. 249�252.
�20� Steiner� K. V.� and Burton� R. P. Hidden
volumes� The 4th dimension. Computer Graphics
World �February 1987�� 71�74.
A 4D Rolling Ball Formula
For completeness� we list the 4D rolling ball for�
mula derived in �10� that is the basis for most of our
4D controls� this natural algorithm for 4D orienta�
tion control requires exactly three control parameters�
thus making it ideally suited to the ��ying mouse� or
CAVE �wand� 3�degree�of�freedom user interface de�
vices. Let � X � �X� Y� Z� be a displacement obtained
from the 3�degree�of�freedom input device� and de�ne
r 2 � X 2�Y 2�Z 2 . Take a constant R with units 10 or
20 times larger than the average value of r� compute
D 2 � R 2 � r 2� compute the fundamental rotation co�
e�cients c � cos � � R�D� s � sin � � r�D� and then
take x � X�r� y � Y �r� z � Z�r� so x 2 � y 2 � z 2 � 1.
Finally� rotate each 4�vector by the following matrix
before reprojecting to the 3D volume image�
2
6
4
1 � x 2 �1 � c� ��1 � c�xy ��1 � c�xz sx
��1 � c�xy 1 � y 2 �1 � c� ��1 � c�yz sy
��1 � c�xz ��1 � c�yz 1 � z 2 �1 � c� sz
�sx �sy �sz c
3
7
5

Figure 3� Closeup of spun trefoil knot in the CAVE
simulator.
Figure 5� 4D control feedback display� single line
shows light direction� wire�frame shows 4D orien�
tation referred to a hypercube.
Figure 7� User view of knotted sphere and controls
inside a rich virtual room.
Figure 4� Closeup of twist�spun trefoil apparent
knot in the CAVE simulator.
Figure 6� The �solid textured beach�ball� view of
the hypersphere.
Figure 8� Adding 3D light and fog to the virtual
workspace.

Figure 3: Closeup of spun trefoil knot in the
CAVE simulator.
Figure 5: 4D control feedback display; single line
shows light direction, wire−frame shows 4D orien−
tation referred to a hypercube.
Figure 7: User view of knotted sphere and controls
inside a rich virtual room.
Figure 4: Closeup of twist−spun trefoil apparent
knot in the CAVE simulator.
Figure 6: The "solid textured beach−ball" view of
the hypersphere.
Figure 8: Adding 3D light and fog to the virtual
workspace.

A Library for Visualizing Combinatorial Structures
Abstract
This paper describes ANIM3D, a 3D animation library
targeted at visualizing combinatorialstructures. In particular,
we are interested in algorithm animation. Constructing a
new view for an algorithm typically takes dozens of design
iterations, and can be very time-consuming. Our library eases
the programmer’s burden by providing high-level constructs
for performing animations, and by offering an interpretive
environment that eliminates the need for recompilations. This
paper also illustrates ANIM3D’s expressiveness by developing
a 3D animation of Dijkstra’s shortest-path algorithm in just
70 lines of code.
1 Background
Algorithm animation is concerned with visualizing the
internal operations of a running program in such a way that
the user gains some understanding of the workings of the algorithm.
Due to lack of adequate hardware, early algorithm animation
systems were restricted to black-and-white animations
at low frame rates [6]. As hardware has improved, smooth
motion [11, 15], color [1], and sound [4] have been used to
increase the level of expressiveness of the visualizations.
Constructing an enlightening visualization of an algorithm
in action is a tricky proposition, involving both artistic
and pedagogical skills of the animator. Most successful
views undergo dozens of design iterations. Based on our experiences
in the 1992 and 1993 SRC Algorithm Animation
Festivals [2, 3] (20 SRC researchers participated each year),
we found that a high-level animation library, coupled with
an interpreted language, was instrumental in developing highquality
views [12].
A high-level animation library allows users to focus on
what they want to animate, without having to spend too much
time on the how. An interpreted language significantly shortens
the time needed for each iteration in the design cycle
because users do not need to recompile the view after mod-
Marc A. Najork Marc H. Brown
DEC Systems Research Center
130 Lytton Ave.
Palo Alto, CA 94301
fnajork,mhbg@src.dec.com
ifying its code. In fact, in our algorithm animation system,
users just need to hit the “run” button in the control panel to
see their changes in action.
In 1992, we began to explore how 3D graphics could
be used to further increase the expressiveness of visualizations
[5]. We identified three fundamental uses of 3D for
algorithm visualization: Expressing fundamental information
about structures that are inherently two-dimensional; uniting
multiple views of the underlying structures; and capturing a
history of a two-dimensional view. Fig. 1 shows snapshots of
some of the 3D animations we developed.
We found that building enlightening 3D views is even
harder than building good 2D views. One obvious reason is
that we (and most people) are much less used to designing in
3D than in 2D. But a more pragmatic problem was that our
3D software infrastructure was quite impoverished: We used
a small, object-oriented graphics library for displaying static
3D scenes. This library (like the rest of our algorithm animation
system) was written in Modula-3 and used PEXlib as
its underlying graphics system. This architecture was limiting
both in terms of turnaround time and in terms of animation
support. Therefore, drawing on our prior experience in 2D
algorithm animation, we built ANIM3D, a 3D object-oriented
animation library.
ANIM3D supports several window systems (X and Trestle
[13]) and several graphics systems (PEX and OpenGL).
The base library is implemented in Modula-3; clients can
either directly call into this base library, or access it through
Obliq [7], an interpreted embedded language. Using ANIM3D,
the size of a prototypical 3D animation (Dijkstra’s shortestpath
algorithm) decreased from about 2000 lines of Modula-3
to 70 lines of Obliq, and the part of the design cycle time
devoted to compiling, linking, and restarting the application
from about 7 minutes to about 10 seconds of reloading by the
Obliq interpreter (on a DECstation 5000/200).
Although ANIM3D was designed with algorithm animation
in mind, it is a general-purpose animation system. We
believe it to be particularly well-suited for visualizing and
animating combinatorial structures.

Figure 1: These snapshots are examples of the type of views for which A NIM3D is very well-suited. Each view requires from 50 to 200 lines of code to produce.
The first snapshot shows a divide-and-conquer algorithm for finding the closest pair of a set of points in the plane. The third dimension is used here to show the
recursive structure of the algorithm. The second snapshot shows a view of Heapsort. Each element of the array is displayed as a stick whose length and color is
proportional to its value. With clever placement, the tree structure of the heap is visible from the front and the the array implementation of the tree is revealed
from the side. The third snapshot shows a k-d tree, for k � 2. When viewed from the top, the walls reveal how the plane has been partitioned by the tree; when
viewed from the front or side, we see the tree. The last snapshot shows a view of Shakersort. The vertical sticks show the current values of elements in the array,
and the plane of “paint chips” underneath provides a history of the execution. The sticks stamp their color onto the chips plane, which is pulled forward as the
execution progresses.
The remainder of this paper is structured as follows. After
presenting an overview of ANIM3D, we show how to use
the library to construct a simple animation. The animation
is of a trivial solar system. We then build a 3D visualization
of Dijkstra’s algorithm for finding the shortest path in a
graph. This animation can also serve as an introduction to our
methodology for animating algorithms. Finally, we discuss
how ANIM3D compares with other general-purpose animation
systems and with other algorithm animation systems.
2 An Overview of Anim3D
ANIM3D is built upon three basic concepts: graphical
objects, properties, and callbacks.
A graphical object, or “GO”, can be a geometric primitive
such as a line, polygon, sphere, or cone, a light source, a
camera, or a group of other GOs. Graphical objects form a
directed acyclic graph; typically, the roots of the DAG are the
top-level windows, the internal nodes are groups of other GOs,
and the leaves are geometric primitives, lights, or cameras.
The GO class hierarchy is as follows:
GO
GroupGO
CameraGO
LightGO
NonSurfaceGO
SurfaceGO
RootGO
OrthoCameraGO
PerspCameraGO
AmbientLightGO
VectorLightGO
PointLightGO
SpotLightGO
LineGO
MarkerGO
PolygonGO
BoxGO
SphereGO
ConeGO
CylinderGO
DiskGO
TorusGO
A property consists of two parts, a name and a value.
Property names are constants, such as “Surface Color” or
“Sphere Radius.” Property values are objects (in an objectoriented
programming sense) representing colors, 3D points,
reals, etc. Because property values are objects, they are both
mutable and can be shared by several GOs. In addition, property
values are time-variant: the actual value encapsulated by
the property value depends on the current animation time, a
system-wide resource.
Associated with each graphical object o is a property
mapping, a partial function from property names to property
values. A property associated with o not only affects the appearance
of o, but also the appearance of all those descendants
of o that do not explicitly override the property.
Although it is legal to associate any property with any
graphical object, the property does not necessarily affect the
object. For example, associating a “Sphere Radius” property
with an ambient light source does not affect the appearance or
behavior of the light. However, associating this property with
a group g potentially affects all spheres contained in g.
Graphical objects are reactive, that is, they can respond to
events. We distinguish three different kinds of events: mouse
events are triggered by pressing or releasing mouse buttons,
position events are triggered by moving the mouse, and key
events are triggered by pressing keyboard keys.
Events are handled by callbacks. There are three types
of callbacks, corresponding to the three kinds of events. Associated
with each graphical object are three callback stacks.
The client can define or redefine the reactive behavior of a
graphical object by pushing a new callback onto the appropriate
stack. The previous behavior of the graphical object can
easily be reestablished by popping the stack.
Consider a mouse event e that occurs within the extent of
a top-level window w. Associated with w is a RootGO r. The

top callback on r’s mouse callback stack will be invoked (if
the callback stack is empty, the event will simply be dropped).
The callback might perform an action, such as starting to spin
the scene, or it might delegate the event to one of r’s children.
3 Using Anim3D
Both Modula-3 and Obliq support the concepts of modules
and classes. (Obliq is based on prototypes and delegation,
not classes and inheritance; however, it is expressive enough
to simulate them.) For each kind of graphical object, there is a
Modula-3 module in ANIM3D. This module contains the class
of the graphical object, and a set of its associated functions and
variables. For each Modula-3 module, there is a “wrapper”
that makes it accessible as a module from Obliq.
The module GO contains the class of all graphical objects.
There are various methods associated with them: methods for
defining, undefining, and accessing properties in the property
mapping of a graphical object, and methods for pushing and
popping the three callback stacks of the graphical object as
well as for dispatching events to their top callback objects.
In addition, there is one property named GO_Transform,
which names the spatial transformation property and is meaningful
for all graphical objects. Unlike other properties, a
transformation property does not “override” other transformations
that are closer to the root, but is rather composed with
them.
The module GroupGO contains the class of all graphical
object groups, i.e. graphical objects which are used to group
other graphical objects together. The GroupGO class has
methods for adding elements to a group and removing them
again. The module also contains a functionNew, which creates
a new group and returns it.
A 3D window is regarded as a special form of group,
which contains all the objects in a scene (we therefore call it
the “root” of the scene), and has some additional properties,
such as the color of the background, whether depth cueing is
in effect, etc. Also associated with each window is the camera
that is currently active, and a “graphics base,” an abstraction
of the underlying windows and graphics system. Finally, the
Figure 2: The ANIM3D Solar System
RootGO module contains functions New and NewStd. The
latter creates a new scene root object with reasonable default
elements, callbacks, and properties (a perspective camera, two
white light sources, top-level reactive behavior that allows
the user to rotate and move the scene, and various surface
properties).
Both PEX and OpenGL distinguish between lines and
surfaces: surfaces are affected by light sources, lines are not.
There are a variety of properties common to all surfaces: their
color, transparency, reflectivity, shading model, and so on. Although
it is legal to attach these properties to non-surfaces, it
will not affect them. In order to emphasize that these properties
are meaningful only for surfaces, we provide a module
SurfaceGO, which contains the superclass of all graphical
objects composed of surfaces, along with their related properties.
We are provide a NonSurfaceGO module for lines and
markers.
The module SphereGO contains the class of spheres,
which is a subclass of the SurfaceGO class, as spheres are
composed of triangles, i.e. surfaces. Apart from the definition
of the sphere class, it contains a function New for creating new
sphere objects, and property names Center and Radius,
which are used to identify the properties determining the center
and the radius of the sphere.
Here is a complete Obliq program to display a planet and
its moon. The user can control the camera using the mouse.
This scene is displayed in the left snapshot of Fig. 2.
let root = RootGO_NewStd();
let planet = SphereGO_New([0,0,0],1);
SurfaceGO_SetColor(planet,"lightblue");
root.add(planet);
let moon = SphereGO_New([3,0,0],0.5);
SurfaceGO_SetColor(moon,"offwhite");
root.add(moon);
Property values can be time-variant; that is, their value depends
on the time of the animation clock. Time-variant property
values can either be unsynchronized or synchronized.
An unsynchronized time-variant property value starts to
change at the moment it is created, and animates the graphical
object o as long as it is attached to o. The animation does not

need to be triggered by any special command. For instance,
unsynchronized property values can be used to rotate the scene
or some part of it for an indefinite period of time.
Synchronized property values, on the other hand, are used
to animate several aspects of a scene in a coordinated fashion.
Each synchronized property value is “tied” to an animation
handle, and many values can be tied to the same handle. A synchronized
property value object accepts animation requests,
messages that ask it to change its current value, beginning at
some starting time and lasting for a certain duration. When
a client sends an animation request to a property value, the
request is not immediately satisfied, but instead stored in a request
queue local to the property value. Sending the message
animate to an animation handle causes all property values
controlled by this handle to be animated in synchrony. The
call to animate returns when all animations are completed.
When added to the above program, the following few
lines create a 25-second animation. The planet rotates six
times about its axis, while the moon revolves once around the
planet. In order to better show the rotation, we add a red torus
around the planet, aligned to the axis of rotation. See Fig. 2,
the three frames at the right.
let torus = TorusGO_New([0,0,0],[1,0,0],1,0.1);
root.add(torus);
SurfaceGO_SetColor(torus,"red");
let ah = AnimHandle_New();
let planettransform = TransformProp_NewSync(ah);
planet.setProp(GO_Transform,planettransform);
torus.setProp(GO_Transform,planettransform);
let moontransform = TransformProp_NewSync(ah);
moon.setProp(GO_Transform,moontransform);
moontransform.getBeh().rotateY(2*PI,0,25);
planettransform.getBeh().rotateY(12*PI,0,25);
ah.animate();
Note that we chose to attach the same transformation property
to both the torus and the planet. Alternatively, we could have
made a group containing both, and attached the transformation
property just to this group.
4 Case Study:
Shortest-Path Algorithm Animation
This section contains a case study of using ANIM3D with
the Zeus algorithm animation system [1] to develop an animation
of Dijkstra’s shortest-path algorithm. We first describe
the algorithm, and then sketch the desired visualization of the
algorithm. Next, we present an overview of the Zeus methodology
and finally, we present the actual implementation of the
animation.
The implementation consists of three elements. First, we
define a set of “interesting events,” used for communication
between the algorithm and the view. Second, we annotate the
algorithm with the events. And finally, we build a view, a
window that displays interesting events graphically.
4.1 The Algorithm
The single-source shortest-path problem can be stated as
follows: given a directed graph G � �V� E� with weighted
edges, and a designated vertex s, called the source, find the
shortest path from s to all other vertices. The length of a path
is defined to be the sum of the weights of the edges along the
path.
The following algorithm, due to Dijkstra [10], solves this
problem (assuming all edge weights are non-negative):
for all v 2 V do D�v� :� 1
D�s� :� 0; S :� �
while V n S 6� � do
let u 2 V n S such that D�u� is minimal
S :� S � fug
for all neighbors v of u do
D�v� :� minfD�v�� D�u� � W�u� v�g
endfor
endwhile
In this pseudo-code, D�v� is the distance from s to v, W�u� v�
is the weight of the edge from u to v, and S is the set of
vertices that have been explored thus far. V n S denotes those
elements in V that are not also in S.
4.2 The Desired Visualization
An interesting 3D animation of this algorithm is shown
in Fig. 3. The vertices of the graph are displayed as white
disks in the xy plane. Above each vertex v is a green column
representing D�v�, the best distance from s to v known so far.
Initially, the columns above each vertex other than s will be
infinitely (or at least quite) high. An edge from u to v with
weight W�u� v� is shown by a white arrow which starts at the
column over u at height 0 and ends at the column over v at
height W�u� v�.
Whenever a vertex u is selected to be added to S, the
color of the corresponding disk changes from white to red.
The addition D�u� � W�u� v� is animated by highlighting the
arrow corresponding to the edge �u� v� and lifting it to the top
of the column (i.e. raising it by D�u�). If D�u� � W�u� v�
is smaller than D�v�, the end of the arrow will still touch the
green column over D�v�, otherwise, it will not. In the former
case, we shrink the column over v to height D�u� � W�u� v�
to reflect the assignment of a new value to D�v�, and color the
arrow red, to indicate that it became part of the shortest-path
tree. Otherwise, the arrow simply disappears.
Upon completion, the 3D view shows a set of red arrows
which form the shortest-path tree, and a set of green columns
which represent the best distance D�v� from s to v.

Figure 3: These snapshots are from the animation of Dijkstra’s shortest-path algorithm described in section 4. The left snapshot shows the data just before entering
the main loop. The next snapshot shows the algorithm about one-third complete. In the third snapshot, the algorithm is about 2/3 complete, and the snapshot at
the right shows the algorithm upon completion.
4.3 Zeus Methodology
In the Zeus framework, strategically important points of
an algorithm are annotated with procedure calls that generate
“interesting events.” These events are reported to the Zeus
event manager, which in turn forwards them to all interested
views. Each view responds to interesting events by drawing
appropriate images. The advantages of this methodology are
described elsewhere [6].
4.4 The Interesting Events
The interesting events for Dijkstra’s shortest-path algorithm
(and many other shortest-path algorithms) are as follows:
� addVertex(u,x,y,d) adds a vertex u (where u is an
integer identifying the vertex) to the graph. The vertex
is shown at position �x� y� in the xy plane. In addition,
D�u� is declared to be d.
� addEdge(u,v,w) adds an edge from u to v with
weight w to the graph.
� selectVertex(u) indicates that u was added to S.
� raiseEdge(u,v,d) visualizes the addition D�u� �
W�u� v� by raising the edge �u� v� by d (where the caller
passes D�u� for d).
� lowerDist(u,d) indicates that D�u� gets lowered to
d.
� promoteEdge(u,v) indicates that the edge �u� v� is
part of the shortest-path tree.
� demoteEdge(u,v) indicates that the edge �u� v� is not
part of the shortest-path tree.
In addition, we need another event for house keeping purposes:
� start(m) is called at the very beginning of an algorithm’s
execution; it initializes the view to hold up to m
vertices and up to m 2 edges.
4.5 Annotating the Algorithm
Here is an annotated version of the algorithm we showed
before:
views.start(jV j)
for all v 2 V do D�v� :� 1
D�s� :� 0; S :� �
for all v 2 V do views.addVertex(v,v x,v y,D�v�)
for all �u� v� 2 E do views.addEdge(u,v,W�u� v�)
while V n S 6� � do
let u 2 V n S such that D�u� is minimal
S :� S � fug
views.selectVertex(u)
for all neighbors v of u do
views.raiseEdge(u,v,D�u�)
if D�v� � D�u� � W�u� v� then
views.demoteEdge(u,v)
else
D�v� :� D�u� � W�u� v�
views.promoteEdge(u,v)
views.lowerDist(v,D�v�)
endif
endfor
endwhile
In this pseudo-code, views is the dispatcher provided by
Zeus. The dispatcher will notify all views the user has selected
for the algorithm.
4.6 The View
A view is an object that has a method corresponding to
each interesting event, and a number of data fields. In this
view, the data fields are as follows: a RootGO object that
contains all graphical objects of the scene, together with a
camera and light sources; arrays of graphical objects holding
the disks (vertices), columns (distances), arrows (graph edges),
and shortest-path tree edges; and an “animation handle” for

w
c
e
b a
Figure 4: The structure of an arrow generated by the addEdge method.
triggering animations. This leads us to a skeletal view:
let view = {
scene => RootGO_NewStd(),
ah => AnimHandle_New(),
verts => ok, (* initialized by method start *)
dists => ok, (* initialized by method start *)
parent => ok, (* initialized by method start *)
edges => ok, (* initialized by method start *)
start => meth(self,m) ... end,
addVertex => meth(self,u,x,y) ... end,
addEdge => meth(self,u,v,w) ... end,
selectVertex => meth(self,u) ... end,
raiseEdge => meth(self,u,v,z) ... end,
lowerDist => meth(self,u,z) ... end,
promoteEdge => meth(self,u,v) ... end,
demoteEdge => meth(self,u,v) ... end,
};
The Zeus system has a control panel that allows the user to
select an algorithm and attach any number of views to it.
Whenever the user creates a new 3D view, a new Obliq interpreter
is started, and reads the view definition. The algorithm
and all views run in the same process, but in different threads;
thread creation is very light-weight. The above expression
creates a new object view, and initializes view.scene to
be a RootGO, and view.ah to be an animation handle.
The remainder of this section fleshes out the 8 methods
of view, which correspond to the 8 interesting events:
� The start method is responsible for initializing
view.verts, view.dists, and view.parent to be
arrays of size m, and view.edges to be an m � m array.
The elements of the newly created arrays are initialized to
the dummy value ok. Here is the code:
start => meth(self,m)
self.verts := array_new(m, ok);
self.dists := array_new(m, ok);
self.parent := array_new(m, ok);
self.edges := array2_new(m, m, ok);
end
� The addVertex method adds a new vertex to the view.
Vertices are represented by white disks that lie in the xy
plane. Above each vertex, we also show a green column of
height d, provided that d is greater than 0. The location of
the cylinder’s base is constant, while its top is controlled by
an animatable point property value.
addVertex => meth(self,u,x,y,d)
self.verts[u] := DiskGO_New(
[x,y,0],
[0,0,1],
0.2);
self.scene.add(self.verts[u]);
if d > 0 then
let top = PointProp_NewSync(self.ah,[x,y,d]);
self.dists[u] := CylinderGO_New(
[x,y,0],
top,
0.1);
SurfaceGO_SetColor(self.dists[u],"green");
self.scene.add(self.dists[u]);
end;
end
� The addEdge method adds an edge (represented by an
arrow) from vertex u to vertex v. The arrow starts at the at
the disk representing u, and ends at the column over v at
height w. An arrow is composed of a cone, a cylinder, and
two disks; its geometry is computed based on the “Center”
property of the disks representing the vertices to which it is
attached. Figure 4 illustrates the relationship.
addEdge => meth(self,u,v,w)
let a = DiskGO_GetCenter(self.verts[u]).get();
let b = DiskGO_GetCenter(self.verts[v]).get();
let c = Point3_Plus(b,[0,0,w]);
let d = Point3_Minus(c,a);
let e = Point3_Minus(c,Point3_Scale(d,0.4));
let grp = GroupGO_New();
grp.setProp(GO_Transform,
TransformProp_NewSync(self.ah));
grp.add(DiskGO_New(a,d,0.1));
grp.add(CylinderGO_New(a,e,0.1));
grp.add(DiskGO_New(e,d,0.2));
grp.add(ConeGO_New(e,c,0.2));
self.edges[u][v] := grp;
self.scene.add(grp);
end
� The selectVertex method indicates that a vertex u has
been added to the set S by coloring u’s disk red:
selectVertex => meth(self,u)
SurfaceGO_SetColor(self.verts[u],"red");
end

� The raiseEdge method highlights the edge from u to v
by coloring it yellow, and then lifting it up by z. The arrow
is moved by sending a “translate” request to its transformation
property. The translation is controlled by the animation
handle self.ah, and shall take 2 seconds to complete.
Calling self.ah.animate() causes all animation requests
controlled by self.ah to be processed.
raiseEdge => meth(self,u,v,z)
SurfaceGO_SetColor(self.edges[u][v],"yellow");
let pv = GO_GetTransform(self.edges[u][v]);
pv.getBeh().translate(0,0,z,0,2);
self.ah.animate();
end
� The method lowerDist indicates that the “cost” D�u� of
vertex u got lowered, by shrinking the green cylinder representing
D�u�. This is done by sending a linMoveTo
(“move over a linear path to”) request to the “Point2” property
of the cylinder.
lowerDist => meth(self,u,z)
let pv = CylinderGO_GetPoint2(self.dists[u]);
let p = pv.get();
pv.getBeh().linMoveTo([p[0], p[1], z], 0, 2);
self.ah.animate();
end
� The method promoteEdge indicates that �u� v�, the edge
that is currently highlighted, shall become part of the
shortest-path tree. This is indicated by coloring the edge
red. If there already was a red edge leading to v, it is
removed from the view.
promoteEdge => meth(self,u,v)
SurfaceGO_SetColor(self.edges[u][v],"red");
if self.parent[v] isnot ok then
self.demoteEdge(self.parent[v],v);
end;
self.parent[v] := u;
end
� Finally, the method demoteEdge removes the edge �u� v�
from the view:
demoteEdge => meth(self,u,v)
self.scene.remove(self.edges[u][v]);
end
This completes our example. The complete view is about
70 lines of code, compared to the roughly 2000 lines of the
PEXlib-based version that generated the animations presented
in [5]. This measure is fairly honest; we did not add any functionality
(such as a new class ArrowGO) to the base library in
order to optimize this example. Furthermore, turnaround time
during the design of this view was limited only by the design
process per se (and our typing speed), whereas compiling a
single file and relinking with the Zeus system takes several
minutes.
5 Related Work
There are two areas that have influenced ANIM3D:
general-purpose 3D animation libraries and algorithm animation
systems that have been used for developing 3D views.
The most closely related general-purpose animation library
is OpenInventor [17, 18], an object-oriented graphics
library with a C++ API. OpenInventor, like ANIM3D, represents
a scene as a DAG of “nodes”. Geometric primitives,
cameras, lights, and groups are all special types of nodes.
However, there are three key differences between ANIM3D
and OpenInventor:
� ANIM3D includes an embedded interpretive language,
which is instrumental for achieving fast turnaround and
short design cycles.
� OpenInventor views properties (such as colors and transformations)
as ordinary nodes in the scene DAG. This
means that the order of nodes in a group becomes important.
In this respect, ANIM3D is more declarative than
OpenInventor: the order in which objects are added to a
group does not matter.
� In a number of aspects, OpenInventor requires the programmer
to do more work than ANIM3D requires. For
example, OpenInventor clients have to explicitly redraw
a scene whereas ANIM3D uses a damage-repair model to
automatically redraw just those primitives that need to be
redrawn.
Nonetheless, OpenInventor is a very impressive commercial
product that greatly simplifies 3D graphics. Many of the
ideas of OpenInventor can be found in work done at Brown
University [16, 19].
There are three algorithm animation systems that have
been used for developing 3D views, Pavane, Polka3D, and
GASP.
In Pavane [8], the computational model is based on tuplespaces
and mappings between them. Entering tuples into
the “animation tuple space” has the side-effect of updating the
view. A small collection of 3D primitives are available (points,
lines, polygons, circles, and spheres), and the only animation
primitives are to change the positions of the primitives.
Polka3D [14], like Zeus, follows the BALSA model for
animating algorithms. Algorithms communicate with views
using “interesting events,” and views draw on the screen in
response to the events. The graphics library is similar to
ANIM3D in goals and features, but it appears to be a bit slimmer
and more focused on algorithm animations. Unlike our
system, views are not interpreted, so turnaround time is not
instantaneous.
The GASP [9] system is tuned for developing animations
of computational geometry algorithms involving three (and
two) dimensions. Because a primary goal of the system is to

isolate the user from details of how the graphics is done, a user
is limited to choosing from among a collection of animation
effects supplied by the system. The viewing choices are typically
stored in a separate “style” file that is read by the system
at runtime; thus, GASP provides rapid turnaround. However,
it does not provide the flexibility to develop arbitrary views
with arbitrary animation effects.
6 Conclusion
The first part of this paper described ANIM3D, an objectoriented
3D animation library targeted at visualizing combinatorial
structures, and in particular at animating algorithms.
The second part presented a case study showing how to use
ANIM3D within the Zeus algorithm animation system for producing
a 3D visualization of a graph-traversal algorithm.
ANIM3D is based on three concepts: scenes are described
by DAGs of graphical objects, time-variant property values
are the basic animation mechanism, and callbacks are the
mechanism by which clients can specify reactive behavior.
These concepts provide a simple, yet powerful framework for
building animations.
ANIM3D provides fast turnaround by incorporating an
interpretive language that allows the user to modify the code
of a program even as it runs. Previous experience has shown
us that powerful animation facilities and fast turnaround time
are crucial for enabling non-expert users to construct new
algorithm animations.
7 Acknowledgments
We are grateful to Allan Heydon and Lucille Glassman
for helping to improve the quality of the presentation.
References
[1] Marc H. Brown. Zeus: A System for Algorithm Animation
and Multi-View Editing. 1991 IEEE Workshop on
Visual Languages (October 1991), 4–9.
[2] Marc H. Brown. The 1992 SRC Algorithm Animation
Festival. 1993 IEEE Symposium on Visual Languages
(August 1993), 116–123.
[3] Marc H. Brown. The 1993 SRC Algorithm Animation
Festival. Research Report 126, Digital Equipment Corp.,
Systems Research Center, Palo Alto, CA (1994).
[4] Marc H. Brown and John Hershberger. Color and Sound
in Algorithm Animation. Computer, 25(12):52–63, December
1992.
[5] Marc H. Brown and Marc Najork. Algorithm Animation
Using 3D Interactive Graphics. ACM Symposium on User
Interface Software and Technology (November 1993),
93–100.
[6] Marc H. Brown and Robert Sedgewick. A System for
Algorithm Animation. Computer Graphics, 18(3):177–
186, July 1984.
[7] Luca Cardelli. Obliq: A language with distributedscope.
Research Report 122, Digital Equipment Corp., Systems
Research Center, Palo Alto, CA (April 1994).
[8] Gruia-Catalin Roman, Kenneth C. Cox, C. Donald
Wilcox, and Jerome Y. Plun. Pavane: A System for
Declarative Visualization of Concurrent Computations.
Journal of Visual Languages and Computing, 3(2):161–
193, June 1992.
[9] David Dobkin and Ayellet Tal. GASP—A System to Facilitate
Animating Geometric Algorithms. Technical Report,
Department of Computer Science, Princeton University,
1994.
[10] E. W. Dijkstra. A note on two problems in connexion
with with graphs. Numerische Mathematik, 1:269–271,
1959.
[11] Robert A. Duisberg. Animated Graphical Interfaces Using
Temporal Constraints. ACM CHI ’86 Conf. on Human
Factors in Computing (April 1986), 131–136.
[12] Steven C. Glassman. A Turbo Environment for Producing
Algorithm Animations. 1993 IEEE Symp. on Visual
Languages (August 1993), 32–36.
[13] Mark S. Manasse and Greg Nelson. Trestle Reference
Manual. Research Report 68, Digital Equipment Corp.,
Systems Research Center, Palo Alto, CA, December
1991.
[14] John T. Stasko and Joseph F. Wehrli. Three-Dimensional
Computation Visualization. 1993 IEEE Symposium on
Visual Languages (August 1993), 100 – 107.
[15] John T. Stasko. TANGO: A Framework and System for
Algorithm Animation. Computer, 23(9):27–39, September
1990.
[16] Paul S. Strauss. BAGS: The Brown Animation Generation
System. Technical Report CS–88–22, Brown University,
May 1988.
[17] Paul S. Strauss. IRIS Inventor, a 3D Graphics Toolkit.
OOPSLA’93 Conf. Proc., (September 1993), 192–200.
[18] Paul S. Strauss and Rikk Carey. An Object-Oriented
3D Graphics Toolkit. ACM Computer Graphics (SIG-
GRAPH ’92) (July 1992), 341–349.
[19] Robert C. Zeleznik et al. An Object-Oriented Framework
for the Integration of Interactive Animation Techniques.
ACM Computer Graphics (SIGGRAPH ’91), (July 1991),
105–111.

Figure 1a: Closest-Pair Figure 1b: Heapsort
Figure 1c: k-d Tree Figure 1d: Shakersort
Figure 1e: Red-Black and 2-3-4 Trees Figure 3c: Shortest-Path

Strata�Various� � Multi�Layer Visualization of Dynamics
in Software System Behavior
Abstract
Current software visualization tools are inadequate
for understanding� debugging� and tuning realistically
complex applications. These tools often present only
static structure� or they present dynamics from only a
few of the many layers of a program and its underly�
ing system. This paper introduces �PV�� a prototype
program visualization system which provides concur�
rent visual presentation of behavior from all layers�
including� the program itself� user�level libraries� the
operating system� and the hardware� as this behavior
unfolds over time. PV juxtaposes views from di�erent
layers in order to facilitate visual correlation� and al�
lows these views to be navigated in a coordinated fash�
ion. This results in an extremely powerful mechanism
for exploring application behavior. Experience is pre�
sented from actual use of PV in production settings
with programmers facing real deadlines and serious
performance problems.
1 Visualization of Software Behavior
To truly understand any realistically complex piece
of software� for purposes of debugging or tuning� one
must consider its execution�time behavior� not just its
static structure. Actual behavior is often far di�er�
ent from expectations� and often results in poor per�
formance and incorrect results. Further� the ultimate
correctness and performance of an application �or lack
thereof� arises not only from the behavior of the pro�
gram itself� but also from activity carried out on its
behalf by underlying system layers. These layers in�
clude user�level libraries� the operating system� and
the hardware. Finally� problems often become appar�
ent only when one considers the interleaving of various
� ���
Doug Kimelman� Bryan Rosenburg� Tova Roth
IBM Thomas J. Watson Research Center
Yorktown Heights� NY 10598
kinds of activity� rather than cumulative activity sum�
maries at the end of a run. Thus� for debugging and
tuning applications in a realistically complex environ�
ment� one must consider behavior at numerous layers
of a system concurrently� as this behavior unfolds over
time.
Clearly� any textual presentation of this amount of
information would be overwhelming. A visual presen�
tation of the information is far more likely to be mean�
ingful. Information is assimilated far more rapidly
when it is presented in a visual fashion� and trends and
anomalies are recognized much more readily. Further�
animations� and views which incorporate time as an
explicit dimension� reveal the interplay among compo�
nents over time.
With an appropriate visual presentation of infor�
mation concerning software behavior over time� one
can �rst survey a program execution broadly using
a large�scale �high�level� coarse�resolution� view� then
narrow the focus as regions of interest are identi�ed�
and descend into �ner�grained �more detailed� views�
until a point is identi�ed for which full detail should
be considered.
Further� displays which juxtapose views from di�er�
ent system layers in order to facilitate visual correla�
tion� and which allow these views to be navigated in a
coordinated fashion� constitute an extremely powerful
mechanism for exploring application behavior.
2 PV � A Program Visualization Sys�
tem
PV� a prototype program visualization system de�
veloped at IBM Research� embodies all of the visual�
ization capabilities proposed above. Success with PV

in production settings and complex large�scale envi�
ronments has veri�ed that these capabilities are indeed
highly e�ective for understanding application behav�
ior for purposes of debugging and tuning.
Users often turn to program visualization when per�
formance is disappointing � either performance does
not match predictions� or it deteriorates as changes
are introduced into the system� or it does not scale up
�and perhaps even worsens�� as processors are added
in a multiprocessor system� or it is simply insu�cient
for the intended application.
With PV� users watch for trends� anomalies� and
interesting correlations� in order to track down press�
ing problems. Behavioral phenomena which one might
never have suspected� or thought to pursue� are often
dramatically revealed. A user continually replays the
execution history� and rearranges the display to dis�
card unnecessary information or to incorporate more
of the relevant information. In this way� users examine
and analyze execution at successively greater levels of
detail� to isolate �aws in an application. Resolution of
the problems thus discovered often leads to signi�cant
improvements in the performance of an application.
PV shows hardware�level performance information
�such as instruction execution rates� cache utilization�
processor element utilization� delays due to branches
and interlocks� if it is available� operating�system�level
activity �such as context switches� address�space ac�
tivity� system calls and interrupts� kernel performance
statistics�� communication�library�level activity �such
as message�passing� inter�processor communication��
language�runtime activity �such as parallel�loop sched�
uling� dynamic memory allocation�� and application�
level�activity �such as algorithm phase transitions� ex�
ecution time pro�les� data structure accesses�.
PV has been targeted to shared�memory paral�
lel machines �the RP3�7��� distributed memory ma�
chines �transputer clusters running Express�� work�
station clusters �RISC System�6000 workstations run�
ning Express�� and superscalar uniprocessor worksta�
tions �RISC System�6000 with AIX�.
PV is structured as an extensible system� with a
framework and a number of plug�in components which
perform analysis and display of event data generated
by a running system. It includes a base set of com�
ponents� and users are encouraged to add their own
and con�gure them into networks with existing com�
ponents. Novice users simply call up pre�established
con�gurations of components in order to use estab�
lished views of program behavior.
Figures 1 through 6 show some of the many views
provided by PV. Section 4 describes some of these
views in detail and discusses their use.
3 AIX Trace
PV is trace�driven. It produces its displays by
continually updating views of program behavior as it
reads through a trace containing an execution history.
A trace consists of a time�ordered sequence of event
records� each describing an individual occurrence of
some event of interest in the execution of the pro�
gram. Typically� an event record consists of an event
type identi�er� a timestamp� and some event�speci�c
data. Events of interest might include� sampling of
a cache miss counter� a page fault� scheduling of a
process� allocation of a memory region� receipt of a
message� or completion of some step of an algorithm.
A trace can be delivered to the visualization system
live �possibly over a network�� as the event records are
being generated� or it can be saved in a �le for later
analysis.
The standard AIX system �IBM�s version of Unix��
as distributed for RS�6000s� includes an embedded
trace facility. AIX Trace�6�� a service provided by the
operating system kernel� accepts event records gener�
ated at any level within the system and collects them
into a central bu�er. As the event bu�er becomes full�
blocks of event records are dumped to a trace �le� or
dumped through a pipe to a process� e.g. for transmis�
sion over a network. Alternatively� the system can be
con�gured to simply maintain a large circular bu�er
of event records that must be explicitly emptied by a
user process.
Comprehensive instrumentation within AIX itself
provides information about activity within the kernel�
and a system call is provided by which user processes
can provide event records concerning activity within
libraries or the application. On machines incorporat�
ing hardware performance monitors� a device driver
can unload hardware performance data� periodically
or at speci�c points during the execution of an appli�
cation� and generate AIX event records containing the
data.
The variant of PV that is targeted to AIX worksta�
tions is based on AIX Trace. Unless otherwise noted�
all of the applications discussed in this paper were
run on AIX RS�6000 workstations with AIX Trace
enabled. Traces were taken during a run of the appli�
cation and saved in �les for later analysis.
For the applications discussed here� tracing over�
head was negligible � less than 5� in most cases. In
no case was perturbation great enough to alter the be�
havior being investigated. Trace �le sizes in all cases

were less than 16 megabytes.
4 Views in Action � Experience with
PV
This section describes some of the many views pro�
vided by PV� and explains their use� by way of exam�
ples of actual experience with PV.
PV has been applied to a number of di�erent types
of application across a number of domains� includ�
ing� interactive graphics applications written in C���
systems programs such as compilers written in C�
computation�intensive scienti�c applications written
in Fortran� I�O�intensive applications written in C�
and a large� complex� heavily�layered� distributed ap�
plication written in Ada.
4.1 Views of Process Scheduling and Sys�
tem Activity
In one example� the developers of �G�� an interac�
tive graphics application� 1 were concerned that it was
taking 12 seconds from the time that the user entered
the command to start the application� until the time
that the main application window would respond to
user input. They suspected that a lot of time was
being lost in the Motif libraries.
End�of�run summaries showed that 51 seconds out
of a 97 second run were spent idle� but these sum�
maries provided no indication of how many of these
idle seconds were in fact warranted� perhaps waiting
for user input� and how many were somehow on the
critical path for the application. Pro�les of time spent
in various functions� and perusal of thousands of lines
of detail in textual reports� would not have been help�
ful.
PV views showing process scheduling alongside op�
erating system activity immediately highlighted the
nature of this performance problem. The schedul�
ing view consists of a strip of color growing to the
right over time� with color used to indicate which
process was running at any instant in time. �Figure 2
shows this view in a window titled �AixProcess j Col�
orStrip�.� The activity view consists of a similar strip
of color� in which color is used to indicate what activ�
ity was taking place at any instant in time. �Figure 2
shows this view in a window titled �AixSystemState j
ColorStrip�.� The two views can show the same time
spans� and they can be aligned so that a point in one
view corresponds to the same instant in time as the
point immediately above or below it in the other view.
Further� the two views aligned in this way can be navi�
gated in a coordinated fashion � when the user zooms
in on either view� expanding a region of interest in or�
der to reveal greater detail� the other view expands
the same region of time automatically. 2
�PV provides a number of other views which can
be aligned and navigated in the same fashion� includ�
ing views showing kernel performance statistics� hard�
ware performance statistics� which loop of a function
is currently active� or which user�de�ned phase of an
algorithm is currently being executed.�
By viewing behavior as it unfolded over time� it was
apparent that the two processes of application �G�
�shown in Figure 2 as light pink and salmon color�
were not even running for much of the 12 seconds that
they should have been rushing to establish the main
application window ���. Further� it wasn�t even the X
server process �light green� that was running instead of
them. In fact the system was idle �dark purple� much
of the time. Thus� 5 of the 51 idle seconds noted above
were occurring during startup and hence were on the
critical path. Finding the point on the scheduling view
where a process of application �G� went idle� zoom�
ing in to show greater detail� and dropping down to
the activity view� revealed the cause of the idle time�
system calls to examine a number of �les were caus�
ing large delays. Having narrowed the focus to a very
small window in time using the graphic views� a view
was opened showing the detailed textual trace report.
As each event is displayed graphically in other views�
this view highlights the corresponding line in the re�
port. With this level of detail it was immediately ob�
vious that startup information had inadvertently been
scattered across a number of �les which might well be
remote�mounted and thereby incur signi�cant access
penalties.
Without the visual correlation facilitated by juxta�
position of views and coordinated navigation� it would
have been much harder to make the connection be�
tween the various aspects of this performance problem.
At the very least� it would have taken much longer by
any less direct means.
4.2 Views of Memory Activity and Appli�
cation Progress
In another example� PV views revealed a number
of memory�related problems in �A�� a compiler. Each
1 �The stories you are about to hear are true. Only the names have been changed to protect the innocent.�
2 Complete detail concerning the contents of these views can be found in a lengthy technical report �8�.

view in this case is rectangular� with each position
along the horizontal axis corresponding to some re�
gion of a linear address space �the size of the region
depends on scale of the display�. In one view� color
is used to represent the size of a block of memory on
the user heap. �Figure 5 shows this view in the upper
window titled �AixMalloc j OneSpace�.� In another
view� color is used to show the source �le name or
line number that allocated the block. �Figure 5 shows
this view in the lower window titled �AixMalloc j One�
Space�.� In a third view� color is used to represent the
state of a page of the data segment of the user address
space. �Figure 5 shows this view in the window titled
�AixDataSeg j OneSpace�.� For purposes of correla�
tion� the views are con�gured to show the same range
of addresses� and they are aligned so that a given ad�
dress occurs at the same horizontal position in each
view. As well� zooming in on a region in one view au�
tomatically causes the corresponding zoom operation
in the other view.
Each of these views in Figure 5 is split into an
upper half and a lower half� each representing part
of the data segment of the address space of com�
piler �A�. The left edge of the upper half represents
address 0x24200000� successive points to the right
along this half represent successively higher addresses�
and the right edge represents address 0x24600000.
Thus� the upper half of these views represent 2MB
of the data segment. Similarly� the lower half of these
views represents an expanded view of the 248KB from
0x244CAE35 to 0x2448E571. The black guidelines
show where the region represented by the lower half
of a view �ts into the region represented by the upper
half.
These views showed a number of wastes of memory�
none of which could technically be classed a �leak�.
Rather� they were �balloons� � still referred to� but
largely full of empty space. In one case� the heap views
showed that every second page of the heap was not be�
ing made available to the end user �shown in Figure
5 �AixMalloc j OneSpace� as alternating green and
white blocks in the lower half of the view�� yet the cor�
responding positions on the data segment view showed
clearly that every page was being faulted in �shown in
the lower half of Figure 5 �AixDataSeg j OneSpace� as
all magenta�. The heap views also showed that all of
the blocks in question were of the same size. Having
identi�ed blocks of a particular size as being prob�
lematic� the source code for the allocator was quickly
inspected� with particular attention to the treatment
of blocks of the problematic size. It rapidly became
apparent that� in certain situations� half of the heap
was being left empty due to an unfortunate interac�
tion between user code� the heap memory allocator�
and the virtual memory system.
In another case� a static array was declared to be
enormous. This was felt to be acceptable because
real memory pages were never faulted in unless they
were required for the size of the program being com�
piled. However� the data segment view emphasized
that the array did occupy address space� and this
became noteworthy when the compiler could not be
loaded on smaller machine con�gurations� even though
only moderate�sized programs needed to be compiled.
Finally� late in the run of this compiler� pages began
�ashing in and out of the data segment view. Glancing
at the system activity view �described earlier�� during
the time that the page �ashing was occurring� allowed
this behavior to be correlated to periods of excessive
disclaiming and subsequent reclaiming of pages by the
compiler.
An application phase view� which provides a
roadmap to the progress of an application� allowed
this thrashing in the address space to be attributed
directly to the o�ending phase of the compiler. The
application phase view consists of a number of strips
of color� as in the process scheduling and system activ�
ity views described earlier. �Figure 5 shows this view
in the window titled �AixPhase j ColorStrip�.� The
strips are stacked one on top of the other� and they
grow to the right together over time. The color of the
top strip shows which user�de�ned phase of the appli�
cation is in progress at any instant in time. The color
of successively lower strips shows successively deeper
sub�phases nested within the phases shown at the cor�
responding positions on the higher strips. �This view
can be driven by instrumentation in the form of sim�
ple event generation statements inserted manually or
automatically into the source� or by procedure entry
and exit events generated using object code insertion
techniques.�
In the case of compiler �A�� correlation in time be�
tween the application phase view and the data seg�
ment view immediately made it clear that a back�
end code generation phase �shown in Figure 5 as light
green� was responsible for the excessive paging activ�
ity.
In another example� these memory�related views
did in fact reveal a number of actual memory leaks
in �F�� a large Ada application. Due to the visual na�
ture of these views� it was immediately apparent that
particular leaks were �ooding the address space �which
was bleeding full of the color of the allocators in ques�
tion� and hence required immediate attention. It was

just as apparent that other leaks were inconsequential
and could be ignored until after a rapidly approaching
deadline. This is something which would not be read�
ily apparent from the textual report of conventional
special�purpose memory leak detectors.
4.3 Views of Hardware Activity and
Source Progress
Finally� in an example involving �T�� a
computation�intensive scienti�c application� a view
showing which loop of a program was active over
time� in conjunction with a view of hardware perfor�
mance statistics over time� highlighted opportunities
for signi�cant improvements in performance.
The program loop view is simply the application
phase view described earlier� with color used to indi�
cate which program loop is active at any instant in
time �rather than which arbitrary user�de�ned phase
is active�. �Figure 6 shows this view in the window
titled �AixPhase j ColorStrip�.� The hardware per�
formance view consists of a stack of linegraphs grow�
ing to the right over time. �Figure 6 shows this view
in the window titled �RS2Pmc j Scale j LineGraph�.�
Hardware�level information� as discussed in Sections 2
and 3� is sampled at loop boundaries and plotted on
the various graphs. In this case� the two views showed
the same time span and were aligned for purposes of
correlation and navigation.
These views allowed programmers to easily iden�
tify the longer�running loops and to correlate execu�
tion of a particular loop with a dramatic decrease in
MFLOPS. The hardware view showed that the loop
was not cache�limited and was not a �xed point loop�
yet one �oating point unit was seldom busy� while the
other was extremely busy but completing very few in�
structions. To understand the behavior of this partic�
ular loop� a number of additional views were opened
to show the program source.
Each source view highlights a line of source at the
beginning of the major loop currently being executed.
One of the views is� in e�ect� a �very high altitude�
view of the source �as in �2��� in which the entire source
of the program �ts within the single window. Al�
though the code is illegible due to the �very small
font�� the overall structure of the program is appar�
ent� and the overall progress of the application can
be tracked easily. The code in the second view of
the source is legible� but the view can only show a
page of source at a time and must be scrolled in order
to view di�erent parts of the program. �These two
source views are shown side by side at the left of Fig�
ure 6� beneath the PV control panel.� The third view
shows the assembly language source� as generated by
the compiler� with the same form of highlighting as
the other two source views. �In �gure 6� this view is
hidden behind the other windows.�
For application �T�� glancing at the source views
con�rmed that� for the loop in question� a divide in�
struction was in fact causing one �oating point unit
to remain fully busy while not completing very many
instructions. The assembly view showed that the rea�
son for the second �oating point unit not even keeping
busy was an unnecessary dependence in the code. Us�
ing these views for feedback� the programmer was able
to experiment rapidly with manual source transforma�
tions� and ultimately to achieve a 12� improvement
in the performance of application �T�.
Overall� through experience with PV in these situ�
ations and many others� the visualization capabilities
proposed above have proven tremendously e�ective for
debugging and tuning� often in cases where traditional
methods have failed.
5 Future Research
The user interface is bound to be a severe limitation
of any current software visualization system. Typical
displays of software are crude approximations� at best�
to the elaborate mental images that most program�
mers have of the software systems they are developing.
Opening� closing� and aligning windows on a relatively
small 2�dimensional screen is a cumbersome means of
manipulating a few small windows onto an elaborate
conceptual world.
With the advent of su�ciently powerful virtual
reality technology� a far more e�ective facility for
software visualization could be achieved by map�
ping multiple�layer software systems onto expansive 3�
dimensional terrains� and providing more direct means
for traversal. Traversal could involve high�level passes
over the terrain to obtain an overview� and descent to
lower�levels over regions of interest for more detailed
views. The system could also provide the ability to
maintain a number of distinct perspectives onto the
terrain. The panorama could include both representa�
tions of the software entities themselves� as well as de�
rived information such as performance measurements�
and more abstract representations of the entities and
the progress of their computation.

6 Related Work
The notion of program visualization per se �15� �18�
�rst appeared in the literature more than ten years
ago �4�. Much of the initial work in program visual�
ization� and many recent e�orts� are concerned solely
with the static structure of a program. They do not
consider dynamics of program behavior at all.
Algorithm animation work �1� �17� has focused
strictly on small algorithms� rather than on actual be�
havior of large applications or on all of the layers of
large underlying systems. Further� algorithm anima�
tions often require large amounts of time to construct
�days� weeks or even months�. This is acceptable in
a teaching environment� where the animations will be
used repeatedly on successive generations of students�
but is unacceptable in a production software develop�
ment environment where it is critical that a tool can
be applied readily to problems as they arise.
Recently� there has been much work in the area
of program visualization for parallel systems �9� This
work has in fact been concerned with dynamics� but
much of it has been con�ned to communication or
other aspects of parallelism. Little consideration has
been given to displaying other aspects of system be�
havior. PIE �11� shows system�level activity over
time� but its displays are limited primarily to con�
text switching. Other system�level activity and activ�
ity from the application and other levels of the system
are not displayed simultaneously for correlation.
The IPS�2 performance measurement system for
parallel and distributed programs �5� �14� does in�
tegrate both application and system based metrics.
However� system metrics are dealt with strictly in the
form of �external time histograms�� each describing
the value of a single performance metric over time�
as opposed to more general event data. Thus� where
non�application data are concerned� IPS�2 is limited
to strictly numeric presentations� such as tables and
linegraphs. Dynamic animated displays of behavior�
such as those showing system activity over time� or
memory state as it evolves� are not possible with IPS�
2. Program hierarchy displays are used primarily only
for showing the overall structure of an application� or
for specifying the program components for which per�
formance measurements are to be presented.
Some vendors provide general facilities for tracing
the system requests made by a given process. How�
ever� these facilities tend to apply to a single process
rather than the system as a whole� and hence are not
useful for showing the interaction between a process
and its surrounding environment. Furthermore� these
facilities tend to have very high overheads.
Pro�ling tools� such as the Unix utilities �prof�
and �gprof�� have existed for some time� but these
utilities simply show cumulative execution time� at the
end of a run� on a function by function basis.
A number of workstation vendors have recently ex�
tended basic pro�ling facilities or debuggers by adding
views to show time consumption and other resource
utilization graphically. Many of these tools now report
utilization with granularity as �ne as a source line�
and many allow sampling during experiments which
can cover some part of a run rather than just an en�
tire run. None of these tools� however� supports the
notion of general visual inspection of continuous be�
havior and system dynamics at multiple levels within
a system.
Some debuggers are now including views of behav�
ior in the memory arena� but none of these tools pro�
vides the power and generality of PV.
The power of PV� and its novelty� lie in its com�
bination of a number of important properties. PV
provides both quantitative and animated displays� and
it presents information from multiple layers of a pro�
gram and its underlying system. Further� PV facil�
itates correlation and coordinated navigation of the
information displayed in its various views. Finally�
PV presents views which address important concerns
for software behavior on mainstream workstation sys�
tems� not just clusters or parallel machines. PV em�
bodies all of these capabilities and it provides e�ec�
tive industrial�strength support of large�scale applica�
tions �even hundreds of megabytes of address space
and hundreds of thousands of lines of code�.
7 Conclusion
In production settings� over a wide range of com�
plex applications� PV has proven invaluable in uncov�
ering the nature and causes of program failures. De�
velopers facing serious performance problems and im�
minent deadlines have found it worthwhile to invest
time to connect PV to their application� and to run
and inspect visualization displays.
Experience with PV indicates that concurrent vi�
sual presentation of behavior from many layers� in�
cluding the program itself� user�level libraries� the op�
erating system� and the hardware� as this behavior
unfolds over time� is essential for understanding� de�
bugging� and tuning realistically complex applications.
Systems that facilitate visual correlation of such infor�
mation� and that provide coordinated navigation of

multi�layer displays� constitute an extremely powerful
mechanism for exploring application behavior.
Acknowledgments
Heartfelt thanks to Keith Shields� Barbara Walters�
and Christina Meyerson for hard work in the trenches�
and to Fran Allen and Emily Plachy for unwavering
support.
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Figure_1

Figure_2
Figure_3

Figure_4

Figure_5

Figure_6

Di�erential Volume Rendering� A Fast Volume Visualization
Technique for Flow Animation
Han�Wei Shen and Christopher R. Johnson
Department of Computer Science
University of Utah
Salt Lake City� UT 84112.
E�mail� hwshen�cs.utah.edu and crj�cs.utah.edu
Abstract
We present a direct volume rendering algorithm
to speed up volume animation for �ow visualizations.
Data coherency between consecutive simulation time
steps is used to avoid casting rays from those pixels
retaining color values assigned to the previous image.
The algorithm calculates the di�erential information
among a sequence of 3D volumetric simulation data.
At each time step the di�erential information is used
to compute the locations of pixels that need updating
and a ray�casting method is utilized to produce the up�
dated image. We illustrate the utility and speed of
the di�erential volume rendering algorithm with sim�
ulation data from computational bioelectric and �uid
dynamics applications. We can achieve considerable
disk�space savings and nearly real�time rendering of
3D �ows using low�cost� single processor workstations �
for models which contain hundreds of thousands of
data points.
Introduction
While there is a rich history of numerical tech�
niques for computing the dynamics of wave propaga�
tion� only recently have researchers been able to vi�
sualize the complex dynamics of large 3D �ow simu�
lations �1� 2� 3�. Visualizations of �ow dynamics typ�
ically involve characterizing relevant features of vec�
tor and scalar �elds. While visualizing vector �elds
is particularly important in many applications� it is
also important to quantify and visually characterize
scalar features of the �ow �eld such as� temperature�
voltage� and magnitudes of vector quantities. Di�
rect volume�rendering techniques� e�ective tools for
exploring 3D scalar data� have been proposed as a
methodology to visualize scalar features in the �ow
� Such as the SGI Indy or comparable workstations
�eld �4� 5� 6�. Unlike surface�rendering methods� direct
volume�rendering methods can be used to visualize 3D
scalar data without converting intermediate geometric
primitives. By assigning appropriate colors and opac�
ities to the scalar data� one can render objects semi�
transparently to expand the amount of 3D information
available at a �xed position. Volume�rendered images
can also be superimposed upon surface�oriented icons
or textures� thus allowing for simultaneous scalar and
vector �eld composite visualizations.
A wide range of volume rendering techniques have
been applied to the problem of �ow visualization. Ma
and Smith �3� introduced a virtual smoke technique
to enhance visualization of gaseous �uid �ows. This
technique allows users to interactively insert a seed
into an interesting location and only the region imme�
diately surrounding the seed is rendered. Max et al.
�5� introduced 3D textures advected by wind �ow upon
volume rendered climate images to visualize both the
scalar and vector �elds of the images. Craw�s and
Max �7� make use of textured splats which utilize vol�
ume splatting and 3D texture mapping techniques to
reveal scalar and vector information simultaneously.
Additionally� Max et al. �8� developed the concept of
�ow volumes� volumetric equivalents of stream lines�
to represent additional information about the vector
�eld.
We are motivated to develop a more e�cient way
to visualize scalar �elds by our attempts to visualize
simulation data from a model of electrical wave prop�
agation within the complex geometry of the heart and
in large scale models of unsteady compressible �uid
�ow �9� 10�. Because of the regular structures used
to characterize the simulation data� we can use direct
volume�rendering techniques to visualize the states at
each time step. We characterize states within the
model by assigning di�erent colors and opacities. By
animating the volume�rendered images at each time

step� we can e�ectively investigate the propagation of
waves throughout the volume.
Direct volume�rendering methods employing ray
casting algorithms have become the standard meth�
ods to visualize 3D scalar data. To characterize the
dynamic behavior of the �ow �eld� one generates a se�
ries of volume rendered images at di�erent time steps
and then records� stores� and animates the sequence.
Because direct volume rendering is very time consum�
ing to realize such animations for models of any sig�
ni�cant size �i.e. realistic problems in science and en�
gineering�� standard volume�rendering techniques are
prohibitive for animating hundreds of time steps inter�
actively. Moreover� the disk space required for storing
hundreds or thousands of sets of volumetric simulation
data could be overwhelming.
The main contribution of this paper is the devel�
opment of an algorithm which signi�cantly reduces
the time to create volume�rendered �ow animations
of scalar �elds. Furthermore� our algorithm reduces
the amount of disk space needed for storing volume
data. We achieve these reductions by implementing
a di�erential volume�rendering method. The method
utilizes data coherency between consecutive time steps
of simulation data to accelerate the volume animation
and to compress the volume data. The method is in�
dependent of speci�c volume�rendering techniques and
can be adapted into a variety of ray casting paradigms
which can be used to further accelerate the visualiza�
tion process �11� 12� 13�.
Di�erential Volume Rendering
From preliminary studies of our wave propagation
simulations� we noticed that the only elements which
changed values between consecutive time steps� when
the time steps were small� were the activated cells and
their neighbors. We hypothesized that only a frac�
tion of elements in the volume change from any given
time step to the next in simulations of physical �ow
phenomena. In addition� when a sequence of prop�
agating images is animated� the viewing parameters
usually don�t change. In our ray casting method� we
are able to cast rays along a path corresponding only
to changed data elements. Therefore� the pixels in
the new image keep the same colors that they had
previously unless they correspond to changed data
elements. Retaining the color values from the non�
changing pixels results in a signi�cant time savings.
The di�erential volume�rendering algorithm thus ex�
ploits the temporal coherence between sets of volume
data from di�erent time steps in order to speed up
volume animation of the 3D �ow.
Simulations
Data
Difference
extractor
Differential File
Figure 1� Visualization Pipeline� Static Phase
The di�erential volume rendering method separates
the data generation and data visualization processes.
Scientists perform simulations to obtain sequences of
data at di�erent time steps. The di�erential volume�
rendering algorithm extracts the di�erential informa�
tion which contains the di�erences between the data
�les at each consecutive time step. According to the
speci�ed viewing direction� the pixel positions where
new rays need to be cast can be computed from the dif�
ferential information and then the ray casting process
is invoked to produce the updated image. Because the
variation between consecutive time steps is small� the
di�erential information �le� which replaces the whole
sequence of volume data� can yield tremendous savings
in terms of disk space.
Visualization Pipeline
The visualization pipeline of the di�erential
volume�rendering method can be divided into two
phases� static and dynamic.
First� data is generated from a simulation which
might typically consist of hundreds or thousands of
time steps worth of information. Second� the di�er�
ence extractor is invoked to compare the simulation
data of consecutive time steps to obtain the positions
of changed data elements between simulation steps.
The positions of those changed elements and their cor�
responding time step values are output into a single
di�erential �le. Because the di�erential �le contains
the state histories of all the data elements through the
whole course of the simulation� the only information
needed for the rendering process� the volume data at
each time step can be then discarded� yielding a con�
siderable savings in terms of disk space. These oper�
ations are classi�ed as the static phase because they
need to be performed only once for a simulation. Fig�
ure 1 illustrates the operations in the static phase.

Differential File
Ray Caster Pixel Calculator
P I X E L S
Figure 2� Visualization Pipeline� Dynamic Phase
The di�erential information obtained over the dura�
tion of the simulations contains the 3D positions and
values of changed data elements. Those positions are
independent of the viewing direction of the rendering.
By pre�processing the simulation data and producing
the di�erential �le in advance� we can avoid the delay
of calculating the di�erential information while per�
forming ray casting.
At each time step� the positions of changed ele�
ments are extracted from the di�erential �le and the
pixels where new rays need to be cast can be computed
according to the viewing direction and the sampling
method. The resultant pixel positions are placed into
a ray casting list to which the ray casting process refers
before �ring new rays to produce the updated image.
We classi�ed these operations as the dynamic phase
because the pixels corresponding to those changed el�
ements are dependent on the viewing direction. The
pixel positions remain undetermined until the user
speci�es the viewing parameters. The operations in
the dynamic phase of the pipeline are illustrated in
Figure 2 and are outlined algorithmically below.
for �each time step t�
�
for �each changed element�x�y�z��
�
calculate the corresponding pixel �u�v��
update volume�x�y�z��
store �u�v� into the ray casting list�
�
for �each pixel �u�v� in the ray casting list�
�
cast a ray from �u�v� into the volume�
update the image value�
�
�
display the image�
Pixel Positions Calculation
There are many interpolation and ray sampling
methods which can be used in the ray casting algo�
rithm. For interactive rendering rates� but coarse im�
age quality� we can use a discrete ray which avoids the
use of interpolation. To obtain higher quality images�
one can increase the sampling rate along the ray and
use� for example� a trilinear interpolation scheme.
Continuous Rays and Trilinear Interpolation�
When using continuous ray sampling along with a tri�
linear interpolation scheme� we compute the value of
any point in the volume by interpolating the values of
the data element�s encompassing eight vertices. If any
of these eight data elements changes its value� the in�
terpolated value of the point inside that cube needs to
be re�computed. We de�ne the interpolation space of
a data element as the space where the values of all the
points located inside are in�uenced by that data ele�
ment while the interpolation is performed. In the case
of a 3D regularly structured grid� a vertex is shared
by its eight adjacent cubes. Therefore� for any data
element� its interpolation space is the volume of its
eight adjacent cubes.
To locate those pixels which will cast a ray through
a data element�s interpolation space� we project the
eight vertices of that cubic volume back to the image
plane according to the viewing parameters �parallel
or perspective�. The pixels bound by the projected
region then need to be updated.
Discrete Rays and Zero�Order Interpolation�
Discrete rays can be obtained by extending the dig�
ital di�erential analyzer �DDA� scan converting line
algorithm �14� into three dimensions. In a zero�order
interpolation scheme� at each forwarding step only the
nearest voxel is sampled. No actual interpolation op�
eration is performed. Suppose the 3D line equations
are y � m1 � x � b and z � m2 � x � c� and both m1
and m2 are less than 1 and greater than �1. Accord�
ing to the discrete ray algorithm� the ray in position
x would be increased by 1 at each forwarding step.
After i times of forwarding� the ray�s discrete position
is �i� Round�m1 � i � b�� Round�m2 � i � c��� and the
voxel located at that position is sampled.
To calculate pixels corresponding to the changed
voxel element� one projects the voxel position back to
the image plane according to the viewing direction.
Most of the time the projected position won�t be lo�
cated exactly at the image plane�s grid point. The four
surrounding pixels of that projected point thus need

to be selected to cast new rays. We cast rays from the
four surrounding pixels instead of choosing only the
nearest pixel to assure that the changed voxel will be
hit and sampled.
Template Based Rays� When parallel projection
is used� all rays �red from the image plane have the
same slope and thus the same incremental form of
forwarding path. Therefore� one can calculate the
incremental form once and store it as a template
�12� 4�. During the ray casting procedure� instead
of computing the ray�s new position at each time
step� which increases computational complexity� we
can use ray templates. Suppose we have the forward�
ing templates in u� v and w directions which have
the forms of ray template�i�.u� ray template�i�.v� and
ray template�i�.w. The ray template�i�.u stores� for in�
stance� the distance the ray should march at the next
step in u direction after the ray has taken i steps of
forwarding. Suppose the current position of a ray is
�u� v� w�� then the next position �u�� v�� w�� of the ray
is calculated as�
u� � u � ray�template�i�.u�
v� � v � ray�template�i�.v�
w� � w � ray�template�i�.w�
To guarantee a complete and uniform tessellation of
the volume by 26�connected rays� sampling� we cast
rays from the base�plane� which is parallel to one of
the volume faces. After having obtained the projected
image on the base�plane� one performs a 2D mapping
from the base�plane to the image plane to obtain the
�nal image.
The templates described above give us the informa�
tion about how far a ray will forward at each step in
each axis direction. From those templates� one com�
putes the displacement a ray has forwarded in each
direction after a particular time step. Inversely� we
can also obtain the number of forwarding steps corre�
sponding to a particular displacement. This informa�
tion gives us two more templates in each axis direc�
tion which can be described as two functions� displace�
ment to step�� and step to displacement��. Given
a displacement from a volume point to the base
plane� the displacement to step�� function will re�
turn the steps taken by a ray to reach that point.
Given a forwarding step number for a ray� the
step to displacement�� will return the displacement
from the base plane.
With these extra templates� the pixel position cal�
culation becomes simple. Given a point in volume
space� we can �nd the distance from that point to the
base plane. By using the displacement to step tem�
plate� we are able to calculate the steps taken by a
ray from the base plane to that point. Thus we can
look up the step to displacement templates to get the
displacement in each direction� and the starting po�
sition of the ray can be calculated. By adopting the
template�based ray casting method and utilizing pre�
computed displacement and step templates� we can
accurately and e�ciently calculate the pixel positions.
Acceleration of Ray Casting
Coordinate Bu�er� Yagel and Shi �11� proposed a
coordinate bu�er to store the �rst� and last�hit voxel
positions for every pixel on the image plane allowing
one to rapidly skip empty space for subsequent render�
ings. The di�erential volume rendering algorithm fur�
ther utilizes this idea for volume animations to speed
up the ray casting process.
If the viewing direction is �xed during the render�
ing of a sequence of volume data� each ray from the
image plane follows exactly the same forwarding path
during the whole course of renderings with di�erent
sets of volume data. If only a small fraction of data
elements change between consecutive simulation steps�
then most of the information stored in the coordinate
bu�er from the previous rendering can be retained.
Initially� the sampling range for each pixel consti�
tutes the full depth of the volume. After the �rst
rendering� the e�ective sampling range for each pixel
can be obtained and stored. At each subsequent time
step� when the di�erential volume rendering algorithm
extracts a changed data element from the di�erential
�le to compute the corresponding pixels� the 3D posi�
tion of that data element can be compared with those
computed pixels� e�ective sampling ranges stored in
the coordinate bu�er to decide if those ranges� i.e.� the
�rst� and last�hit voxel positions� need to be changed.
If the changed element is located outside the speci�ed
ranges� then the stored information needs updating.
The new sampling ranges can then be used to skip
empty space while performing the ray castings and
thus the rendering can be further accelerated.
Sample Caching� During the ray casting� the local
lighting calculation� which includes the trilinear in�
terpolation and shading computation� constitutes the
most expensive part of the process. Ma et al. �13�
proposed a sample caching technique which stores the
interpolated data value and local shading information
at each sample point along a ray. This allows users
to interactively change mapping parameters� such as
color and opacity� so that one only need to composite
the cached information to update the image.
To apply the sample caching technique� initially the
sampled value and lighting information at each sample

point of a ray is saved. For a changed data element
in the subsequent time step� only the sample points
inside the surrounding regions of that new data ele�
ment need to be resampled. The new sampled val�
ues are then used to update the sample cache. The
correct position to insert or replace the new sampled
result can be found from the corresponding pixel and
the distance between the new data element and the
image plane. After resampling the changed data el�
ements and updating the sample cache� one needs to
composite the new sample cache of the changed pix�
els to update the image. By combining this technique
with our di�erential volume rendering algorithm� we
can reduce the complexity of the sampling process and
thus further accelerate the ray casting process.
Results and Discussion
We have implemented our di�erential volume ren�
dering on two sets of simulation data� one from a
biomedical application and the other from a computa�
tional �uid dynamics application. All the comparisons
below consist of the performances of our ray casting
software with and without the di�erential rendering
capability. The performance measurements were eval�
uated on a single 100 MHz MIPS R4000 processor.
Note that the focus should be on the relative perfor�
mances of the ray casting algorithms with and with�
out di�erential capability instead of on any one tech�
nique�s e�ciency.
In the electrical wave propagation simulation� we
attempted to simulate the electrical impulse conduc�
tion in the heart using an anatomically accurate cel�
lular automation model. Simulation data consisted of
the state histories of all the elements in the model over
the duration of the simulation. Scientists were inter�
ested in following the activation wavefront and study�
ing phenomena that facilitate� promote and�or termi�
nate abnormal propagation. Appropriate colors and
opacities were assigned to the di�erent states when
volume rendering was performed. Data is computed
at each time step within 128 � 128 � 128 cubic ele�
ments. To test the di�erential rendering algorithms�
we computed data at 100 time steps. For display we
used a 256�256 image plane. Figure 3 depicts the vol�
ume rendered images of the propagation of electrical
activity within the heart.
In the computational �uid dynamics simulation
�10�� numerical results were computed by software
which uses the MacCormack method to solve the
three dimensional unsteady compressible Navier�
Stokes equations. The results simulated a laminar
�ow entering a rectangular region. The region had
Time Changed Regular Ray Di�erential
Step Elements Casting Ray Casting
0 100� 100� 100�
20 3.505� 100� 3.33�
40 4.531� 100� 4.93�
60 1.369� 100� 1.58�
80 0.379� 100� 0.36�
Table 1� Percentage of rays being cast at selected
time steps for both regular and di�erential ray casting
methods� electrical wave propagation simulation.
a small inlet at one end and a fully wide open outlet
at the other end. Data is computed at 64 � 64 � 64
regular grid points and displayed in a 128 � 128 image
plane. Figure 4 shows the volume rendered images of
the laminar �ow propagation.
On the analysis of our simulation data� we noticed
that a maximum of 4.77� of the elements in the elec�
trical wave propagation simulation� and a maximum of
2.57� in the laminar �ow simulation� changed states
between time steps. Therefore� we could exploit the
data coherency.
The disk space used by the 100 time steps of
128 � 128 � 128 volume data in the electrical wave
propagation simulation was 2.10 M Bytes � 100 �
210 MB. Using the di�erential volume rendering algo�
rithm� which only requires 2.08 MB for a di�erential
�le and 2.1 MB for the �rst time step of volume� we
were able to save more than 95� in storage costs. In
the laminar �ow simulation� the disk space require�
ment was 4.09 MB for a di�erential �le and 0.262 MB
for the �rst 64�64�64 volume. The regular rendering
algorithm for 130 time steps needed 0.262 MB � 130
� 34.06 MB. This amounts to a savings of 88�.
Table 1 lists the average percentages of rays being
cast both with and without use of the di�erential ca�
pability at di�erent time steps of electrical wave prop�
agation data. The algorithm without the di�erential
capability always shot 100� of rays at any time step.
That is� the alogirithm requires every pixel to �re a
ray to sample the volume data. In the di�erential ray
casting method� after the �rst rendering which needs
to cast 100� of rays to obtain the initial image� the
subsequent 99 renderings cast rays only when neces�
sary. This signi�cantly reduced the number of rays
cast. Table 2 shows the results for rendering the lam�
inar �ow propagation data.
Table 3 and Table 4 list the average rendering time
required for both simulations for a single image at se�

Time Changed Regular Ray Di�erential
Step Elements Casting Casting
0 100� 100� 100�
20 1.82� 100� 3.48�
40 2.45� 100� 4.65�
60 1.46� 100� 2.79�
80 0.20� 100� 1.44�
100 0.04� 100� 0.32�
120 0.005� 100� 0.15�
Table 2� Percentage of rays being cast at selected
time steps for both regular and di�erential ray casting
methods� laminar �ow simulation.
Time Changed Regular Ray Di�. Ray Casting
Step Elements Casting Time Pct.
0 100� 13.399 13.399 100�
10 1.148� 13.398 0.446 3.47�
20 3.505� 13.389 0.733 5.47�
30 4.435� 13.347 0.880 6.59�
40 4.531� 13.315 0.924 6.94�
50 3.619� 13.377 0.807 6.03�
60 1.369� 13.390 0.422 3.15�
70 0.607� 13.401 0.291 2.17�
80 0.379� 13.397 0.265 1.97�
90 0.089� 13.398 0.234 1.75�
Table 3� Rendering time �in seconds� at selected time
steps using both the regular and di�erential ray cast�
ing algorithms� electrical propagation simulation data
lected time steps using our ray casting software both
with and without adopting the di�erential capability.
The rendering time for the �rst image �time step 0�
in the di�erential volume rendering was the same as
that in the regular ray casting algorithm. However� for
subsequent renderings� there was a signi�cant reduc�
tion in rendering time. For the electrical wave prop�
agation data� the average time to render 100 images
without adopting the di�erential method was approx�
imately 13�39 � 100 � 1339 seconds. Di�erential vol�
ume rendering achieved the same amount of rendering
and same image quality in only 13.5 � 52 � 65.5 sec�
onds. For the laminar �ow simulation� the di�erential
volume rendering method took 17.98 seconds to render
130 time steps. The rendering algotithm without the
di�erential algorithm needed approximately 2�45�130
Time Changed Regular Ray Di�. Ray Casting
Step Elements Casting Time Pct.
0 100� 2.245 2.245 100�
20 1.82� 2.245 0.23 10.24�
40 2.45� 2.248 0.28 16.27�
60 1.46� 2.239 0.19 8.48�
80 0.20� 2.241 0.08 3.56�
100 0.04� 2.241 0.04 1.78�
120 0.005� 2.246 0.02 0.89�
Table 4� Rendering time�in seconds� at selected time
steps using both the regular and the di�erential ray
casting algorithms� laminar �ow simulation data
Time Changed Di�erential Ray Casting
Step Elements Pixel Ray Total
Calculation Casting
1 0� 0 2.372 2.372
3 10� 0.226 0.173 0.439
5 20� 0.516 0.282 0.798
7 30� 0.765 0.413 1.178
9 40� 1.019 0.529 1.548
11 50� 1.265 0.641 1.906
13 60� 1.515 0.771 2.286
15 70� 1.766 0.884 2.650
17 80� 2.017 1.013 3.030
19 90� 2.266 1.116 3.382
Table 5� Rendering time �in seconds� for di�erent
amount of changed elements in a 64 � 64 � 64 volume
� 318.5 seconds. Our algorithm for rendering both
sets of data achieved a savings of more than 90�.
To understand the limitations and robustness of
the di�erential volume rendering algorithm� we used
a 64 � 64 � 64 volume� 128 � 128 image plane and in�
creased the number of changed elements at each time
step. Table 5 lists the rendering times of our algo�
rithm for di�erent percentages of changed data ele�
ments. The experiment was designed such that the
�ow started propagating from one corner of the vol�
ume and eventually occupied 95� of the whole volume
in 20 time steps. The number of changed elements in�
creased as the propagation proceeded.
Except for the �rst rendering time� which was the
same as that in the regular ray casting� the di�eren�
tial ray casting time at each time step was shorter
than the regular ray casting time. Although our ray

casting time remained smaller than the regular ray
casting time� the pixel calculation time increased with
the increasing number of changed elements. When
the number of changed elements increased� the time of
pixel calculation increased to the point of diminishing
returns. From our experiments� when the percentage
of changed elements was over 50�� the performance
of the di�erential volume rendering method became
worse than the regular ray casting method.
Although our algorithm has limitations when the
number of changed elements exceeds 50�� for most
�ow visualization applications the number of changed
elements during the propagation at each time step
constitutes only a small fraction of the whole volume.
Furthermore� these changed elements tend to cluster
together. Therefore� di�erential volume rendering rep�
resents an attractive technique for scalar �eld �ow vi�
sualization.
Summary
We have presented a di�erential volume rendering
algorithm which exploits data coherency between con�
secutive time steps of data to achieve fast volume ani�
mation. This method can potentially save tremendous
amounts of disk space and CPU time over existing
methods. The algorithm begins by preprocessing the
simulation data over time and extracting the di�er�
ential information between sequential time steps. At
each time step� the pixel locations from which new rays
need to be cast can be calculated and the ray casting
process is invoked to update the image. Our algorithm
has been successfully applied to both biomedical and
computation �uid dynamics applications. We are cur�
rently working on a parallel version of the algorithm.
Acknowledgments
This work was supported in part by the Whitaker
Foundation. The authors would like to thank P.
Gharpure for the heart wave propagation data� and
K. Ma for his CFD simulation software. We would
also like to thank Professors R. Yagel� J. Painter and
K. Coles for their helpful comments and suggestions.
Furthermore� we appreciate access to facilities which
are part of the NSF STC for Computer Graphics and
Scienti�c Visualization.
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Fast Surface Rendering from Raster Data by Voxel Traversal Using
Chessboard Distance
Miloˇs ˇSrámek
Slovak Academy of Sciences, Bratislava, Slovak Republic
miloss@umhp.savba.sk
Abstract
The increasing distinguishingcapability of tomographic
and other 3D scanners as well as the new voxelization algorithms
place new demands on visualization techniques
aimed at interactivity and rendition quality. Among others,
triangulation on a subvoxel level based on the marching
cube algorithm has gained popularity in recent years.
However, without graphics hardware support, rendering
many small triangles could be awkward.
We present a surface rendering approach based on ray
tracing of segmented volumetric data. We show that if a
proper interpolation scheme and voxel traversal algorithm
are used, high quality images can be obtained within an
acceptable time and without hardware support.
1 Introduction
During the last decades we have encountered an immense
boom in the development of computing machinery,
which has enabled us, among others, to generate and manipulate
large matrices of three and even higher dimensional
data. The data results either from of some simulation processes
(e.g. gas flow simulation) or is the product of various
types of 3D scanners, which have found an important
place, e.g. in medical diagnostics (CT, MR, PET scanners).
Caused by the increased resolving ability of the scanners,
the quality of the final 3D reconstruction of the data comes
into prominence. An inevitable condition for the visualization
of details, comparable with the voxel size is the
application of an algorithm that enables us to define the
surface of the scanned object with subvoxel precision.
This development has resulted in a large number of visualization
techniques, among which triangulation on a voxel
level has gained popularity [11]. Its basic idea is to approximate
the object surface within a space, defined by eight
neighboring data samples (cell) by up to four triangles, resulting
in a surface model that can be rendered by standard
tools. The continuous nature of this model enables us to
render the object in various scales without danger of blocky
artifacts caused by the limited sampling frequency of the
scanner. The large number of triangles, typically 10 5 –10 6 ,
causes no problems when rendering through some hardware
renderer. On the other hand, in spite of the rapidly
decreasing hardware prices, many laboratories still exist
with no access to such engines. In this case, rendering a
big amount of primitives can become very awkward.
Another drawback of the surface model approach is the
separation of the surface from the volume data (original
samples) processing. Many of applications exist (e.g. in
medical diagnostics), where just the combined data representation
is important [6] and can grant the operator deeper
insight into the problem.
Direct visualization techniques omitting the intermediate
surface model represent an alternative, which can overcome
the two aforementioned drawbacks of the surface
model approach. Traditionally, they split into two categories:
volume rendering and surface rendering. The first
one represents a class of methods, where all scene voxels
contribute to the resulting image by means of color compositing.
Since no binary decisions between object and
background are done, these approaches are claimed to be
suitable for rendering weak surfaces and thin structures too.
However, interpretation of such images can often lead to
difficulties.
The surface rendering approach lies half way between
the surface model and volume rendering techniques. Its
prerequisite is an object mask, obtained from the original
gray level scene by some kind of segmentation. Only
the boundary mask voxels contribute to the final image by
means of a color value derived from the local properties and
a chosen shading model. Ray tracing can be used to solve
the visibility problem and to enhance the 3D perception by
simulating highlights, shades, reflection and refraction.
In this paper, we would like to present a direct approach
to the visualization of volumetric data based on ray tracing,
with no intermediate surface model. We shall show,
that if a suitable interpolation scheme and a voxel traversal
algorithm are used, high quality renditions can be achieved
with the possibility to magnify details within a reasonable
time, and without hardware support.
1.1 Ray tracing
Ray tracing has been established in the last decades as
a tool for high quality rendering. Its main drawback is
the high computational cost, which can be, among others,
overcome by space subdivision techniques. The scene is
subdivided, either uniformly or hierarchically, into voxels,
containing only fractions of the objects. To reduce the
number of necessary ray-object intersection tests, only the
voxels pierced by a given ray are inspected. Various voxel
traversal algorithms solving this task were proposed [4,
1, 3, 8]. An experiment has shown [1], that the optimal
subdivision rate is relatively low, resulting only in a few
thousands of voxels.
Scanned or simulated volumetric data sets are usually
represented in a similar form, as a 3D raster. The difference
is in the typical size (10 6 –10 7 voxels) and in the fact,
that we have only one kind of object, the voxel itself. The

value assigned to the voxel represents a density of some
measured or simulated primary parametric field. Ray tracing
(or its simplified form ray casting) was used for the
implementation of volume [9] and surface rendering of either
scanned [13] or voxelized data [19, 18]. In the surface
rendering case it is necessary to introduce some interpolation
scheme for the precise detection of the ray-surface
intersection point.
In order to obtain high quality renditions in an acceptable
time, the voxel traversal algorithm should have the
following properties:
1. it should enable rays with arbitrary start point and
direction (perspective views and recursive ray tracing),
2. it should enable ray-surface intersection point with
subvoxel precision (normal computation for shading
and ray refraction/reflection), and
3. it should exploit various kind of coherency for speed
up [12].
1.2 Previous work
The voxel traversal algorithms used in volume visualization
can be categorized as
1. continuous ray generators (floating point 3D DDA),
which define the ray as a sequence of usually equidistant
points on a line. Due to its algorithmic simplicity,
it was used by more authors [9, 14]. However, it is
not possible to apply any of the ray connectivity criteria
[8] to the sequence of voxels thus defined, and
the ray can skip some important voxels. This is usually
overcome by increasing the point density, which
degrades the algorithm performance.
2. discrete ray generators, which generate the sequence
of voxels pierced by a ray directly.
In the latter category we can further distinguish integer
based algorithms, which assume that the ray has start and
end points with integer coordinates [8] and algorithms enabling
arbitrary start point and direction [1, 3].
Not all voxels contribute to the rendered image with the
same weight. Only some of them belong to the interresting
objects, while the others can be traversed rapidly or even totally
skipped. This capability is called space-leaping [18].
The idea to switch between the lower precision but faster
26-connected line generator in the background region and
the precise 6-connected one in the object vicinity was introduced
in [19]. Another popular approach for fast empty
space leaping is based on hierarchical subdivision [10, 15].
Digital distance transforms were used in [21] to speed up
the floating point 3D DDA algorithm as well as in [7]
for 26-connected ray templates traversal. Coherency between
consecutive images was used for minimizing the
background leaping time in [5, 20].
2 Surface rendering of raster data
It is necessary to define some kind of interpolation surface
that is at least C 0 continuous across the voxel boundaries,
in order to detect the object surface with subvoxel
precision. The exact surface point should be then searched
for as an intersection of the ray with that surface. This task
splits into two parts:
1. finding the voxel, where the ray-surface intersection
can be found, i.e. the discrete scene traversal, and
2. exact computation of the intersection position within
the voxel.
First, we shall deal with the suitable surface interpolation
and the exact ray-surface intersection computation and then
with the scene traversal.
2.1 Definitions
Let the 3D image P be a set of K � L � M values,
representing samples of some measured property in the
vertices of a regular unit grid:
P � fpijk 2 R : 0 � i � K� 0 � j � L� 0 � k � M g�
where i, j and k are integers.
Let us segment P into n subsets u l, such that
�
l
u l � P �
�
l
u l � ��
Let u l be the l th object and l its identifier. Let I �
f0� 1� � � �n � 1g be a set of object identifiers and let l 2 I.
The binary image B is the set
B � fbijk 2 I : 0 � i � K� 0 � j � L� 0 � k � M g�
We can assume without any loss of generality, that border
values of B, i.e. those, for which at least one coordinate
i, j or k is either equal 0 or K � 1, L � 1 and M � 1
respectively, belong to the 0 th object.
Let voxel be a tuple Vijk � �vijk� hijk�, where vijk ��
i� i � 1�� � j� j � 1�� � k� k � 1� is voxel volume and
hijk 2 f0� 1g is its value. The point �i� j� k� is a voxel
vertex and similarly the point �i � 0�5� j � 0�5� k � 0�5�
is the voxel center. The voxel value hijk will have the
following meaning:
hijk � 0 means certainty that none of the object surfaces
pass through this voxel, and
hijk � 1 means, that we are not sure about this.
The voxel value depends on the binary image B and on
the choice of the surface approximation. We denote voxels
with values equal to 0 (resp. 1) 0-voxels (resp. 1-voxels).
We denote the set of points D � f�i� j� k� : 0 � i �
p� 0 � j � q� 0 � k � rg voxel scene domain and the set
S�p� q� r� � fVijk : �i� j� k� 2 Dg voxel scene. The voxel
scene S C �M� N� O� is called center representation of the
image P, if the sample pijk is assigned to the center of the
voxel Vijk. Analogously, it is the vertex representation S V ,
if the sample is assigned to the voxel vertex. Apparently
the size of the scene S V domain along each axis is 1 voxel
smaller than that of S C .
Let the voxels whose all corresponding bijk � 0 be
called background voxels (in both representations) and the
rest foreground voxels.

2.2 Surface point definition by ray tracing
Let the surface S ijk
of the object u l
l be defined within
the volume of Vijk by means of an interpolating function
Fl and the threshold value tl: S ijk
l
� fF l�x� y� z� � ijk� � t l : �x� y� z� 2 v ijkg�
The � ijk represent values p ijk or b ijk in some neighborhood
of a voxel V ijk. We speak of gray level interpolating
function if p ijk values are used and we suppose, that the
object l was segmented by thresholding with a threshold t l.
In the case of some other segmentation method we use the
values b ijk in such a way that to samples b ijk � l we assign
value 1 and to samples with b ijk 6� l we assign value 0. In
this case we speak about binary interpolating function and
we use value t l � 0�5 for all objects as the threshold.
We get the exact position of the ray-object surface intersection
by solving the system of equations:
X � A � t�u
Fl�x� y� z� �ijk� � tl� (1)
where A is the eye position and �u is the ray direction vector.
It may happen that a voxel is intersected by the surfaces
of more than one object. In this case, it is necessary to
compute the intersections with all of them and to take the
nearest to the eye point into consideration.
In the following we shall describe some possibilities for
the definition of the function Fl in dependency of �ijk.
2.3 0-order interpolation in the center representation
We shall use this kind of interpolation for faster, less
precise rendering of the 3D image. Let us define Fl within
the voxel volume as
�
1 if bijk � l
Fl�x� y� z� �ijk� �
0 if bijk 6� l
which means that the binary interpolating function is constant
within the voxel volume. The voxel value hijk is in
this case initialized by min�1� bijk� (Figure 1a). To detect
the surface by ray tracing means to find the first voxel of
the ray with a nonzero value without further intersection
computation.
2.4 Trilinear interpolation in the center representation
The trilinear interpolation function is defined as
Fl�x� y� z� �ijk� �
txyzxyz � txyxy � txzxz � tyzyz �
txx � tyy � tzz � t�
The coefficients t � � �txyz depend on the values
v0� v1� � � � � v7, positioned in the voxel Vijk vertices. Since
in the center representation we assume that the samples
of the scanned field are situated in voxel centers, the values
v0� v1� � � � � v7 must be computed as a mean over those
voxels sharing the given vertex.
This approach gives smooth surfaces due to the mean
computation. It has been observed that such surfaces can
be shifted into the object, which may completely “smooth
out” object details comparable with the voxel size.
Another consequence is that the surface can be detected
not only within object voxels but also within background
voxels in their vicinity. Therefore, in this case, we shall
assign the value 0 only to those background voxels of the
scene S C , which are not 26-adjacent to an object voxel. To
all other voxels we shall assign 1 (Figure 1b).
We shall prefer this kind of interpolation, due to its
smoothing ability, for objects segmented by some other
method as the thresholding.
2.5 Trilinear interpolation in the vertex representation
The interpolating function is in this case defined
as in the previous section, with the exception that
the values v0� v1� � � �� v7 are directly given by samples
s ijk � � �s i�1j�1k�1 positioned in the voxel vertices. We
assign value 0 to background voxels and 1 to foreground
voxels. This kind of interpolation is suitable for scenes
with details on a voxel level.
Of course, higher order interpolation schemes considering
a larger environment of the voxel, can be used, too, at
the cost of a higher computational complexity.
3 Discrete ray traversal
In the previous sections we came to the result that, because
of the segmentation and the choice of the interpolation
method, we can partition the scene into 0-voxels
with no object surface and 1-voxels, where a ray-surface
intersection can be found. Let us define a discrete ray as
an ordered sequence of voxels Vi of the scene S C (S V ),
pierced by the ray with the equation X � A � t�r.
The goal of the voxel scene traversal algorithm is to
find the ray’s first 1-voxel. We shall proceed with the
following voxel of the ray only in such a case, that no
ray-surface intersection is found within the volume of the
previous one. If we add to the requirement of correctness of
the surface point detection the requirement of high speed
too, the discrete ray traversal algorithm should have the
following properties:
1. it should pass the region of background voxels usually
surroundingthe object as fast as possible, i.e. it should
utilize the object scene coherency and
2. it must not miss any object voxel, i.e. the voxels of the
ray should fulfill the condition of 6-adjacency [19] at
least in the vicinity of the object.
In the algorithm design, we start from the assumption
that the chessboard distance(CD) [2] to the nearest object
voxels is assigned to each 0-voxel Vijk. Thus a cubic macro
region is created:
O n �i� j� k� � fhpqr � 0 : i � n � p � i � n�
j � n � q � j � n� k � n � r � k � ng
with its center in Vijk and with side size 2n � 1. Since
CD is independent of the projection parameters, we can
calculate it during a preprocessing phase. No extra space is
necessary for the storage of the distance information, if we
use the originally “empty” background voxels. In order to

-
-
-
-
sample b = 0
ijk
sample b = 1,2,3...
ijk
0-voxel
1-voxel
(a)
Figure 1: Scene initialization: (a) S C for 0 order interpolation,
(b) S C for trilinear interpolation (c) S V for trilinear
interpolation
(b)
(c)
t 3
t 2
t 1
t 0
0 1 2 3
Figure 2: Entry point coordinate thresholds
ray
distinguish between the object identifier and the distance,
a flag bit is reserved within each voxel descriptor.
We know that there are no object voxels within
O n �i� j� k�, so we can jump from V ijk directly to the first
ray’s voxel outside of O n �i� j� k�. The traversal speed up
is thus achieved by reducing the number of visited voxels.
Detailed description of the algorithm is in [16], therefore we
limit ourselves only to a brief description of its main loop,
i.e. to the calculation of steps between macro regions and
to the description of the traversal of a single macro region.
Since the algorithm is symmetric with respect to all axes,
we shall present only the relations for the x axis. Further,
we shall assume that the direction vector has only nonnegative
coordinates. Generalization to all possible directions
is done by the proper initialization of some variables.
3.1 Scene traversal
Let us imagine that we have reached the voxel Vijk with
the assigned CD � n at an entry point �p � �px� py� pz�,
whose coordinates are expressed with respect to �i� j� k�.
Let the entry (exit) face be that through which the ray
enters (exits) the voxel.
It is necessary to find the nearest intersection of the line
X � �p � s�r with the planes x � n, y � n and z � n in
order to find the first ray voxel outside of O n �i� j� k�. Let us
suppose that the entry face is of type X, i.e. perpendicular
to the x axis. We can see in Figure 2 that for each n there
exists such a threshold value tx y�n� of the input coordinate
py that (2D case)
1. if py � tx y�n�, then the exit face is also of X type, and
2. if py � tx y�n�, then it is of Y type,
which in the 3D case holds for pz, too. The upper index
denotes the type of the entry face. It can be shown that
t x
y�n� � �n � 1� rx � ry
�
The basic scheme of the algorithm is outlined in Figure 3,
and the scheme for computation of the exit face type in
Figure 4. The expression py � t z
ry
y�n� � pz enables us to
rz distinguish between exit types Y and Z when the type is
not X.
rx

while(in scene and object not found )
n = GetMacroRegionSize();
if(faceType == X)
macroRegionStep(x,y,z,n);
else if(faceType == Y)
macroRegionStep(y,z,x,n);
else if(faceType == Z)
macroRegionStep(z,x,y,n);
end while
Figure 3: CD algorithm: scene traversal
macroRegionStep(x,y,z,n)
f
if(py � t x
y�n� and pz � t x
z�n�)
else
NoChangeMacroRegionStep(x,y,z,n);
if(py � t z
else
ry y�n� � pz ) rz ChangeMacroRegionStep(x,y,z,n);
ChangeMacroRegionStep(x,z,y,n);
g
Figure 4: CD algorithm: macro region step
3.2 Macro region traversal
In this part we describe briefly the calculation of the
next voxel position and the coordinates of the new entry
point. Two kinds of steps across the macro region can be
distinguished as shown in Figure 4:
1. without face type change (NoChangeMacroRegion-
Step, Figure 5a), and
2. with face type change (ChangeMacroRegionStep,Figure
5b).
The first case is a step across a slab defined by two parallel
planes with distance n. If we denote dx y�n� � n � ry , then
rx for the next voxel holds:
i 0
j 0
k 0
� i � n
� j � int�d x
y�n��� p0
y � fract�dx y�n�� � k � int�d x
z �n��� p0
z � fract�dx z �n��
The second case is analogous; only the intersection with
the plane x � 0 must be calculated (Figure 5b):
py � ry
� px
rx
3.3 Further speed up
There are some possibilities for a further speed up of the
CD traversal algorithm:
1. both arrays t�n� and d�n� can be precomputed. However,
in this case the algorithm can be used only for a
parallel projection,
P y
p y
p’ y
p y
y
y
2
2
a
p x b
ray
p’ y
x
d
y
[2]
ray
P y
x
d
y
[2]
Figure 5: Macro region traversal: (a) entry and exit face of
the same type, (b) with different types
2. implementation in the fixed point arithmetics is possible,
and
3. the decision variable triple �px� py� pz� can be replaced
by
�p 0
x� p 0
y� p 0
px
z� � �
rx
� py
ry
� pz
�
rz
As a consequence we get rid of some multiplications.
4 Implementation and results
All algorithms were implemented in the C language on a
DECstation 5000/200 equipped with 48 MB of main memory.
Figure 7 shows the impact of the subvoxel precision surface
detection on the visual quality of the rendered image.
The 64x64x64 data were rendered to a 500x500 image with
either 0-order interpolation (a) or trilinear interpolation(b).
x
x

The scheme based on the vertex scene representation (Section
2.5) was chosen due to the importance of details.
The 500x500 image in Figure 8 was rendered by the
CD voxel traversal algorithm using trilinear interpolation
with recursion depth 4 in 128 seconds. The skull data was
obtained by a CT tomograph,the teapot and the implicit surface
were voxelized from their analytical description [17].
All objects were combined into a 200x200x110 scene.
To obtain information about the behavior of the CD
algorithm over scenes with various complexities, a computer
experiment was set up, based on rendering of scenes
with randomly positioned spheres of various size and number.
Its performance was compared with three similar algorithms
known from the literature, fulfilling the conditions
from the Section 1.1
The first one is Cleary’s and Wyvill’s algorithm [3] for
fast voxel traversal (FVT), generating a sequence of voxels
pierced by the ray in a uniformly subdivided scene. The
decision which voxel will be the next in the sequence is
controlled by three variables dx, dy and dz, recording the
total distances along the ray from some common point to
the last crossings with X, Y and Z type voxel face.
The second Ray Acceleration by Distance Coding
(RADC) algorithm [21] is a modification of the 3D-DDA
algorithm, which takes advantage of a 3D digital distance
to speed up the voxel traversal. Since this algorithm works
with various approximations of the Euclidean distance, we
included tests with the chessboard (Figure 6, RADCchess)
and chamfer (RADCchamf) distances. A sampling rate of
1.4 samples per volume distance has been chosen.
The third one, the Spackman’s and Willis’s SMART
(Spatial Measure for Accelerated Ray Tracing) navigation
oct-tree traversal [15] works on an oct-tree represented as
a breadth first list.
The notion of the spatial coherency leads to the following
idea: The “more” coherent scene contains few larger objects,
while the “less” coherent scene contains many small
objects. The scene with a single spherical object has the
highest spatial coherency. Since the sphere has the smallest
surface-to-volume ratio (SVR), we propose this ratio
to be a measure of the scene coherency for the purpose of
algorithm comparison.
The sequence of scene phantoms was generated in the
following manner. The scene built up of 128x128x128
voxels was subdivided into NxNxN subregions (N �
1� � � � � 10). Within each subregion, a voxelized sphere was
randomly placed (1–1024 spheres), such that total volume
of all spheres was identical for all N. Thus we obtained
scenes with equal number of object voxels but different
SVR. Chessboard and chamfer distance initialization took
from 18 to 21 seconds.
In the experiment, only parallel primary rays were traced
until the first object voxel was found, with no subsequent
shading. The results of the experiment are depicted in
Figure 6. The y axis values represent the pure traversal
time necessary for rendering of a 250x250 image. Values
on the left side of the graph correspond to the simple scenes
with low number of spheres, while those on the right side
belong to the more complex scenes.
We see that the proposed CD traversal algorithm outperforms
all three algorithms over the whole SVR range,
delivering the best results in comparison to FVT for the
less complex scenes. Experiments with tomographic data
traversal time [s]
22
20
18
16
14
12
10
8
6
4
FVT
SMART
RADCchess
RADCchamf
2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
surface/volume ratio
Figure 6: Comparison of various voxel traversal algorithms
gave results similar to those from the left side for most of
the scanned objects. We further see from Figure 6 that
the traversal time for the FVT algorithm is shorter for the
denser populated scenes. In that case the surface is encountered
earlier, which shortens the total distance traversed.
All other algorithms show opposite behavior. In this case
the shorter distance is surpassed by smaller macro regions
which results in shorter mean step.
It is apparent that beyond some scene complexity the
macro region based traversal algorithms are, due to their
larger one step cost, slower than the FVT algorithm. For
example, in semitransparent volume rendering, where the
traversal is not stopped at the object surface, but all or at
least a lot of voxels should be traversed, it proved to be
useful to switch between the CD traversal algorithm in the
background region and the FVT within the object.
5 Conclusion
We concentrated in the paper on the following topics:
1. we showed how, based on the segmentation and choice
of the interpolation function, the voxel values should
be initialized to enable efficient traversal of the scene,
2. we proposed the CD voxel traversal algorithm based
on cubic macro regions defined by the chessboard distance,
and
3. we compared it with other similar algorithms known
from the literature.
Some problems remain open, however, such as the utilization
of the ray-to-ray coherency in the CD traversal
algorithm, or the efficient computation of the ray-surface
CD

intersections. These will be the subject of our further research.
Acknowledgement
This project has been partially supported by the grant
No. 2/999003/92 from the Slovak Grant Agency and by the
grant P8189-MED from the FWF (Austria). The data were
provided by Dr. Serge Weis from the University of Munich,
Institute for Neuropathology (skull) and by University of
North Carolina (molecule). Special thanks are due the
Instituteof InformationProcessing of Austrian Academy of
Sciences for supporting my work and to colleagues at both
institutes in Bratislava and Vienna for helpful comments
and discussions.
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(a) (b)
Figure 7: Electrone density map of High Potential Iron Protein molecule : (a) 0-order interpolation, (b) trilinear interpolation
Figure 8: Scene with voxelized and scanned objects

Parallel Performance Measures for Volume Ray Casting
Abstract
We describe a technique for achieving fast volume
ray casting on parallel machines� using a load balanc�
ing scheme and an e�cient pipelined approach to com�
positing. We propose a new model for measuring the
amount of work one needs to perform in order to ren�
der a given volume� and use this model to obtain a
better load balancing scheme for distributed memory
machines. We also discuss in detail the design trade�
o�s of our technique. In order to validate our model
we have implemented it on the Intel iPSC�860 and the
Intel Paragon� and conducted a detailed performance
analysis.
1 Introduction
As researchers and engineers use volume render�
ing to study complex physical and abstract structures
they need a coherent� powerful� easy to use visualiza�
tion tool� that lets them interactively change all the
necessary parameters. Unfortunately� even with the
latest volume rendering acceleration techniques run�
ning on top�of�the�line workstations� it still takes a
few seconds to a few minutes to volume render images.
This is clearly far from interactive. With the advent
of parallel machines� scanners and instrumentation�
larger and larger datasets �typically from 32MB to
512MB� are being generated that would not even �t
in memory of a workstation class machine. Even if ren�
dering time is not a major concern� big datasets may
be expensive to hold in storage� and extremely slow to
transfer to a typical workstations over network links.
These problems lead to the question of whether the
visualization should be performed directly on the par�
allel machines that generate the simulation data or
sent over to a high performance graphics workstation
for post�processing. First� if the visualization software
was integrated in the simulation software� there would
be no need for extra storage and visualization could
be an active part of the simulation. Second� large par�
allel machines can render these datasets faster than
workstations can� possibly in real�time or at least giv�
ing the possibility of achieving interactive rates. Fi�
nally� if real integration between the simulation and
Cl�audio T. Silva and Arie E. Kaufman
Department of Computer Science
State University of New York at Stony Brook
Stony Brook� NY 11794�4400
the visualization tool is possible� one could interac�
tively �steer� the simulation� and possibly terminate
simulations that are wrong or uninteresting at an ear�
lier stage instead of performing long and expensive
archiving operations for the generated datasets. In
this paper we focus on the architecture and perfor�
mance measures of visualization algorithms that are
running directly on the parallel machines.
Clearly� an algorithm that runs on a parallel ma�
chine has to be e�cient and should be able to make
good use of the computing power. A conserva�
tive tradeo� between scalability and actual process�
ing speed is very important. Also� the algorithm has
to be space e�cient� and for the case of a distributed
memory MIMD machine� memory duplication should
be avoided. In this paper we propose a space e��
cient� fast parallel algorithm that addresses these is�
sues. This algorithm will be the basis of a visualization
library in the molds just described using the VolVis
system �1� as its front end.
A large number of parallel algorithms for volume
rendering have been recently proposed. Schroeder and
Salem �13� have proposed a shear based technique for
the CM�2 that could render 128 3 volumes at multi�
ple frames a second� using a low quality �lter. The
main drawback of their technique is low image qual�
ity. Their algorithm had to redistribute and resam�
ple the dataset for each view change. Montani et al.
�10� developed a distributed memory ray tracer for the
nCUBE� that used a hybrid image�based load balanc�
ing and context sensitive volume distribution. An in�
teresting point of their algorithm is the use of clus�
ters to generate higher drawing rates at the expense
of data replication. However� their rendering times are
well over interactive times. Using a di�erent volume
distribution strategy but still a static data distribu�
tion� Ma et al. �9� have achieved better frame rates on
a CM�5. In their approach the dataset is distributed
in a K�d tree fashion and the compositing is done in a
tree structure. Others �6� 3� 11� have used similar load
balancing schemes using static data distribution� for
either image compositing or ray data�ow compositing.

Nieh and Levoy �12� have parallelized an e�cient vol�
ume ray caster �8� and achieved very impressive per�
formance on a shared memory DASH machine.
In this paper we concentrate on the parallelization
of a simple but fast method for ray casting� called
PARC �polygon assisted ray casting� �2�. Our parallel
implementation uses a static data decomposition and
an image compositing scheme. We have implementa�
tions that work on the Intel iPSC�860 and the Intel
Paragon. In Section 2 we explain the important issues
in designing and writing a parallel ray caster� followed
by Section 3� where we study a new method for mea�
suring the work done by a ray caster. In Section 4 we
describe our algorithm and its implementation.
2 Performance Considerations
In analyzing the performance of parallel algorithms�
there are many considerations related to the machine
limitations� like for instance� communication network
latency and throughput �11�. Latency can be mea�
sured as the time it takes a message to leave the
source processor and be received at the destination
end. Throughput is the amount of data that can be
sent on the connection per unit time. These num�
bers are particularly important for algorithms in dis�
tributed memory architectures. They can change the
behavior of a given algorithm enough to make it com�
pletely impractical.
Throughput is not a big issue for methods based
on volume ray casting that perform static data distri�
bution with ray data�ow as most of the communica�
tion is amortized over time �10� 6� 3�. On the other
hand� methods that perform compositing at the end
of rendering or that have communication scheduled
as an implicit synchronization phase have a higher
chance of experiencing throughput problems. The rea�
son for this is that communication is scheduled all at
the same time� usually exceeding the machines archi�
tectural limits. One should try to avoid synchronized
phases as much as possible.
Latency is always a major concern� any algorithm
that requires communication pays a price for using the
network. The start up time for message communica�
tion is usually long compared to CPU speeds. For
instance� in the iPSC�860 it takes at least 200�s to
complete a round trip message between two proces�
sors. Latency hiding is an important issue in most
algorithms� if an algorithm often blocks waiting for
data on other processors to continue its execution� it
is very likely this algorithm will perform badly. The
classic ways to hide latency is to use pipelining or pre�
fetching �5�.
Even though latency and throughput are very im�
portant issues in the design and implementation of a
parallel algorithm� the most important issue by far is
load balancing. No parallel algorithm can perform well
without a good load balancing scheme.
Again� it is extremely important that the algorithm
has as few inherently sequential parts as possible if at
all. Amadahl�s law �5� shows how speed up depends
on the parallelism available in your particular algo�
rithm and that any� however small� sequential part
will eventually limit the speed up of your algorithm.
Given all the constraints above� it is clear that to
obtain good load balancing one wants an algorithm
that�
� Needs low throughput and spreads communica�
tion well over the course of execution.
� Hides the latency� possibly by pipelining the oper�
ations and working on more than one image over
time.
� Never causes processors to idle and�or wait for
others without doing useful work.
A subtle point in our requirements is in the last
phrase� how do we classify useful work � We de�ne
useful work as the number of instructions Iopt executed
by the best sequential algorithm available to volume
render a dataset. Thus� when a given parallel im�
plementation uses a suboptimal algorithm� it ends up
using a much larger number of instructions than the�
oretically necessary as each processor executes more
instructions than I opt
P
�P denotes the number of pro�
cessors�. Clearly� one needs to compare with the best
sequential algorithm as this is the actual speed up the
user gets by using the parallel algorithm instead of the
sequential one.
The last point on useful work is usually neglected
in papers on parallel volume rendering and we be�
lieve this is a serious �aw in some previous approaches
to the problem. In particular� it is widely known
that given a transfer function and some segmentation
bounds� the amount of useful information in a volume
is only a fraction of its total size. Based on this fact�
we can claim that algorithms that use static data dis�
tribution based only on spatial considerations are pre�
senting �e�ciency� numbers that can be inaccurate�
maybe by a large margin.
To avoid the pitfalls of normal static data distri�
bution� we present in the next section a new way to
achieve realistic load balancing. Our load balancing
scheme� does not scale linearly as others claimed be�
fore� but achieves very fast rendering times while min�
imizing the �work� done by the processors.
3 Load Balancing
This section explains our new approach to load bal�
ancing� which is based on the PARC �polygon assisted
ray casting� algorithm �2�. The section presents a
short description of PARC and describes di�erent ap�
proaches to using it as a load balancing technique.
PARC can be characterized as a presence acceleration

technique �4�� like the octree decompositions of Levoy
�8�. Instead of stepping through the whole volume for
rendering� only the parts that contain relevant data
are used� this can save an enormous amount of ren�
dering time� not only in volume stepping� but also be�
cause it greatly decreases the number of compositing
and shading calculations one needs to perform.
The rational behind PARC is simple. As one needs
to calculate the integral I �R t1
t0 e�R t
��s�ds
t0 I�t�dt dur�
ing rendering� PARC �nds tighter bounds for t0 and t1�
thus� substantially lowering the rendering time. PARC
does this by enclosing the volume with a rough polyg�
onal approximation� which is transformed and scan
converted into front and back Z bu�ers. For each ray
the front one gives us a conservative estimate for t0�
and the back one gives the t1 estimate.
In order to skip over empty space inside volumes�
our implementation of PARC uses pre�calculated
cubes aligned with the primary axes to bound cubes
inside the volume. For each particular view� we scan
convert the cubes into a Z bu�er �implemented in soft�
ware� to obtain closer bounds on the intervals where
the ray integrals need to be calculated. This method
achieves speeds comparable with the fastest high qual�
ity volume renderers.
One can specify the number of cubes in the sub�
division of the original dataset. This determines the
accuracy of the t0 and t1 estimates� the higher the
number of cubes� the closer to the exact intersection
points they are. If the estimates are accurate� we per�
form less work on the ray� but on the other hand the
scan conversion time is higher as the number of cubes
grows very fast. For instance� one can ask for a level 4
PARC approximation� this means the dataset is par�
titioned to 24 intervals in each of the coordinate di�
rections� for a total of 4096 small cubes. Depending
on the low and high threshold speci�ed� one usually
gets a much lower number of such cubes. For instance�
with a level 4 PARC approximation of a CT 3D recon�
structed head at a 20�200 threshold� only 38� of the
cubes are non�empty.
The cubes generated by PARC are the basic units
for our load balancing. As the cubes are very close
approximation of the amount of work one has to per�
form during ray tracing� we use the number of cubes
a processor has as the measure of how much work is
performed by that particular processor. Let P denote
the number of processors� and ci the number of cubes
processor i has. To achieve a good load balancing we
need a scheme that minimizes the following heuristic
function for a partition X � �c1� c2� � � ���
f�X� � max
i6�j jci � cjj� 8i� j � P �1�
The main problem in implementing this approach
is that for ray casting to be e�cient the dataset part
of a particular processor need to be contiguous. Not
only this makes compositing easier but it also reduces
the number of intersection calculations required. Once
one decides what shape to assign to each processor�
one just needs to use either Equation 1 or a variation of
it. For the rest of the paper we describe an implemen�
tation of our load balancing scheme that uses slabs�
which are consecutive slices of the dataset aligned on
two major axes� as the basic partition blocks of the
dataset for load balancing. Slabs are very easy to im�
plement and we show that they provide a good sense of
load balance. In the case of slabs� the PARC algorithm
produces an ordered list of number� b1� b2� � � � � bn�
which are the number of cubes in each slab. We need
to �nd indices pairs �k1 1� k2 1�� �k1 2� k2 2� � � �� �k1 P � k2 P �� that
minimizes the following expression�
f�X� � max
i6�j
������ k2 i X
m�k1 k
bm �
i
2 j X
m�k1 bm
j
������
� 8i� j � P �2�
The problem of computing the optimal �as de�ned
by our heuristic choice� load balance partition indices
can be solved naively as follows. We can compute all
the possible partitions of the integer n� where n is
the number of slabs� into P numbers� where P is the
number of processors. For example� if n � 5� and P �
3� then 1 � 1 � 3 represents the solution that gives the
�rst slab to the �rst processor� the second slab to the
second processor and the remaining three slabs to the
third processor. Enumerating all possible partitioning
to get the optimal one is a feasible solution but can
be very computationally expensive for large n and P .
At this time we have a Prolog implementation of a
slightly revised algorithm. Instead of calculating the
minmax Equation 2� we choose the permutation with
the smallest square di�erence from the average.
In order to show how well our approach works in
practice� let us work out the example of using our
load balancing example to divide the neghip dataset
�the negative potential of a high�potential iron pro�
tein of 66 3 resolution� for four processors� using a
level 4 PARC decomposition with a 10 to 200 value
threshold. After running PARC we get the following
16 numbers� one for each slab� out of the 1570 to�
tal cubes�f12� 28� 61� 138� 149� 154� 139� 104� 106�
139� 156� 151� 129� 62� 29� 13g. The naive ap�
proach of other volume renderers has been to assign
an equal part of the volume to each processor� result�
ing in the following partition� f12�28�61�138�239�
149�154�139�104�546� 106�139�156�151�552�
129�62�29�13�233g� where processors 2 and 3 have
twice as much work than processor 1 and 4. Our ap�
proach based on Equation 2 gives us f388� 397� 401�
384g� clearly a much more balanced solution.
One can see that some con�gurations will yield bet�
ter load balancing than others but this is a limitation

of the particular space subdivision one chooses to im�
plement� the more complex the subdivision one allows�
the better load balancing but the harder it is to im�
plement a suitable load balancing scheme and the as�
sociated ray caster. Figure 1 plots the examples just
described for the naive approach. Figure 2 shows how
well our load balancing scheme works for a broader
set of processor arrangements. By comparing both
plots� one can see that our algorithm generates much
smoother curves� thus leading to better load balanc�
ing.
Number of cubes
600
500
400
300
200
100
out of 4 processors
out of 8 processors
0
1 2 3 4 5 6 7 8
Processor Number
Figure 1� The graph shows the number of cubes per
processor under naive load balancing.
Number of cubes
800
700
600
500
400
300
200
out of 2 processors
out of 3 processors
out of 4 processors
out of 8 processors
out of 10 processors
100
1 2 3 4 5 6 7 8 9 10
Processor Number
Figure 2� Load balancing measures for our algorithm.
The graph shows the number of cubes the processor
receives in our algorithm.
Figures 3 and 4 show the rendering times on the
Intel Paragon� showing the correlation between the
number of cubes a processor has and the amount of
work it has to perform. By comparing these graphs
Time to Render (msec)
11000
10500
10000
9500
9000
8500
8000
7500
7000
6500
time to render with 4 processors
6000
1 2 3 4
Node Number
Figure 3� Naive load balancing on the Paragon. The
graph shows the actual rendering times for 4 proces�
sors using the naive load balancing.
Time to Render (msec)
11000
10500
10000
9500
9000
8500
8000
7500
7000
6500
time to render with 4 processors
6000
1 2 3 4
Node Number
Figure 4� Our load balancing on the Paragon. The
graph shows the actual rendering times for 4 proces�
sors using our load balancing.

and those in Figures 1 and 2� one can observe that our
load balancing is e�ective and accurate� compared to
the naive approach of equally subdividing the dataset.
If one was calculating a single image� the total render�
ing time of the image subparts would be the maximum
of every processor plus the compositing time. As will
be seen in the next section� we use a pipeline approach
to optimize image generation performance� by amor�
tizing compositing over time.
4 Parallel Ray Casting
The version of our parallel PARC�based volume ray
caster described here uses the NX�2 library on the
iPSC�860 and the Paragon� although previous ver�
sions also ran under TCP�IP on workstations. We
plan to release a production level version of this code
on the iPSC�860� Paragon� PVM� network worksta�
tions �TCP�IP� together with a distribution version
of VolVis �1�.
In order to avoid the processors having direct ac�
cess to the dataset description �les� we chose to broad�
cast once the necessary information for the rendering�
like the dataset� processor assignments� transfer func�
tions� and so on� and have all the processors synchro�
nize during this phase. Clearly� this may make our
implementation unsuitable for someone that needs to
generate only a single image� specially because some
machines �like the iPSC�860� have slow processor ac�
cess to NFS mounted �les. The best scenario is one
where the datasets are generated on the parallel ma�
chine� and not moved in and out at all.
After initialization� user commands representing
di�erent viewing angles are sent to all the processors
by broadcast messages. Only information like trans�
formation matrices and image sizes are sent at this
time to minimize the communication cost per image.
In order to avoid �ooding the parallel machine with re�
quests� a feedback synchronization technique is used.
It basically balances the requests rate with the ma�
chine power available. The �ow of messages in the
algorithm is shown in Figure 5.
The feedback synchronization techniques we use are
based on work by Van Jacobson �7�� who designed a
set of techniques to avoid congestion in TCP�IP net�
works. We use a variation of his slow�start and round�
trip�time estimation technique� where the host slowly
sends requests and adaptively changes the rate of re�
quests with the feedback it receives from the network.
This is implemented by having the host keep the num�
ber of outstanding image render requests� and setting
a maximum on this number based on the number of
processors and the amount of memory each has. At
the start of the computation the host begins sending
image requests to the processors� and for every image
received it sends two requests to the processors until
the maximum is achieved. Also the host keeps a run�
ning average of the time taken to compute an image�
Node 1 Node 2 Node 3
Node 0
User Workstation
Node 4
Figure 5� Overview of communication �ow in the al�
gorithm. Arrow width represents the expected band�
width necessary in each communication link.
computed as Tf � �Ti � �1 � ��M� where Tf is the
new estimate� Ti was the initial estimate� M is the
time measured in the last image computation� and �
is an amortization constant. By changing � we can
make the host more or less responsive to changes in
rendering times. By using this procedure� when this
time increases the host can adaptively decrease the
rate of requests or increase the rate if the processors
begin computing images faster.
In the computing processors� a set of working re�
quests are queued and serviced on demand. Basically�
there are two di�erent kinds of requests� rendering re�
quests and compositing requests. The �rst type of re�
quest is received directly from the host �where sup�
posedly the user is waiting for images to show up��
while the second comes from other slab neighboring
processors. The computing processor keeps servicing
both types of requests� by picking a message from each
queue.
While servicing a rendering request the processor
allocates enough memory for it� renders it� and keeps
the rendered image around until a compositing request
for that particular image comes from its respective
neighboring processor� then composites its part of the
image and sends it over to the other neighboring pro�
cessor� and continues working on a new rendering re�
quest. The last processor on a chain� sends the whole
image back to the user�s workstation. If a composit�
ing request is received for an image that is not ren�
dered yet we take the approach of computing it right
away rather then delaying it as this could double our
memory requirements for images. Once requests are
serviced the memory is immediately freed.
This approach is simple and e�ective. One of
the clear advantages is that if we disregard the mes�
sage and synchronization overhead for a moment� we
see that we are maximizing the computation overlap
among processors and getting a much better utiliza�

tion of the communication network as messages are
being sent during the whole course of the image com�
putation time instead of just at a certain point in time.
One may claim that other� tree based� composit�
ing schemes �9� may yield better results� however� the
drawbacks of these schemes �low processor utilization
during compositing and high network utilization dur�
ing the peak of compositing� are major. Even though
the tree approach would give a �nal image in O�log n�
time steps� it still needs asymptotically the same num�
ber of messages. Therefore� it does not save any com�
putation time� but it actually wastes it when some of
the processors become idle.
The use of a pipelined compositing approach� where
images are asynchronously generated and saved in
bu�ers� requires the use of the feedback synchroniza�
tion technique to avoid increasing the memory over�
head without bounds. An interesting side e�ect of this
technique is that our algorithm automatically adjusts
itself to the rendering times of the particular machine
and�or con�guration being used� like the number of
processors and network performance.
5 Performance Analysis
In this section we present a few performance �g�
ures of our algorithm and demonstrate that our ap�
proach is sound and fast. The main points that we
are discussing are� the e�ectiveness of PARC load bal�
ancing� the communication overhead of the composit�
ing scheme� algorithm behavior under di�erent shad�
ing models� and overhead as compared to a sequential
implementation. The e�ectiveness of our PARC load
balancing was studied extensively in the last section�
but to complete our choice of using PARC as our ray
casting algorithm� it is interesting to compare its ad�
vantages to a more naive ray casting approach where
no presence accelerations are adopted.
A conventional ray caster where the rays are cast
from start to end by calculating intersections with the
bounding box of the object is only slightly di�erent
from a PARC ray caster. A PARC ray caster actu�
ally does more work than a naive one� as it needs to
scan convert and to �nd t0 and t1 from the Z bu�er.
The place that a PARC ray caster really gains perfor�
mance is in the fact that it better approximates the
volume bounds. It should be clear that the higher
the cost of the shading function per step� the more
advantageous it is to calculate these bounds well. In
Figure 6� we can see how a PARC based ray caster per�
forms against a naive ray caster under di�erent shad�
ing functions. For our purposes we consider �light�
shading a method that uses 5�10 instructions per sam�
ple� �medium� a method that uses 50 instructions per
sample� and �heavy� shading functions require about
300 instructions per sample. Nieh and Levoy �12� have
reported that trilinear interpolating a ray sample takes
320 instructions. One can see from Figure 6 that not
only times but also the rate of increase of cost de�
creases as one computes more samples.
Rendering Times (sec)
70
60
50
40
30
20
10
naive
parc
0
0 50 100 150 200 250 300
Shading cost in # of instructions.
Figure 6� PARC versus naive ray casting. Times were
calculated on a Sparc1000.
The work performed during rendering each ray can
be broken into Ir� the initialization work� and Wr� the
work performed to calculate and shade the samples
along the ray. If perfect load balancing is achieved for
every ray� each processor will perform W r
P � Ir work
per ray� that is� the initialization time is replicated
for every ray. If Wr � Ir� then we can achieve very
high scalability with the algorithm� otherwise� as the
number of processors increases the amount of work
done on the initialization by all the processors P Ir gets
larger than Wr� thus limiting the performance. This
makes optimization of the initialization time critical
to the performance of the algorithm.
Initialization time is composed of several compo�
nents� the most time consuming being the PARC pro�
jection time and the transformation time. Right now
it takes anywhere from 350 msec to 1600 msec to scan
convert a level 4 PARC approximation on the ma�
chines we used. We believe scan conversion itself can
be done on the order of 20 times faster when the code
is re�written in a more e�cient way� possibly in i860
speci�c code. By broadcasting at the beginning only
the necessary PARC polygons we can also avoid in�
creasing the number of polygons that need to be scan
converted in each processor� and at the same time de�
crease the memory requirement. Until these changes
get incorporated in our code� our timings are going
to be around the 1 second mark� even if the rest of
the algorithm takes no time at all. However� overall
rendering times decreased substantially after we op�
timized our transformation time by factoring out all
common matrix multiplications and inlining the ones
inside tight loops.
All of the performance numbers presented in the
rest of the section are for the Intel Paragon. The Intel

Paragon uses an Intel i860XP� a 50MHz superscalar
microprocessor� and a 2D mesh interconnection net�
work. Every processor of the machine actually con�
tains two i860XPs� but only one is used for computa�
tion.
Figures 7 and 8 show the average time to composite
di�erent image sizes in three di�erent machine con�g�
urations and for �ve di�erent screen sizes� ignoring
completely the rendering time. We use a slow start
technique for these measures �only when pipelining�.
It is interesting to compare the �gures as one can see
that our pipelining method can very well hide the ef�
fects of the network and the work done to composite
the image. For instance� every processor has to spend
around 52 msec to composite a 300 2 image �only com�
pute time�� if we consider 6 processors� it will take
over 300 msec of CPU time to generate this image�
still with our pipelining approach the user only sees
65 msec as opposed to 330 msec a sequential compos�
ite requires.
Time to composite final image (msec)
250
200
150
100
50
0
500x500 image
400x400 image
300x300 image
200x200 image
100x100 image
10 15 20 25 30
Number of processors
Figure 7� Timing as seen by a user of the arriving of
images using our pipelining approach.
In Figure 9� we present some of our rendering times.
These are rendering times for our �rst implementation
and should not be regarded as what we are expecting
for the production level code. One can see from the
graph that our algorithm scales well as the number
of processors increases. Also our prediction that the
higher the shading cost� the better the parallel scala�
bility can be seen from the graphs. We have �ltered
out the PARC rendering time from the numbers. We
expect to speed the PARC projection times up by at
least 20 times with the new scan conversion routines�
and with a new set of fast PARC projection techniques
being designed we anticipate getting scan conversion
to under 25 msec. At this time� the best rates attain�
able by our algorithm are about 1.5 frames�sec on a
32 processor con�guration of our Intel Paragon for a
256 2 image size. This is very competitive and even
Time to composite final image (msec)
6000
5000
4000
3000
2000
1000
0
500x500 image
400x400 image
300x300 image
200x200 image
100x100 image
10 15 20 25 30
Number of processors
Figure 8� Timing as seen by a user of the arriving of
images using sequential composite.
better than other rendering times published for ma�
chines with this number of processors.
Average Rendering Times (msec)
80000
70000
60000
50000
40000
30000
20000
10000
heavy
medium
light
0
0 5 10 15 20 25 30 35
Number of processors
Figure 9� Rendering times on an Intel Paragon.
6 Conclusions and Future Work
We have shown that using PARC cubes for measur�
ing useful work generates an intuitive way to load bal�
ance volume ray casting on distributed memory par�
allel machines. This not only generates a method that
is theoretically sound but its preliminary implementa�
tion seems to present a method that is both e�cient
and scalable.
We have also proposed a new method for com�
positing that achieves better throughput than previ�
ous methods and that can be used to generate better
refresh rates. If one cannot accept the delay pipelin�
ing imposes� one can always make judicious replication
of volume data� for instance� one volume for every 16

processors to avoid long image delay times and still
keep high refresh rates.
We believe our method is simple� fast� uses co�
herency and achieves high resource utilization on a
given machine. As we use PARC� we achieve a
high utilization of the compute processors and thus
a very fast rendering time on every processor. Be�
cause of our pipelined compositing scheme� we achieve
a much higher network utilization than other methods.
Finally� our feedback synchronization image request
technique guarantees a constant �ow of information
that adapts itself to di�erent con�gurations of proces�
sor performance and network utilization.
Our current implementation can be greatly im�
proved and optimized. One of our main concerns is to
smoothly integrate all the parallel code into VolVis�
so our users can take advantage not only of its in�
tuitive and �exible user interface� but also of greater
speed provided by parallel machines. Other plans in�
clude the porting of our algorithm to other architec�
tures and a more detailed performance analysis of the
whole algorithm. We are also planning on introduc�
ing optimization that would allow the system to use
data replication and sharing whenever allowed. This
way users with multiple processor shared�memory ma�
chines� like a network of Sparc1000s would be able to
get better performance.
Another direction of future work is the extension of
our load balancing technique to non�slabs partitions.
The major problem is that computing optimal parti�
tions in one dimention �the slab case� is already hard
and computationally expensive. Another interesting
question is whether this method can be extended to
irregular shaped grids.
Acknowledgments
This research has been supported by the National Sci�
ence Foundation under grants CCR�9205047 and DCA
9303181 and by the Department of Energy under the
PICS grant. Special thanks to Rick Avila and Lisa
Sobierajski for several enlightening discussions about
PARC� volume rendering� and the implementation of
VolVis. We are grateful to Juliana Freire for imple�
menting the e�cient Prolog algorithm described in
Section 3 in the XSB system developed at Stony Brook
by David Warren.
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Spiders� A New User Interface for Rotation and Visualization of
N�dimensional Point Sets
Kirk L. Du�n
Brigham Young University
kirkl�python.cs.byu.edu
Abstract
We present a new method for creating n�
dimensional rotation matrices from manipulating the
projections of n�dimensional data coordinate axes
onto a viewing plane. A user interface for n�
dimensional rotation is implemented. The interface
is shown to have no rotational hysteresis.
1 Introduction
Many techniques for visualizing n�dimensional
data sets separate the data into its component dimen�
sions� allowing the user to look at various coordinate
combinations in a way that hopefully brings under�
standing. These methods do well at avoiding the tra�
ditional projection to two dimensions that hides data.
However the data relationships are not immediately
intuitive to our brains� which are used to transforming
large amounts of information from three dimensional
projections down to two.
On the other hand� projection of n�dimensional
information down to two may be slightly more intu�
itive� but su�ers from the curse of data hiding due to
projection. Moving the data in n�space� by predeter�
mined motion or direct manipulation can help solve
this problem.
Asimov�s �grand tour��Asi85� made it possible
to step through all possible projections of an n�
dimensional data set onto two dimensions in a use�
ful manner. Hurley and Buja introduced a means of
creating �guided tours� of the data by allowing the
user to create two disparate projection plane orienta�
tions and interpolate between them�Hur88�. A good
method of interpolation is to create a n�dimensional
rotation between the two orientations and sample
along the rotation angle�BA86�. Subsequent data ro�
tation tools� while similar� have retained this inter�
polation approach for creating smooth motion in the
projected data�YR91� SC90�.
Here we present a new technique for creating
n�dimensional rotations from information projected
onto the viewing plane. From this technique we
develop an interface for interactively rotating n�
dimensional point sets. User control over the rotation
sequence is �ne enough that no direct interpolation
between projections is needed. Section 2 will review
some of the important principles from matrix algebra.
Section 3 will develop the main algorithm for creating
n�dimensional rotation matrices from manipulationof
data projections in the viewing plane. Section 4 will
William A. Barrett
Brigham Young University
discuss some of the implementation aspects of the al�
gorithm and present an implementation of an interac�
tive n�dimensional rotation interface that is free from
hysteresis e�ects. Section 5 will demonstrate the ma�
nipulation of two 5�dimensional data sets using the
interface� and section 6 will point out some possible
areas of re�nement for the interface.
2 Background
2.1 Notation
In this paper we will hold to an extension of the
notation used in most of the computer graphics liter�
ature� a n�dimensional point is represented by an�
dimensional row vector and is post�multiplied by any
transformation matrices. The vector composed of all
zeros except for a 1 in position i will be denoted ei.
2.2 Coordinate Frames
We represent a n�dimensional data set as a set
of points in an n�dimensional Euclidian space R n .
There are two ways of investigating the projection of
a set of n�dimensional points onto a 2�dimensional
viewing plane. In the �rst� the coordinate system
of the data and the coordinate system of the view�
ing space coincide. A viewing plane is arbitrarily
placed in the viewing space and the data is projected
onto the plane. The second approach to projecting
n�dimensional data onto a viewing plane moves the
coordinate system of the data with respect to the co�
ordinate system of the viewing space. In this latter
approach the viewing plane remains �xed.
Because a rotation leaves the coordinate system
origin invariant� it is possible to focus on the rotation
as a transformation of a vector from the origin to the
data point. This allows the creation of coordinate
frames� a cluster of unit vectors that point down the
positive principal axes of the underlying coordinate
system.
Using coordinate frames gives us some powerful
tools�Piq90�. If we start with an untransformed data
coordinate frame� multiplying each axis vector ei in
turn by the rotation matrix� it can be seen that the
new position in view space of the axis is given by row
iof the rotation matrix.
A corollary to this fact is that if we specify the
new position of the axis vectors such that they remain
orthonormal then the new positions de�ne the rows of

the rotation matrix 1 R.
2.3 Orthogonal Projections in
n Dimensions
We de�ne the orthogonal projection of an n di�
mensional point onto a subspace of lower dimension
�the viewing subspace� as the point in the subspace
closest to the data point. If b1 ���bm�m� nare basis
vectors of the viewing subspace� then the projection
xproj of a data point x is de�ned by
xproj �
mX
i�1
x � bi� �1�
If the b are equivalent to the standard basis vec�
tors ei then the projection of x onto bi is simply the
i�th coordinate of x.
3 Arbitrary Rotations in
n Dimensions
In three dimensions� rotations are commonly
speci�ed in terms of an angle about an arbitrary axis.
However� it is more correct to think of rotation as tak�
ing place in a plane embedded in the space�Nol67�. In
3�dimensional rotations� this plane is the plane per�
pendicular to the axis of rotation. In more than three
dimensions� the idea of rotation about an axis goes
awry because there are an in�nite number of axes that
are perpendicular to any given plane. But as long as
a plane in the space is speci�ed along with a center of
rotation in the plane� the rotation is uniquely de�ned.
The simplest rotation to
describe in n�dimensional space occurs in the plane
formed by anytwo coordinate axes. The rotation ma�
trix Rab��� for the rotation of axis xa in the direction
of xb by the angle � is
Rab��� �
8
�����
�����
rij
�
�
�
�
�
�
��
�
�
rii � 1 i 6� a� i 6� b
raa � cos �
rbb � cos �
rab � � sin �
rba � sin �
rij � 0 elsewhere
9
�����
�
�����
�2�
That is� Rab��� is an identity matrix except for the
entries at the intersection of rows a and b and columns
a and b. Since there are � � n
principal axes planes� n�
2
dimensional rotations are built up as the composition
of speci�ed rotations in each of the principal planes.
This composition is accomplished by multiplying the
corresponding rotation matrices together.
Our goal is to provide an intuitive means of spec�
ifying an n�dimensional interface� hopefully in a con�
cise graphical manner. The key to our approach is
in the observation that if an axis is not contained in
the viewing plane nor is perpendicular to the viewing
plane� then the axis and its projection onto the view�
ing plane de�ne another plane in which rotation can
1 Actually� this is not quite true. The negation of a data
axis is also allowed in this de�nition which corresponds to a
re�ection of the data about that axis. However� the algorithm
presented here will not produce re�ections.
occur. Moreover� by manipulating the projection of
an axis� it is possible to rotate the axis in the rotation
plane such that the axis remains consistent with its
projection. Figure 1 illustrates this observation for
n �3.
Figure 1� A 3�dimensional coordinate frame before
rotation �l� and after rotation �r�. The rotation plane
is de�ned by the rightmost data axis in each diagram
and its projection. The circle at the bottom of each
diagram shows the projection of the data coordinate
axes onto the viewing plane.
3.1 Rotation in the Plane
The problem here is to rotate the selected axis xi
by an unknown angle � to its new position x0 i . All
that is known are the magnitudes of the projections
of the axis.
Let xi be a unit vector representing the positive
direction of the i�th axis of the data coordinate sys�
tem embedded in the n�dimensional viewing coordi�
nate system. The projection of xi onto the viewing
plane is denoted xiproj . The position and projection
of the axis after rotation are denoted x0 i and x0iproj
respectively. See �gure 2.
In the rotation plane� xi can be decomposed
into two vectors� xiproj� and a component orthog�
onal to the viewing plane� xi� such that xi� �
xi � xiproj . These two vectors set up an orthogonal
coordinate system in the rotation plane. Now xi can
be represented by the coordinates �miproj �mi��where � kxi�k.
Since xi and x 0 i are unit vectors� given m0 iproj �
miproj � kxiprojk and mi�
the magnitude of the new projected component can
be determined� namely m 0 i� �
q
0 2
1 � m . Conse�
iproj

x i
x i ’
x i ’
proj
x
’
i
x i proj
x i
Figure 2� Rotation in the plane de�ned by xi and its
projection on the viewing plane. The data coordinate
axis vector xi is rotated to x 0 i.
quently� the parameters for rotation in the plane are
cos � � xi � x 0
i � miproj m0 � mi�m0 iproj i�
�3�
sin � � kxi � x 0
ik � miproj m0i�
� mi�m0 � �4�
iproj
Thus any vector v in the plane can be rotated using
the standard rotation equations
v 0 �
� ��
v � xiproj v � xi� cos � sin �
�
kxiprojk kxi�k � sin � cos �
More importantly�
x 0
i � m 0
xi�
i� kxi�k
� m0
xiproj
iproj
kxiprojk
� �5�
�6�
To determine the n�dimensional rotation matrix
R� all that remains is to �nd the new positions of
each axis vector. This is accomplished by decom�
posing each data space coordinate axis vector into
three components� a vector orthogonal to the rota�
tion plane� and two vector components in the rotation
plane. These last two vectors are the projection of the
data axis vector onto xiproj and xi� respectively. The
rotation is calculated for the rotation plane compo�
nents and the results added to the orthogonal vector
component. This gives the rotated position of the axis
vector.
Let a and b be the coordinates of data axis xj
projected onto the rotation plane� i.e.
a � xjprojxiproj � xj � xiproj
kxiprojk
�7�
b � xjprojxi� � xj � xi�
� �8�
kxi�k
Let xjorth be the orthogonal componentofxjwith
respect to the rotation plane. Then
xjorth � xj � a xiproj xi�
� b � �9�
kxiprojk kxi�k
After rotation� the new position of the data axis
vector x0 j can be expressed
in
x 0
j � xjorth ��acos � � b sin �� xiproj
kxiprojk
��a sin � � b cos �� xi�
� �10�
kxi�k
Substituting �9� into �10� and simplifying results
x 0
j � xj ��a�cos � � 1� � b sin �� xiproj
kxiprojk
��b�cos � � 1� � a sin �� xi�
� �11�
kxi�k
3.2 Algorithm
The foregoing development gives us the following
algorithm for creating an n�dimensional rotation ma�
trix.
Input�
R � the current rotation matrix. The rows
of this matrix are the axis vectors of the
data coordinate system. The elements
of R are denoted rij.
i � the index of the data coordinate axis
m 0
proj
that determines the plane of rotation.
� the desired magnitude of the
projected component of the selected
data axis.
axis1� axis2 � the viewing space axes
de�ning the viewing plane.
Output�
R 0 � the new rotation matrix describing
the transformation from data
coordinate space to viewing
coordinate space.
Variables�
mproj � the current magnitude of the
projected component ofthe
selected data axis.
m� � the current magnitude of the
orthogonal component ofthe
m 0
�
selected data axis.
� the orthogonal component
magnitude of the rotated data axis.
cos� sin � the rotation parameters of the
rotation.
k1� k2� sum �intermediate values
Find magnitude of projected component of selected
axis.

sum � 0
for �1 � � � 2�
sum � sum � r 2
i axis�
mproj � p sum
Find component magnitude of selected axis
perpendicular to viewing plane.
sum � 0
for �1 � � � n�
if �� 6� axis1 and � 6� axis2�
sum � sum � r 2
i�
m� � p sum
m 0
� �
q
0 2
1 � mproj Calculate projection plane parameters.
cos � mproj � m 0
proj � m� � m 0
�
sin � mproj � m 0
� � m� � m 0
proj
Rotate each data space axis.
for �1 � j � n�
sum � 0
for �1 � � � 2�
sum � sum � rj axis� � ri axis�
a � sum�mproj
sum � 0
for �1 � � � n�
if �� 6� axis1 and � 6� axis2�
sum � sum � ri� � rj�
b � sum�m�
k1 � �a � �cos � 1� � b � sin��mproj
k2 � �b � �cos � 1� � a � sin��m�
for �1 � � � n�
if �� � axis1 or � � axis2�
r 0
j� � rj� � k1 � ri�
else
r 0
j� � rj� � k2 � ri�
The simpli�ed formulation of the main inner loop
from �11� is justi�ed by noting that if we limit the
viewing plane to be one of the principal planes in
the viewing coordinate system� then xiproj has non�
zero components only along the axes speci�ed by the
viewing plane. Likewise� xi� will always have 0 coor�
dinates in those two dimensions.
4 Implementation
4.1 Interface
We have used two approaches in applying the
above formulas to the development of user interfaces
for n�dimensional rotation. Each approach allows the
user to select a data coordinate axis and drag the
projected end of the axis in the viewing plane. From
the path traversed in the viewing plane� a sequence of
n�dimensional rotation matrices is created. The dif�
ference in the two approaches is in how the change in
position of a selected projected axis is turned into a
rotation matrix.
In the �rst approach R is composed of two rota�
tions� the �rst occurs in the plane formed by axis xi
and its projection xiproj. The amount of rotation is
determined by the change in length of the projected
axis. The second rotation occurs in the viewing plane
and accounts for the change in projected orientation
of xiproj. Figure 3 illustrates for n �3.
However� rotation in the projection plane pro�
vides no new visual information. In practice� the set
of projected axes tends to spin wildly in the viewing
plane. This in turn makes it di�cult to adjust the
relative positions of the projected axes.
The second approach to the creation of n�
dimensional rotation matrices also decomposes R into
two rotations. The �rst rotation rotates the selected
axis xi in the plane formed by itself and its original
projection so that xi is perpendicular to the viewing
plane. The second rotation rotates xi from its posi�
tion perpendicular to the viewing plane to a position
consistent with the projected position. �Figure 4�.
4.2 Lack of Hysteresis
This latter approach to rotation possesses a nice
theoretical quality. Let the rotation of xi from its
position on the viewing plane� xiproj ��uj�vj�toits
new position x 0
iproj ��uj�1�vj�1� be denoted jRj�1
for any j. But this is the composition of two other
matrices� jRj�1 � jPQj�1 where jP is the rotation
of xi to a position perpendicular to the viewing plane
and Qj�1 is the rotation of xi from the perpendicular
space to its new position corresponding to x 0 iproj .
Now if a user selects an axis xi at position �u0�v0�
on the viewing plane and drags the projected axis
around the viewing plane� then the rotation matrix
of this transformation is the composition of the rota�
tion matrices of every point on the path of the dragged
projected axis in the viewing plane� i.e.
0Rm � 0PQ1 1PQ2 ���jPQj�1 ���m�1PQm �12�
for the path in the viewing plane of �u0�v0�� ����
�um�vm�.
But rotating an axis perpendicular to the viewing
plane and then rotating it back to the same position
is an identity operation. This means that Qj jP�I.
Consequently� �12� collapses to
0Rm � 0PQm� �13�
Thus dragging a projected axis with this method
is a conservative operation. The rotation matrix re�
sulting from dragging xiproj ��u0�v0� to its new po�
sition x0 iproj ��um�vm� is the same� regardless of the
path taken from �u0�v0�to�um�vm�2 .
This lack ofhysteresis is a highly desirable prop�
erty for interactive rotational interfaces for at least
two reasons� First� the user can follow anypath in
the viewing plane when dragging a projected axis and
be guaranteed of receiving the same rotation matrix�
given the same start and end points of the drag. If
2 As long as the path does not pass through the projection
of the origin of the data coordinate viewing system onto the
viewing plane.

Figure 3� Repositioning a projected coordinate axis by 1� rotating for new projected coordinate axis length� and
2� rotating in the viewing plane for new projected coordinate axis orientation.
Figure 4� Repositioning a projected coordinate axis by 1� rotating axis perpendicular to viewing plane� and 2�
rotating out of perpendicular space to new projected axis position.
the desired projected target is overshot or missed� the
axis can be dragged back to the desired position. Sec�
ondly� the interface need not process every point in
the path of the dragged projected axis in order to
maintain consistent interface operation. If the data is
being replotted as the axis is dragged� then a rotation
matrix need be created only from the current viewing
plane position. Any other positions traversed since
the last plotting step can be discarded� giving a great
computational savings if the data set is large.
The unit quaternions also share this lack ofhys�
teresis� which has generated signi�cant interest in
their use in 3�dimensional rotation interfaces�Sho92�.
Now this property can be extended to n�dimensional
rotations as well.
4.3 Orthonormality
As presented� this algorithm is highly dependent
on the fact that the axis vectors are orthonormal. In
practice� as numerical error creeps in� the rotation
matrix R ceases to be orthogonal. This is a standard
problem in 3�d interfaces where the rotation matrix
is occasionally re�orthogonalized. Our experience has
been that renormalizing the rows of the rotation ma�
trix is su�cient to maintain orthogonality. Without
renormalization� numerical error quickly dominates�
making R useless.
Actually� nothing in the derivation of the algo�
rithm depends on the mutual orthogonality of the
coordinate axis vectors. Therefore� the algorithm
given above will properly transform any set of vec�
tors through a rotation speci�ed by an�dimensional
vector and its projection. But in such a case� the
resulting vectors can not be used to form the new ro�
tation matrix.
4.4 Boundary Conditions
Because the algorithm decomposes every n�space
vector into two rotation plane components� it is nec�
essary that the axis xi that determines the rotation
plane be distinct from its projection onto the viewing
plane. Consequently� special measures must be taken
when xi lies in the viewing plane or is perpendicular
to the viewing plane. In practice� due to discretization
error in the interface� conditions when xi is close to
the viewing plane or close to the perpendicular must
also be considered.
In our implementation� when an axis projection
is dragged within a small distance of the center of the
viewing plane� the axis snaps perpendicular to the
viewing plane and stays there. When a user wishes to
drag one axis �of possibly several� out of the space per�
pendicular to the viewing plane� she clicks the mouse
on the center of the projected coordinate frame. A
text menu o�ers a selection of the available perpen�
dicular axes. After a selection is made� a point on the
viewing plane is selected� and the axis is rotated out
to this position. From there the axis can be dragged
like any others visible on the viewing plane.
5 Application
In our work in the Brigham Young University
Computer Vision Laboratory we have implemented
this interface to help visualize images and color
gamuts as 5�dimensional point sets. Each pixel in a
full color image is given �ve spatial coordinates� x� y�
red� green� and blue. Each of these data points is also
given a color corresponding to its red� green� and blue

components. This is done for convenience only and is
not necessary for the functioning of the interface. The
mean of the data set is subtracted from all points so
that rotation will occur about the center of the data
set.
The orthogonal projection of the data set is kept
separate from the rotational interface� which has ac�
quired the appellation of a�spider.� This is due to
the appearance of many moving �legs� on the viewing
plane when many coordinate axes are simultaneously
visible.
Our combining the projected axes into one �gure
is in direct contrast to Hurley�s data viewer�HB90��
which assigns each axis its own interface item. Our
experience seems to indicate that combining the axes
into a single �gure is acceptable when using relatively
low dimension data sets. However� we have imple�
mented the spiders with the facility to display an ar�
bitrary subset of the full data axis complement. We
have also used the powerful concept of linking demon�
strated by Buja� McDonald� et al�BMMS91� to link
several spiders simultaneously to a single data set.
Figure 5 shows an image undergoing 5D rotation.
At �rst only the x and y components are visible. Then
the red axis is dragged out of the space perpendicular
to the viewing plane. Because of the correspondence
between the color attributes and the spatial coordi�
nates� all of the points with high red values appear to
move in the direction of the projected red axis. Note
that as the red axis is brought out slightly� a pseudo
3D e�ect occurs. Next the green axis is dragged out
and the x axis pushed back into the perpendicular
space. Finally� the blue axis is brought out� the y
axis pushed in� and the three remaining color axes ar�
ranged evenly in the projection plane. The points in
the data set realign themselves into a pattern remi�
niscent of a color wheel.
6 Conclusion
We have demonstrated a new method called �spi�
ders� for interactively rotating n�dimensional point
sets. The technique provides n�dimensional rotation
matrices solely from information about the current
data coordinate system and its projection onto the
viewing plane. The interface has no rotational hys�
teresis� similar to the more robust 3D interfaces used
today.
The spiders are not without problems. They do
su�er from the �curse of projection� and data hiding
with dense sets associated with all projective tech�
niques. And like other visualization methods� as more
dimensions are added to the system� the incremental
return in understanding decreases. Nevertheless� we
feel that the interactive nature of this technique pro�
vides a powerful tool to help understand the universe
of data around us.
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�Asi85� Daniel Asimov. The grand tour� A tool for
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Michalak� and Werner Stuetzle. Interactive
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Abstract
Pseudocoloring is a frequently used technique in
scientific visualization for mapping a color to a data
value. When using pseudocolor and animation to
visualize data that contain missing regions displayed as
black or transparent, the missing regions popping in
and out can distract the viewer from the more relevant
information. Filling these gaps with interpolated data
could lead to a misinterpretation of the data. This
paper presents a method for combining pseudo−
coloring and grayscale in the same colormap. Valid
data are mapped to colors in the colormap. The
luminance values of the colors bounding areas of
missing data are used in interpolating over these
regions. The missing data are mapped to the grayscale
portion of the colormap. This approach has the
advantages of eliminating distracting gaps caused by
missing data and distinguishing between those areas
that represent valid data and those areas that do not.
This approach was inspired by a technique used in the
restoration of paintings.
1 Introduction
1.1 Art Restoration
Restorer: A Visualization Technique for Handling Missing Data
In art the term restoration refers to "... the
replacement of missing parts and the filling in of
missing areas in a damaged work of art. [1] In his
book The Materials of the Artist & their Use in
Painting, Max Doerner states: "In the case of valuable
works it is best not to attempt corrections, additions, or
overpainting, but rather to maintain the faulty areas in a
neutral color tone which harmonizes with the general
tone of the painting. The restorer in such cases should
not glory in his skill in making his additions appear to
be parts of the original." [2] Restorers who believe that
Ray Twiddy, John Cavallo, and Shahram M. Shiri
Hughes STX Corporation, NASA Goddard Space Flight Center,
Scientific Visualization Studio, Code 932, Greenbelt, MD 20771
only the original paint is permissible, use neutral tones to
retouch the damaged areas to minimize interference with
the aesthetic appreciation of the painting. This approach
is not without controversy and there are those in the art
world who favor an approach to retouching that deceives
the viewer completely. [3]
1.2 Data Restoration
A similar controversy exists in the realm of scientific
visualization. There are those who believe that missing
data should be indicated either by mapping to a color
outside the valid data domain or by making those areas
completely transparent. Others believe in applying some
method of interpolation that smoothes over the areas of
missing data. Although the latter approach has the
advantage of eliminating distracting gaps caused by
missing data, it does not distinguish between valid data
and interpolated data. The approach we present uses
luminance interpolation to visually blend the missing data
with the valid data while at the same time distinguishing
missing data areas by their lack of color.
1.3 Visual Perception
In the process of seeing, we are dependent on
observing the interactive juxtaposition of gradations of
tone from brightness (or lightness) to darkness. The
presence or absence of color does not affect tonal values.
Tonal values are constant and are infinitely more
important than color in seeing. The contrast of tone
enables us to see patterns that we can simplify into objects
with shape, dimension, and texture. [4]
The term brightness is used to refer to the quantity of
light in the psychological sense of perceived intensity.
The terms luminance and intensity refer to the amount of
light energy reflected or emitted from a surface. The
perception of brightness depends upon the sensitivity of
the eye. The perception of changes in intensity is

nonlinear. For example, if you cycle through the settings
on a three−way 50−100−150−watt light bulb, the step
between 50 to 100 seems much greater than the step
between 100 to 150. [5]
Our perception of brightness of objects often depends
upon the luminance of adjacent objects. This perceptual
effect is called simultaneous brightness contrast. This
effect can be seen when gray squares of the same
luminance are displayed on gray backgrounds that are
lighter and darker (see Figure 1). On the lighter
background the gray appears darker and on the darker
background it appears lighter. Simultaneous color
contrast is the effect in the visual system when the
perception of a color in an area is influenced by the
colors of the adjacent areas. For example, a gray square
surrounded by red will be perceived as tinted green. [6]
Figure 1: When two gray squares of equal luminance
are placed on a background of varying luminance, the
square on the lighter background appears darker and the
square on the darker background appears lighter.
The Restorer technique reduces the effect of
simultaneous brightness contrast by matching the
luminance of the grays used to fill a missing data region
to the luminance of the adjacent colors (see Plate 1).
The effect of simultaneous color contrast is largely
dependent upon the saturation of the colors selected for
the palette. Studies by C. Ware [7] show a colormap
based on the physical spectrum (red+blue, blue,
blue+green, green, green+red, red) offers minimal
contrast distortion.
1.4 Selecting a Color Palette
There are many sources of information on the use of
color for aesthetic purposes [8, 9, 10] and for coding [11,
12, 13]. Although selecting a color palette based upon
constant lightness or value is one of the preferred
methods for providing color harmony, it is not
recommended in this case. A palette based upon light
and dark contrasts takes advantage of the fact that the
eye is more sensitive to spatial variation in intensity than
spatial variation in chromaticity. A palette based upon
the physical spectrum with variation in lightness
provides both aesthetic harmony and extends the range
for coding. Our technique was chosen to compensate for
situations where colors vary greatly in lightness or value.
It is based upon matching each color in the palette to a
gray of equal luminance.
2 The Restorer Technique
The Restorer technique fills in missing data regions
so that the images can either be used in an animation
without distracting gaps, or studied individually with the
missing data regions easily distinguishable. Hence,
Restorer was designed to satisfy three main
considerations in filling missing data areas: to match the
luminance of the adjacent colors, to provide stability for
consistency, and to provide a unique fill solution.
By matching the luminance at the boundary, the fill
will be least intrusive in viewing data sets using a palette
with large variations in luminance. The luminance
associated with a color is dependent on the display
device. In the cases treated so far, we are dealing with
the National Television System Committee (NTSC)
standard, as discussed in Foley, et al [14]. The
luminance (Y) for NTSC is calculated from the red (R),
green (G), and blue (B) components of the color by:
Y = 0.30 R + 0.59 G + 0.11 B.
Since the filled area should not distract the viewer in
an animation, it should be fairly consistent from one
frame to the next while still following the shape of the
data. In a stable fill algorithm, small changes in the
luminance in the vicinity of the missing data will cause
only small changes in the fill. Uniqueness implies that
there should be only one fill that would result from the
algorithm.
In each of the fill algorithms that will be discussed,
the input consists of a two−dimensional array containing
the valid data, a two−dimensional array marking the

missing data, and a colormap. In some implementations,
this can be a single pseudocolored image with the missing
data marked by a unique color. The algorithms will be
applied only to the regions of the image where the data is
missing and will leave the valid regions undisturbed. In
each case a neutral gray is added into the image, with the
luminance of the gray determined by the algorithm.
2.1 Laplacian Fill
In the Laplacian Fill Algorithm, the luminance, Y, in
the interior of the missing data regions, is a solution of
Laplace’s equation,
2 Y(x,y) = 0,
and it matches the luminance of the adjacent valid data.
The general property of Laplace’s equation is that it will
not have a maximum or minimum in the region. It will
also tend to average the luminance of the interior, giving
more weight to the near edges. It has the advantage that
some structures are continued across the missing areas.
This solution will satisfy all of the three conditions stated
earlier. [15]
The Laplacian Fill Algorithm was implemented using
the Gauss−Seidel algorithm [16] for computing the
luminance. The method converged rapidly because the
luminance was represented by a byte array and, generally,
the missing areas were small.
2.2 Rank Fill
The Rank Fill Algorithm applies multiple rank filters
to the missing data. The rank filter is a convolution
process by which the value of a function at a point is
replaced with the maximum value found anywhere within
the small defined region or kernel. This filter will not
affect a function within a region where the function is
uniform or slowly varying. [17]
In order to implement the Rank Fill Algorithm, a
network of modules was generated using IRIS Explorer
on a Silicon Graphics Workstation. The functionality of
the network as shown in Figure 2 is separated into three
categories: Data Input Modules, Filtering Modules, and
Overlay (Filling) Modules.
The Data Input Modules read the physical data and
convert them into Explorer data format. For
three−dimensional data, the OrthoSlice Module is used to
extract two−dimensional slices. A colormap is applied to
the data and a pseudocolor image is generated.
The Filtering Modules generate a grayscale image of
the pseudocolor image and apply two rank filters with
rank levels of 8 and 6, respectively. The result is then
passed through a median filter. The rank filters extend the
luminance of the data in the vicinity of the edges into the
missing areas, and the median filter smoothes the
luminance in the interior regions. In effect, a unique
luminance is generated for the image that covers the
missing areas consistent with the luminance of the data
bounding these regions.
The Overlay Modules use the missing data array as a
mask to insert the filtered luminance into the missing data
regions of the pseudocolor image from the Data Input
Modules.
Data Input
Modules
Grayscale Data
Rank Filter:
Level 8
Rank Filter:
Level 6
Median Filter
Filtering Modules
Input Data
Figure 2: A block diagram of data flow using rank filters
and a median filter applied to the missing regions.
Restored Data combines the grayscale luminance
overlayed on the missing regions of the pseudocolored
data.
2.3 Modified Rank Fill
Generate
Colormap
Pseudocolor
Data
Create A Mask
from
the Missing Data
Apply the Mask
to Filtered Data
Overlay Filter
Result with
Pseudocolor Data
Overlay
Modules
Display
Restored Data
The Modified Rank Fill is a combination of the Rank Fill
and the Laplacian Fill. In some cases, the missing data
regions are too large for the Rank Fill Algorithm to

completely fill. In these cases, the output from the Rank
Fill Algorithm is used as input to the Laplacian Fill
Algorithm in order to fill the remaining areas of the
missing data regions.
3 Results
The technique discussed in this paper has been applied
to several test cases and to atmospheric data sets that
contain missing data. In Plate 2a a test chart of saturated
squares of red, green, blue, and yellow provides a
background for a black linear pattern representing missing
data. Plate 2b shows the result of the Restoration
technique, using the Laplacian Fill. In order to evaluate
the results shown in Plate 2b, the color image was
converted to a grayscale image and is shown in Plate 2c.
Restorer was completely successful in areas where black
was surrounded by a single hue. In those areas where
black overlapped two or four hues of contrasting
luminance, the technique was less successful and we see a
blurring of the gray values as it attempts to interpret the
contour of the underlying colored squares.
The second test illustrates the ability of Restorer to
reduce the distractions caused by missing data when
dynamically slicing through a three−dimensional data set.
Plate 3 shows images from an animation sequence in
which three geometric shapes (a triangle, a square, and a
circle) were placed at fixed positions to indicate missing
data in a data set which contained no missing data. As the
cutting plane moves though the data in the first animation
sequence, the black shapes appear to float in front of the
data while the shapes, restored by the Laplacian Fill, blend
with the background. In the next animation sequence,
Restorer eliminates the distractions caused when the black
shapes appear randomly, popping in and out of the scene..
Restorer was next applied to global column ozone data
collected by the Total Ozone Mapping Spectrometer
(TOMS) on Nimbus−7 for the period January 1, 1992,
through April 30, 1993. Plate 4 illustrates the three basic
steps in the Restoration process to fill the missing areas.
The original data is shown in Plate 4a. Plate 4b shows the
result of using the rank and median filters that correctly
fill for most cases. In this instance the area was too large
for the rank and median filters, so it was completed using
the Laplacian Fill (see Plate 4c).
4 Conclusions
In this paper we have presented a method for filling
regions of missing data by using the luminance of the
colors representing valid data in the neighborhood of the
missing data. The method has the advantages of visually
blending the restored areas with the original data while
remaining distinguishable upon closer inspection. This
approach successfully eliminates distracting gaps when
viewing images in an animation.
The Laplacian fill was found to be adequate for filling
areas where the luminance varies monotonically across the
missing regions. For example, the fills shown in the
geometric shapes in Plate 3 reduce the distractions while
still marking the missing data regions. In cases similar to
the one illustrated in Plate 4, where a narrow missing
region extends between two bright areas across a dark belt,
the Laplacian Fill Algorithm gives an unacceptably bright
fill. The luminance at the center of the fill was
approximately the average of the luminance of the bright
and dark regions. Because of the nonlinearity in visual
perception of intensity, the filled region seemed much
brighter than the dark band and not significantly dimmer
than the bright bands.
The Rank Fill was developed to restore images with
the pathologies just described. Like the Laplacian Fill, the
Rank Fill will fill small areas with the average of the
surrounding luminance. However, in the case of the long
narrow region described above, the luminance value
assigned to the center of the region is determined by the
luminance of the nearest valid data on the left and right
sides. This maintains the pattern of luminance in the
original data.
The rank filter in the Rank Fill Algorithm can be
applied several times to completely fill the missing data
regions. Experimentation with the number of iterations
has shown that if it is applied more than five times, the
luminance will converge to the highest value of the full
image. The result is that the fill will not meet the criteria
of matching the luminance of the adjacent valid data. In
these cases, the Laplacian Fill Algorithm is used to
complete the fill.
It is our intention to further investigate the limitations
of the rank filter after two repetitions. Currently, the
number of repetitions is based upon visual inspection of
the missing data regions and the convergence of the
grayscale with the boundaries of these areas. Ideally, we
would like to automate this process so that the number of
rank filters applied to the missing regions would be
determined by a complete fill of all the missing areas.
Another option that is being added to Restorer is to derive
the luminance of the missing data areas in a
two−dimensional slice by interpolating the luminance of
the slices adjacent to it. We also intend to apply the

Restorer technique to other types of physical data such as
oceanography, wind, and fluid flow data sets.
5 Acknowledgments
This work is being done under NASA Contract Number
NAS5−32350. We would like to thank Dr. Horace Mitchell
and Dr. James Strong of the Scientific Visualization and
Applications Branch, and Dr. Mark Schoeberl of the
Atmospheric Chemistry and Dynamics Branch at NASA
Goddard Space Flight Center for their support.
We are very grateful to Pamela O’Neil for her hard work
and excellent video production. We also wish to thank Houra
Rais for proofreading this paper.
References
[1] R. Mayer, A Dictionary of Art Terms and Techniques, Thomas
Y. Crowell Co., New York, 1969, p. 90.
[2] M. Doerner, The Materials of the Artist & their Use in
Painting, Revised Edition, Harcourt, Brace & World, Inc.,
New York, 1962, p. 310.
[3] F. Kelly, Art Restoration: A Guide to the Care and
Preservation of Works of Art, McGraw−Hill Book Co., New
York, 1972, pp. 181−192.
[4] D. A. Dondis, A Primer of Visual Literacy, The MIT Press,
Cambridge, Mass, 1973, pp. 85−103.
[5] J. D. Foley et al., Computer Graphics: Principles and
Practices, 2nd. edition, Addison−Wesley Publishing
Company, Reading, Mass., 1990, pp. 563−564.
[6] S. Coren, C. Porac, L.M. Ward, Sensation and Perception,
Academic Press, New York, 1979, pp. 150−155.
[7] C. Ware, "Color Sequences for Univariate Maps: Theory,
Experiments, and Principles", IEEE Computer Graphics &
Applications, Vol. 8, No. 5, Sept. 1988, pp. 41−49.
[8] J. Itten, The Art of Color: The subjective experience and
objective rationale of color, Van Nostrand Reinhold Co.,
New York, 1973.
[9] J. Albers, Interaction of Color, Revised Edition, Yale
University Press, New Haven, Conn. 1976.
[10] E. R. Tufte, Envisioning Information, Graphics Press,
Cheshire, Conn., 1990, pp 81−95.
[11] J. D. Foley and J. Grimes, "Using Color in Computer
Graphics," IEEE Computer Graphics and Applications, Vol.
8, No. 5, Sept. 1988, pp. 25−27.
[12] R. E. Christ, "Review and Analysis of Color Coding Research
for Visual Displays," Human Factors, Vol. 17, No. 6, June
1975, pp. 542−570.
[13] W. Cleveland and R. McGill, "A Color−Caused Optical
Illusion on a Statistical Graph," American Statistician, Vol.
37, May 1983, pp. 101−105.
[14] J. D. Foley et al., Computer Graphics: Principles and
Practices, 2nd. edition, Addison−Wesley, Reading, Mass.,
1990, p. 589.
[15] W. H. Press et al., Numerical Recipes in C, Cambridge
University Press, Cambridge, Mass., 1988, pp. 636−643.
[16] W. H. Press et al., Numerical Recipes in C, Cambridge
University Press, Cambridge, Mass., 1988, pp. 673−676.
[17] R. C. Gonzalez, P. Wintz, Digital Image Processing, 2nd
edition, Addison−Wesley Publishing Company, Reading,
Mass., 1987, pp. 162−163.

Plate 1: When two gray squares of equal luminance are
placed on a background of varying luminance, the square
on the lighter background appears darker and vice versa.
If the luminance of the gray squares is equal to the
luminance of the background, the squares blend into the
background.
Plate 2: The black areas in a. represent the areas of
missing data. The missing areas have been filled in b.
based upon the luminance of the adjacent colors. The
corresponding grayscale version of b. is shown in c.
Plate 3: Three black geometric shapes are used to
simulate areas of missing data in a 2D slice through a 3D
data set which contains no missing data The Restored
version appears above the un−Restored.
a.
b.
c.
Plate 4: The Restoration sequence on Total Ozone
data from Nimbus−7 TOMS for Feb. 13, 1993: a. the
original data; b. using Rank and Median filters; c.
final step after using the Laplacian fill.

Please reference the following QuickTime movie located in the MOV
directory:
QTOZONEJ.MOV
Copyright © 1994 by NASA Goddard Space Flight Center
QuickTime is a trademark of Apple Computer, Inc.

Abstract
Meaningful scientific visualizations benefit the interpretation
of scientific data, concepts and processes.
To ensure meaningful visualizations, the visualization
system needs to adapt to desires, disabilities and abilities
of the user, interpretation aim, resources (hardware,
software) available, and the form and content of
the data to be visualized. We suggest to describe these
characteristics by four models: user model, problem
domain/task model, resource model and data model.
This paper makes suggestions for the generation of a
user model as a basis for an adaptive visualization
system.
We propose to extract information about the user
by involving the user in interactive computer tests and
games. Relevant abilities tested are color perception,
color memory, color ranking, mental rotation, and fine
motor coordination.
1. Introduction
User Modeling for Adaptive Visualization Systems
G. O. Domik B. Gutkauf
University of Paderborn University of Colorado
D-33095 Paderborn, Germany Boulder, CO. 80309-0430, USA
domik@uni-paderborn.de gutkauf@cs.colorado.edu
The transformation of physical phenomena (denoted
as “Reality” in Figure 1) into computer readable
numbers is a prerequisite to computer visualization
([1]). Visualization denotes the actual process of mapping
numbers to pictures. The viewer’s interpretation
of a picture is the final stage of visualization. Figure 1
distinguishes between two goals of visualization: gaining
new insight into the numbers represented in the
picture (e.g. interpreting the quality of a computer
model representing a physical phenomena) or gaining
a better understanding of the real phenomena itself.
Meaningful computer visualizations benefit the interpretation
of (scientific) data, concepts and processes.
Qualitative measurements for the meaningfulness of
pictures have been defined as “expressive” and “effective”
by [2]. Such systems are also called “graphically
articulate” [3]. Often a trial-and-error approach leads
to finding the most expressive and effective (graphically
articulate) visualizations. Ineffective, inexpressive
(graphically inarticulate) computer visualization is
at best useless and at worst misleading. However, the
value of a picture for the purpose of a particular interpretation
is often not obvious to the viewer before its
use for interpretation. The same picture might bring
about new insights to one user, but not to another; the
same picture might be effective for one scientific problem,
but not for another; the same animation might be
adequate to understand a problem on one type of hardware,
but not on another. In order to generate the most
meaningful picture for a specific instance, a careful
mapping process from “numbers to pictures” is necessary.
The next section discusses the factors influencing
the quality of a visualization.
2. Factors influencing the quality of visualizations:
data, user, task, problem domain
and resources
Past scientific visualization literature, e.g. [4], [5],
[6], [7], [8], [9], indicate the need for a careful mapping
process from numerical data to visual attributes
of a picture. A meaningful visualization must fit syn-
Reality Numbers Pictures Viewer
Figure 1: The essence of visualization is a mapping process from numbers to pictures.

tax and semantics of the numerical data values to be
represented, support the interpretation aim (task), conform
to the problem domain, adapt to the user as well
as be adequately supported by available computer resources.
Large scientific institutions, e.g. the National
Center for Supercomputing Applications (NCSA),
have in the past provided visualization experts to aid
scientists with the mapping process on an individual
basis. Visualization experts collect all background information
necessary to generate meaningful pictures
before starting the actual visualization process, therefore
minimizing trial-and-error time. In lieu of help
from a visualization expert, the visualization system itself
should be “knowledgeable” about background information.
We suggest to formally describe the knowledge
of the system by four models: user model, data
model, resource model, and problem domain/task
model. The more knowledge the visualization system
possesses about user, data, resources and problem domain/task,
the better it can adapt to one specific visualization
instance. Lack of knowledge in any of the
above named areas will lead to a trial-and-error approach
in finding meaningful visualizations, or in the
worst case, will result in misleading visualizations. We
call visualization systems that are able to conform to a
specific visualization instance “adaptive visualization
systems”. Adaptive visualization systems belong to a
new class of “intelligent visualization systems”.
2.1 Data model
The “data model” serves to organize information in
the numbers to be visually represented. The potential
of visual indicators in a picture to express information
as well as artifacts is dependent on syntactic and semantic
characteristics of these numbers, such as dynamic
range, data type, dimensions, structures and dependencies.
The need for data models to organize complex data
sets for visualization has recently been recognized, e.g.
[10], [11], [12], [13],[14]. However, the use of data
models for adaptive or intelligent visualization systems
is still rare. Mickus-Miceli ([15]) suggests the use
of an object-oriented data model to model both the
structural and behavioral components of the data as a
basis for an intelligent visualization system.
2.2 Resource model
The “Resource model” identifies the hardware and
software environment a user works in. It should contain
information on the display hardware, input and
output devices, processors, special graphics extensions,
as well as software modules available, and any
other factors influencing capabilities and limitations of
the computer. If relevant, this also includes lightening
of the room the user works in, or other factors influencing
the performance of the system without actually
being part of the system.
2.3 Problem domain/task model
Certain problem domains prefer specific pictorial
representations. Often this preference is rooted in tradition,
and emphasized through education. Sometimes
the use of particular visual cues seems “common
sense”, such as the use of blue for lower elevations (in
particular for water), or for cold temperatures. However,
if the objects to be represented are abstract, artificial
color mapping is established. In electrical engineering,
positive charges are encoded in red and negative
charges in blue or black. While such color mapping
is not “common sense”, it is taught in schools
universally throughout the world, and becomes “common
sense” for electrical engineers.
In a similar way, different interpretation aims (such
as “locating”, “visually correlate”, or “identify”) demand
different visual expressions for quick and accurate
interpretation. While problem domain and interpretation
aim are very different factors influencing the
quality of visualizations, we here suggest the use of
one model to represent information on both. This has
no deeper purpose than to minimize the number of
models in an adaptive visualization system.
The fourth model, the user model, is described in
more detail in the next section.
3. User model
The “user model” describes the collective information
the system has of a particular user (viewer). A picture
is subjectively interpreted by the viewer in dependency
of past experiences, education, gender, culture,
and individual limitations, abilities and requirements.
E.g., color deficient viewers are limited in interpreting
color pictures; a person with deficient fine motor skills
will have problems accurately pointing at small objects
on the screen. In order to create a user model, the system
needs to learn facts about the user. Most of these
facts can be extracted from observing the user performing
special tasks. In the following sections we
will describe the generation of a user model in more
detail.

3.1 User modeling
User modeling denotes the generation of the user
model by extracting information from the user ([16]).
Information can be extracted from the user in one of
three ways:
a) Through explicit modeling: the user typically fills
out a form and answers to direct questions.
b) Through implicit modeling: the user is being observed
in his/her use of the system.
c) The user is asked to solve special tasks and is being
observed in doing so.
A complete user model evolves in several stages,
whereby each style of user modeling is being used.
Typically, the extraction of information starts with explicit
modeling to inquire about gender, age, or education.
Subsequently, the user has to complete special
tasks that reveal limitations of his/her vision and/or
hand-eye coordination. By continuously observing the
user in his/her use of the visualization system, the user
model can be improved over time. Significant information
for the user model is expected from the completion
of special tasks. We have therefore focused on
implementing special tests that reveal abilities and disabilities
of the user. Most of the tests are currently implemented
at the motivational level of educational
games: users enjoy performing the tests and may call
them up repeatedly. Motivation in performing these
tests is important both for the purpose of user modeling
as well as for training certain abilities through repetitive
performance of the tests.
3.2. Parameters of the user model
The following five abilities describing the performance
of a user as relevant for visualization have been
investigated in more detail: color perception, color
memory, color ranking, mental rotation, and fine
motor coordination. Additionally we have found several
abilities that might be of influence, but have not
been investigated in any detail so far: reasoning, size
recognition, perceptual speed, visual versus verbal recognition,
and embedded figure recognition.
Following is a description of each of the five user
abilities investigated, explaining its relationship to visualization
and the special tasks which extract information
about this ability from the user. Between fifteen
and twenty users were tested for each of the abilities.
In three of the five abilities (perception, mental
rotation, fine motor coordination) we were able to derive
quantitative evaluations of the results. In two of
the five abilities (color ranking and color memory) the
test results are described in a qualitative manner.
Color perception: Color perception is defined as
the process of distinguishing points or homogeneous
patches of light by a subject. It is a cognitive ability
and varies among people. Color perception depends on
a person’s color deficiency, gender, ethnological information,
quality of monitor used and the lightening situation
in the room. No general assumption of color
perception for one visualization instance can therefore
be made a priori.
Since color graphics workstations have become a
standard equipment at scientific institutions, color coding
is one of the most commonly used visualization
techniques to represent data. Colors used must be set
to values that are easily distinguished by the viewer.
Therefore the goal for the “color perception test” is to
find a set of discriminatory colors. A standardized test
optometrists use to determine color deficiencies, the
“Farnsworth Hue 100” test ([17]), has been used as a
basis. In a very similar manner, the essence of our test
is to sort forty pastel (low saturation) color chips so
that every chip has a minimal color difference to its
neighbor. Figure 2 (upper row) shows the arrangement
of color chips at the start of the test, Figure 2
(lower row) shows the correct result after rearranging
the chips. In order to use perceptually similar leaps between
the forty color chips, the CIELUV color space
([18]) is used. Colors that can not be distinguished
would appear out of order in the test results. We use
only forty colors instead of one hundred or more to
keep the user motivated to perform the test. Our first
results indicate that these forty colors give us enough
information to eliminate colors that would cause problems
for color coding.
For twenty users two parameters each were measured:
a) the time it took the user to perform the color
perception test, and b) an error factor describing the errors
in the final arrangement of the color chips. In
order to compare measured time between slow and
quick users, we used the time difference between the
actual timing measurement of the color perception test
and similar tasks (e.g. sorting numbers, sorting angles).
A value of zero on the time difference axis in
Figure 3 (a) specifies that the color perception test was
finished in the same or less amount of time than other
similar tasks. Specifically user 9 and user 18 show significantly
long times to perform the test. Both users
have been diagnosed as color deficient in competent
optometric tests.
The error factor e c calculated for each test result is

Time Difference (sec.)
Error Factor (e c )
Error Factor (e c )
700
600
500
400
300
200
100
2
1
5
4
3
2
1
1 5 10 15
User Number
Figure 3(a): Timing for color perception
test
green
dependent on the misplacements of chips relative to
their neighbors:
e c = abs(c c -c l ) + abs(c r -c c )-2,
with c c ... current chip number, c l ... number of left
chip, c r ... number of right chip. Figure 3 (b) reflects
the error factor for each of the forty chips for a typical
non-color deficient user who misplaced one color chip.
Figure 3 (c) shows the error factor for each chip for
user 9 (color deficient).
20
Color ranking: Color ranking is the association of
color with ordinal or quantitative data items. If a temperature
scale is to be displayed, standard color mapping
(low temperature mapped to blue, high temperatures
mapped to red, yellow and white) may be used. If
no “standard mapping” is known for specific data
characteristics, an artificial ranking of colors from
“low” to “high” is necessary. Because human perception
has no inherent ranking of colors, the association
of color with ordinal or quantitative data items is individual,
depending on the problem domain as well as
the viewer’s education, color deficiencies and preference.
Interpreting the meaning of a color picture often involves
instant recognition of low values versus high
values. The presence of a color scale showing the used
1 5 10 15 20
Chip Position
25 30 35 40
Figure 3(b): Error Factor of non color deficient user
yellow-green cyan-blue
1 5 10 15 20 25 30 35 40
Chip Position
Figure 3(c): Error factor of color deficient user
color range and its associations is extremely important.
Still, misinterpretation can only be avoided, if the
viewer is in agreement with the association. The goal
of the “color ranking test” is therefore to have the
viewer reveal their own intuition on color sequences.
This is done by having the user match colors of an
available color gamut (see Figure 4) to a number scale
from one to ten. Typical color scales expected from
this test were:

lack (0) - purple - blue - cyan - dark-green - lightgreen
- yellow - orange - red - white (10)
or
black(0) - purple - blue - cyan - dark-green - lightgreen
- red - orange - yellow - white (10)
Results from the color ranking test proved very individual.
Even though spectral scales and increasing
perceptual brightness were among the fifteen test results,
most users lacked a specific strategy. We are
currently devising the test to receive more meaningful
results. The user chosen color scale should be used
preferably for visualization of ordinal data items.
Color memory: Color memory is the ability of a
person to recall a color from memory. Even though we
are able to distinguish between numerous colors [19]),
we can only do so when comparing one color to another.
If color samples are not directly compared to each
other we can only process a small fraction of colors
(around ten). Interpreting the meaning of a color picture
series often involves recognizing (and therefore
remembering) the same color in several pictures. Specifically
for the distinction of nominal data items, we
prefer to use colors the user can easily distinguish.
The idea for the corresponding “color memory”
test has been borrowed from the well known game
“Memory”. Nine color cards (Figure 5) are shown to
the user for ten seconds. When they are turned facedown,
the user needs to pick the correct color corresponding
to one color card that will be shown to
him/her. The user may guess as many times as needed,
but the amount of guesses is entered in a log file. The
game can vary in difficulty by changing the timing allowed
to view the nine cards. As in all other tests and
games introduced in this paper, the results (what colors
are memorized best) are not only dependent on user’s
abilities but also on the quality of hardware and lightning
environment. For our purposes we see this as an
advantage of modeling the current environment as perfectly
as possible rather than a lack of separation between
the content of different visualization models
(e.g. user model and resource model).
Mental rotation: Mental rotation is the ability to
rotate mental representations of two and three dimensional
objects. The response time required by degree
of rotation is called “mental rotation rate”. Determination
of the mental rotation rate is used to study the
speed of spatial information processing and as such
part of intelligence/ability tests.
One of the most interesting and pioneering re-
search areas in visualization today is the use of glyphs
to represent multi-variate data (e.g. [20], [21], [22],
[23]). In some cases, rotation of glyphs (in two or
three dimensions) denote changes of variables. In
order to interpret rotated glyphs, mental rotation needs
to be applied. The Joslyn-Glyph ([23], representation
of a cube with variable height, width, depth, rotation
angles in three dimensions, and color) is a good example
for the three-dimensional case: an interpretation of
an individual glyph is only possible by performing
mental rotation. The “mental rotation test” consists of
a set of twenty pairs of rotated objects; the viewer has
to decide if the represented objects are the same or
mirror images of each other. A standard set of 3-dimensional
objects ([24]) was used. Both error rate and
response time are measured and recorded. “Response
time” of the mental rotation test was again normalized
similarly to the user perception test. A value of zero
expressed (almost) no hesitation between deciding on
mirror or rotated image. The results of fifteen test
Error (# out of 20)
11
10
9
8
7
6
5
4
3
2
1
III
I
IV
1 2 3 4 5 6 7 8 9
Time (sec.)
Figure 6: Result of mental rotation test
users in Figure 6 are divided into four quadrants: quadrant
I shows users that are both fast and accurate in
their mental rotation ability; quadrant IV shows users
that are both slow and error prone. The divisions between
the quadrants are established by the mean values
over all users. For users inside quadrant IV, interpretation
of glyphs that encode rotation should be
avoided.
Fine motor coordination: Fine motor coordination
is the user’s ability to perform precision manual
tasks, demanding good eye-hand coordination and
motor speed. This ability is influenced by age, gender,
experience, motor (dis)abilities of the user as well as
the input device used. With increasing age the ability
to coordinate the interaction between eye and hand de-
II

creases ([25]). While in general men are faster in tapping
a single key, women are faster in moving small
objects to a specific destination. Players of computer
games are an excellent example to show the increase
of fine motor coordination with experience. Naturally,
motor disabilities impair fine motor coordination. Additionally
to these individual characteristics of a user,
type and quality of the input device used plays a large
role in the accuracy of fine motor coordination.
Interactive visualization requires fine motor coordination,
e.g. to point at small objects or to trace structures
on the screen. The “fine motor coordination test”
as currently implemented determines the user’s speed
and error rate to trace a predefined path (see Figure 7).
Error rate, as an indicator for accuracy, is measured by
counting the number of times the user leaves the predefined
path (green path in Figure 7).
Error Factor *
10
9
8
7
6
5
4
3
2
1
The result of fifteen test users in Figure 8 compares
time and error rate: as in the previous user test, the
quadrants are separated by the mean error rate and
mean user’s speed, respectively. Quadrant I describes
the best candidates for a quick and accurate access via
mouse. Quadrant IV describes users with a certain
handicap for these tasks. Initial tests showed, that the
type of pointing device used was a major influence to
the results. In order to observe differences in the user’s
ability alone, the same mouse was used for all tests.
4. Outlook
III
Meaningful visualizations have often been created
through trial-and-error in the past. The generation of a
priori meaningful visualizations will effectively increase
scientific productivity. Such visualizations can
be supported by adapting visual representations to the
characteristics of user, data, resources, and problem to
be solved. In this paper we have explained the use of
tests and games to extract information characterizing
the user. It was our goal to implement tests as well as
games in a motivational format.
Ability tests and their implementation are explained
in full detail in [26]. We are planning the following
improvements for the future:
Several additional user abilities, such as reasoning,
size recognition, perceptual speed, visual versus verbal
recognition, and embedded figure recognition, seem to
be of importance but their ability tests have not been
designed yet. Additionally, the user’s knowledge about
visualization should influence the use of visual representations.
Some of the implemented ability tests need
to be extended: The mental rotation test is specifically
developed for three dimensional objects, however, a 2-
1 5 10 15 20 25 30
Time (sec.)
* Number of times predefined path was accidently left
I
IV
Figure 8: Results of fine motor coordination test
II
dimensional rotation test would also seem of importance.
The fine motor coordination test is specifically
developed to test the user’s abilities with a mouse;
alternative tests to determine strengths/weaknesses of
other input devices should be developed. While the results
of our ability tests are well understood, they have
not been implemented yet into a visualization system,
which will be an important step for the future.
Acknowledgement
Thanks to Dr. Stephen Franklin, University of California,
Irvine, for very useful criticism to a previous
version of Figure 1. The authors also gratefully acknowledge
the conversion of the “color perception
test” from HSV to CIELUV (Figure 2) by M. Riese
and M. Roland and the development of the game
“Color memory” (Figure 5) by J. Blume and M.

Buschmann-Raczynski.
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Figure 2: (upper row) Arrangements of color chips at start of color perception test.
(lower row) Correct result after rearranging chips.
Figure 4: Color gamut for color ranking test.
Figure 5: Color Memory game. Nine color cards are
uncovered for ten seconds at the beginning of the game.
Figure 7: Fine motor coordination test. User traces green path from white rectangle
to black rectangle. Errors (leaving green path) are counted and speed is measured.

Streamball Techniques for Flow Visualization
Manfred Brill
Hans Hagen
Hans�Christian Rodrian
Computer Science Department
University of Kaiserslautern
Germany
Abstract
We introduce the concept of streamballs for �ow visu�
alization. Streamballs are based upon implicit surface
generation techniques adopted from the well�known
metaballs. Their property to split or merge automati�
cally in areas of signi�cant divergence or convergence
makes them an ideal tool for the visualization of arbi�
trary complex �ow �elds. Using convolution surfaces
generated by continuous skeletons for streamball con�
struction o�ers the possibility to visualize even tensor
�elds.
1 Introduction
1.1 Streamlines and stream surfaces
Streamlines� streaklines� pathlines and timelines play
an important role in �ow visualization. Most of these
terms are directly derived from experimental �ow vi�
sualization� where the corresponding phenomena are
generated by inserting foreign material into the �ow
and observing it while it moves through the �eld.
� Streaklines are produced by continuously inject�
ing material like smoke or little hydrogen bubbles
into the �ow at certain points and watching the
resulting clouds of particles.
� Pathlines can be obtained by putting small ob�
jects into the �ow �eld and exposing a photo�
graph for a longer time� thus depicting the traces
of these objects over time.
� Timelines are given by observing a line of particles
�owing with the stream and making snapshots at
several time steps.
Wladimir Djatschin
Stanislav V. Klimenko
Computer Science Department
Institute for High Energy Physics �IHEP�
Russia
1
� Streamlines �nally are de�ned as curves tangent
to the velocity �eld in every point.
For steady �ows streaklines� pathlines and stream�
lines obviously coincide �7�.
In computational �ow visualization� streak�� path��
time�� and streamlines are often simulated to get an
insight into the structure of a �ow �eld. Though these
constructions are powerful tools for the investigation
of two�dimensional �elds� they are not very well suited
for the visualization of three�dimensional �eld data as
they heavily su�er from display ambiguities when be�
ing displayed in two dimensions on a computer screen.
Therefore� streak�� path�� stream�� or timelines are of�
ten used to build time surfaces or streak�� path�� and
stream ribbons� �tubes� or �surfaces� which in conjunc�
tion with standard lighting and shading techniques
can provide a much better idea of the overall topology
of a 3D �ow �eld. Furthermore� local parameters of
the �eld can be mapped onto these surfaces and thus
be displayed together with the �eld�s velocity struc�
ture.
1.2 Previous work
Many new techniques for �ow visualization have been
presented in the last few years �7�.
Schroeder et. al. �8� introduced the stream polygon
technique� where n�sided� regular polygons perpen�
dicular to the local velocity vector are placed along
a streamline. E�ects like twist or scalar parameters
of the �eld are displayed by accordingly rotating and
shearing the polygons or changing attributes like ra�
dius or color. By sweeping stream polygons along
streamlines� three�dimensional stream tubes can be
built.

Another method is the generation of stream sur�
faces by connecting adjacent streamlines with poly�
gons. Special care has to be taken whenever diver�
gence of the �ow causes adjacent streamlines to sep�
arate or convergence causes streamlines to come very
close to each other� as the polygonal approximation
may become poor in these cases.
Helman and Hesselink �4� proposed an algorithm
which connects the critical points of the vector �eld
on the surface of an object to form a two�dimensional
skeleton. This skeleton represents the global topology
of the �ow on the surface. Starting from points on
the skeleton� streamlines in the �ow around the object
are calculated. By tesselating adjacent streamlines�
stream surfaces are built. To avoid splitting of the
stream surfaces in areas of divergence� the surfaces are
recursively re�ned by introducing additional starting
points for the streamline calculations.
Hultquist �5� introduced an algorithm which simul�
taneously traces a set of particles originating from dis�
crete points on some curve through the �eld and con�
nects the resulting paths with triangles. In this way�
an advancing front of a steadily growing stream sur�
face is obtained. Whenever the divergence between
two of these particles becomes too big� new particles
are inserted into the front� when particles come too
close to each other� some of them are removed.
Van Wijk �9� proposed the usage of so�called surface
particles for �ow visualization. With this technique� a
big number of particles released from a number of par�
ticle sources are traced through the �ow �eld. By po�
sitioning the particle sources on a curve and displaying
all particles as small geometric primitives shaped and
coloured in dependency of certain �eld parameters�
the impression of stream surfaces textured according
to local parameters can be given.
A di�erent approach which guarantees the genera�
tion of smooth stream surfaces was also introduced by
Van Wijk �10�. The central concept of this method is
the representation of stream surfaces as implicit sur�
faces f�x� � C representing the sweep of an initial
curve through the �eld. The shape of the initial curve
is de�ned by the value of f at the in�ow boundaries.
To calculate f� Van Wijk proposes two methods� solv�
ing the convection equation or tracing backwards the
trajectories of grid points. The same technique can be
used for the construction of time surfaces or stream
volumes.
1.3 Overview
In this article we present a new method for �ow visu�
alization based upon implicit surface generation tech�
2
niques adopted from metaballs. We call the resulting
objects streamballs.
In particular� streamballs are distinguished by their
ability to split or merge with each other automatically
depending on their distances. By advancing appropri�
ate skeletons through the �eld and displaying the re�
sulting streamballs� streak�� stream�� path�� and time�
lines as well as �surfaces� or �volumes can easily be vi�
sualized� no matter how complex the given �eld may
be. The mathematical representation of streamballs
o�ers a variety of mapping possibilities for parame�
ters of the �ow �eld.
Section 2.1 introduces the concept of streamballs
de�ned by a set of discrete centerpoints and their
usage in �ow visualization. In Section 2.2� the con�
cept of streamballs constructed from continuous two�
dimensional skeletons� which open up a wider range
of visualization possibilities� is introduced. In Sec�
tion 2.3� some mapping techniques for streamballs are
presented. The rendering method that we used is de�
scribed in Section 2.4. Finally� Section 3 contains a
short summary and concluding remarks.
2 Streamballs
2.1 Streamballs with discrete skeletons
2.1.1 Basic concept
In 1982� Blinn �1� introduced the usage of implicit
surfaces to display molecular compounds. With his
method� a potential �eld F de�ned by a �nite set S of
centerpoints si is used to generate an implicit surface
which represents the molecules.
At a given point x in space� F �S� x� is given as the
sum of weighted in�uence functions Ii�x� generated by
each of these centers�
F �S� x� � X
i
wiIi�x� � X
wie �aifi�x� � �1�
where fi�x� describes the shape� ai the size� and wi
the strength of the potential �eld.
Based on this �eld an isosurface F �S� x� � C is
constructed.
For example� if there is only one centerpoint s1 and
if a1 � 1
R2 and f1�x� � jjx � s1jj 2� the resulting iso�
surface will be a sphere whose radius depends on R.
G. Wyvill et. al. �11� used a similar technique to
construct what they called soft objects. To localize
i

the in�uence of the centerpoints and to avoid the com�
putation of the exponential function� they applied the
following polynomial approximation�
Ii�x� �
8
����
����
a fi�x�6
R 6
� b fi�x�4
R 4
� c fi�x�2
R 2
�
� 1 � fi�x� � R
0 � fi�x� � R
�2�
with fi�x� � jjx � sijj and a� b� and c chosen to satisfy
Ii�0� � 1 Ii�R�2� � 0�5 Ii�R� � 0
I 0 i�0� � 0 I 0 i�R� � 0
�3�
The described primitives are commonly known as
metaballs or blobby objects. Metaballs are distin�
guished by numerous useful properties�
� A single centerpoint generates a single� spherical
surface.
� As two centerpoints come close� their correspond�
ing shells blend smoothly� i.e. the resulting surface
is C 1 �continuous.
� If two or more centerpoints coincide� a single�
larger sphere is produced �in fact� if the value of
C is chosen properly� the sphere generated by two
of such centers will have exactly twice the volume
as a sphere produced by one single centerpoint�.
� as two centerpoints separate� the blending process
is reversed.
2.1.2 Discrete streamballs
The basic idea for visualizing �ow data with stream�
balls is to use the positions of particles in the
�ow as centerpoints for implicit surfaces� which then
by blending with each other form three�dimensional
streamlines� stream surfaces etc. The centerpoints can
be looked upon as a discrete skeleton of the surface
constructed in this way. Referring both to the term
metaballs and to the usage of discrete skeletons we
call the resulting three�dimensional objects discrete
streamballs.
To represent a streamline using discrete stream�
balls we simply distribute a number of centerpoints
along this streamline close enough to each other to
let the surrounding isosurfaces blend. This blend�
ing process is shown in Figure 1. By increasing the
number of centerpoints si step by step� a continu�
ous� three�dimensional representation of a streamline
is produced.
3
Figure 1� The blending process of the streamballs.
To construct stream surfaces� a number of particles
originating from di�erent positions on some starting
curve are advanced through the �ow �eld. Their posi�
tions at several time steps are used as a skeleton for the
streamballs. When the particles initially are close to
each other� they will produce a continuous and smooth
surface� which will split automatically in areas where
divergence occurs� and merge automatically in areas of
convergence. An example for this can be seen in Fig�
ure 2� where the �ow around an obstacle� simulated
by the combination of a source and a sink� is shown.
Notice how the streamballs split around the obstacle
and merge again behind it. Color is used to map the
velocity of the �ow.
Figure 2� Discrete streamballs �owing around an ob�
stacle.
Time surfaces for any given time t � t0 � �t are
built by distributing skeleton points on a starting sur�
face at a time t � t0 and letting them �ow with the

Figure 3� Three snapshots of a time surface hitting an
obstacle.
�eld for a time �t. Figure 3 simultaneously shows
three snapshots of a time surface hitting the same ob�
stacle as in Figure 2. Though the time surface splits
on the obstacle� the obstacle�s shape can clearly be
seen.
Stream volumes of arbitrary initial shape are gen�
erated by advancing a cloud of particles through the
�ow �eld� which are initially arranged to form the de�
sired shape� and using their positions over time as a
skeleton for the streamballs.
As can be seen from the �gures� streamballs have
the convenient property to split automatically in areas
of signi�cant divergence and to merge with each other
in areas where convergence occurs. This behavior is a
natural consequence of the properties of metaballs.
Thus� streamballs will not necessarily produce
closed stream surfaces. The way in which streamballs
behave in such cases� however� can give valuable in�
formation on the structure of the �ow. In order to
produce closed surfaces nevertheless� one can simply
release additional particles in areas of high divergence.
2.2 Streamballs with continuous skele�
tons
2.2.1 Basic concept
Bloomenthal and Shoemake �2� generalized the idea of
metaballs proposing the usage of an arbitrary skeleton
consisting of a continuum of points �i.e. lines� curves
etc.� instead of a limited number of centerpoints to
generate the in�uence function.
The �eld function F is given by the convolution of
the skeleton�s characteristic function � S�x� with the
weighted in�uence function I�x��
4
F �S� x� � � S�x� � �I�x� �
Z
�2S
����I��� x�d� �4�
Using an exponential in�uence function� we get
F �S� x� �
Z
�2S
jjx��jj2
�
����e 2 d� �5�
The convolution surface is given by building an iso�
surface F �S� x� � C.
To get reasonable computation times� we used an
in�uence function similar to the one we already used
for the streamballs with discrete skeletons�
I��� x� �
and
8
���
���
af��� x� 3 � bf��� x� 2 �
� cf��� x� � 1 � f��� x� � 1
f��� x� �
jjx � �jj2
R 2
0 � f��� x� � 1
�6�
�7�
again with a� b� and c chosen to satisfy the conditions
�3�.
The objects generated in this way preserve all useful
properties of Blinn�s implicit surfaces.
2.2.2 Continuous streamballs
Convolution surfaces with continuous skeletons are a
powerful tool for �ow visualization. They provide the
ability to produce perfectly smooth surfaces around
their skeletons.
With the discrete streamballs� we used a set of par�
ticle positions as a skeleton for an implicit surface.
Now we use these points to construct a continuous
skeleton which in turn generates the implicit surface.
To represent a streamline� for example� we trace a
particle along this streamline through several discrete
time steps and connect the single particle positions to
build the skeleton of the streamball. The resulting
three�dimensional streamline generally will be thin�
ner and more regular than one produced by discrete
streamballs using the same points as a skeleton.
Stream surfaces again are constructed from a set of
three�dimensional streamlines which are very close to
each other �Figure 4�.
Similarly� time surfaces are built of a number of
three�dimensional timelines.

Figure 4� A stream surface produced by a rake of 100
3D�streamlines in a �ow �eld containing a vortex.
When time surfaces are traced through the �eld�
special attention has to be paid to obstacles to prevent
the time surfaces from being pulled �through� the ob�
stacle. This problem can be overcome by controlling
the length of the skeleton segments and dividing them
if necessary.
2.3 Mapping of local �eld parameters
Streamballs o�er a variety of possibilities for the map�
ping of local �eld parameters. Besides standard map�
ping techniques� new mapping techniques based on the
mathematical representation of the streamballs can be
applied.
With discrete streamballs� for example� an easy
method to map the value of a local parameter along
a streamline is to choose the radius of the in�uence
functions of each skeleton point according to the pa�
rameter�s value at that point. The result is a three�
dimensional streamline whose diameter corresponds to
the magnitude of the parameter to map. In Figure 5
we used this technique to map the velocity of the �ow.
A similar method has been used for the upper layer
of the streamballs shown in Figure 6. The radius of the
in�uence function of these streamballs was increased
at several discrete positions along the streamline. By
choosing the distances of these positions dependent
on the absolute value of the velocity of the �eld� a
good idea of this parameter�s value along the stream�
line is given. It can be seen clearly that velocity is
lower in front of the obstacle and higher on the side
of it. To increase the radius of the in�uence functions
at certain positions� we just placed discrete stream�
balls� each with a skeleton consisting of exactly one
centerpoint� along the streamline.
5
Figure 5� Discrete streamballs in a �ow �eld contain�
ing a vortex. Both radius and color show velocity.
A di�erent method has been used to map the ve�
locity on the surface of the lower layer of streamballs
in Figure 6.
Figure 6� Di�erent mapping methods using stream�
balls.
With this mapping technique local scalar parame�
ters are mapped as roughness of the surface. For this�
the amplitude of a three�dimensional oscillating func�
tion F �x� is modulated by the value of velocity. The
potential function of the streamballs then is superim�
posed by F �x�. For simplicity� we choose F to be
F �x� � sin�f1x1�sin�f2x2�sin�f3x3�� �8�
where the fi are �not necessarily di�erent� frequencies.
As the described mapping techniques in�uence only
the geometry of our streamballs� more common map�
ping techniques� using e.g. material properties of the

surface� can be applied simultaneously. In all our �g�
ures we additionally used simple color mapping to de�
pict velocity.
The streamballs in the middle of Figure 6 show a
di�erent color mapping method. The component of
the velocity which describes the deviation of the �ow
from the central axis is mapped as color spots on the
surface of these streamballs. The density of the color
spots depends on the absolute value of the considered
velocity component.
Similar to a technique introduced by Hesselink and
Delmarcelle �3�� continuous streamballs can be used for
the representation of tensor data. For this purpose the
skeleton is directed along a so�called hyperstreamline
�a curve tangent to the main eigenvector of a tensor
�eld�. At every point of the skeleton a local coordinate
system ��1� �2� �3� is used with �1 tangent to the main
eigenvector e1 at this point and �2 and �3 oriented in
the directions of the two eigenvectors e2 and e3 which
are perpendicular to the main eigenvector. Choosing
the radius of the in�uence function in the directions
of �2 and �3 corresponding to the absolute values of
the two eigenvectors e2 and e3� an asymmetrical in�u�
ence function can be constructed. The resulting iso�
surface will have an asymmetrical cross section with
orientation and diameter dependent on the directions
and eigenvalues of these two eigenvectors. Using some
mapping technique to show the eigenvalue of the main
eigenvector� it is possible to represent not only direc�
tion� but even the magnitude of all three eigenvectors
at the same time.
2.4 Rendering
For rendering� the �eld function F �S� x� of the stream�
balls is evaluated on a regular grid. The isosurface
F �S� x� � C is extracted from this grid by a simpli�ed
marching cubes algorithm� and the resulting triangles
are rendered using Iris Explorer.
The modi�ed marching cubes algorithm processes
the grid slice by slice� so only two slices have to be
held in memory at one time �6�. As the polynomial in�
�uence functions �2� or �6� are used� each part of the
skeleton has only a local in�uence on the �eld func�
tion. Therefore� when computing the values of the
�eld function for a grid point x� F �S� x� has to be
evaluated only for those parts of the skeleton which
are close enough to that grid point to in�uence the
potential �eld at x. This greatly reduces computation
costs.
For high grid resolutions� as they may be neces�
sary to see �ne details� the huge number of triangles
generated by the marching cubes algorithm is a con�
6
siderable drawback. For this reason we are working
on a fast adaptive triangulation algorithm� which will
both reduce the number of �eld function evaluations
and the number of triangles produced.
3 Summary
The proposed technique proves to be useful for 3D �ow
visualization in several ways�
� The representation of streamlines� stream sur�
faces and stream volumes as well as time surfaces
is possible in a quite easy and natural way.
� Streamballs split or merge automatically in areas
of signi�cant divergence or convergence. Valu�
able information on the �ow is given by the way
in which the streamballs divide or blend in these
cases.
� Due to the underlying mathematical representa�
tion� streamballs provide powerful mapping pos�
sibilities for �ow�related parameters. Hence� they
are not only suited for the examination of vector
�elds� but can even be used for the exploration of
the complex structure of tensor �elds.
� Streamballs can be applied even in cases of very
complex �ow �elds.
Acknowledgements
The research for this project is funded by a grant of the
�Stiftung Innovation f�ur Rheinland�Pfalz� awarded to
the University of Kaiserslautern. Wladimir Djatschin
is supported by the DAAD.
Many thanks to Henrik Weimer for programming
and helpful discussion.
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Volume Rendering Methods for Computational Fluid Dynamics Visualization
David S. Eberty Roni Yagelz Jim Scottx Yair Kurzionz
yComputer Science Department, University of Maryland Baltimore County, Baltimore, MD 21228 �
zDepartment of Computer and Information Science, The Ohio State University, Columbus, Ohio 43210
xDepartment of Aeronautical and Astronautical Engineering, The Ohio State University, Columbus, Ohio 43210
Abstract
This paper describes three alternative volume rendering
approaches to visualizing computational fluid dynamics
(CFD) data. One new approach uses realistic volumetric
gas rendering techniques to produce photo-realistic images
and animationsfrom scalar CFD data. The second uses ray
casting that is based on a simpler illuminationmodel and is
mainly centered around a versatile new tool for the design
of transfer functions. The third method employs a simple
illumination model and rapid rendering mechanisms to
provide efficient preview capabilities. These tools provide
a large range of volume rendering capabilities to be used by
the CFD explorer to render rapidly for navigation through
the data, to emphasize data features (e.g., shock waves)
with a specific transfer function, or to present a realistic
rendition of the model.
1 Introduction
This paper describes three alternative approaches for
volumetric visualization of CFD data. One new approach
uses a realistic surface and volumetric rendering and animation
system which will allow scientists to obtain accurate
photo-realistic images and animations of their simulations,
containing more information in a more comprehensible
format than current tools available to the researchers. The
second approach is based on ray casting without the modeling
of self shadows. It is mainly intended to support the
interactive design of transfer function. Currently this tool
supports transfer function mapping voxel value (density)
and gradient to opacity and mapping of voxel values to
colors. The third approach is mainly intended for interactive
preview of the data. It is based on an extended
implementation of the template based ray casting [1] that
supports screen and volume supersampling. To support
interactivity, this renderer is based on a very simple illumi-
� e-mail: ebert@cs.umbc.edu
nation model. Combined, these tools are shown to provide
effective visualization of scalar CFD data.
1.1 Background and the state of CFD
visualization
Computationalfluid dynamics is an active research area.
CFD research involves large three-dimensional volumes of
data. The data from CFD simulations often contains many
data values per three-dimensional location (e.g., velocity,
pressure, density, energy, temperature). Recently, scientific
visualization systems have been developed to aid CFD
researchers in the interpretation of these computational
data sets, including commercial systems such as Data Visualizer,
AVS, and Iris Explorer T M , and non-commercial
systems such as FAST by NASA Ames Research Center.
Techniques used in these systems include isosurface rendering,
stream-lines, contour plots, and three-dimensional
volume visualization.
Current visualization systems for CFD simulations are
lacking in the following way:
Many current systems fail to accurately display the entirety
of the three-dimensional data from the CFD simulations.
CFD simulations model three-dimensional
flow phenomena. Surface based rendering techniques,
stream-lines, and contour plots can only capture a
small fraction of the data in a three-dimensional simulation,
resulting in important information being hidden
or not easily discernible in the resulting images. Volume
visualization is an important tool for discerning
three-dimensional data, especially three-dimensional
scalar data. A comparison of Figures 1, 3 and Figure
2 reveals the quantity of information that can be lost
in traditional visualization systems. Figure 2 uses
a standard ray-marching volume rendering algorithm
[2] to produce results similar to a isosurface renderer;
whereas, Figure 1 uses our gaseous CFD visualization
system and Figure 3 is rendered by our ray caster
which employs a specially designed opacity transfer
function.

There has been some recent work on addressing some
of these shortcomings. However, these systems only
overcome some of the shortcomings of current CFD visualization
systems. Surface particles [3], and a combination
of volume rendering, vector fields, and texturing [4] have
been used to capture the dynamics and massive information
content in a CFD simulation.
While visualization systems for CFD have concentrated
on simple illumination and shading models, recent advances
in computer graphics for realistic rendering of gases
and fluids have produced dramatic and near photo-realistic
images of water, steam, fog, and smoke [5, 6, 7, 8, 9]. These
techniques use physically-based rendering techniques to
display these natural phenomena realistically. These photorealistic
rendering techniques can be incorporated into volume
visualization to provide important perceptual cues for
comprehending complex three-dimensional data.
Our systems for CFD visualization combine several
volume rendering techniques which provide the scientist
with the option to trade time for image quality. One
technique employs a sophisticated illumination model to
render our test data in approximately 83 seconds. The
second technique simplifies illumination and emphasizes
material modeling (via transfer functions) to render the
data in approximately 64 seconds. The third technique
is mainly intended for speedy rendering based on a basic
illumination model and material modeling which produce
an image in approximately 27 seconds. All reported times
in this paper where taken on a HP 9000 Series Workstation
model 735.
1.2 Turbo-jet compressor visualization
The development of advanced aerospace propulsionsystems
relies heavily upon the management of critical flow
features which influence performance and operating characteristics.
In order to understand the critical flow behavior,
numerical simulation techniques must be used to complement
experimental studies. Among the propulsion system
components in which some of the most complex flow behavior
is encountered are the turbomachinery components.
In the compressor, the flow may be entering a stage at a supersonic
velocity in which case a complex system of shocks
and expansion waves form through the blade passage. This
system of shocks and expansions not only produce losses
themselves, but also contribute to other flow features which
degrade performance. In order to control such flow features
in a manner to optimize performance and minimize losses,
numerical solution of the time-dependent Navier-Stokes
equations is necessary. Such solutions provide the velocity
field, along with pressure, density and temperature variation
in time and space through the compressor rotor blade
passages. To understand the nature of these phenomena
and their influence on performance, it is desirable to be
able to have a visual representation of the computed flow
data which can be compared directly with experimental
flow visualization. This type of data is particularly valuable
in the analysis of shock waves for which the temporal
and spatial variation can have a significant impact on the
mass flow rate of air passing through the compressor. The
use of interactive visualization will provide the engineer a
means of analyzing the behavior of such critical flow features
as he controls the flow entering the compressor stage
under investigation. Specifically, this represents control
of the operating condition through interactive variation of
the flow pressure, velocity, or temperature, all of which
influence performance characteristics. Other flow features
which are of specific interest and play crucial roles in the
overall performance and operation of turbomachinery devices
include: vortices which form around the blade leading
edges and along the corners at the blade - hub junction,
and boundary layer growth and separation. The analysis
of these features and their influence is also significantly
enhanced through the use of interactive flow visualization
of the type demonstrated here.
We have applied the three visualization techniques of
our system to this problem of visualizing the flow between
two blades of the compressor of a turbo-jet engine [10].
The computational mesh for the simulation is a threedimensional
curvilinear grid with dimensions of 64 � 46 �
18. For visualization purposes, the computational grid
was resampled to a rectilinear grid with dimensions of
64 � 64 � 32.
2 The gas visualization system
Our gaseous visualization system demonstrates the benefits
of realistic volumetric rendering techniques for flow
visualization.
A more complete description of the system can be found
in [5, 11]. The system has the following features:
Fast combination of surface-based objects and volume
data.
Fast, accurate, physically-based atmospheric attenuation
and illumination models for low-albedo gases.
Fast volumetric shadowing algorithm.
The system is a hybrid surface and volume rendering
system, which uses a fast scanline a-buffer rendering
algorithm for the surface-defined objects in the scene,
while volume modeled objects are volume rendered. The
algorithm first creates the a-buffer for a scanline containing
a list for each pixel of all the fragments that partially or

fully cover the pixel. Then, if a volume is active for a pixel,
the extent of volume rendering needed is determined.
The volume rendering is performed next, creating a-buffer
fragments for the separate sections of the volumes. Volume
rendering ceases once full coverage of the pixel by volume
or surfaced-defined elements is achieved. Finally, these
volume a-buffer fragments are sorted into the a-buffer
fragment list based on their average Z-depth values and
the a-buffer fragment list is rendered to produce the final
color of the pixel.
The system supports volume rendering of both procedural
and scientific data-defined volumes. The rendering
techniques currently available in this system for volumes
include gas-based rendering methods and traditional
density-based volumetric rendering [2].
2.1 Gaseous volume rendering algorithm
The volume rendering technique used for gases in this
system is similar to the one discussed in [2]. The ray from
the eye through the pixel is traced through the defining
geometry of the volume. For each increment through the
volume sections, the density is determined from the CFD
data. The color, density, opacity, shadowing, and illumination
of each sample is then calculated. The illumination
and densities are accumulated based on a low-albedo illumination
model for gases and atmospheric attenuation. The
following algorithm is the basic gas rendering algorithm:
for each section of gas
for each increment along the ray
get color, density, & opacity of
this element
if self_shadowing
retrieve shadowing of this element
from the solid shadow table
color = calculate gas illumination
using opacity, density and the
appropriate model
final_color = final_color + color;
sum_density =sum_density +density;
if( transparency < 0.01)
stop tracing
increment sample_point
create the a_buffer fragment
In sampling along the ray, a Monte Carlo method is used
to choose the sample point to reduce aliasing artifacts. In
the gaseous model, we are approximating an integral to calculate
the opacity along the ray [12]. Therefore, the opacity
is the density obtained from evaluating a volume density
function multiplied by the step-size. The approximation
used is
opacity � 1 � e �� Ptfar ��x�t��y�t��z�t���t
tnear
where � is the optical depth of the material, ��� is the
density of the material, tnear is the starting point for the
volume tracing, and tfar is the ending point. The final
increment along the ray may be smaller, so its opacity is
scaled proportionally [13].
The volume density functions interpolate the values
stored in the CFD grid and allow density scalars and
power functions to be applied to enhance the visualization
results. By applying a density scalar, more volumetric
information can be seen. Applying a power function to the
density allows the contrast in change of the densities to be
increased. The following function is used to achieve these
results:
power exponent
density � �density � density scalar�
2.2 Illumination algorithm
For the gaseous rendering, the following low-albedo
illumination model is used, where the phase-function is
based on the summation of Henyey-Greenstein functions
as described in [14]:
B �
where I is
tfar X
tnear
e ���
X
i
P t
tnear
��x�u��y�u��z�u���u � I �
��x�t�� y�t�� z�t�� � �t�
Ii�x�t�� y�t�� z�t��phase����
P hase��� is the phase function, the function characterizing
the total brightness of a particle as a function of the
angle between the light and the eye [14]. Ii�x�t�� y�t�� z�t��
is the amount of light from light source i reflected from
this element.
The system also features a fast, accurate table-based
volume shadowing technique [5] to increase the visual
realism of the resulting images. The system can, therefore,
accurately display three-dimensional gases and the
shadows they cast.
2.3 Results for gaseous rendering
A comparison of the results achievable through the
three-dimensional gas rendering of our system and more
traditional volumetric rendering techniques can be seen in
Figures 1 and 2. Both figures show the visualization of the

static pressure for the computation, illuminated with two
light sources. Figure 1 shows the data visualized using the
gaseous volumetric rendering techniques. Figure 2 shows
the results achievable using more traditional volumetric
rendering techniques, where the gradient of the change in
the data is used to imply surfaces and a traditional Phong
surface illumination model is used. Figure 2 is lacking
in several respects. First, the lack of shadowing makes
the understanding of the image more difficult. Second, the
implied surfaces obscure the details of the flow data within.
Third, additional flow detail is lost because of the small
degree of change in the values in the data. In contrast, notice
the details in the changes in the pressure that can be seen in
Figure 1. Also notice the depth of information that can be
discerned in this image. The image uses self-shadowing of
the volume data, as well as shadowing of the volume onto
the walls to increase the understandability of the data. As
you can see, much more of the three-dimensional flow can
be seen using our gas visualization technique.
The advantages of the gas visualization prototype system
can be clearly seen from comparing these two images.
These images were computed at a resolution of 512�341
and took 83 seconds each to compute. Lower resolution
images can be combined with rendering optimizations and
more powerful graphics workstations to achieve interactivity
of the visualizations.
Figure 5 shows the flexibility of the system for visualization.
This figure contains 9 images with varying density
scaling and power exponent values used in the visualization.
The following table has the parameter values for each
image in the figure (image 1 is the top-left image, image 9
is the bottom-right image).
Image Density Scalar Exponent
1 1.0 3.0
2 1.0 1.84
3 1.0 0.98
4 0.91 3.0
5 0.91 1.84
6 0.91 0.98
7 0.73 3.0
8 0.73 1.84
9 0.73 0.98
From left to right in the image, the power exponent is
decreased, increasing the density of the data. From top to
bottom, the density scaling factor is decreased, increasing
the transparency of the flow.
This sequence of images demonstrates how the variation
of transparency/opacity can be used to investigate the
spatial gradients of a particular flow quantity. In this
case, the static pressure varies significantly throughout the
flowfield and among the critical flow features. The most
critical flow features, which arise as gradients in the static
pressure, are shock and expansion waves. These waves
appear as abrupt changes or nearly discontinuous changes
in the static pressure. In the sequence of renderings shown
here, the shock can be seen as the abrupt change from blue
to white along the vertical part of what appears as a cutout
region on the left hand side of the image. Once again, the
shock is caused by the turning of the flow as it enters the
blade passage. It is readily apparent that the strength of the
shock varies significantly across the flowfield or along the
blade leading edge (the blade’s geometry-angle thickness
and twist vary in the radial direction from hub to tip). This
sequence allows the viewer to look into the flowfield and
see where the most intense or sharpest gradients exist and
what their influence is on the subsequent (downstream)
flow. In this figure, the greatest pressure gradient exists
at the region which appears as the cutout in the upper left
hand region of the flowfield just after the flow enters the
blade passage. This is indicative of the greatest shock
strength. Examination of the successive images across and
down the figure reveals that the shock strength decreases
as it approaches the adjacent blade surface. This sequence
also gives an indication of the three-dimensionality of
the shock geometry and shock strength which can have a
significant influence on the operating characteristics of the
compressor. Such analysis is especially valuable when it is
incorporated into animations which will provide a means
of analyzing the time variation as well.
3 The ray caster
This renderer is a ray caster that is equipped with
a versatile new mechanism for the design of color and
opacity transfer functions. These are interactively defined
by the user prior to rendering.
Processing commences with some initialization computations.
We start by computing the normal vector at
each voxel. Unlike traditional 6-neighborhood central
difference based gradient calculation, our computation
considers all 26 neighbors, attenuating by p 2 the 18neighbors
and by p 3 the 26-neighbors. The normal vector
�Nx� Ny� Nz� p is used in shading and its magnitude
r � Nx 2 � Nx 2 � Nx 2 , is utilized to index our 2D
opacity transfer function. We calculate the magnitude of
the gradient at each voxel location and store them in a 3D
byte array of the same size as the original data set.
Given a data set and light source information, we duplicate
the original data set and shade the newly generated
one. We apply diffuse lighting on the data set using the
computed normal at each voxel �Nx� Ny� Nz� and the
raw densities as intensity values �. Given m light sources
located at direction Li � �lxi� lyi� lzi�� i � 1� � � � � m,

we set the shaded voxel intensity S to:
S �
mX
i�1
� � �lxi � Nx � lyi � Ny � lzi � Nz � 1��2
which is simply the intensity times cosine the angle between
Li and N (normalized to the range �0� 1�). Note that we
assume that all light sources have intensity 1.
These preliminary calculations require two additional
memory banks the size of the original data set for storing the
shaded data set and the gradient information. Alternatively,
these computations can be performed on-the-fly, which
will save the need for the extra storage and allow viewdependent
light effects such as specular light. Once this
initialization is completed, the data is ready for rendering.
The ray caster traverses image space. For each subsample
a ray is shot. Along the ray, at user controlled distances
�t, a trilinearilyinterpolated sample is taken in each
of the three datasets at the sample points x�t�� y�t�� z�t�.
These sampled values participate in the illumination calculation
to produce the final intensity for that sample point,
as described in the next section. After all the samples along
a ray are collected and composited, the compositing buffer
contains a floating point number representing the intensity
at the corresponding image sub-sample. When all image
rays are shot, the final image is derived by averaging all
sub-samples in a pixel and transforming the floating point
intensities into the integer range [0,255].
The ray caster is equipped with an interactive tool, as
well as an interpreter that can be used to build 1D and 2D
transfer functions. The language supported by this simple
modeler contains various boolean operators (min, max,
add, subtract, etc.) between several 1D and 2D linear and
exponential primitives. Using this tool, the user can define
such transfer functions as threshold rendering, iso-value
rendering, gradient emphasis by opacity and so on. The
user defined transfer functions are evaluated and converted
into 1D and 2D tables that are used by the renderer as color
and opacity lookup tables. These tables are indexed by
raw voxel value, gradient magnitude, and the like, as we
describe in the next section.
3.1 Illumination algorithm
Our illumination procedure takes as input a dataset
consisting of one byte per voxel intensity and is capable
of generating one or three band images. Each band in the
image has a pair of transfer functions associated with it:
1. Voxel value to intensity transfer function which maps
original intensities to desired ones (implemented as a
1D lookup table).
2. Voxel value and gradient to opacity which sets the
transparency of each voxel during the ray casting
process (implemented as a 2D lookup table).
Given a sample triple ��� S� r�, where � is the raw
intensity, S is the shaded intensity, and r is the gradient,
all sampled at x�t�� y�t�� z�t�, we calculate two new values
for each color band �: ��, the transparency value and ���,
the intensity value:
�� � T F 2D�r� ��
��� � T F 1D�S�
where TF are 1D and 2D transfer functions designed by
the user. Although we currently use the above 1D and 2D
functions, the same mechanism we developed can be used
to functionally bind reflectivity to gradient, for example,
and to develop transfer function of higher dimensionality.
The compositing scratchpad consists of a pair �I�� O��
of float numbers, where I� stands for intensity and O� for
opacity (in each color band �). Compositing of ���� ����
onto the compositing scratchpad is performed as follows :
I� � I� � I� � �� � O�
O� � O� � �1 � ���
After we composite a ray, I� contains the final pixel
intensity.
3.2 Results for the ray caster
The dataset was shaded in 5 seconds and gradient
calculation took 5 additional seconds. Rendering time for
all images (256 2 resolution) is 35 seconds when one ray
is traced from each pixel. The transfer function used for
rendering Figure 3 was
T F 1D�S� � S �
T F 2D�r� �� � �r � �� �
In Figure 3a we assigned � � 0�3 and � � 1�1. In Figure 3b
we assigned � � 0�3 and � � 0�5.
Figure 3 shows a typical rendering of the flowfield static
pressure generated using the computed Navier-Stokes flowfield
data between two adjacent compressor rotor blades.
In this figure, the flowfield is oriented such that the flow
is entering the blade passage from the left and proceeds
to the right and out the back of the page as it exits the
blade passage. It should be noted that the data set used
in this rendering has been truncated in the vicinity of the
blade trailing edge or in the region where the flow leaves
the blade passage. The specific transfer function used in
this rendering reveals various regions in the flow in which

large gradients exist. These are the regions in which abrupt
changes are present. Specifically, the rapid change from
dark to light near the left hand boundary appears to capture
the general shape of the shock wave produced as the
supersonic flow is turned when it enters the blade passage.
As the flow proceeds toward the exit, the variation in the
shading along the boundaries bounded by solid surfaces
might be associated with the presence of the boundary layer
along the solid surfaces. However, there are other flow
quantities, such as velocity, which will reveal much more
of the detail of the boundary layer behavior, including its
growth and separation.
In the vicinity of the blade trailing edge, there appears to
be a region of separated flow which may well be associated
with the important phenomena of vortex shedding and
interaction. These flow features have been subjected to
a preliminary examination using an interactive animation
procedure to look at the three-dimensional nature of their
behavior at a single instant in time. The extension of
the three-dimensional rendering procedure to include the
temporal variation will provide even greater detail which
will be of value in understanding these flow features.
4 Template-based rendering
In parallel viewing, where the observer is placed at
infinity, all rays have exactly the same form. Therefore,
there is no need to reactivate a line algorithm for each ray.
Instead, we compute the form of the ray once and store it
in a data structure called a ray-template. All rays can then
be generated by following the ray template. This approach
has great advantages both in terms of performance and in
terms of accuracy, as we show later.
The rays, however, differ in the appropriate portion of
the template that should be repeated. We choose a plane,
parallel to one of the volume faces, to serve as a base-plane
for the template placement. The image is computed by
sliding the template along that plane, emitting a ray at each
of its pixels. This placement guarantees a complete and
uniform tessellation of the volume by 26-rays.
The proposed algorithm is composed of three phases:
initialization, ray casting, and 2D mapping. In the first
phase, the base-plane is computed and a template in the
viewing direction is constructed. In the second phase, a
ray is traversed from each pixel inside the image extent by
repeating the sequence of steps stored in the ray template.
In the last phase, the projected image is mapped from
the base-plane onto the screen-plane by employing a 2D
image transformation. The proposed algorithm guarantees
uniform sampling and is based on simple calculations that
form the basis for an extremely efficient implementation.
We now turn to describe, in more detail, each of the
algorithm’s three phases.
The basic algorithm can be slightly modified to support
multiple rays emitted from the same pixel, but from different
sub-pixel addresses [15]. Instead of having one type
of ray (i.e., one template), we now have several templates,
one for each relative position of the ray-origin in a pixel.
For example, if we want a supersampling rate of four rays
per pixel, for the pixel at coordinate �i� j�, these four rays
will originate from �i � 1�4� j � 1�4�. We observe that for
all �i� j� the rays �i � 1�4� j � 1�4� have the same form.
Therefore we need only four different templates - one for
each relative displacement from the pixel origin.
The template-based algorithm can also be modified to
efficiently support multiple samples per voxel [15]. If we
assume that rays are emitted from the centers of the pixels
in a plane that is parallel to one of the volume faces (such
as the base-plane), then all these sample points are in the
same relative position inside the voxel; that is, they have
exactly the same set of distances to the eight corners of
the voxel. Therefore, we can employ a continuous line
algorithm to generate a template of floating point steps.
For each sample point, we can pre-compute the weight
to be assigned to each of the voxel values participating
in the sampling operation. This will save the need to
compute, for each sample, the weights of all eight voxel
values participating in the trilinear interpolation. This
template based ray casting algorithm uses very few (eight,
on the average) integer additions per sample, instead of the
naive implementation which requires tens of floating-point
additions and multiplications.
4.1 Illumination algorithm
To benefit from the template algorithm, illumination
capabilities were kept to a minimum. Illumination is based
on the Phong illumination model with one light source.
Samples are shaded on-the-fly and all samples along a ray
are incrementally composited. The ray stops when a user
defined opacity threshold is reached. Transfer function
for color assignment to raw voxel value is a simple linear
palette of grey (i.e., ��� � �). Transfer function for
opacity assignment is also a simple linear palette assigning
�� � ��6 so that the rays would accumulate less opacity
and will not stop in one or two steps. Normal is calculated
by central difference in a 6-neighborhood. Shading includes
ambient, diffuse, and specular shading.
4.2 Results for the template-based rendering
The gas images shown in Figure 4 were generated
by an adaptive variation of the ray template algorithm.
This approach renders the empty space using one template

(discrete) and another template at the region of the data
values. In this area, samples are taken at �t � 0�7.
The generation of this 140 2 resolution image, took 3.01
seconds for one ray per-pixel, which is approximately 3
times faster than the previous algorithm. Supersampling
increases rendering time linearly (e.g., 4-rays per pixel
takes 12.1 seconds). At �t � 0�3, the image is rendered
in 4.11 seconds.
5 Future extensions
There are several extensions that can be made to the
current system. First of all, further optimizations of the
rendering algorithms are being explored by the authors to
increase the performance of the system.
Second, we are developing techniques to support the
visualization of unstructured, multi-level, and adaptive
grid data [16]. The current system will only allow the
display of structured (rectilinear) grid data. Note that
illumination models and transfer function design are the
same for all types of grids, only stepping along the ray
and sampling methods change. Techniques have been
developed to resample structured and unstructured grid
data to a rectilinear grid for visualization; however, this
step may introduce aliasing artifacts into the resulting
images. Resampling unstructured or multi-resolution data
is also too time consuming to be done in an interactive
system. For instance, resampling the curvilinear grid data
used in Figures 1 and 2 took several minutes. Direct
support for unstructured grids and multi-level solution of
adaptive grids is needed. The use of adaptive, multidimensional,
and unstructured grids has increased greatly
in the recent past [17, 18]. These grid structures allow
for better utilization of computational power during the
CFD simulations and need to be supported in an interactive
visualization system. We plan to explore incorporating
and enhancing these recent algorithms for unstructured
grids into our visualization system. We will also develop
algorithms for the direct support of multi-level grids and
multi-level solution of adaptive grids.
Finally, a graphical user-interface for the visualization
system needs to be developed to increase the usability
and interactivity of the system. A significant part of this
task involves the development of the interface between the
computational model and the visualization system to allow
direct interaction with the computations to aid debugging.
6 Conclusion
We have described and compared three new efficient
volume rendering techniques for flow visualization and
demonstrated their value for visualizing the flow between
two blades of a turbo-jet compressor. As can be seen
from the resulting images in this paper, these techniques
provide higher quality images than most CFD visualization
systems. Combining these techniques in a single system
can provide CFD researchers with a powerful visualization
system. With these tools, researchers can see photorealistic
images of their flows for detailed analysis and also
produce high-quality images at interactive rates for testing
and debugging of the computational model.
While the renderings we generated of the static pressure
provide a good representation of flow features caused
by pressure gradients, numerous other flow quantities will
provide even better representations of other significant flow
features. Thus, these renderings represent a significant step
in the development of an important, enabling technology
for the analysis of complex flow phenomena encountered
in advanced aerospace propulsion systems as well as a wide
range of other CFD visualization applications.
References
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Figure 1: Gas Rendering of static pressure. Figure 2: Traditional isosurface volume
rendering of static pressure.
Figure 3: Changing opacity transfer function
to emphasize gradient
Figure 4: Adaptive template−based rendering
Figure 5: Varying density scaling and power exponent values.

Abstract
Line Integral Convolution (LIC), introduced by Cabral
and Leedom in Siggraph '93, is a powerful technique for
imaging and animating vector fields. We extend the LIC
paradigm in three ways:
1. The existing technique is limited to vector fields over
a regular Cartesian grid. We extend it to vector fields over
parametric surfaces, specifically those found in curvilinear
grids, used in computational fluid dynamics simulations.
2. Periodic motion filters can be used to animate the flow
visualization. When the flow lies on a parametric surface,
however, the motion appears misleading. We explain why
this problem arises and show how to adjust the LIC
algorithm to handle it.
3. We introduce a technique to visualize vector
magnitude as well as vector direction. Cabral and Leedom
have suggested a method for variable-speed animation,
which is based on varying the frequency of the filter
function. We develop a different technique based on kernel
phase shifts which we have found to show substantially
better results.
Our implementation of these algorithms utilizes texturemapping
hardware to run in real time, which allows them to
be included in interactive applications.
1. Introduction
Visualizing Flow Over Curvilinear Grid Surfaces
Using Line Integral Convolution
Lisa K. Forssell
Computer Sciences Corporation, NASA Ames Research Center
and Stanford University
lisaf@cs.stanford.edu
Providing an effective visualization of a vector field is
a challenging problem. Large vector fields, vector fields
with wide dynamic ranges in magnitude, and vector fields
representing turbulent flows can be difficult to visualize
effectively using common techniques such as drawing
arrows or other icons at each data point, or drawing
streamlines[2]. Drawing arrows of length proportional to
vector magnitude at every data point can produce cluttered
and confusing images. In areas of turbulence, arrows and
streamlines can be difficult to interpret.
Various techniques have been developed which attempt
to address some of these problems. Max, Becker, and
Crawfis[13], Ma and Smith[12], and Max, Crawfis, and
Williams[14] have implemented systems which advect
clouds, smoke, and flow volumes. These techniques show
the flow on a coarse level but do not highlight finer details.
Hin and Post[11] and van Wijk[16] have visualized flows
with particle-based techniques, which show local aspects of
the flow. Bryson and Levit [4] have used an immersive
virtual environment for the exploration of flows. Helman
and Hesselink[10] have generated representations of the
vector field topology, which use glyphs to show critical
points in the flow.
In this paper we discuss a new technique for visualizing
vector fields which provides an attractive alternative to
existing techniques. Our technique makes use of Line
Integral Convolution (LIC)[5], which is a powerful
technique for imaging and animating vector fields. The
image of a vector field produced with LIC is a dense display
of information, and flow features on the surface are clearly
evident.
The LIC algorithm as presented by Cabral and Leedom
in [5] is applicable only to vector fields over regular 2dimensional
Cartesian grids. However, the grids used in
computational fluid dynamics simulations are often
curvilinear. In this paper we show how to extend the LIC
algorithm to visualize vector fields over parametric surfaces.
Thus, for example, our extended algorithm allows us to
visualize the flow over the surface of an aircraft or turbine.
In the original work on LIC, a technique for animation
of vector field visualizations is presented. Our work extends
this animation technique to apply to the parametric surfaces
found in curvilinear grids as well.
Lastly, we present a new technique for displaying
vector magnitude which can be applied to both 2dimensional
regular grids and parametric surfaces. Our
method varies the speed of the flow animation to give an
intuitive representation of vector magnitude.
In the next section we discuss the basic LIC algorithm.
In section 3 we describe our extension to curvilinear
surfaces. In section 4, we discuss the implementation of
animation for curvilinear grid surfaces. In section 5, we
introduce our technique for displaying vector magnitude. In
section 6, we describe our implementation of all the
algorithms in the paper. We conclude with a brief discussion
of directions for further applications of the LIC algorithm in
vector field visualization.

2. Background
The Line Integral Convolution (LIC) algorithm takes as
input a vector field lying on a Cartesian grid and a texture
bitmap of the same dimensions as the grid, and outputs an
image wherein the texture has been “locally blurred”
according to the vector field. There is a one-to-one
correspondence between grid cells in the vector field, and
pixels in the input and output image. Each pixel in the output
image is determined by the one-dimensional convolution of
a filter kernel and the texture pixels along the local
streamline indicated by the vector field, according to the
following formula:
∑
Cout ( i, j)
= Cin ( p)
⋅ h ( p)
p ⊂ τ
where
τ = the set of grid cells along the streamline within a set
distance 0.5 l from the point (i, j), shown as the shaded cells
in Figure 1.
l = the length of the convolution kernel
C in (p) = input texture pixel at grid cell p
β
∫
h ( p)
= k ( w)
dw
α
where
α = the arclength of the streamline from the point (i, j) to where
the streamline enters cell p
β = the arclength of the streamline from the point (i,j) to where
the streamline exits cell p
k(w) = the convolution filter function
Figure 1.A vector field where the streamline through point (i,j)
is shaded.
Thus each pixel of the output image is a weighted average
of all the pixels corresponding to grid cells along the
streamline which passes through that pixel’s cell. Section 4
of [5] provides the complete details of the algorithm. When
this algorithm is applied at every pixel, the resulting image
appears as if the texture were “smeared” in the direction of
the vector field.
i,j
Figure 2.The input white noise bitmap on left is smeared using
LIC on a circular vector field to produce the output image on
the right.
3. Curvilinear Grid Surfaces
Because of the one-to-one correspondence between grid
cells and pixels in the input/output images, the algorithm
described above requires that the vector field lie on a
regular, Cartesian grid. Here we show how to use the
algorithm on 2-dimensional slices of structured curvilinear
grids, which describe parametric surfaces.
We denote the curvilinear space coordinates of a point as
ξ = ( ξ, η, ζ)
and the physical space coordinates as
x = ( x, y, z)
. The vector which describes the velocity of
the flow at each point is
∂x
∂t
∂x
∂t
∂y
∂t
∂z
∂t
We transform the vector field to the coordinate system of
the curvilinear grid, hereafter called “computational space.”
The transformation from physical space to computational
space is performed by multiplying the physical-space
velocity vectors by the inverse Jacobian matrix s.t.
∂ξ
∂t
∂η
∂t
∂ζ
∂t
=
The computational-space vectors give velocity in gridcells
per unit time. Because the data points are given for
integer coordinates in computational space, this constitutes
a regular Cartesian grid.
We can compute a LIC-image of any 2-dimensional
=
∂x ∂x ∂x
∂ξ
∂η
∂ζ
∂y ∂y ∂y
∂ξ
∂η
∂ζ
∂z ∂z ∂z
∂ξ
∂η
∂ζ
−1
∂x
∂t
⋅
∂y
∂t
∂z
∂t

slice of the grid by projecting the vector field onto it. For
example, if we want to examine the k=1 plane of the
computational grid, which in many CFD data formats
usually lies on the surface of the object about which the flow
is being simulated, we drop the (∂ζ/∂t) term and use the 2dimensional
vector [∂ξ/∂t, ∂η/∂t] T in the LIC algorithm.
The resulting image, which is a visualization of the
vector field in computational space (see Figure 3) is then
mapped onto the surface in physical space using a standard
inverse mapping algorithm, such as that described in [9].
The inverse mapping converts the vector field
representation back into physical space (Figure 3). The final
result is a visualization of the flow which is dense, easily
interpreted, and effectively handles the complicated areas of
the flow.
Figure 3. Top: a LIC image of the computational-space velocity
field over the surface of the space shuttle. The input texture is white
noise, and the convolution kernel is a simple box filter. Below: the
LIC image above texture mapped over the space shuttle in physical
space. Note the separation apparent on the fuselage and the vortices
at the wingtip.
4. Animation
While the image described above and shown in Figure
3 correctly shows the streamline direction of the vector field,
the visualization is ambiguous in regards to whether the flow
is moving forward or backward along the lines indicated. To
disambiguate the direction of flow, animation is useful.
Also, animating a flow visualization is physically
meaningful.
As Cabral and Leedom discuss in [5], periodic motion
filters [6] can be used together with LIC to create the
impression of motion, such that a flow appears to be moving
in the direction of the vector field. A small number n of LIC
images are computed, where in frame i the filter kernel is
phase shifted by is/n, where s is the period of the filter
function. When played back, these images cause the
appearance of ripples moving in the direction of the vector
field. Because the filter kernel is periodic, the n frames can
be cycled through continually for smooth motion.
On a parametric surface, the images are ‘played’ by
texture mapping each in turn onto the surface. However,
additional steps must be taken to ensure that the animation
does not introduce misleading information into the
visualization. The conversion from computational space to
physical space maps square grid cells into quadrilaterals of
varying dimensions. Therefore, the length of the
convolution filter, which is measured in computational
space units, is mapped to varying lengths in physical space.
The length of the periodic filter determines the size and
speed of the “ripples” in the animation. The speed is given
by the amount of phase shift in physical space per unit time.
Thus, if the period of one filter function is longer in physical
space than another, that ripple appears to move faster than a
shorter filter.
As a result of the warping that occurs in the mapping
from computational to physical space, the animation appears
uneven and erratic. In areas where the grid is sparse, the flow
appears as little ripples moving fast, because the convolution
kernel has been compressed, and in areas where the grid is
dense, the flow appears as large ripples moving slowly.
Since there is no correlation between apparent speed and
actual speed of the flow, this motion is highly misleading.
The situation can be corrected by varying the length of
the convolution filter while computing the LIC image. The
length of the convolution filter must vary inversely with the
grid density in the direction of the flow. Where the grid is
sparse in physical space, we want to use a narrow
convolution filter in computational space, as it will be
stretched out when mapped. Likewise, where the grid is
dense in physical space, we want to use a wide filter in
computational space, as it will be compressed when
mapped. See figure 4, next page.

We compute frames where the length of the convolution
kernel used in the LIC algorithm at each grid cell p is given
by
where
Physical Space
Dense
Area of
Grid
Sparse
Area of
Grid
a is the minimum length of the kernel, measured in grid
cells
b controls the range of possible kernel lengths, and
r is the grid density at grid cell p in the direction of the
flow.
a must greater than 1; if the length of the filter is 1 or less,
the LIC algorithm simply returns the input texture pixel,
unaffected by the vector field.
b must be set to a finite length which will vary with the
particular grid.
r(p) for each grid cell is given by
Computational Space
Figure 4. A convolution filter function in physical space and
computational space. The goal is for the filter to be of uniform length in
physical space, regardless of grid density. Therefore we stretch the filter in
computational space in areas where the grid is dense, and compress the
filter in computational space in areas where the grid is sparse.
b
l ( p)
= a +
r ( p)
∂x
∂ξ
∂y
∂ξ
∂z
∂ξ
∂x
∂η
∂y
∂η
∂z
∂η
∂x
∂ζ
∂y
∂ζ
∂z
∂ζ
r(p) for the entire grid is computed by the following steps:
−1
dx
dt
dx
dt
1) Normalize the vector field to unity in physical space.
2) Convert to computational space using the inverse
•
Jacobian as described in section 3.
3) Take the magnitude of the computational-space
vectors.
Figure 5 shows a single texture frame of an animation
sequence computed in this way. When the 10 texture
frames are played back, the flow appears smooth and even
everywhere on the surface, rather than uneven and erratic.
Thus we are able to use periodic motion filters even on
parametric surfaces from curvilinear grids.
Figure 5a. A single texture frame from an animated sequence computed
using the LIC algorithm with a raised cosine filter. The length of the filter
varies inversely with the grid density in the direction of the flow. Therefore
the ripples appear stretched in some areas and compressed in others.
Details in section 4.
5b. Close-up of the texture frame from above mapped onto the curvilinear
grid, used to simulate flow around a post. The bottom edge of the texture
maps around the post. When mapped onto the grid, the ripples are of uniform
size. When animated, the ripples give the impression of smooth flow.
5. Variable Speed
The next step in flow animation, whether on a regular
grid or on a parametric surface, is to give a visualization of
vector magnitude as well as vector direction. Thus, in a CFD
flow visualization, the periodic motion should be slow
where the flow has low velocity and quick where the flow
has high velocity. Cabral and Leedom [5] suggest achieving
this effect by varying the frequency of the filter function,
while keeping its length constant. However, the limited
dynamic range (experimentation shows only between 2 and
4 ripples per kernel are interpretable) and the artifacts
caused by changing the shape of the filter make it difficult to

use this approach for meaningful results. We have found that
a better solution is to vary the amount of filter function phase
shift at each grid cell in proportion to the physical-space
vector magnitude.
The amount of phase shift is what determines the
apparent speed, given a uniform-length filter kernel. An
infinitesimally small phase shift will appear not to move at
all. Likewise, a 90-degree phase shift in every frame will
produce a full cycle in four frames, which appears to move
very quickly. (At anything greater than 180 degrees,
temporal aliasing occurs.) Phase shifts ranging from 0 to 90
degrees can be mapped to the actual range of physical vector
magnitude for a convincing variable-speed animation.
In a frame from a variable-speed animation sequence,
each pixel will be computed with a convolution kernel that
has a phase shift proportional to the corresponding grid
cell’s physical vector magnitude. Therefore the period of the
filter function is different at each pixel, and there is no fixed
number of frames that can be used in a cyclic animation.
Therefore, we adopt the following strategy of sampling the
“real” solution and interpolating to find the pixel values
which we will display.
In practice, the texture frames are computed as follows:
1) First compute N LIC images, such that in image i,
where θi is the amount of filter phase shift, θι = is/N.
As in section 4 s is the period of the filter function.
The larger N, the more accurate the visualization. The
intensity of pixel p in image i is defined as T(i, p).
2) For each grid cell p , let
y − min ( y)
q =
max ( y)
− min ( y)
where y denotes physical vector magnitude at grid
cell p. q is a real number in [0,1] that gives the vector
magnitude in cell p relative to the magnitudes in the
whole grid.
3) The intensity of pixel p in frame j of the displayed
image, I(j,p), is found by interpolating linearly
between the two LIC images from step (1) closest to
it:
Let
α q
at cell p. Then
N
N
jmodN q j modN
4 4 −
=
I ( j, p)
( 1 − α)
T q N
j modN p
4 ,
( )
=
αT q N
j modN p
4 ,
( )
+
6. Implementation
We use the texture-mapping capabilities of a high-end
workstation [8] to display surfaces with LIC-images
mapped onto them in an interactive program. All LIC
images are computed prior to running the interactive
program, since the LIC algorithm is fairly computeintensive
(images take on the order of several seconds to
minutes to compute). The hardware is capable of switching
between pre-loaded textures quickly enough that we are able
to run animation in real time, while the user manipulates the
surface.
For single-speed periodic motion, we find that 5 - 12
texture frames is sufficient for smooth animation.
For variable-speed animation, we are no longer able to
precompute a finite number of frames and cycle through
them, because the amount of phase shift varies at every grid
cell. However, steps 2 and 3 of the algorithm described in
section 5 are not compute intensive, once the N LIC-images
of step 1 have been precomputed. This implies that a real
time implementation of variable-speed animation should be
possible. Unfortunately, we have not been able to achieve
this with our hardware, an SGI Reality Engine, because the
time required to load a new texture into the texture cache is
too long to permit good frame rates. However, we expect
that within the foreseeable future texture-mapping hardware
will allow fast texture definition and this feature will be
possible.
In the meantime, we have experimented with two
alternative solutions.
(1) Calculate and store a large number of texture frames
using a real phase shift at every grid cell which is a linear
function of the physical vector magnitude in that grid cell.
The number of frames required for anything more than a few
seconds of animation using this solution is so large that
storage requirements quickly exceed the memory
capabilities of a workstation. Therefore either (a) the short
sequence of animation must be continually restarted, which
causes the flow to appear to “jump” every few seconds, or
(b) the animation must be stored on video or another digital
playback device, and the real time interactivity possible with
all other techniques described in this paper are forfeited. We
show an implementation of (a). The animation is smooth in
spurts of 5 seconds, and the speed of the flow is clearly
varying across the surface (see video).
(2) Approximate the continuous solution by choosing a
minimum phase shift, φ, and quantizing all phase shifts as
integer multiples of φ. In this solution, only M = s/φ frames
need to be precomputed, because the M frames will form a
complete cycle. The drawback of this solution is that it is
susceptible to aliasing and rasterization caused by the
sampling and quantization of phase shifts. The advantage is

that it can be played continually in real time on a
workstation, without the jumps of solution (1).
To compute the frames for this discretized solution, we
follow the same steps as described for the continuous
solution in section 5, but round q to the nearest multiple of
1/M. In this case there is no interpolation and M frames
suffice to form a cycle of animation.
In our implementation of solution (2), we see that while
the speed of the flow is clearly varying, aliasing and
rasterization artifacts do appear.
7. Future Work
There are several promising directions for future work.
First, in order for this technique to become useful for practical
applications, a number of extensions must be implemented.
Foremost among these are multigrid solutions,
unsteady flows, and unstructured grids.
Also, we hope to extend this technique to the visualization
of 3-dimensional vector fields. While the LIC algorithm
in itself extends easily to a 3-dimensional Cartesian
grid, the output image data requires additional processing
before a useful image is produced. Cutting planes, isosurfaces,
or volume rendering techniques will be necessary for
this extension. Experimentation with input textures and
convolution filters will be needed to achieve effective images.
Furthermore, new algorithms will be required to handle
curvilinear grids in this situation as well.
8. Summary
We have presented several extensions to Line Integral
Convolution. First, we have described how to use the LIC
algorithm on curvilinear grid surfaces. We have shown how
to solve the problems that arise when using periodic motion
filters in LIC on a curvilinear surface. Lastly, we have introduced
a method of incorporating visualization of vector
magnitude into the LIC algorithm, by showing the animation
at variable speeds. All algorithms are designed such
that with modern graphics hardware the surfaces can be displayed,
animated, and manipulated in real time.
Our visualization technique provides intuitive and accurate
information about the vector field, and thus is a useful
complement to other visualization techniques.
Acknowledgments
The author wishes to thank David Yip, who provided
support of many forms throughout this research, the Applied
Research Branch staff of NASA Ames, who offered
helpful discussions, Marc Levoy, who gave advice on the
research and on the writing of this paper, and Brian Cabral
and Casey Leedom, who made their LIC code publicly
available.
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1a. The flow over the surface of the space shuttle visualized in
FAST with arrow icons for velocity vectors and glyphs for critical
points in the topology.
2a. Detail of the top of the fuselage of the space shuttle, visualized
in FAST.
3a. Detail of the wing of the shuttle.
1b. The flow over the surface of the space shuttle visualized using
Line Integral Convolution on the computational space vector field
and texture mapped onto the shuttle surface.
2b. Detail of the top of the fuselage of the space shuttle, visualized
using LIC for curvilinear surfaces.
3b. Detail of the wing of the shuttle.

Abstract
Vector field rendering is difficult in 3D because the
vector icons overlap and hide each other. We propose four
different techniques for visualizing vector fields only near
surfaces. The first uses motion blurred particles in a thickened
region around the surface. The second uses a voxel
grid to contain integral curves of the vector field. The third
uses many antialiased lines through the surface, and the
fourth uses hairs sprouting from the surface and then
bending in the direction of the vector field. All the methods
use the graphics pipeline, allowing real time rotation and
interaction, and the first two methods can animate the texture
to move in the flow determined by the velocity field.
Introduction
There are many representations of velocity fields:
streamlines, stream surfaces, particle traces, simulated
smoke, ..., etc. One of the simplest is to scatter vector
icons, for example, small line segments, throughout the
volume. There are two fundamental problems with such an
approach. First, the 2D projection of a line segment is ambiguous;
many 3D segments can have the same projection.
Second, densely scattered icons can overlap and obscure
each other, leading to a confusing image. These problems
can be partially solved with real time (or playback) animation,
since motion parallax can resolve the projection ambiguities.
In addition, icon motion in the velocity direction
can give added visual information.
The second problem can also be resolved by restricting
the icons to special regions of interest. For example, in
[Crawfis92] and [Crawfis93], the vectors’ opacity depended
on their magnitude, so only the regions of highest velocity
were emphasized. In this paper we take the region of
interest to be on or near a contour surface. The scalar function
being contoured can come directly from the vector
field, for example, the vector magnitude or a vector com-
Visualizing 3D Velocity Fields Near
Contour Surfaces
Nelson Max
Roger Crawfis
Charles Grant
Lawrence Livermore National Laboratory
Livermore, California 94551
ponent. It can also be an independent scalar field defined
on the same volume, for example, porosity in a flow simulation,
or a linear or quadratic function for interactive slicing.
We report here on four different techniques for
visualizing velocity fields near contour surfaces.
Spot Noise
Van Wijk [vanWijk91] generated a directional texture
by superimposing many oriented shapes such as ellipses.
For texture on curved surfaces, the texture plane is
mapped to the surface, and the texture generation accounts
for the stretching induced by the mapping. Van Wijk visualized
a tangential velocity field on a ship hull with this
method. Cabral [Cabral93] has generated similar but more
accurate 2D velocity textures by “Line Integral Convolution”
of random noise, and Forssell [Forssell94] extended
this with mapping to visualize velocity near an airplane
surface. Both of these techniques can be animated to make
the texture flow. However, they are not applicable to contour
surfaces, which cannot easily be parameterized.
Stolk and van Wijk [Stolk92, vanWijk93] have also
visualized flows with surface particles: individual spots
motion-blurred to elliptical shapes and composited separately
onto the image in software. Each spot has a surface
normal, which is carried along appropriately by the flow,
and used in the shading. These particles can also move in
animation, but only on surfaces related to the flow, not on
fixed contour surfaces of an unrelated function. Here we
use the same sort of spots, but take advantage of hardware
rendering for interactive speed. We do not restrict the particles
to lie on a surface, and therefore do not use normals
in the shading. Instead, we think of the particles as
spheres, which are motion blurred to ellipsoids. We only
draw particles which lie within a specified distance D of
the contour surface of interest.
We should emphasize that most of the previous work
represents tangential flows on a surface, for example, a
stream surface, but the problem we are trying to address

uses contour surfaces of functions possibly unrelated to
the velocity vectors, which are therefore not usually tangent
to the surface.
We assume that the vector field V(x, y, z) and the scalar
function f(x, y, z) are defined on the same rectilinear lattice.
To estimate the distance d(x, y, z) of a particle at (x, y,
z) from the contour surface f(x, y, z) = C, we use a technique
described by Levoy [Levoy88]. We approximate the
gradient ∇f(x,
y, z) by finite differences of the neighboring
vertex values of f, and store the magnitude | ∇f(x,
y, z)|.
Then
d ( x, y, z)
=
f ( x, y, z)
− C
∇f ( x, y, z)
where the quantities f(x, y, z) and | ∇f(x,
y, z)| are trilinearly
interpolated from the eight surrounding lattice vertices.
We wish to randomly deposit particles uniformly into
the region R where d(x, y, z) < D. Assuming the specified
distance D is larger than the cell size in the lattice, the following
procedure does this efficiently. We mark all lattice
vertices which are within D of the contour surface, and
then mark all cells which have at least one marked vertex.
These markings are updated whenever the user changes C
or D. We randomly insert a specified number N of particles
in each marked cell, and render a particle at (x, y, z) only if
d(x, y, z) < D.
The particles are kept in a linked list, which also contains
their ages. They are moved in each time step by second
order Runge-Kutta integration, using velocities
trilinearly interpolated from the lattice vertices. If a particle
moves into an unmarked cell or out of the data volume,
it is scheduled for deletion from the linked list. At each
time step, we also check that all marked cells still have N
particles, and add or delete particles as necessary.
The particle size is proportional to a factor b = s(d),
which is equal to 1 for small d and decreases continuously
to zero when d reaches D. Thus the particles fade in as
they first cross into R, and fade out as they leave. Since a
particle could randomly be created near the contour surface
to replace another particle leaving a cell, we actually
take b = minimum(s(d), ka), where k is a constant and a is
the age of the particle. This makes new particles fade in at
birth. They are similarly faded out when deleted.
As in [vanWijk93], we draw the particles as blurred
ellipses, stretched out in the direction of motion. However
we do the compositing using the hardware in our SGI
workstations. A single blurred disk is used as the texture,
and is mapped to a 3D rectangle. If r is the radius of the
particle, P is its position vector relative to the viewpoint,
and V is its velocity vector, then the rectangle vertices are
at P + S + T, P + S - T, P - S - T, and P - S + T, where
Figure 1. Spot noise near contour surface
V P
S r and .
×
S P
= T V r
V × P
×
= +
S × P
Basically, T is along the velocity direction, but the second
term is added so that the particle will shrink to a small
round dot of radius r when the velocity approaches zero or
is oriented near the viewing direction. Because these semitransparent
particles are sent through the graphics pipeline
after the opaque objects, they can be combined with
opaque contour surfaces in the z-buffer and be appropriately
hidden. Currently they are all the same color, so they
do not need to be sorted. (See [Max93].) Figure 1 shows a
collection of ellipsoidal motion-blurred particles near a
contour of velocity magnitude on a “tornado” velocity data
set. Figure 2 includes a contour surface of velocity magnitude,
which hides some of the dots. Figure 3 represents a
0.5 micron simulation of the airflow through a HEPA filter.
Figure 2. Spots noise with contour surface.

An animation has been produced showing the flow at varying
velocity contours, from high to low. A contour region
of moderate flows is illustrated here. Figure 4 illustrates a
velocity contour near zero and close to the metallic fibers.
The implementation is in C++, using SGI’s Inventor
for the user viewing interface. The contour value C and
width D are controlled by sliders. On our office workstations
with Elan graphics, the actual texture mapping is
done in software, but the same code automatically calls the
hardware texture mapping on the Onyx workstation in
our Graphics Lab. We can thus get real time rotation and
particle motion on small data sets, and interactive performance
on large data sets.
Particle traces on 2D surfaces
The goal of this technique is to represent the magnitude
and direction of a vector field at each point on an arbitrary
set of surfaces (not necessarily a contour surface),
allowing long flowlines to be easily understood, but without
having the visualization become too dense or too
sparse at any point.
To do this we try to draw long, evenly spaced particle
traces. The beginning and ends of these traces do not necessarily
indicate a source or sink in the vector field. Traces
begin and end over the entire surface in order to keep a
nearly constant density of lines in the image. The particle
trace lines are broken into small segments of contrasting
color. The length of the segments represents the magnitude
of the vector field (i.e. a constant time interval). The direction
of the segments is the direction of the vector field at
that point projected onto the 2D surface. In still pictures,
using a sawtooth shaped color map across each segment
resolves the directional ambiguity of the lines. We experimented
with several different projections of the vector
field to the surfaces.
Once the particle traces are calculated and rendered,
color table animation can be used to add motion to the display
so that the lines “flow” in the direction of the vector
field with a velocity proportional to the magnitude of the
vector field at each point. This flow animation was a primary
goal of this technique. Color table animation is an
old technique [Shoup79] which has been applied to flow
visualization by Van Gelder and Wilhelms [VanGelder92].
The first step in this technique is to scan convert the
surfaces into an octree. A piecewise linear representation
of the 2D surface(s) is stored in the octree. Each cell holds
a plane equation (four numbers). The octree allows us to
use any kind of surface (as long as the surface does not
pass through any leaf cell twice, which is unavoidable for
self-intersecting surfaces). The octree allows fast access to
adjacent cells but is much more memory efficient than a
full 3D grid. Only those cells that intersect the surfaces are
present in the octree. In this implementation, the surfaces
are subdivided down to a constant size cell.
A seed point is then chosen to try to place the first particle
trace. The particle is advected along the surface in
both directions, forward and backwards, using a variable
step size Euler’s method. We continue to advect the particle
until it reaches a stagnation zone, reaches an edge of
the surface, or becomes too close to its own trace or that of
another particle. The particle trace is considered acceptable
if it is longer than the current length threshold. If the
trace is too short, it is erased and a new seed point is tried.
The length of the erased trace is preserved in the cells in
which it passed. This value is used to prevent extensive recalculation
while backtracking.
The seed points are chosen on an integer lattice in a
spatially hierarchial manner so that the first particle traces
will start well away from each other and are likely to be
traced for long distances before getting too close to other
lines. Random placement of seed points would also be
likely to yield long lines for the first points chosen, but
would not guarantee that some seed point was near every
point on the surfaces. All seed points, at some particular
resolution, are tried first using a large length threshold.
Then the length threshold is reduced and the process is repeated.
Gradually reducing the length threshold from
some maximum to minimum produces the most esthetically
pleasing distribution of particle traces, but takes longer
to calculate than using a single length threshold value.
Projections
Four techniques were tried for projecting a 3D vector
onto the 2D surface. Three techniques, normal, xy normal
and cylinder projections, are viewpoint independent. The
eye projection technique is viewpoint dependent. Viewpoint
dependent techniques require the particle traces to be
recalculated each time the viewpoint is changed, while
viewpoint independent techniques do not require this recalculation.
To compare these four projections and their
ability to indicate the 3D flow, we have used a simple test
case: a constant velocity field.
Normal Projection
With the normal projection technique, the 3D vector
at a point on the surface is projected onto the surface in a
direction parallel to the surface normal at that point. This
projection, while being very straightforward, can yield
very nonintuitive particle traces. Figure 5 shows the tornado
surface in a vector field in which all vectors are in exactly
the same direction, pointing to the upper right at 45

degrees. The uniformity of the vector field is not at all apparent
with this projection.
XY Normal Projection
Figure 5. Normal Projection
The xy normal projection technique is designed for
producing film loops where the viewpoint is rotated about
the z (vertical) axis. In this technique the 3D vector is projected
onto the surface in the direction of a vector which
consists of only the x and y components of the surface normal
at that point. As can be seen in figure 6, this preserves
the z component of the vector field and gives a somewhat
more intuitive visualization, but the uniformity of the vector
field is still not readily apparent.
Figure 6. XY Projection
Cylinder Projection
The cylinder projection technique is also designed for
producing film loops where the viewpoint is rotated about
the z (vertical) axis. In this technique the 3D vector is projected
onto the surface in the direction of a vector which
points away from the rotational axis. As can be seen in figure
7, this gives results very similar to the xy normal projection.
This projection is only suitable for simple surfaces
which are centered on the rotational axis.
Eye Projection
Figure 7. Cylinder Projection
In the eye projection technique, the 3D vectors are
projected onto the surfaces in a direction parallel to the
viewing direction. As shown in figure 8, for a single image
this technique produces the most intuitive representation
of the constant direction vector field. The uniform direc-
Figure 8. Eye Projection

tion of the vector field is readily apparent. When changing
the viewpoint for a film loop, the projection of the 3D vectors
to the surfaces changes, resulting in a different set of
particle traces. We attempt to minimize the visual effect of
these changes by using the same set of trial seed points as
the previous frame, and by starting each particle trace with
the same “phase” as a nearby trace in the previous frame.
Figure 9. is a color reprint of Figure 6..
In conclusion, the uniformly spaced, animated particle
traces are an effective means of visualizing a 2D vector
field, but the projection of a 3D vector field onto a 2D
curved surface loses and distorts some of the information
from the 3D field, limiting the usefulness of these techniques
for 3D. This is a difficulty for our goal of representing
flows non-tangential to the surface. It is still an
effective technique for stream surfaces or other tangential
flows.
Line Bundles
Line bundles use the back-to-front compositing and
overlapping of splatting to construct a volume of tiny line
segments. Taken as a whole, these line segments construct
the appearance of a fibrous volume. While drawing many
tiny vectors to represent a vector field is not new, we have
combined this idea with back-to-front compositing and
techniques to generate anisotropic textures (see Figure
10). Our basic implementation plan is to extend the concept
of splats - each data point is composited into the
frame buffer in a back-to-front manner [Crawfis93]. Rather
than trying to reconstruct a C 1 3D signal, we want a
very discontinuous, yet antialiased, 3D volume representation.
At each data point, a collection of antialiased lines is
splatted. The lines are randomly scattered within a 3D
bounding box associated with each splat. The hue, saturation,
value and center position within the box are all ran-
Figure 10. Line bundle tornado with magnification
Figure 11 Line bundle near aerogel surface
domly perturbed. The direction of each line is in the
direction of the flow field. The jittering of the color and
position produce a nice anisotropic texture oriented in the
direction if the flow field, even when the lines are so dense
that there is no space between them. The primary color
about which we jitter can be different for each splat or can
be fixed for the entire volume. Having different colors per
splat allows us to encode additional information about the
volume, either the vector magnitude, some separate scalar
field, or a positional indicator. Using a single primary color
allows us to precompute the line bundle into a GL object,
which is then very rapidly reoriented and redrawn at
each data point. Line bundles in excess of over 300 line
segments can be used for each data point with no degradation
in real-time performance.
Three key issues are addressed in these fibrous volumes:
back-to-front compositing for a thin wispy-like appearance,
antialiased lines with transparent heads and tails
to avoid unwanted edges, and controlled jittering of colors
and positions to avoid regular patterns or completely filled
regions. Figure 10 shows a sample tornado data set using a
homogenous color that is heavily jittered. A zoomed in
portion of the image is to the left of it. Notice the antialiasing
and lack of any hard edges. Figure 12 shows the wind
field over North America in a simulated global climate
model. Data points close to the isocontour surface of a particular
wind velocity magnitude were chosen for the line
bundle splatting. Figure 11 represents the airflow through
a filter substrate known as aerogel.The data points were
chosen to lie close to the surface of the filter particles. This
provides a nice mossy appearance. In Figure 13, points
were chosen near a velocity contour of the HEPA filter
simulation, and are color coded by velocity, giving a solid
volume with a fibrous texture. These images can be generated
in real time on a mid- to high-end graphical workstation.

Hairs
We tried to use the line bundles to represent the flow
around or near a surface. This broke down when the flow
was slightly into the surface. A solution is to grow tiny
hairs coming out of the surface. We draw the line segments
out of the surface first and then have them bend in
the flow field, much like normal hair. Several controls over
this behavior are offered. The physical layout of the hairs
are specified by a number of connected line segments, by
an interpolation of the normal vector and the velocity vector,
and by a stiffness or weighting factor per segment. The
default number of segments is six, and all of the images
here were generated using only six segments. The interpolation
is completely specified by the user with coefficients
t i at each line segment point. A new directional vector is
generated using the formula: Direction = w i * ( t i * Normal
+ (1-t i ) * Velocity * Velocity_Scale). The Normal vector
is normalized, and the velocity is scaled by the user
parameter Velocity_Scale. Smooth curves or sharp bends
can be specified with the proper t i ’s. The Direction vector
is then added to the endpoint of the last line segment. The
weighting by w i is useful to control the apparent stiffness
of the hair. Greater weights can be given to the initial segment
in the direction of the normal, lower weights to the
middle segments and finally, large weights can be given to
the last one or two line segments to produce longer hairs in
the velocity direction. This defines an individual hair. A
number of hairs are scattered throughout the splat volume
and jittered from one splat to the next. The splats are positioned
near the contour surface by the method of Levoy
discussed above. They are moved towards the contour surface
by a multiple of the gradient.
The color and transparency of the hairs are controlled
by a weighted function of the hair’s root color, a specified
splat color, a vector head color, and an HSV space jittering.
The user specifies a root color and a vector head color.
A splat color is derived from a scalar field to color table
mapping. At each segment, the user can specify the fraction
of the root color, the fraction of the splat color, and the
fraction of the vector head color that the endpoint of that
segment should be. A random jitter is added to this final
color. The jittering is controlled by a scale factor for each
component. All computations are performed in HSV
space, with the hue wrapping around from one to zero, and
the saturation and value components clamped at zero and
one. A transparency value is also specified at each segment.
No jittering is applied to this.
Figure 14 shows our tornado with very sparse and
opaque hairs. With these settings you can clearly see the
hairs coming from the normal direction and bending into
the velocity direction. Figure 15 has more hairs that are
much more transparent.
Conclusions
How do these techniques relate to each other and previously
developed techniques? Table 1. correlates 3D vector
field visualization techniques to various attributes and
problem tasks. Our motto throughout this research is that
we are not developing better techniques, but expanding on
the set of tools available. Different scientists can gain insight
better with different tools and many tools are usually
required in an analysis. We have not tried to be all encompassing
or thorough. There are many other techniques that
should be added: stream tubes, topology extraction, shaded
particles, etc. There are also many other characteristics
that should be considered. Hopefully it is a useful starting
point.
Acknowledgments
This work was performed under the auspices of the
U.S. Department of Energy by Lawrence Livermore National
Laboratory under contract number W-7405-ENG-
48, with specific support from an internal LDRD grant.
Barry Becker helped with some of the programming and
debugging, and Jan Nunes helped with the video production.
Th HEPA filter data is courtesry of Bob Corey at
LLNL. The Aerogel data is courtesy of Tony Ladd and
Elaine Chandler, both at LLNL. We would also like to
thank the reviewers for their valuable comments.

Hedgehogs
Particle
Trace
Stream
Line
Stream
Ribbon
Flow
Volume Textured
Splats
Stream
Surface
LIC
Spot
Noise
Line
Bundles
Constrai
ned
Stream
Lines
Dimensionality 0D/1D 0D 1D 2D 3D 3D 2D/3D 2D/3D 3D 3D 2D 2D
Hardware
Accelerated
Doesn’t
Require
Advection
Dynamic
Motion
Global
Representation
● ● ● ● ● ● ❍ ❍ ● ● ● ◗
● ❍ ❍ ❍ ❍ ● ❍ ❍ ❍ ● ❍ ●
❍ ● ❍ ❍ ◗ ● ❍ ● ● ❍ ● ❍
◗ ❍ ❍ ❍ ● ● ◗ ● ● ● ❍ ❍
Near Surfaces ❍ ❍ ❍ ❍ ❍ ◗ ❍ ◗ ● ◗ ● ●
Near
Tangential
Surfaces
Unsteady
Flows
❍ ❍ ❍ ❍ ❍ ❍ ● ● ● ● ● ❍
❍ ● ❍ ❍ ◗ ● ❍ ◗ ❍ ● ❍ ◗
User Probing ❍ ● ● ● ● ❍ ◗ ❍ ❍ ❍ ❍ ❍
Interactive
Rendering
References
◗ ● ● ● ● ◗ ◗ ❍ ◗ ● ❍ ●
Table 1: Comparison of 3D Vector Field Visualization Techniques
[Cabral93] Brian Cabral and Lieth Leedom, “Imaging vector
fields using line integral convolution,” Computer Graphics
Proceedings, Annual Conference Series, ACM Siggraph,
New York (1993) pp. 263 - 270
[Crawfis92] Roger Crawfis and Nelson Max, “Direct volume visualization
of three dimensional vector fields,” Proceedings,
1992 Workshop on Volume Visualization, ACM Siggraph,
New York (1992) pp. 261 - 266
[Crawfis93] Roger Crawfis and Nelson Max, “Texture splats for
3D scalar and vector field visualization,” Proceedings, Visualization
’93, IEEE Computer Society Press, Los Alamitos,
CA (1993) pp. 261 - 266
[Forsell94] Lisa Forssell, “Visualizing flow over curvilinear grid
surfaces using line integral convolution,” these proceedings.
Levoy88] Mark Levoy, “Display of surfaces from volume data,”
IEEE Computer Graphics and Applications Vol. 8, No. 5
(May 1988) pp. 29 - 37
[Shoup79] Richard Shoup “Color table animation,” Computer
Graphics Vol. 13 No. 4 (August 1979) pp. 8 - 13
[Stolk92] J. Stolk and J. J. van Wijk “Surface particles for 3D
flow visualization,” in “Advances in Scientific Visualiza-
Hairs
tion,” F. H. Post and A. J. Hin, eds., Springer, Berlin (1992)
pp. 119 - 130
[VanGelder92] Allen Van Gelder and Jane Wilhelms, “Interactive
animated visualization of flow fields,” Proceedings, 1992
Workshop on Volume Visualization, ACM, New York (1992)
pp. 47 - 54
[vanWijk91] J. J. van Wijk “Spot noise: texture synthesis for data
visualization,” Computer Graphics Vol. 25 No. 4 (July 1991)
pp. 309 - 318
[vanWijk93] J. J. van Wijk “Flow visualization with surface particles,”
IEEE Computer Graphics and Applications Vol. 13
No. 4 (July 1993) pp. 18 - 24

Figure 3.Spot noise rendering of HEPA filter..
Figure 9. Projected Streamlines - XY Projection.
Figure 13. Line bundle rendering of HEPA filter.
Figure 4. Spot noise near filter fibers.
Figure 12. Line bundle near aerogel surface.
Figure 15. Finer hair on tornado velocity contour.

Abstract
Time�dependent �unsteady� �ow �elds are com�
monly generated in Computational Fluid Dynamics
�CFD� simulations� however� there are very few �ow
visualization systems that generate particle traces in
unsteady �ow �elds. Most existing systems generate
particle traces in time�independent �ow �elds. A par�
ticle tracing system has been developed to generate par�
ticle traces in unsteady �ow �elds. The system was
used to visualize several 3D unsteady �ow �elds from
real�world problems� and it has provided useful insights
into the time�varying phenomena in the �ow �elds.
In this paper� the design requirements and the archi�
tecture of the system are described. Some examples of
particle traces computed by the system are also shown.
1 Introduction
Particle systems were introduced in �14� to model
fuzzy objects like �re� clouds� and water. For this type
of particle system� the motion of the particle is based
on some stochastic model. Extensions to this type
of particle system have included modeling of snow�
grass� smoke� and �reworks. In CFD� particle traces
can be used to visualize several time�varying phenom�
ena in the �ow �eld. For example� vortex shedding�
formation� and separation �15�. When particle traces
are used in this context� the motion of the particle is
based on the physical velocity from the �ow �eld.
An instantaneous streamline is a curve that is tan�
gent to the vector �eld at an instant in time. In time�
independent �steady� �ow� instantaneous streamlines
are computed from the �ow �eld at an instant in time.
A streakline is a line joining the positions at an instant
in time of all particles that have been released from
a �xed location� called the seed location. In unsteady
�ow� streaklines are computed from several thousand
UFAT � A Particle Tracer
for Time�Dependent Flow Fields
David A. Lane
Computer Sciences Corporation
NASA Ames Research Center
M�S T27A�2
Mo�ett Field� CA 94035
time steps. Streaklines are commonly simulated by
releasing particles continuously from the seed loca�
tions at each time step. In hydrodynamics� streaklines
are simulated by releasing hydrogen bubbles rapidly
from the seed locations. Instantaneous streamlines
and streaklines are identical in steady �ow �elds.
In this paper� I introduce a particle tracing sys�
tem called Unsteady Flow Analysis Toolkit �UFAT��
which generates particle traces in unsteady �ow �elds.
UFAT di�ers from existing systems in that it com�
putes streaklines from a large number of time steps�
performs particle tracing in �ow �elds with moving
grids� provides a save�restore option� and supports
playback. Preliminary results of UFAT were presented
in �10�. This paper describes the design requirements
and the architecture of UFAT. First� the basic problem
of particle tracing in unsteady �ow �elds is described.
The design requirements and the particle tracing al�
gorithms of UFAT are then described. Examples of
streaklines computed for three real�world problems are
shown� and the performance of UFAT is analyzed. Fi�
nally� future enhancements for UFAT are discussed.
2 Particle Tracing
The basic problem of particle tracing can be stated
as follows� assume that a vector function � V �p� t� is
de�ned for all p in the domain D and t 2 �t1� t n��
where n is the number of time steps in the unsteady
�ow. For any particle p 2 D� �nd the path of p. The
path of p is governed by the following equation�
dp
dt � � V �p� t�� �1�
The path of p can be found by numerically integrating
Equation �1�. Several schemes can be used to inte�
grate the above equation. A common scheme is the

second�order Runge�Kutta integration with adaptive
stepsizing. Let p0 be the initial point �the seed loca�
tion� of the particle p and k � 0. Then�
p � � p k � h � V �p k� t��
p k�1 � p k � h� � V �p k� t� � � V �p � � t � h���2�
t � t � h and k � k � 1� �2�
where h � c�max� � V �p k��� max�� is the maximum ve�
locity component of � V �p k�� and 0 � c � 1� The con�
stant c controls the step size of the particle. If c is
small� then the particle will traverse many steps in
the grid cell. Small values of c should be used for grid
regions with rapidly varying velocity. Otherwise� the
particle p may advance out of the domain in just a
few steps. The integration scheme stated above can
be performed in the physical coordinate space or in
the computational coordinate space. If the integra�
tion is performed in computational space� then it is
simple and fast. During the integration� a cell search
operation is performed to determine the grid cell that
p k�1 lies in. Since the grid domain D is rectilinear
in computational space� the grid cell can be easily de�
termined by taking the integer computational coor�
dinates of p k�1. For example� if the computational
coordinates of p k�1 are ��� �� ��� then p k�1 lies in grid
cell �int���� int���� int����. In the physical coor�
dinate space� the grid domain D is curvilinear. The
cell search operation usually requires performing an
iterative algorithm to �nd the cell that p k�1 lies in.
A common algorithm used is the Newton�Raphson
method. Although particle integration can be done
faster in the computational coordinate space than in
the physical coordinate space� integrating in compu�
tational space may be inaccurate if there are singular�
ities in the grid. For computational space integration�
physical velocities are transformed into computational
velocities. Singularities in the grid could result in in��
nite transformed velocities �4�. For this reason� UFAT
performs particle integration in physical space.
If the grid is moving in time� then p k�1 is likely
to be in a grid cell di�erent from the cell that p k lies
in. To determine the cell that p k�1 lies in� the cell
search operation discussed above is performed. If the
grid consists of several blocks �a type of grid known
as a multi�block grid�� then p k�1 may lie in a block
di�erent from the block that p k lies in. If p k is near
the boundary of a block� then it is necessary to check
if p k�1 will be in a di�erent block. This also requires a
cell search operation. For a detailed discussion of the
basic problems in particle tracing� see �5� and �13�.
3 Related Work
Presently� many systems are available for steady
�ow visualization. However� most of these systems
only provide instantaneous visualization of the �ow
data. For example� instantaneous streamlines� isosur�
faces� and slicing planes. Several e�ective techniques
were recently developed for interactive interrogation
of instantaneous �ow �elds. Some of these techniques
are described in �6�8�11�. To date� there are very few
particle tracing systems that can generate streaklines
using a large number of time steps from unsteady �ow
�elds. Two of these are Virtual Wind Tunnel �VWT�
�3� and pV3 �7�. VWT provides interactive visualiza�
tion of particle traces in a virtual environment using
a stereo head�tracked display and a data glove. Al�
though VWT is an e�ective interactive tool for un�
steady �ow visualization� it requires a preprocessing
of the �ow data� and the number of time steps that
the user can visualize is determined by the memory
size of the system. pV3 allows interactive animation
of unsteady �ow data by looping through an input �le
that contains the names of the �ow data �les. This is
similar to an interactive scripting approach. pV3 does
not save the visualization results� hence� playback is
not supported and re�calculation is required to repeat
the animation.
4 Requirements
It is common to generate several thousand time
steps of �ow data in a CFD simulation� however� it
is presently impossible to visualize �ow data from all
these time steps at one time. Scientists sometimes
use one of the following approaches� �1� visualize the
data at some snapshots in time or �2� save every nth
time step of the data and then visualize the subset of
data. Regardless of the approach used� there are usu�
ally hundreds of time steps that need to be visualized
�10�. A requirement for UFAT is that it must be able
to compute streaklines from a large number of time
steps.
A complex grid usually consists of several grid
blocks. For some grids� one or more grid blocks may
move as a function of time� a characteristic of grids
with rigid�body motion. Moving grids are commonly
used in pitching airfoils� oscillating �aps� rotating tur�
bine fans of combustion engines� and rotating heli�
copter blades. Another requirement for UFAT is that

it must be able to compute particle traces in unsteady
�ow with moving grids.
The ability to visualize a scalar quantity in the �ow
data can be crucial for some �ow analysis. Quantities
that are commonly computed are temperature� pres�
sure� mach number� and density. A requirement for
UFAT is that it must assign a color to each particle
based on the value of a speci�ed quantity sampled at
the particle�s location. The color of the particle can
also be based on its position� the time at which it was
released� or the seed location where it was released.
Interactive visualization of large time�dependent
�ow �elds is di�cult or nearly impossible due to the
data size. Sometimes� a scripting approach is used
to save the visualization results from each time step�
and the visualization results are then played back at a
later time. In some visualization systems� interactive
visualization is feasible by using one of the following
approaches� �1� preprocess the data so that a number
of time steps can be stored in memory or �2� sample
the �ow data at a lower resolution so that the data
can be stored in memory. By storing the �ow data
in memory� the data can be interactively visualized.
The latter approach is usually not desirable because
the accuracy of the �ow data is lost when the data is
sampled at a lower resolution. With either approach�
the size of the physical memory dictates how much
�ow data can be visualized interactively. Although
these two approaches can provide interactive visual�
ization� important features may not be detected be�
cause of the reduced representation of the �ow data.
Using a scripting approach� the entire �ow data can be
analyzed. Furthermore� once the visualization results
have been saved� the scientist can play back the results
repeatedly without any additional computation. Visu�
alization playback is another requirement for UFAT.
5 Unsteady Flow Analysis Toolkit
UFAT was developed to compute streaklines using a
large number of time steps in 3D unsteady �ow �elds.
It handles single and multi� block curvilinear grids�
and the grid may have rigid�body motion. Particles
are released continuously from the speci�ed seed lo�
cations at each time step. The particles are advected
through all time steps until they leave the grid domain.
UFAT saves the current positions of the particles at
each time step� thus� the particle traces can be played
back at a later time. Particles are colored according
to a scalar quantity. The quantity may be a physical
quantity of the �ow �e.g. pressure� temperature� and
density�� a position coordinate �x� y� or z� of the parti�
cle� the time at which the particle was released� or the
seed location where the particle was released. UFAT
uses an adaptive�time integration scheme to advect
the particles in the physical coordinate space. The
integration scheme can be of second or fourth order.
UFAT also allows particles to be traced along the grid
surface. This type of particle trace simulates oil �ow
on a surface. Sometimes� the available disk on a sys�
tem may not be able to store all time steps of �ow
data. UFAT provides a save�restore option so that
particle tracing can be performed in several run ses�
sions. This allows particle traces to be computed from
many time steps without requiring all time steps to be
online at one time.
5.1 Data Structure
In order to advect particles through all time steps�
UFAT stores the two most recent time steps of the �ow
data in memory. Particles are successively advected
from the current time step to the next time step. The
particle traces are stored in a two�dimensional array
of size N s � N t� where N s is the number of seed lo�
cations and N t is the number of time steps. Each
entry in the array is a structure that contains the
physical and computational coordinates of the parti�
cle and the time at which the particle was released
from the seed location. There is also an array of size
N s that stores the number of particles in each trace.
Let T race Length�s� denote the number of particles in
trace s� and it is initialized to zero. At each time step�
a new particle is released from the seed location and
T race Length�s� is incremented by one. When a parti�
cle in trace s leaves the grid domain� T race Length�s�
is decremented by one.
5.2 Algorithm
This section outlines the particle tracing al�
gorithm in UFAT. The following procedures in
the algorithm are described� Step Through Time���
Advect Trace��� and Advect Particle��. Proce�
dure Step Through Time�� steps through all time
steps in the given �ow data and calls Advect Trace��.
The main task of procedure Advect Trace�� is to
advect the active particles in all traces from the
current time step to the next time step. The ac�
tual particle integration is performed in procedure
Advect Particle��. For brevity� let current time de�
note the current time step and next time denote the
next time step. For each procedure� a description is
given followed by the pseudocode of the procedure.
Procedure Step Through Time�� begins by loading
the �rst two time steps of the �ow and grid data

into memory. If the grid is �xed� then only one
grid is loaded into memory. Then� it steps through
all time steps in the �ow data. For each time step�
the following tasks are performed� �1� Call procedure
Advect Trace�� to advect particles in every trace
from current time to next time. �2� Write the cur�
rent particle traces to the trace �le. A frame marker
is also written to denote the end of each time step.
�3� Read the next time step�s �ow. If the grid is mov�
ing in time� then read the next time step�s grid. The
pseudocode for this procedure is given below�
Procedure Step Through Time��
Read first two time steps of flow and grid data
For t � 1 to Nt � 1 do
Advect Trace�t� t � 1�
Write current traces to the trace file
Read the next time step�s flow data
If moving grid then
Read the next time step�s grid
End for
Procedure Advect Trace�� advects particles in ev�
ery trace from current time to next time. The pro�
cedure performs the following steps for each trace�
�1� Copy all particles in the trace to a working trace
array w. �2� Call procedure Advect Particle�� to
advect each particle in the working trace w from
current time to next time. If the particle is inside
the grid domain D after the advection� then the par�
ticle is saved to the trace. Otherwise� the particle is
considered to be inactive and it is discarded. �3� Re�
lease a new particle from the trace�s seed location and
save the particle in the trace. The pseudocode for this
procedure is as follows�
Procedure Advect Trace�current time� next time�
For s � 1 to Ns do
Copy trace s to working trace w
W Length � T race Length�s�
Remove all particles in trace s
T race Length�s� � 0
For i � 1 to W Length do
p � the ith particle in working trace w
Advect Particle�current time� next time� p�
If p 2 D then
Store p in trace s
T race Length�s� � T race Length�s� � 1
End if
End for
Release a new particle from seed s and
store it in trace s
T race Length�s� � T race Length�s� � 1
End for
Procedure Advect Particle�� advects the given
particle p from current time to next time. The pseu�
docode shown below uses the second�order Runge�
Kutta integration scheme given in Equation �2� and
is based on a predictor�corrector algorithm used in
PLOT3D �5�. The �ow data is only given at some
number of time steps. If t 6� t i for i � 1� � � � � N t�
then an interpolation in time is performed. Since the
velocity is known only at discrete points in the grid�
when p does not coincide with a grid point� a trilinear
interpolation in physical space is also performed. Fol�
lowing are the steps in procedure Advect Particle���
�1� Initialize t to current time. The variable t is in�
cremented at each advection and the procedure exits
when t � next time or when the particle has left the
grid domain D. �2� Interpolate the velocity � V at p. �3�
Compute the time increment h� where h � c�max� � V �
and c is a fraction of the grid cell that each particle
must take inside the cell. The constant c can be con�
sidered as a normalized stepsize and 0 � c � 1. For
example� if the particle must traverse �ve steps in a
cell� then let c � 0�2. �4� Increment t by h. �5� Com�
pute the predictor p � . �6� Interpolate the velocity � V �
at p � . �7� Compute the corrector� which is the posi�
tion of p after the advection. Below is the pseudocode
for procedure Advect Particle��.
Procedure Advect Particle�current time� next time� p�
t � current time
While �t � next time AND p 2 D � do
�V � Interpolate Velocity�p� t� current time�
next time�
Adjust�
h � c�max��V �
If �t � h � next time� h � next time � t
t � t � h
f Predictor step g
p � � p � h � � V
�V � � Interpolate Velocity�p � � t � h�
current time� next time�
f Adaptive stepsizing g
�Vtotal � ��V � �V � ��2
If �h � max��Vtotal� � c� then
�V � �Vtotal
t � t � h
Goto Adjust
Endif
f Corrector step g
p � p � h � � � V � � V � ��2
End while
Although the pseudocode shown above only pro�
vides a second�order integration scheme� a fourth�
order integration scheme has also been implemented in

UFAT. Procedure Interpolate Velocity��� which is
not shown� interpolates velocity in time followed by a
trilinear interpolation in the physical space. When a
new position for p is computed� a cell search step is
performed to determine the grid cell that p lies in �see
Section 2�.
5.3 Animation
The most e�ective method to view streaklines is to
animate the particle traces. UFAT saves streaklines
at each given time step to a trace �le� which can then
be animated with a visualization system. Although
particles may traverse in non�uniform time steps� the
positions of the particles are sampled at uniform time
steps �i.e. the given time steps�. The particle trace
�le contains basic graphics primitives such as points
and lines. These basic primitives can be written in
a format so that they can be easily read by other vi�
sualization systems such as AVS� IRIS Explorer� and
FAST �2�. The only requirement for the visualization
system is that it must be able to animate the streak�
lines through a given sequence of time steps.
6 Distributed Visualization
It is common that an unsteady �ow data set is too
large to be stored locally on a graphics workstation.
The �ow data set is often stored on a remote system
with a large disk capacity. The computation is then
performed on the remote system while the results are
sent over the network to the graphics workstation for
interactive visualization. The data transfer rate on
the network must be fast enough so that the image
on the graphics workstation can be updated at least
15 frames per second for a reasonable animation. The
size of the particle traces at each time step is relatively
small compared to the size of the grid �le. Thus� the
particle traces at each time step can be sent over the
network in a reasonably short period of time. If the
grid is moving in time� then a new grid must also
be sent over the network at each time step. It may
take several seconds to transfer a grid consisting of
several million grid points� depending on the speed of
the network. Hence� distributed �ow visualization is
practical if the data transfer rate of the network is fast
enough to handle the amount of data that will be sent
over the network.
7 Results
This section shows some streaklines that were com�
puted by UFAT for three unsteady �ow data sets. Al�
though the examples shown in this section are only
from CFD applications� other applications with time�
dependent �ow data can easily use UFAT to gener�
ate streaklines. The input data must consist of the
grid geometry and the �ow quantities sampled at the
grid points. Currently� UFAT only supports curvilin�
ear grids.
The �rst data set is a clipped Delta Wing with con�
trol surfaces� which oscillate at a frequency of eight
Hertz �Hz� and with an amplitude of 6.65 degrees.
Each oscillation cycle consists of 5�000 time steps. For
visualization purpose every 50th time step is saved�
for a total of 100 time steps per cycle. The clipped
Delta Wing grid consists of 250 thousand points in
seven blocks. For this CFD simulation� the scientists
evaluated a new zoning method called �virtual zones��
which is used for grids with time�varying boundary
conditions. Virtual zones simplify the grid genera�
tion problem for complex geometries and for time�
dependent geometries �9�. Figure 1 shows the seed
locations� which are colored by position� at the lead�
ing edge of the clipped Delta Wing. Figure 2 shows
streaklines at time step 7� where the control surfaces
have de�ected 2 degrees up. The evenly spaced parti�
cle traces �colored in cyan� near the center of the wing
indicates that the �ow is relatively steady in that re�
gion. However� the �ow is very turbulent near the
tip of the wing. Figure 3 shows the streaklines after
the control surfaces have completed one oscillation. At
this time� the control surfaces have de�ected 5 degrees
up. This �gure shows that some particles �colored in
red�� which were released from the outer part of the
leading edge of the wing� have moved toward the tip
of the wing due to the control surfaces. This behavior
can be seen clearly in an animation of the streaklines.
The second data set is an arrow wing con�guration
of a supersonic transport in transonic regime. Tran�
sonic �utter is known as a design problem on this con�
�guration. Scientists want to develop a computational
tool to examine the in�uence of control surface oscil�
lations on the lift of the transport for the suppression
of the �utter. The arrow wing grid consists of ap�
proximately one million points in four blocks. The
control surfaces oscillate at a frequency of 15 Hz and
with an amplitude of 8 degrees. Figure 4 shows streak�
lines surrounding the transport at time steps 25 and
175. The particles are colored by their seed locations�
where the particles were released. It can be seen that
there is vortical separation from the leading edges of
the wing. From the simulation� it was found that
the symmetric oscillation produces higher lift than the
anti�symmetric oscillation �12�.

The third data set is the proposed airborne observa�
tory known as the Stratospheric Observatory For In�
frared Astronomy �SOFIA�. SOFIA is a modi�ed Boe�
ing 747SP transport with a large cavity that holds a
three�meter class telescope. The CFD scientists want
to assess the safety and optical performance of a large
cavity in the 747SP �1�. SOFIA would be the successor
to the Kuiper Airborne Observatory �KAO�� which is
the only aircraft in the world that currently provides
this type of infrared observing capability. SOFIA con�
sists of approximately four million points in 41 grids.
A total of 50 time steps were saved for the visual�
ization. Figure 5 shows streaklines surrounding the
SOFIA airborne observatory at time step 40. In the
�gure� the telescope �partially visible� inside the cav�
ity of the jet is colored in cyan. The particles are
colored by the time of their release from a rake posi�
tioned in the aperture of the cavity. Blue represents
the earliest time and orange represents the most re�
cent time. Figure 6 shows a close�up view of the tele�
scope without the aircraft body. The �oating object
�colored in gray� above the telescope is the secondary
mirror of the telescope. The cavity is represented by
the semitransparent surface enclosing the telescope.
Note that some particles are trapped inside the cavity�
while some have escaped and passed the empennage.
8 Performance
The performance of UFAT depends on three fac�
tors� �1� the grid size� �2� the number of time steps�
and �3� the number of seed locations. At each time
step� UFAT reads the �ow data �le �and the grid �le
if the grid is moving in time�. Depending on the disk
I�O rate and the grid size� it could take from several
seconds up to several minutes to read the �ow and
grid �les at each time step. For example� if the disk
I�O rate is 10 megabytes per second� then it would
take approximately 1.2 seconds to read a grid �le with
one million grid points� assuming that there are three
physical coordinates �x� y� and z� for each grid point.
If the �ow data �le �solution �le� contains �ve scalar
quantities� then it would take approximately 2.0 sec�
onds to read the �le. Thus� it would require a total of
3.2 seconds per time step to read the grid and solu�
tion data. The number of particles that UFAT advects
at each time step increases linearly. If there are 100
seed locations and 1�000 time steps� then the maxi�
mum number of particles that UFAT can advect at
time step 1�000 is 100�000 particles. Using the clipped
Delta Wing as an example� it took approximately 2.3
seconds to read a grid �le and 3.0 seconds to read
a solution �le at each time step on a Silicon Graph�
ics 320 VGX graphics workstation. The size of the
grid �le is four megabytes and the solution �le is �ve
megabytes. It took approximately 21 minutes to com�
pute streaklines from 100 time steps and with 36 seed
locations using a single 33�megahertz processor on the
VGX graphics workstation. This includes the time
to read the grid and solution �les at each time step.
The size of the trace �le generated by UFAT is 2.5
megabytes.
9 Future Work
A disadvantage of the current version of UFAT is
that it performs particle tracing sequentially. Particle
tracing is an �embarrassingly� parallel application. It
would be ideal to take advantage of multiple proces�
sors to perform particle tracing in parallel since each
particle trace can be computed independently. A par�
allel version of UFAT has been developed on the Cray
C90� Convex C3240� and SGI systems. The initial re�
sults indicate that the performance can be improved
by several factors� depending on the number of proces�
sors used. Another enhancement that is currently be�
ing investigated is how to distribute the particle trac�
ing task to a cluster of heterogeneous systems using a
message passing library. An issue to be worked out is
how to minimize the amount of data that each system
needs for particle trace computation. The goal is to
have a distributed parallel version of UFAT that would
provide interactive particle tracing in large�scale un�
steady �ow �elds.
Acknowledgments
The �ow data sets were provided by Chris Atwood�
Goetz Klopfer� Steve Klotz� and Shigeru Obayashi.
This work was supported by NASA under contract
NAS 2�12961.
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The Design and Implementation of the Cortex Visualization System
Deb Banerjee Chris Morley Wayne Smith
Abstract
Cortex has been designed for interactive analysis
and display of simulation data generated by CFD ap�
plications based on unstructured�grid solvers. Unlike
post�processing visualization environments� Cortex is
designed to work in co�processing mode with the CFD
application. This signi�cantly reduces data storage
and data movement requirements for visualization and
also allows users to interactively steer the application.
Further� Cortex supports high�performance by running
on massively parallel computers and workstation clus�
ters.
An important goal for Cortex is to provide visual�
ization to a variety of solvers which di�er in their so�
lution methodologies and supported �ow models. Cou�
pled with the co�processing requirement� this has re�
quired the development of a well de�ned programming
interface to the CFD solver that lets the the visualiza�
tion system communicate e�ciently with the solver�
and requires minimal programming e�ort for porting
to new solvers. Further� the requirement for targeting
multiple solvers and application niches demands that
the visualization system be rapidly and easily modi��
able. Such �exibility is attained in Cortex by using
the high�level� interpreted language Scheme for im�
plementing user�interfaces and high�level visualization
functions. By making the Scheme interpreter avail�
able from the Cortex text interface� the user can also
customize and extend the visualization system.
1 Introduction
Cortex is a visualization system developed for in�
teractive control and visualization of simulations per�
formed with a variety of unstructured CFD solver
codes. Most current visualization systems for CFD
are either post�processing systems�1� 6� or data�ow
Fluent Inc.
Centerra Resource Park
Lebanon� NH 03766
fdeb�cmm�wasg��uent.com
visualization�13� systems. Post�processing systems
read in data which the CFD application has written
out to disk in a format recognizable by the visualiza�
tion system. A drawback of this paradigm is that it
hinders interactivity. The user cannot conveniently
observe the solution as it evolves or modify �ow�
solution parameters in response to those observations.
Data�ow visualizations systems allow co�processing
where the CFD application can run as a module in
the network. This lets the user observe the �ow��eld
at frequent intervals as the solution evolves. However�
the visualization modules in the network may require
a copy of the �ow��eld to perform the visualization. In
addition to increasing data storage requirements� such
data movement can cause signi�cant network commu�
nication when the CFD application runs on a remote
compute�server and the rendering modules run on the
users local workstation. Additionally� in data�ow sys�
tems the �ow of data is directed from the applica�
tion towards the display. Features such as application
steering and data probes require data��ow in the re�
verse direction � from the user�interface or display to
the CFD application. Cortex is a co�processing visu�
alizing system which has a well�de�ned programming
interface to the CFD application. This programming
interface was designed to minimize data storage and
data movement requirements. This programming in�
terface has been ported to a variety of unstructured�
grid solvers that di�er in their solution methodologies
and supported �ow models. Currently� Cortex is avail�
able as the visualization system for the following CFD
codes� �1� Rampant�10� designed for solving transonic
compressible �ows� �2� Fluent�UNS designed for solv�
ing low�speed incompressible �ows and �3� Nekton �8�
for very low�speed creeping �ows. These solvers have
2D versions based on triangular and quadrilateral cells
and 3D versions based on tetrahedral and hexahedral
cells.
Cortex has been designed for providing interactive

visualization to a wide variety of solvers and applica�
tion niches. This requires that the visualization sys�
tem be able to provide user�interfaces that are cus�
tomized to the application. In Cortex� both graphi�
cal and text interfaces are implemented in the high�
level� interpreted language Scheme. This lets devel�
opers and users interactively and rapidly modify the
user�interface. Similarly� most high�level visualization
functions are also implemented in Scheme on top of
low�level functions that are implemented in C. The
advantage of using Scheme is that it supports rapid�
interactive development lending �exibility to the visu�
alization system.
Currently CFD solvers are solving increasingly
larger problems. For example� Cortex has been used
to visualize a Rampant simulation over a 3D tetrahe�
dral automobile intake cylinder head mesh consisting
of 800�000 cells. This has driven the implementation of
those codes on massively parallel computers and work�
station clusters. In addition to reducing run�times�
larger problems can be run only on parallel computers
due to larger available memory. Often� these problems
cannot even be run on a serial computer. Domain de�
composition is a commonly used programming model
for parallelizing solvers � the �ow�domain is parti�
tioned among processors and an instance of the solver
is run on each node. A serial visualization system for a
parallel CFD solver would require that the entire �ow�
domain be copied over across the network. It is more
e�cient if the visualization system� or at least those
portions of it that require access to the �ow�domain�
also be implemented in parallel and run on the same
processors as the solver. Cortex has been parallelized
on a variety of parallel architectures including Intel
IPSC�860� Intel Paragon and on workstation clusters.
Cortex and the CFD application exist as either a
single process� two processes or multiple processes for
the parallel version. The Cortex implementation has
various forms� �1� a library for a single process ap�
plication� �2� a visualization process and a client li�
brary that links in the CFD application to form the
solver process� �3� a visualization process �same as
in �2�� and multiple instances of client libraries that
communicate with the visualization process through
a portable communication library for the parallel ver�
sions. However� all these implementations are derived
from a single set of sources �les by using di�erent C
pre�processor switches during compilation and linking.
The rest of the paper is organized as follows. In
Section 2� we describe the primary goals that have
driven the design of Cortex. The software architec�
ture of Cortex is outlined in Section 3. Among the
software modules described in Section 3� the data
mapping module is of special interest since it accesses
solver data through an API�application programming
interface�. This module and its integration with the
solver is described in Section 4. Further� the data
mapping module is that portion of Cortex that runs
on massively parallel computers and on workstation
clusters. The parallel implementation is described in
Section 5. The ability to customize and extend Cor�
tex using Scheme is described in Section 6 and� �nally�
Cortex is compared to other visualization systems in
Section 7.
2 Design Goals
The Cortex visualization system has been designed
to provide advanced visualization techniques� a mod�
ern graphical interface� and a programmable textual
interface to a family of unstructured CFD codes which
di�er in their solution methodologies and supported
�ow models. The major design goals of Cortex are�
1. Interactivity� User interaction must be sup�
ported by a variety of means including direct ma�
nipulation of view parameters via a virtual track�
ball method and interactive modi�cation of color
maps. In addition� users should be able to query
data values on a displayed object. For example�
users can ask for cell coordinates and velocity val�
ues at any point of an iso�surface by simply click�
ing at that point. Finally� users can interactively
steer the simulation by modifying solution param�
eters of the �ow�solver or by adapting the grid.
2. Scalability� The visualization system should
support CFD applications running on work�
station clusters and massively parallel comput�
ers. The visualization system should minimize
data storage and network communication require�
ments. It should store only that part of the �ow�
�eld that makes up the displayed object which�
in general� could be orders of magnitudes smaller
than the entire �ow��eld data. In addition to re�
ducing computation time� an important reason
for running simulations in parallel is the large
available memory which makes it feasible to run
very large problems involving millions of cells.
Since the �ow��eld can not even be stored on a
serial computer� the portion of the visualization
system that requires access to �ow data must also
be executed in parallel with the solver.

3. Solver Portability� The visualization applica�
tion should provide user�interfaces and visualiza�
tion functions to a variety of CFD solvers which
are organized around potentially distinct domain
data structures and are targeted towards distinct
application areas. Integration with new solvers
should require minimal additional programming
e�ort. Access from the visualization system to
solver data structures should be through clear�
well�de�ned interfaces.
4. Flexibility� Since the visualization system is tar�
geted towards multiple solvers and application
niches� it should be �exible� i.e.� extensible and
customizable. For example� implementing appli�
cation steering requires that solvers provide user�
interfaces that allow modi�cation of �ow�solution
parameters. Both the user�interfaces and �ow�
solution parameters are solver�speci�c. The vi�
sualization system should allow rapid interactive
development and modi�cation of user�interfaces
and visualization functions when porting to new
solvers.
3 Software Architecture
3.1 Overview
The serial implementation of Cortex has the follow�
ing software modules as shown in Figure 1.
1. Data Mapping� Data mapping functions in�
cluding iso�surface extraction� particle tracking�
generation of contours and velocity vectors are
implemented here. This is the only portion of
Cortex that runs as part of the CFD application
process. It has has access to the solver through a
well�de�ned API �Application Programming In�
terface�. This module is described in greater de�
tail in Section 4. It is also that portion of Cortex
that has been implemented to run on massively
parallel computers and workstation clusters as de�
scribed in Section 5.
2. Graphics� Display and view manipulation rou�
tines are implemented in this module. All low�
level graphics subroutines are implemented by
calls to the HOOPS�14� graphics library to pro�
vide platform independent graphics. This module
receives data from the data mapping module via
RPC�s.
3. User Interface� Cortex provides a graphical
point�and�click user interface and a command�line
text interface. The Cortex graphical user inter�
face is based on Motif. Routines for implementing
various widgets including tables and lists are de�
�ned in this module.
4. Scheme Interpreter� A Scheme interpreter is
available from the Cortex text interface. This lo�
cal interpreter which runs as part of the visual�
ization process has access to a wide variety of C
functions implemented in other Cortex modules.
Graphical and text interfaces have been imple�
mented as Scheme functions that call Cortex C
routines which have been made accessible to the
interpreter. Similarly� there is a Scheme inter�
preter which runs as part of the remote solver
process.
When a serial CFD solver runs under the Cortex
visualization system� the resulting application has one
of the following structures.
1. Single Process� The solver and Cortex runs as a
single process. In this case Cortex is implemented
as a library that is linked in with the solver.
2. Two Process� The solver and Cortex runs as
two processes. In this case� the Cortex modules
�2�� �3� and �4� make up the visualization pro�
cess that runs on the user�s local work�station.
All graphics operations such as rotations� pan�
ning� color�map editing etc. are performed in
this process thereby providing greater interactiv�
ity and frees the compute�server concentrate on
running the simulation only. The Cortex data
mapping module� is linked with the solver to form
the solver process that may possibly run on a
remote high�performance compute�server. The
solver process communicates with the visualiza�
tion process through sockets and RPC�s. User
commands are sent from the user�interface to the
solver process via sockets and display data is
transmitted from the solver process to the visual�
ization process via RPC�s.
An interesting feature in Cortex is the use of
Scheme interpreters in the visualization process and
the solver process. C functions in each process can be
made accessible to the respective Scheme interpreters.
This is done at run�time by registering the C function
and its Scheme name with the interpreter. This al�
lows the interpreter to evaluate the scheme function by
simply calling the relevant C function. Cortex trans�
forms all user commands from the GUI and text inter�
face into a collection of Scheme function calls. These
Scheme functions are either evaluated locally in the

User Interface
Toolkit
Visualization Process
Rendering
Engine
Scheme Interpreter (GUI and Text user interface)
RPC’s
Data
Mapping
Scheme Interpreter
(API)
Solver Process
(Sockets)
CFD
Application
Figure 1� Software Architecture of Cortex �Two�Process Implementation�
visualization process or transmitted to the Scheme in�
terpreter on the solver process via sockets.
Application steering is implemented in Cortex by
de�ning C functions in the CFD application that mod�
ify appropriate solution parameters or adapt the grid.
Next these C functions are bound to Scheme function
names and made available to the solver Scheme in�
terpreter. Finally� the GUI and text user interfaces
are implemented for this feature. These user�interface
functions are converted into a collection of Scheme
function calls by the local Scheme interpreter. Some
of these calls will be evaluated remotely in the solver
Scheme interpreter by calling relevant C functions.
Section 6 provides more details on using Scheme for
interactively de�ning user�interfaces.
3.2 Parallel Implementation
The parallel implementation of Cortex is based on
domain decomposition. This model was chosen since
this is a commonly used model for parallelizing CFD
solvers. In fact� both the CFD solvers� Rampant and
Nekton� have been parallelized�11� 2� using this model.
In this model� the �ow domain is partitioned among
multiple processors and an instance of the solver is
run on each processor or node. In parallel Cortex�
the data mapping module is partitioned into a host
program and a node program as shown in Figure 2.
The node module is linked in with the solver node
program and executes on each processor. The visu�
alization process remains unchanged from the serial
implementation since the host makes it appear as if
the data is coming from a serial solver process. De�
tails are provided in Section 5.
4 Data Mapping in Cortex
Data mapping�12� consists of transforming appli�
cation data into renderable objects. Functionally� this
module receives data from the the solver� converts it to
desired geometric objects� and transmits the converted
data to the rendering engine. For example� during con�
touring� a 2D triangular cell from the grid is mapped
into a 2D triangle with color map indices �based on
�eld values� at each node and is then sent to the
renderer for display. The module handles visualiza�
tion functions such as iso�surfacing� particle tracking�
contouring and velocity vector generation. Section4.1
describes how this module uses surfaces to minimize

User Interface
Toolkit
Visualization Process
Rendering
Engine
(RPC’s)
network communication. Data mapping requires in�
tensive access to the application�s data structures. In
traditional post�processing visualization systems� this
module loads in the entire grid along with required
�eld�values at each node. In Section 4.2� we show how
Cortex reduces its memory overhead by having the
data mapping module execute with the solver as part
of the same process.
4.1 Surfaces
In Cortex� a visualization may be performed by
de�ning portions of the domain that are of interest.
These portions are represented and stored as surfaces
in Cortex. For example� users may create an iso�
surface of any of the stored or derived solution quan�
tities� or use quadratic functions in x� y� z to de�ne
lines� planes or other geometry. They may then ob�
serve �ow��eld variables such as density� pressure and
velocity vectors on that surface. Such surfaces are cre�
ated� typically� at the beginning of the simulation and
the �ow��eld is observed on it at frequent intervals as
the simulation unfolds through time. This makes it
useful to store surfaces during creation so that suc�
cessive displays of �ow��eld values on it will avoid re�
Scheme Interpreter (GUI and Text interface)
Scheme Inter.
Data Map
Node program
CFD App.
Node Program
(Sockets)
Scheme Interpreter
Host Program =
Data Map Host + CFD Host
Multiport (Portable Communication Library)
Node Program 1 Node Program 2 Node program n
Figure 2� Software Architecture of Parallel Cortex
computation of the surface grid. Further� storing such
surfaces on the visualization process which executes
on the local graphics workstation results in reduction
in network tra�c. This is because only �eld�values
and not the surface grid needs to be transported from
the solver process to the visualization process across
the network during� for example� contour and velocity�
vector displays. Since network data movement can
bottleneck distributed visualizations� such techniques
are important for enhancing performance.
4.2 Data Mapping API
Cortex has been designed to be integrated with the
CFD application and has access to the solver�s data
structures through a well�de�ned API. However� it is
important that the solver�speci�c components of the
API be as small as possible� thereby minimizing pro�
gramming e�ort in porting Cortex to a new solver.
Only the data mapping module in Cortex accesses
solver functions and data through the API. This is
achieved by having the data mapping module share
the solver process�s address space. The rest of Cor�
tex may run as a separate process as is the case in
the two�process and parallel implementations. The

float SV�Cell�Coordinates PROTO��CX�Cell�ID c� int dim���
void SV�Cell�Values PROTO��CX�Cell�Id c� float �val���
float SV�Node�Value PROTO��CX�Node�Id n���
void CX�Contour�Poly PROTO��int npts� float �points� float �vals���
separate Cortex process communicates with the data
mapping module through sockets and RPC�s.
The data mapping API of Cortex has two compo�
nents�
1. Solver De�ned� These functions are provided
by the solver. The data mapper uses them to
query the solver about data values including cell
and node values and coordinates� and cell con�
nectivity. Cortex can handle triangular� quadri�
lateral� tetrahedral and hexahedral cells. Fur�
ther� these cells may have sub�cells de�ned within
them. E�ort has been made to keep the solver
API concise and simple since it has to be imple�
mented separately for each speci�c solver.
2. Cortex De�ned� These functions are exported
by Cortex for use directly in CFD applications.
Figure 3 provides some of the API functions.Cortex
has been integrated with three separate solvers�
�1�Rampant� �2� Nekton and �3� Fluent�UNS. The
only additional programming e�ort for integration was
in implementing the solver�de�ned portion of the data
mapping API in each solver.
5 Parallel Implementation of Cortex
In this section we describe the implementation of
Cortex on massively parallel architectures and work�
station clusters. The need for parallelization was
driven by the fact that the solver may be running in
parallel. Running a parallel solver under serial Cor�
tex requires signi�cant data movement from the solver
to Cortex though slow communication networks. Fur�
ther� Cortex requires large amounts of memory� often
not available on serial workstations� to store the re�
sulting data. Our approach was to parallelize the data
mapping module which� as described earlier� runs as
part of the solver process in the serial environment.
An additional possibility is to parallelize the renderer.
Currently� we have not pursued this possibility. It is
not clear whether parallel renderers can out�perform
workstations with hardware support for rendering. In
fact in �3�� it is reported that it would take between
Figure 3� Functions in the Data Mapping Module API
10 and 20 high�end RISC workstations to equal the
performance of the SGI VGX graphics system. The
data mapping module performs functions that include
iso�surface extraction� particle tracking� surface con�
touring� and generation of velocity vectors. Most vi�
sualization functions except particle tracking are em�
barassingly parallel. Particle tracking is communica�
tion intensive since particles frequently migrate across
processors.
Parallel implementations of CFD solvers are usually
based on a domain decomposition strategy in which
the spatial domain for a problem is partitioned into
smaller sub�domains or partitions� and a separate in�
stance of the solver is invoked to simulate the �ow
within each partition. Information is transferred be�
tween neighboring partitions along partition bound�
aries. Communication is proportional to the perimeter
of the partition which is usually an order of magnitude
smaller than the size of the partition� and hence does
not slow down the simulation. We have parallelized
the data mapping module using domain decomposi�
tion. A separate instance of the module runs in each
processor or node and performs the visualization on
its portion of the data. These sub�visualizations are
sent to a host instance of the module which combines
the sub�instances and sends the data over to the visu�
alization process for display.
Our parallel implementation of iso�surfacing� based
on the marching cubes algorithm�7�� is similar to that
reported in �4�. Sub�surfaces generated on each node
must be communicated to the host which then trans�
fers it to the visualization process. The visualization
process is the same as in the serial implementation
� the host process makes it appear as if there is a
serial solver process generating the data. The cre�
ation of each local iso�surface does not require any
communication since all cells are stored locally. This
is achieved by having the solver store cells lying on
the boundary of domain partitions on both proces�
sors. As described before� representations of surfaces
are maintained both in the visualization process and
the solver process. The reduces network communica�
tion requirements� e.g.� when a surface is contoured
only the �eld�values at each point need to be trans�
mitted form the nodes to the host and then onto the

visualization process for display.
We have observed that network communication is
usually the slowest component in parallel visualiza�
tion. In the above example� the iso�surfaces can be
generated quite quickly� but they have to be transmit�
ted to the host process� one at a time. Other visual�
ization functions such as particle tracking have even
higher communication overhead. In particle tracking
the track of a massless particle in the �ow��eld is gen�
erated and displayed. The user speci�es the set of
points from where the particles are released. Parti�
cles keep falling o� sub�domains and require to be
restarted at a sub�domain stored in a remote parti�
tion causing large communication overheads.
6 Extensibility and Customizability
through Scheme
User�interfaces and high�level visualization func�
tions have been implemented in Scheme in Cortex.
Scheme is a high�level interpreted language with fea�
tures such as dynamic typing� automatic garbage col�
lection and has functions as �rst class objects. These
features make Scheme highly suited for rapid inter�
active prototyping. User�interfaces and visualization
functions can be modi�ed dynamically in the same
Cortex session. Such customization is performed not
only by developers but also by users. This �exibil�
ity is achieved by making a Scheme interpreter avail�
able from the text interface. Examples of such Scheme
functions are provided in Figure 4.
cx�clear�menubar clears the menu bar. menus is
a Scheme variable that contains the menu�names with
their descriptors. �cx�delete�menu is implemented
in C in the user�interface module. cx�add�item adds
a new menu�item. �cx�add�item is de�ned in C in
the user�interface module.
In Cortex� Scheme is used primarily for two pur�
poses.
1. User interfaces� All GUI panels� application
menubars and callbacks are de�ned in Scheme in
terms of functions implemented in a low�level Mo�
tif toolkit in C. CFD application developers and
users can add or modify existing interfaces and
add new ones interactively without quitting the
session. This is particularly important for Cortex
since it has been to targeted to a number of dis�
tinct solvers which support di�erent �ow�models.
2. Modify and de�ne new visualization func�
tions� Visualization functions that have been im�
plemented in Cortex in C are accessible from the
Scheme interpreter. Users can de�ne new visu�
alization functions using Scheme with these pre�
de�ned visualization functions.
7 Comparisons With Other Visualiza�
tion Systems
Cortex provides an easy way for CFD developers to
integrate advanced� parallel and distributed visualiza�
tion techniques and modern graphical interfaces into
their applications. Di�erent aspects of Cortex such as
integration with the CFD application� use of Scheme
as an extension language and parallel and distributed
execution have been implemented in other visualiza�
tion systems. To our knowledge� Cortex is the �rst
visualization application that integrates them into a
coherent whole and has been implemented on a wide
variety of platforms for 3 di�erent CFD solvers.
General purpose visualization systems such as
Fieldview�6�� FAST�1� and PV�WAVE are designed for
post�processing which requires the CFD application
tp write out one or more �les containing the complete
results of the simulation. Cortex� however� provides
an integrated visualization environment for CFD ap�
plications where simulation data is accessed directly
from the application through a well�de�ned interface.
The lack of integration implies that the stand alone
visualization systems must read in the entire �ow�
�eld to process it� and users cannot steer the solu�
tion process. The Cortex visualization environment is
quite �exible � developers and users can rapidly and
interactively modify user�interfaces and visualization
functions through a high�level interpreted language.
The extension language in Cortex is a full�featured�
standard high level programming language� rather
than an ad�hoc scripting language. Cortex resembles
SuperGlue�5� in providing Scheme as an extension lan�
guage. SuperGlue is designed for post�processing and
o�ers object�oriented extensions to Scheme. Integra�
tion of the visualization system with the CFD applica�
tion has also been implemented successfully in pV3�3�.
There are a number of visualization packages such
as AVS�13�� Explorer and Khoros�9� which are orga�
nized around the data �ow model and allow users to
interactively create customized visualization systems
through a visual programming environment. While
Cortex does not provide such an appealing interface
to customize the visualizations� it also does not ex�
hibit many of the problems of the data �ow approach.
There is a single visualization process� rather than the
multiple processes common in data �ow systems. This

�define �cx�clear�menubar�
�for�each �lambda �m� ��cx�delete�menu �menu��id m��� menus�
�set� menus �����
�define �cx�add�item menu item accel mnemonic test callback�
�let ��m �if �string� menu� �name��menu menu� �id��menu menu���
�id �f��
�if �not m� �error �cx�add�item� no such menu.� menu��
�set� id ��cx�add�item �menu��id m� item accel mnemonic callback��
�menu��item m item id test� id��
Figure 4� Examples of Scheme Functions for User Interfaces
reduces the consumption of system resources such as
�le descriptors and memory. In addition� data��ow vi�
sualization system are designed for data�ow in the for�
ward direction � from application to display. Features
such as interactive querying of simulation data values
from an image are di�cult to implement in such sys�
tems. On the other hand� Cortex is directed towards
the CFD domain� and is therefore not as general as
some existing visualization systems.
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Figure 5� Unstructured tetrahedral mesh for Pampa jet in Parallel Rampant on workstation clusters� Panels for
interactive addition of compute�nodes and optimizing communication are shown.

An Annotation System for 3D Fluid Flow Visualization
Maria M. Loughlin John F. Hughes
Cambridge Research Lab Department of Computer Science
Digital Equipment Corporation Brown University� Box 1910
One Kendall Sq.� Cambridge� MA 02139 Providence� RI 02812
loughlin�crl.dec.com jfh�cs.brown.edu
Abstract
Annotation is a key activity of data analysis. How�
ever� current data analysis systems focus almost exclu�
sively on visualization. We propose a system which in�
tegrates annotations into a visualization system. An�
notations are embedded in 3D data space� using the
Post�it 1 metaphor. This embedding allows contextual�
based information storage and retrieval� and facil�
itates information sharing in collaborative environ�
ments. We provide a traditional database �lter and
a Magic Lens 2 �lter to create specialized views of the
data. The system is customized for �uid �ow applica�
tions� with features which allow users to store param�
eters of visualization tools and sketch 3D volumes.
1 Introduction
In a study to characterize the data analysis process�
Springmeyer et al. �13� observed scientists analyzing
scienti�c data. They found that recording results and
histories of analysis sessions is a key activity of data
analysis. Two types of annotating were observed�
� recording� or preserving contextual information
throughout an investigation
� describing� or capturing conclusions of the analy�
sis sessions.
Despite the importance of annotation� current systems
for data analysis emphasize visualization and provide
little or no annotation support.
In this paper� we describe a system that supports
annotation as an integrated part of a �uid �ow visu�
alization system. Unlike typical annotations on static
2D images� our system embeds annotations in 3D data
space. This immersion makes it easy to associate user
1 Post�it is a registered trademark of 3M.
2 Magic Lens is a trademark of Xerox Corporation.
comments with the features they describe. To avoid
clutter and data hiding� annotations are represented
by graphical annotation markers that have associated
information. Therefore� graphical attributes of the
markers� such as size and color� can di�erentiate an�
notations with di�erent functions� authors� etc.
Annotations can easily be added� edited and
deleted. Also� many sets of annotations can simultane�
ously be loaded into a visualization. This allows scien�
tists� collaborating on a data set� to use annotations as
a form of communication� as well as a history of data
analysis sessions. Annotation markers also aid scien�
tists in navigating through the data space by providing
landmarks at interesting positions. Figures 1�a���c�
show the visualization environment� annotation mark�
ers� and the annotation content panel. Figure 1�d�
shows a Magic Lens �lter which hides the annota�
tion markers and widget handles. The implementa�
tion has been applied to three�dimensional Computa�
tional Fluid Dynamics �CFD� applications. However�
the techniques can be used in visualization systems of
many disciplines. The design can also be extended to
3D stereo and virtual�reality environments.
This paper is organized in six sections. Section 2
reviews previous approaches to annotation. Section
3 describes design guidelines for annotation systems.
Section 4 details our implementation of annotation
within a visualization system. In the last two sections�
we discuss possible future work and conclusions.
2 Background
Scienti�c visualization systems provide little� if any�
support for annotation. For example� the Application
Visualization System �AVS� �15� and the Flow Analy�
sis Software Toolkit �FAST� �1�� two software environ�
ments for visualizing scienti�c data� facilitate attach�
ment of labels to static 2D images. These systems
also allow a user to record and playback a sequence

�a� hedgehog and streamlines showing �b� annotation markers �small geometric
3D �uid �ow objects� placed at points of high velocity
�c� annotation content panel �d� Magic Lens �lter hiding annotation
markers and widget handles
Figure 1� The visualization and annotation system.

of interactions with the visualization. This support
is useful for generating presentations from the data�
but does not facilitate the recording and describing
operations observed by Springmeyer et al.
Outside the scienti�c visualization domain� anno�
tations have been integrated in di�erent applications.
MacDraw� a 2D paint program� introduced a notes
feature� which allows static 2D annotations using the
Post�it metaphor. Media View �11�� a multi�media
publication system� allows annotations in all media
components including text� line art� images� sound�
video� and computer animations. The format of anno�
tations has been expanded� but their use is still limited
to presentation of information in a static environment.
Document annotation is used as a means of commu�
nication in the Wang Laboratories multi�media com�
munication system� Freestyle �7�. Freestyle�s multi�
media messages are based on images� including screen
snapshots and hand�drawn sketches. Freestyle ad�
vances the concept of annotations as communicators�
but does not address the issues of clutter and manage�
ment of annotations in the environment.
Verlinden et al. �16� developed an annotation sys�
tem to explore communication in Virtual Reality �VR�
environments. In general� annotation in immersive
VR systems is restricted� as the user must interrupt
the session to interact with objects in the real world�
such as notebooks and computer monitors. Verlin�
den�s system overcomes this problem by embedding
verbal annotations in the VR space. The annotations
are represented as visual 3D markers. When the user
activates a marker� the verbal message stored with
that marker is played. This system is unique in that
it embeds annotations in 3D scenes� but it is limited
to verbal annotations and provides no support for an�
notation �ltering. It also limits annotations to a �xed
position in a time�based environment.
3 Design Issues
We have extracted� both from the Springmeyer et
al. study and from our own experience with scien�
ti�c visualization� a set of three design guidelines that
seem appropriate for an annotation system. These
guidelines� discussed below� formed the basis for the
design of our system.
Guideline 1� To support ongoing recording of con�
textual information� an annotation system must be
an integral part of a visualization system. E�ective
placement and storage of annotations are required.
Traditionally� annotations to scienti�c visualiza�
tions are recorded on paper or in electronic �les� and
both the dataset and the �les are labeled to mark their
association. This separation of data and annotations
means that some e�ort is required to �nd the data
features described by annotations. The 3D data space
of many scienti�c applications provides the context in
which annotations should be placed. Recording anno�
tations in this space capitalizes on a human�s ability
to locate information based on its spatial location.
However� inserting annotations in the data space
creates an immediate con�ict between the annotation
and visualization functions� both compete for screen
territory. We do not wish to impose restrictions on the
amount of information that can be recorded. At the
same time� we do not wish the annotations to obscure
data� since information is contained in the data itself.
Our approach is to decompose an annotation into
� an annotation marker� or small geometric object
that identi�es the position of the annotation in
the data space
� an annotation content in which a user stores in�
formation.
By clicking on a marker� a user can expand the
associated annotation to read or edit its content. Sep�
arating the annotation�s content from the annotation
marker in this way allows direct insertion of arbitrar�
ily large annotations.
Guideline 2� Annotations must be powerful enough
to capture information considered important by the
user.
There are di�erent types of information. Tanimoto
�14� distinguishes between data �raw �gures and mea�
surements�� information �re�ned data which may an�
swer the users� questions� and knowledge �information
in context�. Similarly� Bertin �3� considers informa�
tion as a relationship which can exist between ele�
ments� subsets or sets. The broader the relationship�
the higher the level of information.
In our annotation system� we provide support for
di�erent levels of information in two ways. First�
within each annotation� scientists can record both nu�
merical and textual details� and high�level information
speci�c to �uid �ow. This is discussed in section 4.4.
Second� the system supports hierarchically organized
annotations. The hierarchical structure allows scien�
tists to record facts in separate annotations� and group
related annotations in sets that describe broader ob�
servations.

Although some data� such as date of creation and
author� are likely to be relevant to all applications� it
is possible that knowledge can be captured only when
an annotation system is customized for a speci�c ap�
plication. The customization would ensure that anno�
tations can represent information relevant in the con�
text of the application. For example� if the data of a
particular application is time�varying� the annotation
system should provide time�varying annotations that
can track the features being described.
Guideline 3� The user interface �UI� of an anno�
tation system will play a key role in determining its
acceptance �or lack thereof� by scientists.
We considered many established UI rules �6� and
designed our annotation system accordingly. One rule
states that a UI should allow users to work with min�
imal conscious attention to its tools. We achieve this
goal by using a direct manipulation interface� that is�
an interface in which the objects that can be manipu�
lated are represented physically. For example� the vol�
ume of data a�ected by the Magic Lens �lter can be
controlled directly by moving and resizing the phys�
ical representation of the lens. Another design rule
states that an interface should provide feedback� e.g.�
on the current settings of domain variables. In our sys�
tem� annotation markers give visual feedback on the
location of annotations� and marker geometry gives
feedback on annotation content.
Because the geometric data space of �uid �ow ap�
plications has three dimensions� we considered design
issues speci�c to 3D graphical user interfaces �5�. One
issue is the complexity introduced by 3D viewing pro�
jections� visibility determination� etc. A second issue
is that the degrees of freedom in the 3D world are not
easily speci�ed with common hardware input devices.
A third issue is that a 3D interface can easily obscure
itself. We use guidelines outlined by Snibbe et al. �12�
to deal with these problems. For example� we provide
shadows� constrained to move in a plane� to simplify
positioning of annotation markers �see section 4.3.2�.
We provide feedback on the orientation of the data by
optionally drawing the principal axes and planes. We
also ensure that annotations do not obscure data� by
making it easy for a user to change the viewpoint and
resize or hide annotation markers.
4 Implementation
This section describes the implemented annotation
system. We �rst set the context by describing �uid
�ow visualization and the development environment.
Then we discuss the main components of the anno�
tation system� the annotation markers� support for
information capture� and interaction techniques.
4.1 Fluid Flow Visualizations
Computational �uid dynamics �CFD� uses com�
puters to simulate the characteristics of �ow physics.
Computed �ow data is typically stored as a 3D grid
of vector and scalar values �e.g.� velocity� temperature�
and vorticity values�� which are static in a steady �ow�
and change over time in an unsteady �ow. CFD visu�
alization tools allow a scientist to examine the char�
acteristics of the data with 3D computer displays.
Interaction with the visual representation is essen�
tial in the exploration and analysis of the data� and has
three goals� feature identi�cation� scanning� and prob�
ing �8�. Feature identi�cation techniques help �nd �ow
features over the entire domain� and give the scientist
a feel for the position of interesting parts of the �ow
volume. Scanning techniques are used to interactively
search the domain� by varying one or more parame�
ters� through space or through scalar and vector �eld
values. Probing techniques are localized visualization
tools� typically used to gather quantitative informa�
tion in the �nal step of investigating a �ow feature.
The Computer Graphics Group at Brown Univer�
sity has developed a �ow visualization system� to
study modes of interaction with �ow tools. The an�
notation system was built as part of this visualization
system. This provided a test�bed for techniques to
integrate visualization and annotation functionality.
4.2 The Development Environment
The annotation system was developed using C��
and FLESH� an object oriented animation and mod�
eling scripting language �10�. The FLESH objects de�
�ned for the annotation system include annotation
markers� lenses� and �lters. Some of these FLESH
classes have corresponding C�� classes� in which data
is stored and compute�intensive operations performed.
This allows us to bene�t from the power of an in�
terpreted interactive prototyping system and the e��
ciency of a compiled language.
4.3 Annotation Markers
Annotations are represented in the 3D data space
by small geometric markers. Each marker has an as�
sociated content which the user can edit at any time.

4.3.1 Marker Graphical Attributes
The geometry of a marker gives visual feedback on the
content of the annotation. In the �uid �ow visualiza�
tions system� the user can de�ne annotation keywords
�e.g.� plume� vortex�� and select a geometry to asso�
ciate with each keyword. Then� when the user assigns
a keyword to an annotation in the system� the anno�
tation�s marker takes the associated shape. It is likely
that other mappings between graphical attributes of
markers and annotation content would also be useful.
For example� the color saturation of a marker could
depend on the age or priority of the annotation.
The graphical attributes of annotations are also
user�customizable. The size and color of all markers
in one level of hierarchy can be changed. We predict
that this feature would be useful if many scientists
work collaboratively on a data set� and each scientist
de�nes a unique color and size for his markers.
4.3.2 Marker Behavior
Since the function of a marker is simply to identify
points of interest in the visualization� its behavior
is quite simple. A marker is created when the user
presses the annotation push�button. It appears at the
point on which the user is focussed� making it easy for
the user to position it near the feature of interest.
Using the mouse� a scientist can translate and ro�
tate markers. He can also project interactive shadows
of the marker on the planes de�ned by the principal
axes �9�. Each shadow is constrained to move in the
plane in which it lies. If a user moves a shadow� the
marker moves in a parallel plane. This constrained
translation helps to precisely position a marker.
Markers can be highlighted in response to a �lter
request. In the current system� the color of a marker
changes to a bright yellow when highlighted. This
simple approach seems adequate.
Since the features of unsteady �uid �ows change
over time� we would like the annotation describing a
particular feature to follow the feature�s movement in
the visualization. The current annotation system pro�
vides partial support for this by allowing the user to
specify the position of an annotation at any number of
points in time. The annotation markers then linearly
interpolate between the speci�ed positions in time.
4.4 Knowledge Stored
Our annotations can store generic information� as
well as information speci�c to �uid �ow applications.
The generic information includes keyword� textual
summary and description� author� and date. We
consulted with �uid �ow experts to understand how
the information content of annotations could be cus�
tomized for �uid �ow applications.
4.4.1 Parameters of Visualization Tools
One of the additions to the annotation system sug�
gested by the �uid �ow experts results from the in�
teractive nature of �uid �ow analysis. As described
earlier� a scientist must insert �ow visualization tools
�such as streamlines and iso�surfaces� in the data space
to see the underlying data. Much time is spent deter�
mining which tools most e�ectively highlight a feature�
and positioning and orienting both the tools and view�
point to best show o� the feature being described.
To support this activity� our concept of an annota�
tion was expanded to include parameters of �ow vi�
sualization tools. When a user wishes to store the
parameters of a set of tools� he or she presses a button
to indicate that a set of tools is being saved� and then
clicks on the tools of interest. The time�varying lo�
cation� orientation� size� and other parameters of the
tools are saved with the annotation. This can be re�
peated any number of times for di�erent groupings of
tools with di�erent parameters. When an annotation
is restored� the user is presented with a list of all saved
sets of tools� and can recover each set of tools to see
how they illustrate the annotated feature.
4.4.2 3D Volume Descriptions
It also became obvious that annotation markers� which
are appropriate for locating point features in a visual�
ization� are not su�cient for CFD applications. Fluid
�ows contain volume features� such as vortices �masses
of �ow with a whirling or circular motion� and plumes
�mobile columns of �ow�. We therefore allow users
to associate an annotation with a volume of the data
space� rather than a single point in the space.
To specify a volume� the user positions �pegs� that
de�ne the region�s extreme vertices. The convex hull
of the pegs is computed using the quickhull algorithm
�2� and is rendered in either wireframe or transpar�
ent mode. Vertices can be added� deleted and moved�
and the volume redrawn repeatedly. Figure 2 shows a
volume which has been de�ned in this way.
This implementation provides a simple way to draw
volumes. However� since it uses a convex hull� certain
shapes� such as a 3D �L� shape� cannot be sketched.

Figure 2� A volume de�ned as the convex hull of pegs.
4.5 Retrieving the Annotations
E�ective information retrieval and communication
requires that a user can easily identify annotations
relating to a speci�c topic� by a speci�c author� etc.
The system facilitates such data �ltering in two ways.
First� a traditional database �lter allows users to
specify selection criteria �such as the annotation au�
thor or keyword� via a Motif panel. Markers of anno�
tations that satisfy the search criteria are highlighted.
A second �lter uses the Magic Lens metaphor in�
troduced by Bier et al. �4�. A Magic Lens �lter is a
rectangular frame� placed in front of the visualization�
that appears as if it moves on a sheet of glass between
the cursor and the display. The lens performs some
function on the application objects behind it.
Four functions are de�ned for the lens in the anno�
tation system. The �rst sets the color of all objects�
except annotation markers� to gray. This helps users
�nd markers in a cluttered scene. The second displays
only annotations that satisfy the criteria speci�ed in
the Motif database �lter. The third lens function hides
all annotation markers behind the lens. Finally� the
default function hides all annotation markers and all
interaction handles on the visualization tools behind
the lens. Other lens functions could be de�ned� for ex�
ample� a lens could remove all �uid �ow tools except
those in the user�sketched volume behind the lens.
We believe that the Magic Lens �lter alleviates
the problem of visualization and annotation functions
sharing the same screen space. Using the lens� a sci�
entist can choose either to tightly integrate the two
functions or to focus exclusively on either visualiza�
tion or annotation.
5 Future Work
The work described in this paper could be expanded
in a number of ways.
For the �uid �ow application� the facility for record�
ing visualization tool parameters could be extended to
record view parameters. Annotations could also be�
come more active in the data investigation process.
For example� annotation markers could be used as
seed points for automatic �ow feature�characterization
code. The output of the feature�characterization
code �i.e.� speci�cations of the feature found� could
then be added to the annotation content. Feature�
characterization code could also be used to improve
support for time�varying annotations. If the location
of an annotation marker were constrained to the fea�
ture�s position �as found by feature�characterization
code�� the marker would follow the movement of the
feature over time.
We would also like to implement annotations in
other applications and environments. For example�
virtual reality environments pose many new research
problems. User studies would have to be performed to
determine which annotation modalities would be ap�
propriate in this space. If textual annotations were ap�
propriate� we would have to determine where to place
the text� �oating in space near the marker� or on 2D
panels which exist in the virtual space� or perhaps in
some other place. New interaction mechanisms for an�
notation markers and �lters should also be developed.
Finally� we would like to expand the scope of an�
notations. Springmeyer et al. noted that scientists
record their interactions with visualization systems.
Perhaps the annotation system could help in record�
ing and examining these edit trails. Also� scientists
routinely compare di�erent data sets. The current
annotation system could be redesigned to �t in the
context of more than one data set.
We hope that further experience with the current
system and its extension to other applications and en�
vironments will allow us to evaluate our design guide�
lines� and develop principles for customization of a
general�purpose annotation system.
6 Conclusion
The importance of annotation and the lack of anno�
tation support in data analysis tools led us to develop
a system that integrates annotation and visualization.
We hope our system will help scientists by
� storing annotations with the correct data set

� providing powerful �lters to sort annotations
� making it easy to relate a comment to a data
feature �both are located in the 3D data space�
� giving team members ready access to the deci�
sions and judgements of other scientists
� reducing session setup time by easy restoration of
visualization tools
� providing a means of communication between col�
laborating scientists.
Initial feedback from scientists indicates that the
integration of annotation and visualization facilitates
the ongoing recording activity observed by Spring�
meyer et al. At the same time� the ability to group and
�lter annotations supports the organization of analysis
conclusions� i.e.� the describing activity.
Acknowledgments
The authors thank the members of the Graphics
Group at Brown and the Visualization group at CRL
for their support. The paper is based on the Master�s
thesis of the �rst author� whose attendance at Brown
University was made possible by Digital Equipment
Corporation�s Graduate Engineering Education Pro�
gram. The work was supported in part by grants from
Digital Equipment Corporation� NSF� DARPA� IBM 3 �
NCR 4 � Sun 5 � and HP 6 .
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�a� Hedgehog and streamlines showing �b� Annotation markers �small geometric
3D �uid �ow objects� placed at points of high velocity
�c� Annotation content panel �d� Magic Lens �lter hiding annotation
markers and widget handles

Abstract
Discretized Marching Cubes
C. Montani ‡ , R. Scateni ⋆ , R. Scopigno †
‡ I.E.I. – Consiglio Nazionale delle Ricerche, Via S. Maria 46, 56126 Pisa, ITALY
⋆ Centro di Ricerca, Sviluppo e Studi Superiori Sardegna (CRS4), Cagliari, ITALY
† CNUCE – Consiglio Nazionale delle Ricerche , Via S. Maria 36, 56126 Pisa, ITALY
Since the introduction of standard techniques for isosurface
extraction from volumetric datasets, one of the hardest
problems has been to reduce the number of triangles (or
polygons) generated.
This paper presents an algorithm that considerably reduces
the number of polygons generated by a Marching Cubes-like
scheme without excessively increasing the overall computational
complexity. The algorithm assumes discretization of
the dataset space and replaces cell edge interpolation by
midpoint selection. Under these assumptions, the extracted
surfaces are composed of polygons lying within a finite number
of incidences, thus allowing simple merging of the output
facets into large coplanar polygons.
An experimental evaluation of the proposed approach on
datasets related to biomedical imaging and chemical modelling
is reported.
1 Introduction
The use of the Marching Cubes (MC) technique, originally
proposed by W. Lorensen and H. Cline [7], is considered to
be a standard approach to the problem of extracting isosurfaces
from a volumetric dataset. Marching Cubes is a very
practical and simple algorithm and many implementations
are available both as part of commercial systems or as public
domain software.
Despite its extensive use in many applications, it does have
some particular shortcomings: topological inconsistency [1],
algorithm computational efficiency and excessive output data
fragmentation. Standard MC produces no consistent notion
of object connectivity; the local surface reconstruction criterion
used give rise to a number of topological ambiguities,
and therefore MC may output surfaces which are not necessarily
coherent. These shortcomings have been extensively
studied [11] and solutions have been proposed [12, 15, 8].
MC computational efficiency can be increased by exploiting
implicit parallelism (each cell can be independently processed)
[4] and by avoiding the visiting and testing of empty
cells or regions of the volume [14].
Excessive fragmentation of the output data can prevent interactive
rendering when high resolution datasets are processed.
What has changed since the technique was introduced
seven years ago, has been the amount of data to be
processed while extracting such surfaces. Equipments that
can generate volumetric datasets as large as 512∗512∗[≤ 512]
are now generally available, and we are on the way to achieving
machines capable of producing 1024 ∗ 1024 ∗ [≤ 1024]
datasets or, in other words, 1 Gigavoxel per dataset. Although
an isosurface does not usually cross all the voxels,
we can understand how easy it is to generate more than one
million triangles per surface. State-of-the-art hardware is
not yet fast enough to manipulate such masses of data in
real time.
These obstacles gave rise to substantial research aimed at
reducing the number of triangles generated by MC. The solutions
proposed can be classified into adaptive techniques,
where the cell size is locally adapted to the shape of the surface
[10] or the dataset is organized into high and low interest
areas and more primitives are produced in selected areas
only; and filtering techniques, where facet meshes returned
by a surface fitting algorithm are filtered in order to merge
or eliminate part of them.
Filtering–based approaches can be classified as:
a) coplanar facets merging, in which facets are filtered by
searching for and merging coplanar and adjacent facets [6];
b) elimination of tiny facets, where the irregularity of the
surface produced is reduced by eliminating the tiny triangles
produced when the iso-surface passes near a vertex or
an edge of a cubic cell; this is accomplished by bending the
mesh so that a number of selected mesh nodes will lie on the
iso–surface and the tiny triangles will degenerate into single
vertices. The solution is based on a modified iso-surface fitting
algorithm and a filtering phase; 40% reductions in the
number of triangles are reported [9];
c) approximated surface fitting, based on trading off data
reduction for a reduction in the precision of the representation
generated, using error criteria to measure the suitability
of the approximated surfaces.
Schroeder et al. [13] proposed an algorithm based on multiple
filtering passes, that by analysing locally the geometry
and topology of a triangle mesh removes vertices that pass
a minimal distance or curvature angle criterion. The advantage
of this approach is that any level of reduction can be
obtained, on the condition that a sufficiently coarse approx-

Figure 1: The set of different vertex locations produced by
DiscMC.
imation threshold is set; reductions up to 90% have been
obtained with an approximation error lower than the voxel
size.
In another approach, by Hoppe et al. [5], mesh optimization
is achieved by evaluating an energy function over the
mesh, and then minimizing this function by either removing/moving
vertices or collapsing/swapping edges.
Both approaches require a topological representation of the
mesh to be decimated.
In this work we propose Discretized Marching Cubes
(DiscMC), an algorithm situated half-way between the cuberille
method, which assumes constant value voxels and
directly returns the voxels faces (orthogonal to the volume
axes) [3], and the cell interpolation approach of MC. On the
basis of two simple considerations, which both relate to data
characteristics and visualization requirements, our solution
leads to interesting reductions in output fragmentation by
applying a very simple filtering approach. Moreover, the
use of an unambiguous triangulation scheme [8] allows isosurfaces
without topological anomalies to be obtained.
2 The Discretized Marching Cubes Algorithm
Given a binary dataset, linear interpolation is not needed
to extract isosurfaces. When a cell edge in a binary dataset
has both on and off corners, the midpoint of the edge is the
intersection being looked for.
In a number of applications where approximated isosurfaces
might be acceptable, the former assumption can be reasonably
extended to n–value high resolution datasets. The maximal
approximation error involved by adopting midpoint interpolation
is 1/2 of the cell size, and in some applications
the resolution of the dataset justifies such a lack of precision.
Considering a 512 ∗ 512 ∗ [≤ 512] resolution, rendering
the isosurface generated produces approximately the same
images whether linear interpolation or midpoint selection is
used.
Discretized Marching Cubes (DiscMC) is here proposed
as an evolution of MC based on midpoint selection. The
set of vertices that can be generated by DiscMC are shown
in Figure 1: there are only 13 different spatial locations on
which new vertices can be created (12 cell-edge midpoints
plus the cell centroid). Moreover, applying midpoint selection
in MC allows for a finite set of planes where the generated
facets lie. We have only 13 different plane incidences
onto which a facet can lie, and these are described by the
following equations:
Figure 2: The facets returned by DiscMC for each different
plane incidence.

Figure 3: The sets of facets returned by DiscMC for each
cell vertex configuration.
x = c, y = c, z = c,
x ± y = c, x ± z = c, y ± z = c,
x ± y ± z = c.
As shown in Figure 2, for each incidence the algorithm generates
a limited number of different facets.
The following considerations are the basis of our DiscMC
algorithm:
a) each facet can be simply classified in terms of its shape
and plane incidence;
b) the limited number of different plane incidences increases
the percentage of coplanar adjacent facets and therefore
drastically reduces the number of polygons returned while
preserving small, but possibly significant, roughnesses;
c) the algorithm does not require interpolation of the surface
intersections along the edges of the cells; this implies that it
works in integer arithmetic (except for the computation of
normals) at a higher speed than standard methods.
2.1 A new lookup table
For each on-off combination of the cell vertices (there are
256 different combinations), the standard MC lookup table
(lut) codes the number of triangles produced and the cell
edges on which these vertices lie.
DiscMC requires a simple reorganization of the standard MC
lut. Midpoint selection means that the number of different
facets returned by DiscMC is fixed, and we only have a constant
number of different output primitives for each plane
Figure 4: Some cell configurations and related DiscMC
lookup table entries.
incidence: only right triangles are generated on planes x = c,
y = c and z = c (Figures 2.1, 2.2 and 2.3); only rectangles on
planes x ± y = c, x ± z = c and y ± z = c (Figures 2.4, 2.5,
2.6, 2.7, 2.8 and 2.9); only equilateral triangles on planes
x ± y ± z = c (Figures 2.10, 2.11, 2.12 and 2.13). Moreover,
using midpoint interpolation means that the geometrical location
of facet vertices depends solely on the vertices configuration
and the position of the cell in the dataset mesh.
Under these assumptions, the resulting facet set returned
by DiscMC for each of the canonical MC configurations is
reported in Figure 3. With respect to the original proposal
by Lorensen and Cline we omit configuration 14 (this can be
obtained by reflection from configuration 11, i.e. configuration
k in Figure 3.). Furthermore, three more configurations
have to be managed in order to prevent topological ambiguity
(configurations n, o and p in Figure 3 [8]).
Each facet is coded in the DiscMC lut by using a shape code,
which codifies the shape and position of the facet (1..4 for
right triangles, 1..2 for rectangles and 1..8 for equilateral triangles),
and an incidence code, i.e. the plane on which the
facet lies. Geometrical information on the facet vertices is
not explicitly stored in the DiscMC lut.
For each cell vertex configuration, DiscMC lut stores from
zero up to seven facets, each represented by a shape code
(1..8) and an incidence code (-13..13). We use signed incidences
to store separately facets which lie on the same plane
and have opposite normals direction (both in order to give
an implicit representation of facet orientation and to fast
facet search in the postprocessing merging phase).
Some cell configurations are graphically represented in Figure
4, together with the corresponding DiscMC lut entries.
2.2 Isosurface extraction
The isosurface reconstruction process returns intermediate
results using a set of indexed data structures. The volume
dataset is processed slice by slice. For each cell traversed
by an isosurface, the DiscMC produces a set of facets by

means of the DiscMC lut. Each facet is coded by its shape,
incidence and the index of the cell in which it lies (i.e. its
geometical position).
In order to optimize the merging phase the facets produced
are stored in a number of hash tables, one for each different
incidence of the facets. Thus, 26 hash tables are used, and
hash indexes are computed in terms of shape code and cell
index.
2.3 Post-processing merging phase
The merging phase begins when the isosurfaces have been
fitted. Each hash table is analyzed in order to search for
adjacent faces, which by construction of the hash tables will
also be iso-oriented and mergeable. Hash coding is chosen
to allow a rapid search for adjacent facets (a nearly constant
mean access time has been measured in a number of algorithm
runs).
The merging algorithm does not work with the vertex coordinates
of each merging polygon, but adopts Freeman’s
chains [2] as an intermediate representation scheme. In this
scheme, a polygonal line is represented by the coordinates
of the starting point of the chain and a set of directed links,
that is, a set of relative displacements. This solution allows
the unnecessary vertices to be rapidly eliminated.
The merging algorithm is simple and efficient. Due to
the limited number of facet shapes and orientations, for each
facet f and for each edge e of f the facet f ′ which might be
adjacent on e to f is univocally determined. The algorithm
is outlined in Figure 5 (an example is shown in Figure 6).
PUSH ∗ verifies, for each edge pushed onto the edgestack,
if an opposite edge exists on the stack, i.e. an edge with the
same geometrical position but moving in the opposite direction.
If this edge exists, mark both the edges as connecting
edges. Marked edges will produce either connecting links
(i.e. links which connect the starting point of the chain to
the boundary of the region, or the boundary of the region
to the boundary of the holes; see links 2 and 7 in the 15th
tiles triple of Figure 6), or consecutive opposite links that
have to be eliminated due to the reconstruction algorithm
adopted.
The Merge algorithm main loop iterates until hash tables
are empty. For each iteration of the first while loop, Merge
produces the boundary of a region (anticlockwise in our implementation)
and the boundaries of the holes (clockwise),
if any. At the end of each iteration the boundaries of regions
and holes are reconstructed by eliminating the marked links
and, if necessary, by splitting the chain. Chains are then
converted in the usual vertex–based representation.
The Merge algorithm uses a set of simple lookup tables
which permit a general procedure to be designed irrespective
of the type of the facets and the plane they belong to. These
lookup tables store:
• the edges to be pushed onto the edgestack (depending
on the starting point chosen);
• the edges to be pushed onto the edgestack when an
adjacent facet has been found, or otherwise the link to
be added to the Freeman’s chain;
• the position (with respect to the current cell) of the
cells to be inspected for adjacent facets.
In addition, through lookup tables we convert the chain links
into relative displacements depending on the incidence plane
we are examining.
Figure 7: The links of Freeman’s chains for (a) right triangles
belonging to plane 3, (b) rectangles of plane 9, and (c)
equilateral triangles of plane 12.
As previously introduced, with Freeman’s chain representation
scheme (Figure 7 shows the links used for three types of
elementary primitives) unnecessary vertices can be removed
by simply converting equal consecutive links into a single
segment.
The worst case computational complexity of the merging
phase is linear to the number of facets returned by the isosurface
reconstructor. For each edge, the merger computes
the potential adjacent facet and searches for such a facet in
the hash table (a nearly constant time operation). In the
worst case, when no mergeable facet pairs exist, the test is
repeated e times for each facet f, with e the number of edges
of facet f.
2.4 Vertex normals computation
Normals on the vertices of the isosurfaces extracted are
needed in order to compute Gouraud or Phong shading.
Normals can be computed during isosurface extraction (in
terms of gradients [16], as in standard MC) or after the
merging phase. In the current DiscMC implementation we
computed vertex normals at the end of the merging process
in order to avoid computing and storing a lot of unnecessary
vertex normals. However, further processing of the volume
data is thus needed.
3 Evaluation of results and conclusions
We tested DiscMC on a series of different datasets and compared
results with a classic MC implementation. Table 1 reports
the number of polygons generated and it refers to three
datasets: Sphere is a voxelized sphere, Buckyball is the
electron density around a molecule of C60 (courtesy of AVS
International Centre) and Head is a CAT scanned dataset
(courtesy of of Niguarda Hospital, Milan, Italy). The numbers
of facets and vertices returned by Classic MC and DiscMC
are reported in Table 1. DiscMC returns triangular
(3 − facets), quadrilateral (4 − facets) or n-sided facets

Algorithm MERGE
input HT1, ..., HT26:facet hash tables;
output F :facet list;
begin
for each hash table HTi do
while HTi is not empty do
• extract a facet f from hash table HTi;
• select one of the vertices of f as the starting point of the
current Freeman chain;
• push the edges of the facet onto the edgestack (LIFO);
{each edge is coded in the edgestack by the shape code of the current facet
and the shape code and the cell coordinates of the potential adjacent facet.
This notation will indicate, for each edge extracted from edgestack,
the source facet and the adjacent facet to be searched for.}
while edgestack is not empty do
er:=POP(edgestack); {er:edge record}
fadj:=er.adjacent facet;
if facet fadj is contained in HTi
then
• extract the facet fadj;
for each edge ej ∈ fadj such that ej �= er do
PUSH ∗ (edgestack, ej); {PUSH ∗ : see text in Section 2.3}
else
• add a link to the chain which is directed according to the current edge er;
• if the edge is a connecting edge, add a marked link to the current chain
(e.g. the link with a white arrow head in the 16th tile triple in Figure 6);
• insert the current Freeman chain into F ;
end algorithm.
(n − facets); the respective numbers are in the rightmost
three columns in Table 1.
Time comparison needs to be split into three steps: facet
extraction, merging and generation of normals. The percentage
of time spent in each stage of the computation varies
from dataset to dataset; on average, it takes about 10% of
the total time to extract facets, 85% to merge polygons, and
about 5% to generate normals. The buckyball dataset and
the head dataset, which are comparable in terms of voxel
number, took around 2-3 minutes and 6-7 minutes, respectively,
on an IBM RISC6000/550 workstation.
It is difficult to make a time comparison with other filtering
approaches, because most of them do not report the
running times but only the simplification percentages obtained.
The mesh optimization approach by Hoppe et al.
[5] is the only alternative technique which reports running
times; the simplification of meshes (8000-18000 facets) with
this method, which produces very good results indeed, took
tens of minutes on a DEC Alpha workstation.
In the proposal by Schroeder et al. [13] running times are
not reported, but the decimation phase is a much more complex
task than the simple merging phase of DiscMC. In fact,
the simplification of the mesh is obtained by multiple passes
over the mesh. At each pass a vertex is selected for removal,
all triangles that are incident on that vertex are removed,
and the resulting hole is patched by computing a new local
triangulation.
On the other hand, in the worst case, DiscMC has a complexity
linear to the number of edges: for each edge of each
facet, it searches for the adjacent facet on a hash list (a
constant time and cheap operation), and makes an insertion/removal
onto/from the edge stack.
The reduction in time complexity is significant, because the
Figure 5: Pseudocode of the Merge algorithm
design goal of DiscMC was to give simplified meshes with
high efficiency, to be used, for example, while searching for
the correct threshold. Once this threshold has been selected,
a more sophisticated method such as [13] can be used to obtain
the best approximated mesh.
Another characteristic which differentiates DiscMC from
other simplification approaches is that it does not entail
managing a geo-topological representation of the triangle
mesh. The topological relations are implicitly stored in the
coding scheme used (facets shape and incidence) and this
simplifies the implementation at the cost of single constant–
time search into hash lists.
The results obtained and the good quality of the output
images (the colour plates in Figures 9 and 11 were obtained
with our algorithm, while the ones on Figures 8 and 10 were
obtained with classic MC without mesh simplification) support
our claim that Discretized Marching Cubes represents
a valid tool for the rapid reconstruction and visualization of
isosurfaces from medium and high resolution 3D datasets.
One of the most salient characteristics of the algorithm is
that integer arithmetic is sufficient, and restricts the use of
floating point computations to normals only. This is an important
factor which enhances the overall performance.
Discretized Marching Cubes is both a valid solution for applications
where the precision of the result is not critical or
also as an intermediate solution to speed up the time needed
to tune parameters, relegating to the final stage alone the
use of techniques that are more precise in terms of visual
results or geometrical approximation, such as ray tracing or
standard MC.

Figure 6: Steps required to merge a number of adjacent facets: for each iteration, the figure represents the facets remaining
in the facet list (left tile), the edges present on the edge stack (centre tile) and the current Freeman chain (right tile). In the
edge stack tiles, the label associated with the edges represents the order of insertion in the stack (1 is the top edge); edges
which have a circled label represent connecting edges. In the chain tiles, arrows with a white end represent connecting links.
Classic MC DiscMC
# facets # vertices # facets # vertices # 3-facet # 4-facet # n-facet
Sphere (100 3 ) 37,784 18,556 5,501 9,594 0 5,167 334
Buckyball (128 3 ) 204,408 103,072 17,039 28,528 1,238 12,200 3,601
Head (256 2 x33) 428,181 216,431 57,413 77,712 13,005 34,856 9,552
Table 1: The number of facets returned by the Discretized Marching Cubes and classic Marching Cubes algorithms on three
different datasets.

4 Acknowledgements
This work has been partially carried out with the financial
contribution of the Sardinian Regional Authorities.
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Figure 8� Isosurface reconstruction from the Buckyball dataset using standard
MC �no mesh simpli�cation�.
Figure 9� Isosurface reconstruction from the Buckyball dataset using DiscMC.

Figure 10� Isosurface reconstruction from the Head dataset using standard MC
�no mesh simpli�cation�.
Figure 11� Isosurface reconstruction from the Head dataset using DiscMC.

Approximation of Isosurface in the Marching Cube�
Ambiguity Problem.
Abstract
The purpose of the present article is the considera�
tion of the problem of ambiguity over the faces arising
in the Marching Cube algorithm. The article shows
that for unambiguous choice of the sequence of the
points of intersection of the isosurface with edges con�
�ning the face it is su�cient to sort them along one
of the coordinates. It also presents the solution of this
problem inside the cube. The graph theory methods are
used to approximate the isosurface inside the cell.
Introduction
Let there be a rectilinear volume grid whose nodes
contain the values of the function Fijk � F �x� y� z�.
The problem is to approximate the isosurface
S� � f�x� y� z� � F �x� y� z� � �g� �1�
In the MC algorithm �1�� �2� the isosurface is ap�
proximated sequentially in all cells comprising the vol�
ume grid and intersecting the speci�ed surface.
In this case the coordinates of the points of of edges
intersecting the isosurface are computed. Then the
part of the surface intersecting the given cell is con�
structed at the points obtained.
In virtue of symmetry there are only 15 possible
types of intersection of the isosurface and the cubic
cell.
The problem is that during approximation of the
isosurface there is a possibility for �holes� to appear
inside the cells of the volume grid as a result of the
wrong connection of the points on the edges of the
cells �see Figure 1�. Here and in what follows the
black points denote the edges outside the isosurface
Fnode � �. In the MC method this problem is solved
for each cell separately without taking into account
the e�ect of the adjacent cells. Now the problem is to
connect the points at the cell edges correctly� in which
case the problem of �holes� appearing at the cell edges
is solved.
Sergey V. Matveyev
Computer Science Department
Institute for High Energy Physics
142284� Protvino� Moscow Region� Russia
E�mail� matveyev�desert.ihep.su
Figure 1� Ambiguity at the Edge
To connect the points correctly one may use the
value of the function at the edge center �2�.
The comparison of the value at the point with the
one on the isosurface allows one to conclude whether
the given point is inside or outside the isosurface.
However this solution does not always yield the correct
result �see Figure 2�.
Figure 2� Example of wrong connection
For the solution of this problem Nielson and
Hamann �4� proposed to use a bilinear representation
of the function. In this case the curve describing the

intersection of the isosurface with the edge will be a
hyperbolic one.
De�ning the value of the function at the point of the
intersection of the hyperbola asymptotes we may tell
in what sequence it is necessary to connect points at
the edge �see Figure 3� because this point lies between
the isolines at the edge.
Figure 3� Two Ways of Connecting the Points at the
Edge
Another problem is that inside the cell it is neces�
sary to obtain the isosurface topologically equivalent
to the given one �3�� �4�. The solution to this prob�
lem consists in the correct connection of the points
in the cell volume and in separating triangles �in the
general case of polygons� approximating the isosurface
correctly. The possible cases are analyzed in the paper
by Nielson and Hamann �4�.
Figure 4� Cell
Let 8 values of the function Bi �see Figure 4� be
speci�ed in the cube nodes. A trilinear interpolation
will be a natural description of the function inside the
cell. In this case when going over to the face we ob�
tain a bilinear description and when going over to the
edge we obtain a linear one. Then using a unity cube
for the description of the cell we obtain the following
equation�
F �x� y� z� � a � bx � cy � dz � �2�
� exy � fxz � gyz � hxyz�
where 0 � x � 1� 0 � y � 1� 0 � z � 1.
The constants are de�ned as
a � B1�
b � B2 � B1�
c � B4 � B1�
d � B5 � B1� �3�
e � B3 � B1 � B2 � B4�
f � B6 � B1 � B2 � B5�
g � B8 � B1 � B4 � B5�
h � B7 � B5 � B4 � B2 � B1 � B3 � B6 � B8�
Solution at the Cell Edges
To determine the point of the intersection of the
isosurface with the cell edge the MC method uses the
linear interpolation. The bilinear interpolation will be
a natural representation of the function at the edge.
Figure 5� Sorting points at the edge
When analyzing the function behaviour at the edge
we will use the edge projection onto a unity square.
Then the function behaviour at the edge is described
by the equation in local coordinates u� v�
F �u� v� � a � bu � cv � duv� �4�
where 0 � u � 1� 0 � v � 1.
For the straight line u � const equation �2� will
depend only on one variable
F �u � const� v� � F �v�� �5�
and� hence� will have not more than one solution on
the section from 0 to 1� i.e. not more than one inter�
section with the isosurface.

Let us sort the points of the intersection of the edges
with the isosurface with respect to u and connect them
in pairs �see Figure 5�� in which case the condition of
�one intersection� will be satis�ed.
In this case for v � const this rule is satis�ed auto�
matically.
Figure 6� Inadmissible intersection of the isosurface
with edge
Let us assume that this is not so and the case shown
in Figure 6 is possible. Let the isosurface with the
function value S0 � f�x� y� z� � F �x� y� z� � 0g be
approximated. As to other cases� they are reduced by
the transfer of the coordinates. Then for the points of
the intersection of the isosurface with the face edges
the following inequalities should hold true�
B01
B01 � B11
B00
B00 � B01
B10
B10 � B00
B11
B11 � B10
�
�
�
�
We obtain from the 1st pair that
B01
B10
and from the 2nd one that
B01
B10
B00
�
B00 � B10
B10
�
B10 � B11
B11
B11 � B01
B01
�
B01 � B00
� �6�
� �B00 � B01��B01 � B11�
� �7�
�B00 � B10��B10 � B11�
� �B01 � B00��B11 � B01�
� �8�
�B10 � B00��B11 � B10�
So� we have proved that this case is impossible.
Consequently� to connect the points at the edge cor�
rectly it is su�cient to sort them along one of the
coordinates.
Obtaining Points inside the Cell
Now let us consider the function behaviour inside
the cell. As a complexity criteria we use the number
of the isosurface intersections with the cube diagonals
�see Figure 7�.
In the case of one intersection it is su�cient to have
the values of the function in the cell nodes� whereas
in the case of two intersections one may introduce an
additional point inside the cell that can be the point
of the intersection of the diagonals as it was o�ered
in the work by E. Chernyaev and S. Matveyev �5�.
However� using the technique o�ered it is impossible
to determine the intersection points for the case pre�
sented in Figure 2. The technique o�ered in the paper
by Nielson and Hamann �4� that consists in obtaining
an additional point belonging to the isosurface inside
the cell will yield the correct result in cases 7a� b� c
Figure 7� Types of intersections with the diagonal
In the case of three intersections it becomes impos�
sible to reconstruct the topology using the technique
o�ered. The point is that it is necessary to obtain ad�
ditional points belonging to the isosurface inside the
cube.
Let us construct on the diagonals of the cube six
rectilinear slices� each determined by two diagonals
�see Figure 8�� S1476 �points B1� B4� B7� B6�� S2385�
S1278� S3456� S1573 S2684.
Let us introduce a local variable � connected with
the position of the point on the diagonal. Then the
equations describing the function behaviour on the di�
agonals B1 � B7� B2 � B8� B4 � B6� B5 � B3 are equal
to
F ��� �� �� � a � �b � c � d�� � �9�
��e � f � g�� 2 � h� 3 �
F �1 � �� �� �� � a � b � ��b � c � d � e � f�� �
���e � f � g � h�� 2 � h� 3 �
F ��� 1 � �� �� � a � c � �b � c � d � e � g�� �
���e � f � g � h�� 2 � h� 3 �
F ��� �� 1 � �� � a � d � �b � c � d � f � g�� �
respectively� where 0 � � � 1.
��e � f � g � h�� 2 � h� 3 �

Figure 8� Con�guration of slices
Hence� on the diagonals the function is speci�ed by
a cubic equation and there is a possibility to �nd the
coordinates �i of three intersections with the diagonal
of the surface approximated.
Two cube edges and two diagonals of the cube faces
form the edges con�ning each slice �see Figure 8�. The
points of the isosurface intersection with the diagonals
of the cube edges are found from equations �2� be�
coming a bilinear one on the edge by introducing the
parameter �. On the diagonals of the cube edges the
function will have a quadratic dependence on � and�
hence� will have not more than two intersections with
the isosurface.
The next step is the correct connection of the ob�
tained points in the planes of the slices.
Figure 9� Slice S3456 in local coordinates
Let us consider the slice S3456 shown in Figure 9.
Let us go over to the local coordinate system �u�v�
related to this slice. Then for the straight line u �
const equation �2� will have a linear dependence�
F ��� const� const� � F ���� �10�
and for the line v � const it will have a quadratic one�
F �const� 1 � �� �� � F �� 2 �� �11�
Figure 10� Graph for the cell points
Hence� the points sorting on the slice with respect
to u speci�es the correct sequence for their connection.
In this case the points lying on the edges �boundary�
of the slice should be marked as boundary ones at
which there is the transition from one isoline to an�
other one. On the sorted list these points should be
at the adjacent places. In the general case� the list is
as follows�
fb1� i1� ���� b2� b3� ik� ���� im� b4� ���� bn�1� ���� bng�
where �b�oundary points� �i�nner points�
and the set of the isolines is presented in the form �see
Figure 9�
fb1� i1� ���� b2g� fb3� ik� ���� im� b4g� ���� fbn�1� ���� bng�
Approximation of Isosurface inside the
Cube
As a result of previous steps we obtained a large set
of points and segments constructed at these points.
We will consider these points to be the nodes of
a graph and segments to be its arcs �see Figure 10�.
Here bj denote the points belonging to the boundary
of the cell �lying on the faces� and ij denote those
inside it.
Let us construct the adjacency matrix of this graph�
adj �
0
B
�
0 � � � 1 � � � 1 0
0 0 � � � 1 � � � 1
0 0 0 � � � 1 0
. . . . . .
0 0 0 � � � � � � 0
1
C
A
�12�

Here 0 means the absence of the connection between
points i and j and 1 presence of such a connection
or the path equal to 1. In this case the elements of
the main diagonal and those below it are equal to 0
because it is su�cient for us to take into account only
once that the node i is connected with that of j.
Now consider the expression �6�
�adj�i� 1� and adj�1� j�� or
�adj�i� 2� and adj�2� j�� or � � � �13�
or �adj�m� 1� and adj�m� j���
The value of this expression is equal to 1 if there
is a path of length 2 from the node i to the node j.
From here one can �nd out through which node they
are connected.
It is seen from the matrix that element �ij� of ma�
trix adj2 is equal to Boolean product of adjacency ma�
trix by itself�
adj2 � adj � adj
Now let us �nd the logical sum of matrixes adj and
adj2�
adj12 � adj and adj2
The element �ij� of the matrix adj12 obtained is
equal to 1 if the nodes i and j are the vertices of
a triangle. The third node can be found from the
expression �13� used for the construction of the matrix
adj2.
The matrix of paths equal to 3 is obtained as a
result of Boolean product of the adjacency matrix by
the one of paths equal to 2�
If matrix adj13�
adj3 � adj � adj2�
adj13 � adj and adj3
contains elements equal to 1 we may mark the nodes
approximating the isosurface by quadrangles. Their
triangulation is carried out following� for example� the
criterion o�ered by Choi and his co�authors �7�. If all
elements of matrix adj13 are equal to 0 this means that
triangles are su�cient for approximation and there is
no need in further calculations.
The given procedure will be carried out m times
until all the elements of the matrix adj1m are equal to
0.
Conclusions and Acknowledgments
The present work o�ers a new approach to the so�
lution of the problem of ambiguity at the cell edge
using the MC algorithm. It has been shown that for
such a solution it is su�cient to sort the points of the
intersection of the isosurface with the edges con�ning
the given face along one of the coordinates and then
to connect them in pairs.
The procedure of approximating the surfaces of a
complicated con�gurations inside a volume cell was
presented. It also shows the technique for obtaining
inside it the points lying on the surface and for con�
necting them in the correct sequence. The obtained
points and connections between them are presented in
the form of a graph. To approximate the isosurface�
the graph theory methods are used.
To conclude� the author would like to express his
sincere gratitude to V. Gusev for fruitful discussions
and also to L. Milichenko and V. Yankova for them
assistance in the preparation of the text of this paper.
References
�1� Lorensen W.E. and Cline H.E. �Marching Cubes�
A High�Resolution 3D Surface Construction Al�
gorithm�� SIGGRAPH 87 Conference Proceed�
ings�Computer Graphics� Vol. 21� No. 4� pp. 163�
169� July 1987.
�2� Wyvill G.� McPheeters C.� Wyvill B. �Data struc�
tures for soft objects�� The Visual Computer� Vol.
2� No. 4� pp. 227�234� 1986.
�3� J. Wilhelms� A.Van Gelder� �Topological Con�
siderations in Isosurface Generation Extended
Abstract�� Computer Graphics� Vol. 24� No. 5�
pp. 79�86� 1990.
�4� G.M. Nielson� B. Hamann� �The Asymptotic
Decider� Resolving the Ambiquity in Marching
Cubes�� Proceedings of Visualization�91� IEEE
Computer Society Press� pp. 83�90� 1991.
�5� E.V. Chernyaev� S.V. Matveyev� �The Main As�
pects of Visualization of Isosurfaces of Implic�
itly Speci�ed Function�� International Confer�
ence GraphiCon�92� Theses� Moscow� P. 15�17�
1992.
�6� Y. Langsam� M. Angenstein� A. Tenenbaum�
�Data Structures for Personal Computers��
Prentice�Hall� Inc� 1985.
�7� B.K. Choi� H.Y. Shin� Y.I.Yoon and J.W. Lee�
�Triangulation of scattered data in 3D Space��
CAD� Vol. 20� No. 5� pp. 239�248� 1988.

Nonpolygonal Isosurface Rendering for Large Volume Datasets
James W. Durkin John F. Hughes
Program of Computer Graphics Computer Science Department
Cornell University Brown University
Ithaca, NY 14853 Providence, RI 02912
Abstract
Surface-based rendering techniques, particularly those
that extract a polygonal approximation of an isosurface,
are widely used in volume visualization. As dataset size
increases though, the computational demands of these
methods can overwhelm typically available computing resources.
Recent work on accelerating such techniques has
focused on preprocessing the volume data or postprocessing
the extracted polygonization. Our new algorithm concentrates
instead on streamlining the surface extraction
process itself so as to accelerate the rendering of large volumes.
The technique shortens the conventional isosurface
visualization pipeline by eliminating the intermediate polygonization.
We compute the contribution of the isosurface
within a volume cell to the resulting image directly from a
simplified numerical description of the cell/surface intersection.
Our approach also reduces the work in the remaining
stages of the visualization process. By quantizing the
volume data, we exploit precomputed and cached data at
key processing steps to improve rendering efficiency. The
resulting implementation provides comparatively fast renderings
with reasonable image quality.
1 Introduction
1.1 Background
Increasingly complex environments present an ongoing
challenge to computer graphics. A dominant source of increased
complexity in volume visualization is growth in
data size. Early volume datasets typically ranged from 64 3
to 128 3 voxels, while many of today’s volumes are reaching
the 512 3 to 1024 3 voxel range. The introduction of higher
resolution data acquisition devices and more complex simulations
suggests this growth trend will continue.
The essential difficulty such growth poses is that the computational
complexity of most volume rendering algorithms
is O�n 3 � for a dataset of size n� n� n. 1 Thus doubling volume
dimension, say from 256 3 to 512 3 , yields an eightfold
increase in computational cost. Even today’s fastest workstations
are, at best, barely keeping pace with the demands
of rendering large volumes in reasonable amounts of time.
1.2 Prior work
Volume data has, by itself, no visible manifestation. Implicit
in its visualization is the creation of an intermediate
representation, some visible object or phenomenon, that
can be rendered. Levoy [3] classifies volume rendering algorithms
by the intermediate representation they employ.
1 The notable exception is frequency-domain volume rendering [5, 9],
with a complexity of O�n 2 logn�.
Among the classes are surface-based techniques, those using
polygons or surface patches as the representation. Such
techniques have proved popular due to their ease of use,
range of applicability, and comparatively fast execution.
Surface-based techniques are characterized by the application
of a surface detector to the data, followed by a fitting
of geometric primitives to the detected surface, and the rendering
of the resulting geometric representation. The techniques
differ primarily in their choice of primitives and the
scale at which they are defined. The primitives are typically
fitted to an approximation of an isosurface of the continuous
scalar field within cells of the volume. 2
The best known of these techniques is the Marching
Cubes algorithm [4]. Processing the volume cell by cell,
the algorithm classifies each cell based on the value of its
voxels relative to that of the isosurface being reconstructed.
The classification yields a binary encoding that provides an
index into a table describing the polygonal approximation
of the isosurface within the cell. Polygon vertex positions
are computed by interpolating voxel values, as specified by
the indexed table entry. The generated polygons are transferred
to a hardware or software polygon renderer for display.
Gouraud shading is often used to achieve a smoother
image. To do this, the algorithm approximates the volume
gradient at voxel positions and interpolates these gradient
vectors to produce normals at polygon vertices.
Wyvill et al. [12] present a very similar technique. They
too classify cell voxels relative to isosurface value and calculate
polygon vertex positions by voxel value interpolation.
Their technique differs from Marching Cubes in that
it uses an approximate value at the center of a cell face to
select among alternate polygon configurations.
An alternative to isosurface polygonization is the pointbased
Dividing Cubes algorithm [1]. It subdivides volume
cells into sub-cells with lattice spacing equal to the image
grid spacing. Data values for sub-cell vertices are interpolated
from the divided cell’s vertex voxels. Sub-cells intersecting
the surface are identified as those having values both
above and below the isosurface value. For these sub-cells,
a normal vector is interpolated from volume gradients as in
Marching Cubes. This normal is used to shade the intersection
point, considered to lie at the sub-cell center, which is
then projected onto the image plane where the computed intensity
is assigned to the appropriate pixel.
Recent work has focused on improving the performance
of such techniques. Wilhelms and Van Gelder [11] use spa-
2 We adopt the terminology of Wilhelms [10], referring to individual
volume data points as voxels, and to a region of space bounded by a set of
voxels (typically eight for regular volumes) as acell.

tial data structures as a preprocess to reduce the work devoted
to regions within the volume of little or no interest.
Schroeder et al. [8] reduce the number of triangles required
for the polygonal representation of objects through a postprocess,
making the extracted representation renderable on
typical graphics hardware.
1.3 Motivation
As dataset size grows, the processing demands of conventional
techniques can severely tax even the fastest workstations.
Consider an example. The industrial CT dataset of
the turbine blade in Figure 6 (also illustrated by Schroeder
et al. [8]) contains 300 slices, each of size 512�512. The
isosurface created from this data by Marching Cubes contains
approximately 1.7 million triangles. Several stages in
the algorithm are particularly expensive when processing
such complex surfaces. The number of floating point operations
required to calculate position, normal, and color information
for the 5.1 million triangle vertices is enormous,
even when reusing data at shared vertices. The amount of
data transferred to the display system is also enormous: at
50 bytes per vertex, it is roughly 250 megabytes of information.
Finally, rendering 1.7 million polygons is beyond the
capabilities of all but the most advanced workstations.
The necessity of such expensive processing is an open
question. In a typical image, say 512�512 pixels, these 1.7
million polygons are each rendered at sub-pixel size. One
can well suggest that, given the small contribution of each
polygon to the final image, the tremendous work involved
in processing polygons for such a surface is probably excessive.
An alternative to preprocessing the volume data or
post-processing the polygonal surface is to concentrate instead
on the surface extraction process itself. Our technique
streamlines the isosurface visualization pipeline by eliminating
the intermediate polygonization stage and reducing
the work required at the remaining stages.
2 Foundations
Our algorithm is based, in part, on three observations
about the surface-based rendering of large volumes:
Cell projections are small
In a ‘complete’ image of a large volume, a cell projects
to about the size of a pixel. If we make the correspondence
exact (i.e., volume inter-voxel spacing equals image interpixel
spacing), an orthographic projection of an n 3 volume
has a maximum image size of p 3 n � p 3 n pixels. For n
between 512 and 1024, the image occupies from 60–240%
of a typical workstation screen, suggesting that using such
a correspondence produces sufficiently large images.
Isosurfaces are locally almost planar
The intersection of an isosurface with a cell is almost always
well approximated by a plane. The function whose
isosurface we are reconstructing was sampled in some way
to generate the volume data. Unless the original function
was band-limited before sampling, the data will contain
aliasing. We therefore assume we are reconstructing the
isosurface of the (unique) band-limited function f whose
samples constitute our volume. Such band-limited func-
11
10
11
10
12 12
Original
2
2
6 1
4
10
2
2
Quantized
Figure 1: Geometric change resulting from data quantization.
Left: original values and placement of planar approximation of the
level 6 isosurface. Right: data quantized to a range of 3 centered
on the original isolevel, and resulting shift in isosurface position.
tions are always C ∞ . Sard’s Theorem [6] guarantees that
for almost every (in the measure-theoretic sense) isosurface
value v, the isosurface f �1 �v� contains no zeroes of the gradient
of f . Thus almost every isosurface is locally smooth
(by the implicit function theorem). Smooth surfaces can be
approximated by their tangent planes, to an accuracy that
depends on the surface curvature. The inaccuracy of our
method is thus quantified by the local curvature of the isosurface.
Near singular points, this can become arbitrarily
large, but surface-based methods typically suffer from this:
cells containing singular points are generally ambiguous.
Data can be quantized
To reduce rendering expense, we want to exploit precomputed
and cached data whenever possible (e.g., using a precomputed
approximation of the isosurface within a cell). To
keep such data of reasonable size, we need to index it not
by full-range volume data, but by a quantized representation
of a cell’s voxel values. Quantized data can produce
a reasonably accurate isosurface approximation. Figure 1
shows that quantizing the data introduces some error; increasing
the allowable range for the quantized representation
reduces the error, but cannot eliminate it. Although we
have not formally analyzed the error due to quantization,
our empirical results suggest that the visual artifacts that result
are acceptable. In any event, the quantization error is
in the sub-cell placement of the isosurface, and hence (after
projection) in the sub-pixel placement of the surface image.
If the original data indicates the presence of surface within a
cell, so too will the quantized data (if processed properly):
isosurface topology is not altered, so no spurious surfaces
or erroneous holes are introduced.
3 Algorithm overview
To render an image, we start with a volume dataset, an
isolevel �, a viewing direction, and lighting information.
As we precompute an approximation of the isosurface
within a cell and index it by voxel values, our first step is to
limit the data range within the volume. We choose a range
r and quantize voxel values to an interval of length r that
contains �. To illustrate, let us assume that � lies halfway
along this interval (in practice this is not a requirement).
The quantization may be as simple as clamping values to
the range ��r�2 � � � ��r�2, or may involve a more complicated
scaling of values from a larger range, encompassing
both r and �, into the range � � r�2 � � � � � r�2.
2
2
0
2

Next is the computation of the planar isosurface approximation
data, the isoplane table (see Section 4.1). This table
depends only on r and �, and can therefore be precomputed.
For each possible eight-tuple of values at the cell vertices,
the table contains a description of the planar approximation
of the isosurface within that cell, including the area of the
plane within the cell and the plane normal.
The second step of the algorithm proper is initialization
of the image, corresponding to a region on the projection
plane. The inter-pixel spacing on this plane is made equal to
the inter-voxel spacing in the volume. Thus the projection
of a cell overlaps at most nine image pixels. The color and
α channels of the image are initialized to zero.
The third step in the algorithm uses the isoplane table to
compute a rendering of the isosurface. The volume is traversed
from front to back and each cell is examined. The
quantized voxel values at a cell’s vertices are used as an index
into the isoplane table, which contains the area and normal
vector for the isosurface approximation within the cell.
If the area is zero (the isosurface does not intersect the
cell), or if the dot product of the normal and the projection
direction is negative (the surface is back-facing), the cell is
ignored. Otherwise, the area is multiplied by this dot product
to find the projected area of the cell’s isosurface on the
image plane. We compute the light reflected from the surface
fragment (if not previously computed) and record it in
a cached data structure called the intensity table (see Section
4.2), indexed by the cell’s eight voxel values.
We accumulate into the image the light reflected from the
surface fragment towards the image plane. Lacking precise
geometric information describing the position of the surface
within the cell, we assume that the projection of the surface
fragment is evenly distributed across that of the entire cell
onto the image plane. We can therefore clip the cell’s projection
against the pixels in the image plane to compute the
fraction of the reflected light that should be composited into
each of the nine pixels the cell projection may overlap. We
avoid repeated clipping by precomputing a table describing
the projection/pixel overlap for a representative set of the
possible projection positions (see Section 4.3).
Using a modification of standard compositing (see Section
4.4) we accumulate values in the image until the α
value for a pixel is 1.0, after which no more light is composited
into the pixel.
4 Algorithm details
This algorithm exploits precomputed and cached data
wherever possible, an approach that might well be termed
look-up tables everywhere. We discuss the most important
of these tables, and other implementation details, below.
4.1 Isoplane table
The precomputed planar approximation of the isosurface
within volume cells depends on the quantized data range r
and the isosurface level � — more precisely, on the location
of � within an interval of length r. We therefore create
a table whose entries are indexed by eight-tuples of values
�v0 � � � � �v7� (vi in the range 0 � � � r�1) and associated with a
level � 0 between 0 and r�1.
Suppose the vertices of the unit cube are labeled by bi-
nary numbers so that, for example, vertex 6 has coordinates
�x�y�z� � �1�1�0� (as 610 � 1102). Table entry �v0�v1� � � ��
corresponds to a cube whose vertex 0 has value v0, vertex 1
has value v1, and so on. Given the values v0� � � � �v7 at these
corners (whose positions we denote p0� � � � � p7), we approximate
the isosurface by a plane determined by these values.
We do this by one of three methods, all variants of the
same technique. Certain vertices of the cube are marked,
and those vertices alone are used to find a least-squares bestfit
plane. That is to say, we seek the function
such that
f �x�y�z� � Ax � By �Cz � D
∑ � f �pi� � vi�
i2M
2 � where M � fmarked verticesg
is minimized. This is a straightforward least-squares problem
in the unknowns A, B, C, and D.
The three methods differ in the choice of which cell vertices
to mark. The first method marks all vertices, yielding
a least-squares solution using all available data. We call
this method all-voxels. In the second, we mark only cube
edge endpoints with values on opposite sides of the isosurface
level. The method, called edge-crossings, is analogous
to Marching Cubes’ identification of polygon vertex locations.
In the third method, if any vertex has a value at either
extreme of the quantized data range, and all three neighbor
vertices (those connected to it by an edge) share the same
value, then that vertex is unmarked; all others vertices are
marked. This approach reduces the error in the approximation
of the plane equation by eliminating data values that
violate the assumption of linearity due to the limited range
of the table. We term this technique sans-clamped. For
any cell intersecting the isosurface, there are at least four
marked voxels for any of the three methods, so the leastsquares
problem always has a unique solution.
Having found the function f above, we consider the
plane f �x�y�z� � � 0 to be our ‘best linear approximation’
to the isosurface. We clip this plane to the bounds of the
cell, compute the area remaining, and record this in the table
along with A, B, and C, which constitute the plane normal.
The table as described has r 8 entries. We store the area
and normal data as floating-point numbers. At four bytes
per value, we have sixteen bytes per table entry. Using ten
megabytes as a rough limit for such precomputed data, the
maximum allowable r is 5.
Fortunately, we can exploit the symmetry of the cube to
give us a table compression scheme. A cube centered at the
origin has many geometric symmetries: rotations about the
x-, y- and z-axes, reflections in the xy-, yz-, and xz-planes,
and combinations of these. For any eight-tuple of values labeling
the cube’s vertices, we consider the labelings derived
from it by applying such symmetries to the cube as equivalent.
That is, the area of the planar isosurface approximation
for each is the same, and the surface normals are simple
transformations of one another. We wish to map each
such equivalence class to a single entry in our compressed
isoplane table. Doing so requires that we produce just one

Data Uncompressed Compressed Compression
range size size factor
2 256 65 3.94
4 65,536 5,995 10.93
6 1,679,916 100,446 16.72
8 16,777,216 793,650 21.14
10 100,000,000 4,076,215 24.53
Table 1: Isoplane table information forr in the range 2–10. Shown
are the number of entries for the uncompressed and compressed
forms, and the compression factor for each r value.
entry for the equivalence class in generating the table, and
that we can identify that entry and appropriately transform
the stored data when accessing the table.
To identify a canonical element for the equivalence class,
we permute the eight-tuple of cell vertex values �v0� � � � �v7�
(using only permutations allowable under rotation and reflection)
so that the value at p0 is the smallest of the eight
and the values at p1, p2, and p4 satisfy the relation v1 � v2 �
v4. In generating the table, we limit ourselves to one entry
per equivalence class by iterating over values that obey the
stated restrictions. Using this scheme, the size of the isoplane
table for a range r is
r�1
∑
a�0
r�1
∑
b�a
r�1
∑
c�b
r�1
∑ �r � a�
d�c
4 �
an eighth-degree polynomial in r. In the limit, as r � ∞, the
size approaches 1�48r 8 . Table 1 gives size information for
representative values of r.
In accessing the compressed table, the voxel values at cell
vertices are permuted according to the above scheme, and
the permuted tuple is used for table look-up. The permutation
is a linear transformation of the cell, so we apply the
inverse adjoint of this transformation in extracting the normal
vector. The processing required to access data from the
compressed table is approximately double that for the uncompressed
table. Only with such compression though, is
this precomputation technique feasible for larger values of
r. In practice, we observe an actual increase in rendering
time with compressed isoplane tables of only 1–10%. That
this is lower than the raw increase in access time suggests,
makes sense; isoplane table access is not the only step in
rendering, nor do all cells intersect the isosurface (access
time for ‘empty’ cells is the same for both table forms).
The incentive to use as large an r as possible is strong,
as the quantized data range is a major factor in determining
image quality. Figure 2 shows the result of using tables
of varying r value. As expected, image quality improves
with larger tables. 3 We typically keep precomputed tables
for ranges of 2, 4, 6, and 8, with � 0 at the range midpoint.
The first two tables are usually kept uncompressed, and the
latter two in compressed form.
4.2 Intensity table
In examining each cell, we look up its isosurface approximation
in the isoplane table and apply the user-defined
3 The test sphere, with its large, constant-curvature surface, highlights
the effects of the quantization and linear approximation; lowerr values frequently
produce satisfactory images on real-world datasets.
Figure 2: Test volume containing a ‘spherical’ isosurface, rendered
using isoplane tables with different values of r. Clockwise
from upper left, values for r are 2, 4, 6, and 8.
lighting definition to the data found there, computing the
color of light reflected from the cell’s isosurface fragment.
To avoid repetitive calculations, we cache this information
in the intensity table, indexed by the cell’s voxel values.
With an uncompressed isoplane table, we could add the information
to that table at run-time, but for a compressed table,
the need to permute the surface normal makes this impossible.
In principle, this table is of size r 8 . To limit the size in
practice, we allocate space for the intensity table in parts.
We first allocate a table of pointers of size r 4 , indexed by
the cell’s first four voxel values. This part of the table is
quite small, requiring only 4,096 pointers for r � 8. We allocate
‘data pages’ only when we must compute the light
intensity for a given eight-tuple of values on that page. The
data pages are of size r 4 , and are indexed by the cell’s last
four voxel values. Individual entries on the data pages are
computed only as needed, and are reused by later cells with
the same eight voxel values.
To minimize space requirements, we use only four bytes
per entry, one byte each for the red, green, and blue color
values plus a byte for flags used in table management. This
follows the heuristic that eight bit color values are adequate
in image compositing, but higher accuracy is required for
the α-channel. We compute the α value elsewhere in the
rendering process (using the projected area of the cell’s isosurface)
and do not cache it in the intensity table.
This approach provides what we feel is the best tradeoff
among access time, memory usage, and time spent computing
intensity values. In practice, cell voxel values are not
evenly distributed (they tend to cluster), so for typical r values
(4 and above) we rarely allocate all data pages or compute
a large percentage of the entries on allocated pages.
4.3 Cell projection table
As we restrict ourselves to parallel projections, the projections
of any two cells onto the image plane are congruent.
By precomputing a limited set of cell projections, we can

approximate the projection of any volume cell and avoid the
expense of repeated projection operations. We exploit this
in computing the contribution to the image of a cell’s projected
isosurface fragment.
At the start of rendering, we project the vertices of a single
cell onto the image plane and compute their convex hull,
thus providing the polygonal projection of the whole cell.
We calculate the position of the lower left corner of the projected
polygon’s bounding rectangle and designate it as the
projection marker (see Figure 3). The polygon is translated
so that the projection marker lies at the origin of the image
plane: the polygon now lies on a 3�3 pixel grid whose
lower left corner has coordinates �0�0�. We clip the polygon
to each of the nine pixels and record the fraction of its area
lying within each. If we offset the position of the polygon
so that its marker lies at each of a discrete set of sub-pixel
positions in the region �0�1���0�1�, performing the clip and
record operation each time, we then have a reasonable approximation
of the cell’s projected area over the 3�3 grid
for any projection (and hence any cell) position. We use a
20�20 array of sub-pixel positions, which can be computed
quickly and provides reasonable accuracy.
We can compute the position of the projection marker for
each cell intersecting the isosurface, and use the fractional
part of that position (modulo the discretization rate) as an
index into the cell projection table. The corresponding table
entry gives the fraction of the light contributed by the
cell to be composited into each of the nine pixels its projection
may overlap. Exploiting the congruency further, we
need perform the complete projection marker calculation
only for the first cell visited; the position of any other projection
marker can be computed via a simple offset from the
first cell’s marker, based on the x, y, and z offset of the cell’s
position relative to that of the initial cell.
Distributing the light contributed by a surface fragment
evenly across the cell’s projection is an approximation.
Some approximation is necessary to avoid the prohibitive
expense of storing in the isoplane table a precise geometric
description of the isosurface intersection. Using an even
distribution is roughly equivalent to averaging over all positions
within the cell of a fragment with that area and normal.
Clipping the cell’s projection to pixel boundaries is
equivalent to convolving the projection with a box filter and
point sampling at pixel centers. It would be straightforward
to extend the method to use arbitrary filters, and as the cell
projection table is precomputed the cost would be negligible.
Our experience, however, indicates that box filtering,
in conjunction with the averaging operation just discussed,
yields sufficiently good results.
4.4 Compositing
As described above, for each cell containing a frontfacing
piece of the isosurface we compute the amount of
light reflected toward the image plane, and then distribute
that light to the nine pixels onto which the cell projects.
In essence we are compositing a series of 3�3 pixel subimages
into an accumulating larger image. Porter and
Duff’s image compositing algebra [7] assumes that the contents
of two pixels being composited are randomly dis-
Figure 3: Projection of a cell onto the image plane. The fraction of
the projected area lying within each of the nine pixels is stored in
the cell projection table, which is indexed by the sub-pixel location
of the projection marker.
tributed. In our situation this assumption is almost always
false, so we extend the algebra by a new operator to compensate.
Consider the case where adjacent cells both project onto
the same image pixel and contain adjoining bits of the isosurface
(see Figure 4). As the first cell’s surface fragment is
composited onto the target pixel, the pixel becomes partly
covered; as the second cell’s surface fragment is composited,
the previously uncovered part of the pixel becomes
completely covered. Let us call the first cell’s projection
the foreground pixel A and the second cell’s projection the
background pixel B, and assume that the α value for each is
0.5. Using the Porter-Duff over operator, where
FA � 1�0 and FB � 1�0 � αA �
the α value of the composited pixel (A over B) is
�FA�αA���FB�αB� � �1�0�0�5���0�5�0�5� � 0�75 �
The composited pixel has less coverage than expected and
appears too dark.
In this case, the contents were not at all independently
distributed. To address this situation we replace over with
a new compositing operator, add. For add the values of FA
and FB are
FA � 1�0 and FB � min� �1�0 � αA��αB� 1�0 � �
The α value of the composited pixel (A add B) is now
�FA�αA���FB�αB� � �1�0�0�5���1�0�0�5� � 1�0 �
This gives the composited pixel the expected (full) coverage
and the correct intensity.
The assumption that the contributions from multiple cells
to a single pixel is not independent fails in some cases.
If non-neighboring cells contribute to the same pixel, the
Porter-Duff independence assumption is valid. We therefore
expect slightly-too-bright edges when multiple silhouette
edges project onto the same pixel. We have not observed
this in practice, nor do we expect to: a pixel generically
is the projection of the interior of a surface, and being
on the projection of a silhouette is unusual. Being on

Figure 4: The isosurface projected from adjacent cells is not independently
distributed within the image pixel.
the projection of two silhouettes is very unlikely, and should
happen only at isolated pixels. The error is dual to that made
by a Z-buffer renderer, in which a nearby polygon that partially
covers a pixel can totally obscure a distant polygon
that completely covers the pixel, yielding a too-dim image.
5 Algorithm extensions
The algorithm as described makes every effort to reduce
the work at key steps in the rendering process. The characteristics
of the volume data and its rendered image that
make the acceleration possible also limit the types of data
we can process and the kinds of image we can render. Limitations
include quantizing the volume as a whole prior to
rendering, restricting processing to regular-uniform data 4 ,
and rendering only orthographic views.
These limitations are not, however, fundamental: each
can be relaxed or eliminated, at some additional cost. We
have extended the algorithm in various ways. The more important
extensions implemented so far are described below.
We give the user control in enabling these extensions, so
that the visualization needs can dictate the tradeoff between
flexibility and speed.
5.1 Cell-based quantization
Our extensive use of precomputed and cached data requires
restricting the volume’s data range, so that we can use
the cell voxel values as a look-up table index. Normally this
involves processing the volume once per isosurface level,
prior to the start of rendering. This quantization requires
visiting each voxel only once, and is comparatively fast.
Quantizing prior to rendering gives the best speed-up, but
limits our accuracy in approximating the isosurface within
a cell. The range of values across cells intersecting the
isosurface is not always the same, and can be much less
than the data range over which we quantize the volume as
a whole. We can therefore achieve a more accurate approximation
if we quantize voxels on a cell-by-cell basis rather
than once per volume. Unfortunately, cell-by-cell quanti-
4 Regular volumes are those with voxels arranged on a regular lattice,
with constant spacing along each axis. Uniform refers to equal spacing
along all axes.
Figure 5: Test volume illustrating the potential difference between
quantization strategies. Left: volume quantized as a whole prior to
rendering. Right: volume processed during rendering using cellbased
quantization.
zation may give a voxel different quantized values depending
on the cell being evaluated. To do this as a preprocess
would require storing multiple values per voxel, which is
prohibitively expensive for large volumes.
We do, however, allow cell-based quantization. When
specified, we skip the quantization preprocess and quantize
during the rendering. For each cell that contains isosurface,
we copy its voxel values to local storage and quantize using
the data range over just that cell. We then use these quantized
values as our table look-up index.
For certain data, this processing can produce a marked
improvement in image quality, particularly surface smoothness.
The dataset for Figure 5 is a 256 3 volume, with fullrange
eight bit data. The volume contains a ‘spherical’ isosurface,
of radius 120, at a level of 127.5. The data in the
vicinity of the surface ranges from 0 to 255, via a linear
ramp, over a distance of eight voxels. Prequantizing the
volume from the full range of 256 into a range of 8 (corresponding
to the isoplane table range used for rendering)
leaves us roughly one bit of accuracy per voxel value, for
cells intersecting the surface. Using cell-based quantization
allows us to achieve almost the full three bits allowed by the
isoplane table. The resulting improvement in image quality
is clear. While in practice, only a limited number of volumes
actually exhibit such characteristics, the feature is always
available for use as desired.
5.2 Non-uniform volumes
The isoplane table data is computed for cubical cells. If
the voxel spacing in the volume is not uniform, so that the
cells are parallelepipeds, we typically resample the volume
to make it uniform. By modifying our algorithm slightly we
can avoid this resampling. The transformation from a cube
to a general parallelepiped is just a non-uniform scaling, so
we can still use the precomputed isoplane table data. The
area and normal information extracted from the table must,
however, be transformed prior to its use.
Let us denote the inter-voxel spacing along the three axes
of the non-uniform volume by Sx, Sy, and Sz. We index
into the isoplane table using the quantized cell values, as described
previously. Before using the isosurface fragment’s
area and normal though, we must first compute its ‘scaletransformed’
normal and area, N 0 and A 0 respectively.
We transform the normal vector N by the scale transfor-

mation, saving the result as the vector U to be used in computing
N 0 and A 0 . U is calculated as
U � � �Nx �Sx�� �Ny �Sy�� �Nz �Sz� � �
To compute N 0 , we normalize U,
The transformed area A 0 is
A 0 � A
N 0 � U �jjU jj �
�
�N 0
x Sy Sz� 2 � �N 0
y Sx Sz� 2 � �N 0
z Sx Sy� 2� 1�2
� A �Sx Sy Sz� jjU jj �
The factor �Sx Sy Sz� can be computed once for the entire
volume. The net expense of this non-uniform scaling is
therefore three divides, one vector normalization, and two
multiplications per cell.
We must also adjust the size of our pixel grid in the cell
projection table, since the projection of a cell in the nonuniform
volume may not always lie within a 3�3 pixel area
(for a 1�1�2 inter-voxel spacing, the projection may cover
any of the pixels in a 4�4 region).
6 Results
We illustrate the algorithm’s use with datasets from the
medical and industrial communities. The data was processed
and rendered on Hewlett-Packard Series 700 workstations,
typically with machines having sufficient real
memory to contain both the volume data and the precomputed/cached
tables. No graphics hardware acceleration
was used. All images were rendered with a r � 8, edgecrossings,
compressed isoplane table.
Figure 6 shows a CT scan of a turbine blade. The original
dataset comprised 300 slices, each of size 512�512, with a
1�1�2 cell aspect ratio. To obtain cubical cells, we resampled
the volume to size 512�512�600,using trilinear interpolation.
The image shows fine detail and smooth shading.
The holes on the leading edge of the blade, the slots along its
tail, and the serial number on the base are all clearly visible.
The concave surface of the blade is smoothly shaded. The
slightly rough texture on the surface of the base results from
features in the original data (either noise or actual surface
information) that are visible when viewing slices in isolation.
The few minor artifacts visible along some flat surfaces
are caused by the limited range of normals available
due to data quantization. The image took 10 minutes, 9.6
seconds to render.
Figure 7 shows a human pelvis CT study. The original
data contained 56 slices, each of size 256�256. To obtain
cubical cells we again interpolated intermediate slices, this
time using a cubic B-spline. The resulting 256�256�111
volume was rendered in 18.1 seconds. Notice the smooth
shading along bone surfaces and the fine detail visible on
the spinal column.
Figure 8 is an image of an angiography dataset showing
vasculature in the pelvic region. The dataset was 80 slices,
with 256�256 samples each. We resampled the volume to
384�384�240 for rendering, both to obtain cubical cells
and provide a larger volume for testing. The original data
Figure 6: Industrial CT scan of a turbine blade [volume size: 512�
512�600 — isoplane table range: 8].
is fairly noisy, but the algorithm nonetheless extracts sufficient
fine detail to interpret the key elements of the blood
vessel structure. Although rendered with a r � 8 isoplane
table, for data of this type smaller ranges (e.g., r � 4) produce
nearly identical results. Rendering time for the image
was 86.5 seconds.
In considering the rendering performance, note that the
algorithm currently make no use of spatial data structures,
such as octrees or bounding volumes, to accelerate the rendering.
As such, every cell in the volume is examined in
generating the surface representation. Clearly spatial data
structures can reduce the effort expended on regions of the
volume not intersecting the isosurface, yielding a corresponding
improvement in performance. We chose to focus
on reducing the work at cells containing isosurface, knowing
that spatial acceleration techniques could be integrated
later. Based on existing work using such approaches (e.g.,
Wilhelms and Van Gelder [11], Laur and Hanrahan [2]) and
our own observations on the small percentage of cells that
intersect a typical isosurface, we expect a substantial speedup
from using such a technique, quite likely a factor of ten
or more.
7 Future work
After integrating a spatial acceleration mechanism, next
on our list of future work items is an error analysis of the
algorithm. The approximations made at various stages of
the rendering process to minimize cost introduce some ‘errors’
into the resulting image. The main sources of error are
data quantization, the approximation of the surface within a
cell by a plane, and the projection of that surface approximation
onto the image plane. It is fairly straightforward to
quantify the error at each stage. The more difficult problem
is relating the separate error metrics in a way relevant to actual
image quality.

Figure 7: CT study of a human pelvis [volume size: 256�256�
111 — isoplane table range: 8].
We could use such error measures to improve accuracy,
even if we cannot develop a single quality metric. For instance,
we could store an error value based on the closeness
of the planar surface approximation, for each isoplane table
entry. We could use this value, alone or in combination
with some quantization error measure, to decide whether
the surface approximation for a given cell is sufficiently accurate.
If not, we could subdivide the cell, interpolating interior
values as necessary, and recursively apply our surface
approximation method to the resulting sub-cells.
We are also interested in examining higher-order isosurface
approximations that could be stored in a minimum of
space. One motivation for this stems from what we term
ambiguous cells. For a cell intersecting the isosurface, it is
possible for the least-squares solution in Section 4.1 to have
A�B�C�0. Such ambiguous cases have reflectance functions
that are not well approximated by the reflection function
of a plane. An extended algorithm might enhance the
isoplane table in these cases by storing a second-order approximation
of the reflectance function.
Finally, we hope to develop a parallel or distributed implementation
of the algorithm. The minimal processing in
the main rendering loop, mostly table index generation and
look-up table access, makes the algorithm comparatively
simple to implement in a multiprocessor environment. Its
object-order nature also provides for convenient data partitioning
without replication across processing nodes. The
only communications-intensive step would be compositing
sub-images to produce a complete image of the isosurface.
8 Acknowledgements
This work was supported by the NSF/ARPA Science and
Technology Center for Computer Graphics and Scientific
Visualization (ASC-8920219). The first author also received
support as an ONR-NDSEG fellow. We gratefully
acknowledge the equipment grants from Hewlett-Packard
Company and Sun Microsystems, on whose workstations
this research was conducted. We also wish to thank those
who provided us with volume data: Terry Yoo of the University
of North Carolina at Chapel Hill, William Schroeder
of General Electric Company Corporate Research and De-
Figure 8: Angiography dataset showing pelvic vasculature [volume
size: 384�384�240 — isoplane table range: 8].
velopment, and Dr. E. Kent Yucel of Massachusetts General
Hospital – Harvard Medical School. Finally, special
thanks to Richard Lobb for his encouragement during the
writing and his assistance with volume resampling.
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234, April 1986.

Mix�Match� A Construction Kit for Visualization
Abstract
We present an environment in which users can in�
teractively create di�erent visualization methods. This
modular and extensible environment encapsulates most
of the existing visualization algorithms. Users can eas�
ily construct new visualization methods by combining
simple� �ne grain building blocks. These components
operate on a local subset of the data and generally ei�
ther look for target features or produce visual objects.
Intermediate compositions may also be used to build
more complex visualizations. This environment pro�
vides a foundation for building and exploring novel vi�
sualization methods.
Key Words and Phrases� interactive� extensible�
spray rendering� smart particles� visualization environ�
ment.
1 Introduction
The diverse needs of scientists demand the devel�
opment of general purpose� �exible and extensible vi�
sualization environments. Flexibility and extensibility
are particularly important since no monolithic pack�
age can be expected to satisfy every need. Users often
need variations on a particular technique and there are
always new techniques being developed. In this pa�
per� we present an environment for the �exible creation
of visualization techniques from basic building blocks.
Designing a technique involves identifying the tasks as�
sociated with target feature detection and behavioral
responses for displaying those features. This process is
simpli�ed by the categorization of these tasks into dif�
ferent classes and the formalization of what constitutes
a valid construction. A complete and valid construc�
tion de�nes a new visualization method. Each com�
ponent of a construction is usually very simple and
operates in a local subset of the data space. One of
these components speci�es which local subset of data
to process next. Hence� we can think of these construc�
tions as active processes that can be replicated and sent
to work on di�erent parts of the data. In fact� these
Alex Pang and Naim Alper
Baskin Center for
Computer Engineering � Information Sciences
University of California� Santa Cruz
Santa Cruz� CA 95064 USA
processes embody particle systems that interact with
the data they encounter. Other traditional algorithms
can also be decomposed and reconstructed with similar
components using this environment.
In the next section� we describe how our work dif�
fers from related work. We then describe the Spray
Rendering framework and how Mix�Match enriches it.
This is followed by detailed description of the internals
of Mix�Match. Finally� we show a couple of construc�
tions and their e�ects.
2 Related Work
In the last few years the data �ow paradigm has
become popular in scienti�c visualization. Visualiza�
tion environments such as AVS �17�� Iris Explorer �15��
Khoros �12�� apE �3�� and IBM Data Explorer �10� o�er
many modules that perform �ltering� mapping and ren�
dering tasks that can be combined to achieve a desired
visualization goal. These systems o�er generality� �ex�
ibility� modularity and extensibility. They address the
needs of novice� intermediate and expert users. Novices
merely load and execute previously constructed net�
works. Intermediate users use a network editor to con�
struct such a network from existing modules while ex�
pert users extend the system by adding modules.
All of these systems can be classi�ed as large grain
data �ow systems. Data �ow refers to the production
and consumption of blocks of data as they �ow through
modules in a network. Modules are required to ��re�
as new data arrive. The granularity refers to the size
of the data block that the module processes. In these
systems� it is the same size as the data model �hence
large� rather than being an atomic element of the data
model �18�. Granularity may also refer to the size
and complexity of the modules. Once again� in these
systems they are large in the sense that they implement
complete algorithms �e.g. mapping or �lter modules�.
A drawback with this approach is that memory re�
quirements become prohibitive and cause performance
degradation when the data set and the network are
large. Performance also su�ers when there is a lot of

interaction or when the data is dynamic and contin�
ually changing. Recently a �ne grain data �ow envi�
ronment has been proposed to overcome some of these
problems �16�. In this approach� the algorithms are re�
designed to work locally on incoming chunks of data
where the chunks are a few slices. However� visual�
ization algorithms that require random access to the
data set� such as streamlines for �ow visualization� are
di�cult to convert.
In spite of such shortcomings� these systems en�
joy a large following mostly because of their �exibil�
ity and extensibility to meet new user demands. The
importance of these qualities have been recognized in
other work. In ConMan� users constructed networks
for dynamically building and modifying graphics ap�
plications �5�. Abram and Whitted used an interactive
network based system for constructing shaders from
building blocks �1�. Kass used an interactive data �ow
programming environment to tackle many computer
graphics problems �7�. Corrie and Mackerras recently
extended the Renderman shading language to provide
a modular and extensible volume rendering system
based on programmable data shaders �2�.
Our approach strives to maintain the extensibility
and enhance the �exibility and interactivity of modu�
lar visualization environments at the expense of some
e�ciency. Instead of modules grinding on entire data
sets that �ow through them� we send or assign light
weight processes to work on a small subset of the data.
Thus the two main di�erentiating points are the gran�
ularity of both the modules and size of working data
set� and the execution style. Although the visualiza�
tion of the whole of the data set would computationally
be more expensive with this approach� the �ne�grained
nature of our components which work locally on parts
of the data allow quick� interactive exploration. The
components are conceptually simple and can be net�
worked in a very �exible way to create more complex
components. Because of our choice of execution style�
large and dynamic data sets can be handled by local�
izing these visualization components only to regions of
interest.
3 Spray Rendering
In this section� we brie�y describe the Spray Ren�
dering �11� framework which we use for the construc�
tion and application of visualization methods using
Mix�Match. Spray Rendering uses the metaphor of
spray cans �lled with smart paint particles. These par�
ticles are sprayed or delivered into the data set to high�
light features of interest. Features can be displayed in
a variety of ways depending on how the paint particles
have been de�ned. To get di�erent visual e�ects� users
simply choose di�erent spray cans from a �shelf�. The
regions that are displayed depend primarily on the po�
sition and the direction of the spray can. Cans also
have nozzles that can train the particles into a focused
beam or distribute them across a wider swath. The
number of paint particles and the distribution of these
particles can also be varied.
The key ingredient of Spray Rendering are the smart
particles or sparts. These sparts are reminiscent of the
Particle Systems introduced by Reeves �13� but also
possess some of the features of boids in �14� and �8�.
Sparts are born and have a �nite life time. As they
travel through the data space� they interact with the
data� and perhaps among themselves� leaving behind
them visual e�ects for the users. These particle be�
haviors can be roughly classi�ed into two categories�
targets and visual behaviors. Targets are features in
the data set that the sparts are hunting for �e.g. iso�
values� gradients� combination of two �elds� etc.� while
behaviors specify how sparts manifest themselves visu�
ally or non�visually �e.g. leaving a polygon or an in�
visible marker behind� attaching color attributes� etc.�.
Some of these e�ects can be seen in Figure 1.
Figure 1� Spray Rendering workspace show�
ing e�ects of di�erent types of smart particles
�sparts�. Users control viewing and spraying
through either graphics window. The lower left
graphics window shows the view from the cur�
rent can.
While sparts are conveniently portrayed to live in
3D space and handle 3D data sets� they can also be
designed to operate in lower or higher dimensional
space. For example� to track data values from a

stationary sensor� one can imagine the spart as sitting
on the sensor and producing glyphs �e.g. polylines�
according to changes in sensor readings. Or a spart
can be called upon to handle time dependent �ow �elds
where the spart is required to travel through time.
Eventually� a spart may also map and travel through
any N�parameter space. However� we still need to
investigate this further since mapping parameter values
to Euclidean space will generally produce scattered
data sets. This also complicates the point location test
for a spart.
In earlier implementations of Spray Rendering �11��
we mentioned the idea of mixing di�erent targets and
behaviors together to form new sparts. However� we
only had prede�ned sparts in the sense that each spart
on the shelf was a complete spart and could not be
altered. It was evident that since these sparts shared
some common characteristics� they could be decom�
posed into simpler components and reorganized almost
arbitrarily. The next section discusses the issues and
implementation details of how this is done.
Note that the idea of visualization processes com�
posed of basic building blocks moving through data
does not require spray cans as a launching pad. Indeed�
we have a mode where the processes are executed at
each grid location.
4 Mix�Match
Here we analyze the structure and components of a
spart and how they can be categorized. We then dis�
cuss the construction rules for building new sparts out
of these components. We also address issues such as
macro facilities� multi�stage spawning� handling multi�
ple data sets simultaneously and e�ciency.
4.1 Building blocks of a spart
As can be seen in Figure 1 each spart type produces
a di�erent visual e�ect. Sparts can be programmed
to generate iso�surfaces� they can be asked to trace
through �ow �elds and leave vector glyphs� ribbons
or stream lines� or generate a quick�and�dirty volume
rendering e�ect by mapping the data values to colored
points or spheres. Thus� each spart can be regarded as
a di�erent visualization method.
The goal of this research is to provide users the capa�
bility to interactively create new visualization methods
�or sparts�. We achieve this by providing a construc�
tion kit� made up of an extensible list of spart building
blocks� and allowing users to �exibly combine di�erent
pieces together.
As noted earlier� prede�ned sparts have two general
types of components� target detection and behavioral
expression. We can further re�ne this analysis by not�
ing that sparts are based on particle systems. They
therefore have rules regarding when they are born and
when they die. In addition� since these sparts are to
be sent into the data space� they also have position
update rules that may be di�erent from those found in
behavioral animation �collision avoidance� group cen�
tering� etc.�. We have grouped these spart components
into four categories�
1. Targets. These are feature detection compo�
nents. They operate on the data locally and check
to see whether a boolean condition is satis�ed.
Components in this category may include local
pre�processing operations such as smoothing or
gradient operators but not global ones such as
Fourier transforms. Relational operators� such as
And�Or� are also implemented as target functions
and can be used to combine any functions that
output a boolean.
2. Behaviors. These are components that depend
on a boolean condition� usually a target being
satis�ed� and may produce abstract visualization
objects �AVOs� to be rendered.
3. Position update. These are components that up�
date the current position of a spart. For example�
position changes may depend on the initial spray
direction or may be dictated by a �ow �eld.
4. Birth�Death. These components decide whether
the spart should die or spawn new sparts. For
example� a spart may be terminated as soon as a
target is found or wait until it has exited the data
space.
Figure 2� Components browser showing the list
of functions categorized under targets� behav�
iors �visuals�� position update and death func�
tions.
Figure 2 shows a growing list of components under
each category. Each element in the list is a building
block that can be used in the creation of a spart. By

eaking down the spart into components� we allow
the components to be used in the rapid prototyping of
other sparts.
Each building block in the construction kit is a
regular C function with variable number and type of
inputs and outputs. The input and output ports can
be connected together interactively. There is strong
type checking at the I�O connectivity but no type
coercion. Apart from the number and types of inputs
and outputs� components may also have parameters
that can be set by the user through widgets �e.g.
threshold value� step size� etc�. A spart is therefore a
set of functions grouped together to carry out a speci�c
visualization method.
4.2 Putting them together
False
Birth
Target
Function
Death
Function
True
Behavior
Function
Position
Update
Death
Figure 3� Flow chart illustrating the life�time of
a typical spart.
The process of creating the spart corresponding to a
visualization method can be seen as the mixing of dif�
ferent pigments on a palette to obtain a desired color.
In this analogy� the building blocks are the pigments.
We call this process of mixing di�erent building blocks
to obtain a desired visual e�ect Mix�Match. The rules
for putting the building blocks together are quite sim�
ple. The basic pattern follows the typical operations
over the lifetime of a spart as illustrated in Figure 3.
Note that Figure 3 is merely illustrative. For instance�
there may be sparts that do not have a target function
and whose behavior function executes unconditionally�
or there may be death functions that depend on mul�
tiple conditions.
True
False
We provide both a textual and a graphical interface
to the process of composing a spart and users can
switch freely between the two. Spart construction
starts with the selection of components to be included
in the editor from the components browser �Figure 2�.
As building blocks are included in the construction�
users must manipulate the input and output �elds
of each component to establish connections. In the
textual interface� this is done by editing the inputs and
outputs such that a name given to an output �eld of a
component and the input �eld of another component
indicates a connection between them. In the graphical
editor �Figure 4�� one merely selects the input and
output names from the menus of the components in
question. We have avoided the use of a programming
language for the de�nition of a spart to make the task
simpler for the scientist.
Figure 4� The graphical spart editor showing
the iso�surface spart composition. The IsoSurf
component is shown expanded to reveal the
types and the status of connections. Circles
indicate connections of the �elds while colors
of the triangles represent the types of the �elds.
Before enumerating the rules for constructing a
spart� let us look at how a common visualization
method can be expressed in the form of a spart as it
would appear in the textual spart editor. The March�
ing Cubes algorithm �9� can be converted to a localized
spart construction using three building blocks as illus�
trated below. Instead of looking at every cell in the
volume� individual sparts handle a local subset of the
data. In this particular example� it would be those
cells that the spart visited as it travels through the
data space.
Iso�surface spart�
IsoThresh � S1 � � Found � � Tag � � IsoVal �
IsoSurf � S1 � � Found � � Tag � � IsoVal � � Obj �
RegStep � S1 �

The above construction consists of a target function
IsoThresh� a behavior function IsoSurf� and a posi�
tion update function RegStep. The Streams compo�
nent �Figure 4� exists in all networks for binding data
streams to the input ports of the other components.
There is also a default death function that kills the
sparts once they exit the bounding box of the data set.
Input �elds are identi�ed with � � while output �elds
are identi�ed with � �. IsoThresh is a simple function
that examines the cell the spart is in within the input
stream S1. It sets the boolean Found if there exists
a surface at the given iso�value. In this case� IsoSurf
will generate one or more polygon visualization objects
Obj in that cell. The spart then advances a �xed step
size according to the parameter set in RegStep. These
functions are repeated until the spart is terminated af�
ter exiting the bounding box of the data volume.
The power of Mix�Match becomes evident when the
user has the �exibility of modifying sparts to produce
di�erent e�ects. For example� in the construction
above� the death function could be made conditional
on an iso�surface being found. This slight modi�cation
will produce iso�surfaces that are visible only from the
spray can�s perspective. The position update function
may be modi�ed to follow surface gradients. Likewise�
the behavior function may be substituted with one that
paints the entire cell achieving a cuberille e�ect�6�.
For some sparts it may be undesirable to rely on
position update and death functions that sample the
data. In the example above� cells that are missed will
not produce polygons and may result in discontinuous
surfaces. For this reason� we provide a mode where
the spart visits all the cells and only the target and
behavior functions are executed.
There are a few simple rules for constructing sparts
which are enforced either during construction or during
parsing�
1. Strong typing. The types of the input and output
�elds of a connection must match. Type checking
is done at the time the connection is being es�
tablished in the graphical editor and is postponed
until parsing time in the textual editor.
2. No optional inputs. All function input �elds must
either be connected to an output �eld or have a
constant value associated with it. Output �elds
can be left �oating.
3. Fan out but no fan in. There can only be a single
connection into an input �eld. The same output
�eld can be connected to multiple input �elds
however.
4. Acyclic graph. A directed graph where the edges
denote the dependency between components has
to be acyclic.
5. Execution order. The components of a spart ex�
ecute according to a speci�c order� target func�
tions �rst� then behavior functions� position up�
date functions and �nally birth�death functions.
Topological sorting ensures correct dependency or�
dering within each category. However� a com�
ponent from a category that will execute earlier
should not depend on another component that will
execute later.
The environment also provides a macro facility to
build a component from a collection of other com�
ponents allowing more succinct compositions. The
macros can be nested and more than one macro can
appear in a composition. Note that macros are like
procedures and can be saved and used in compositions.
They are not merely a temporary visual grouping of
components.
4.3 Handling multi�parameter data sets
A spart can handle multi�parameter data sets. Each
parameter of a multiple parameter data set is treated
as a separate data stream. The spart composition
then contains separate components to handle di�erent
data streams individually. The stream identi�ers� e.g.
S1� saved with the spart composition are bound by
the user to the actual data streams at the time the
spart is loaded into a can. When two di�erent streams
appear as input in the composition� it may mean that
they are the two parameters of a data set with the
same bounding volume� or two di�erent data sets with
di�erent bounding volumes. The latter implies that
there may be multiple incarnations of the spart� one
in each stream. An incarnation in one stream may be
dead but another may still be alive and the spart will
continue executing its program until all incarnations
are dead. This allows us to look for relationships
between parameters of the same data set or between
parameters in di�erent data sets that have overlapping
bounding volumes.
Relational expressions used in combining di�erent
targets� whether from the same data stream or not�
are also implemented as target functions. For example�
if the target functions TargetA and TargetB have
boolean outputs A and B� they could be combined as
follows�
TargetA ... � A � ...
TargetB ... � B � ...
And � A � � B � � AandB �

4.4 Multi�stage spawning
Sparts are initially spawned as they are sprayed from
the cans. Each time the user sprays or holds the spray
button down� sparts are continually being spawned and
added to the can�s pool of sparts to be executed. These
sparts are eventually executed and terminate when
they have satis�ed the death function.
New sparts may also be spawned during the life span
of a spart. This is achieved by including a spawn
function in the construction. The spawn function takes
the name of the spart to be spawned as an argument.
The new spart does not have to be the same as the
parent spart. The spawn function is handy in certain
situations. For instance� new sparts may be spawned
in the vicinity where iso�surface sparts have located a
surface. This will �ll in the surface more quickly than
relying on the spraying marksmanship of the user.
4.5 Extensibility
One advantage of Mix�Match over other systems is
the relative ease of writing small �ne�grained functions
that perform very speci�c tasks �e.g. update position
in certain way or produce certain AVOs�. In compar�
ison� coarse�grained modules are typically larger and
also have some degree of code replication since some
modules may be very similar in certain tasks but di�er
on details.
Extending the functionality of the environment in�
volves adding more functions to the browser. New
functions must be registered so that they can be in�
cluded in the browser. A con�guration manager pro�
vides a graphical user interface for this task. The user
de�nes the number and types of the inputs and out�
puts and graphically designs the control widgets for
the component. The con�guration manager then gen�
erates appropriate wrapper code. The new component
is integrated into the system by compilation and link�
ing.
4.6 E�ciency and object compaction
There is a tradeo� between �exibility and e�ciency.
If components exist at a low level� there is greater �ex�
ibility in composition but one su�ers higher costs in
execution overhead. Inversely� a high level component
results in loss of �exibility. At the cost of code replica�
tion and program size� one can include both the high
level module and its components. At the extreme� one
could have the spart be a single module. We call these
prede�ned sparts. If a certain spart is to be used of�
ten it may be worth the e�ort to re�implement it as a
prede�ned spart.
The building blocks are written independently from
each other and hence have to determine at run time
where to �nd the inputs and the parameters they need
and where to send the outputs. This is handled by
the components looking for their inputs and parame�
ters from �xed places in their own structure. During
parsing� memory is allocated for the addresses of input
and output �elds of each component. These addresses
are �lled according to the connections in the compo�
sition. All that the function does when called is to
dereference the pointers from the component structure
passed to it. Multiple instances can thus coexist in a
composition.
The main reason for the cost in execution is not so
much the extra function calls and pointer dereferences
but the fact that those functions that produce AVOs
have to generate them at each call that satis�es the
target function. For instance� a prede�ned streamline
spart would accumulate vertices that de�ne a single
multi�segmented line object �polyline�. A Mix�Match
streamline spart� on the other hand� would de�ne a
simple line segment consisting of the present and the
previous vertex each time it is called. This causes inef�
�ciency both in execution �many more calls to malloc�
and in storage �inner vertices are replicated�. The ren�
dering time also su�ers because of the greater number
of AVOs generated that need to be traversed. To alle�
viate the latter problem� objects of similar attributes
are compacted periodically into a single object. In
the above example� all the simple line segment objects
would be compacted into a single polyline object.
5 Examples
In this section� we give some examples of spart com�
positions. By changing single lines of these composi�
tions� di�erent visualizations can be achieved. Users
can experiment with the di�erent compositions and
save those that they are likely to use again.
5.1 Flow visualization
Showing streamlines is a typical �ow visualization
method for displaying vector �elds �Figure 5�. In this
technique� the path of a massless particle through the
�ow �eld is traced assuming that the vector at the
current location is tangential to the path. The new
position is calculated by forward integration using the
vector at the current location. Such a spart can be
constructed as follows�
Streamline spart�
StreamLine � S1� � �TRUE � � Vec � � Obj �
VecForwInteg � S1 � � Vec �

Figure 5� A spart that generates streamlines
from a vector �eld. Iso�surfaces from a scalar
�eld are also shown in the background.
This is a spart without a target function. The �rst
component is a behavior function that uncondition�
ally outputs objects while the position update func�
tion VecForwInteg calculates the new position. The
�rst component also outputs the calculated vector at
the current location so that it can be used by the fol�
lowing component. It is a good idea to pass interme�
diate values that may require expensive computation
so that other components can use them without re�
computation.
Another technique for vector �eld visualization is to
use vector glyphs. Usually� the glyphs are placed at
some sub�sampling of the grid but in spray rendering�
we can place them at intervals along the path of the
spart. By replacing the behavior function in the com�
position above with a behavior function that produces
vector glyphs� we can place glyphs at intervals along a
streamline. Alternatively� we can include both behav�
ior functions and obtain streamlines and glyphs along
the streamline.
5.2 Iso�surfaces
We can make some minor variations to the iso�
surface spart described in section 4.2. For example�
we can combine two or more iso�surface seeking sparts
within one construction. The target functions may be
bound to the same input stream �i.e. looking for di�er�
ent iso�values� or they may be bound to di�erent input
streams. The target function of the iso�surface spart
can be used in a spart that does not actually gener�
ate an iso�surface� but merely uses this component as
a �ltering operation. Another behavior that takes in
a geometry�the iso�surface� as input and colors it ac�
cording to a stream value can be used to investigate a
relationship between two parameters of a data set by
showing the variation of one parameter over a surface
on which the other parameter is constant as in �4�. The
following composition illustrates these ideas�
A spart with four streams�
IsoThresh � S1 � � Fnd1 � � Tag1 � � Val1 �
IsoSurf � S1 � � Fnd1 � � Tag1 � � Val1 � � Obj1 �
AddColSurf � S2 � � Fnd1 � � Obj1 � � Obj2 �
IsoThresh � S3 � � Fnd2 � � Tag2 � � Val2 �
VecGlyph � S4 � � Fnd2 � � Vec �
RegStep � S1 �
In this example� an iso�surface is created based
on one stream �S1�geopotential height� and the val�
ues of another stream �S2�humidity� are mapped onto
the generated surface as color. A third stream
�S3�temperature� is �ltered based on a threshold value
�S4�wind �eld� and vector glyphs are placed at those
locations that satisfy this condition.
Figure 6� A spart that shows the relationship
between four input streams of a climate model.
An iso�surface is generated from the geopoten�
tial height �eld and the relative humidity is
mapped onto this surface. The temperature
�eld is thresholded and wind vectors placed at
the locations that would have produced an iso�
surface.
6 Conclusions
Mix�Match is an extension to Spray Rendering

which allows composition of visualization techniques
from simple� �ne grain building blocks. Unlike the
prede�ned sparts presented in our earlier work� the
Mix�Match sparts are made up of elementary com�
ponents and users are allowed to edit them by adding�
removing or changing di�erent components with the
aid of a textual or graphical spart editor. This capa�
bility encourages the users to experiment with di�erent
ways of visualizing their data. In contrast to data �ow
networks� the execution model used here sends mul�
tiple independent agents to di�erent localities of the
data space. Its strengths are its extensibility and the
fact that users can create their own visualization meth�
ods interactively. On the other hand� its weaknesses
are primarily e�ciency and the duplication of e�ort by
multiple sparts that enter the same data space.
The current work opens up the proverbial Pandora�s
box. There are many issues that need to be resolved
to fully exploit the capabilities of sparts. Among these
are the traversal through unstructured grids and scat�
tered data� mapping to parallel architectures� inter�
spart communication and letting sparts query scienti�c
databases. Whether our approach o�ers advantages in
massively parallel environments is something that we
will be investigating in the near term.
Acknowledgements
We would like to thank the other members of the
spray team� Je� Furman� Tom Goodman� Elijah
Saxon� and Craig Wittenbrink. We would also like
to thank Dr. Teddy Holt and Dr. Paul Hirschberg for
kindly providing us the meteorological data sets used
in the �gures. Support for this work is partly funded
by NSF grant CDA�9115268 and ONR grant N00014�
92�J�1807.
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Figure 1� A spart that produces contours and pseudo�
color mapping of the cut plane of a humidity �eld.
Figure 3� A spart that fuses 4 input streams� geopo�
tential height� temperature� humidity and wind �eld.
Figure 5� This spart maps wind directions to surface
normals. Vorticity is used to color the bumpy surface.
Figure 2� A slight variation of the same spart that dis�
places the pseudocolored cut plane and contour lines.
Figure 4� An isosurface spart applied to a temperature
�eld� and a streamline spart applied to a wind �eld.
Figure 6� An isosurface spart probe. The threshold
value is determined interactively from the probe tip.

A Lattice Model for Data Display
William L. Hibbard 1&2 , Charles R. Dyer 2 and Brian E. Paul 1
1 Space Science and Engineering Center
2 Computer Sciences Department
University of Wisconsin - Madison
Abstract
In order to develop a foundation for visualization,
we develop lattice models for data objects and displays
that focus on the fact that data objects are
approximations to mathematical objects and real
displays are approximations to ideal displays. These
lattice models give us a way to quantize the information
content of data and displays and to define conditions on
the visualization mappings from data to displays.
Mappings satisfy these conditions if and only if they are
lattice isomorphisms. We show how to apply this result
to scientific data and display models, and discuss how it
might be applied to recursively defined data types
appropriate for complex information processing.
1 Introduction
Robertson et.al. have described the need for formal
models that can serve as a foundation for visualization
techniques and systems [13]. Models can be developed
for data (e.g., the fiber bundle data model [4] describes
the data objects that computational scientists use to
approximate functions between differentiable manifolds),
displays (e.g., Bertin's detailed analysis of static 2-D
displays [1]), users (i.e., their tasks and capabilities),
computations (i.e., how computations are expressed and
executed), and hardware devices (i.e., their capabilities).
Here we focus on the process of transforming data
into displays. We define a data model as a set U of data
objects, a display model as a set V of displays, and a
visualization process as a function D: U → V. The usual
approach to visualization is synthetic, constructing the
function D from simpler functions. The function may be
synthesized using rendering pipelines [5, 11, 12],
defining different pipelines appropriate for different types
of data objects within U. Object oriented programming
may be used to synthesize a polymorphic function D [9,
15] that applies to multiple data types within U.
We will try to address the need for a formal
foundation for visualization by taking an analytic
approach to defining D. Since an arbitrary function
D: U → V will not produce displays D(u) that effectively
communicate the information content of data objects
u ∈ U, we seek to define conditions on D to ensure that it
does. For example, we may require that D be injective
(i.e., one-to-one), so that no two data objects have the
same display. However, this is clearly not enough. If we
let U and V both be the set of images of 512 by 512 pixels
with 24 bits of color per pixel, then any permutation of U
can be interpreted as an injective function D from U to V.
But an arbitrary permutation of images will not
effectively communicate information. Thus we need to
define stronger conditions on the function D. Our
investigation depends on some complex mathematics,
although we will only present the conclusions in this
paper. The details are available in [7].
2 Lattices as data and display models
The purpose of data visualization is to communicate
the information content of data objects in displays. Thus
if we can quantify the information content of data objects
and displays this may give us a way to define conditions
on the visualization function D. The issue of information
content has already been addressed in the study of
programming language semantics [14], which seeks to
assign meanings to programs. This issue arises because
there is no algorithmic way to separate non-terminating
programs from terminating programs, so the set of
meanings of programs must include an undefined value
for non-terminating programs. This value contains less
information (i.e., is less precise) than any of the values
that a program might produce if it terminates, and thus
introduces an order relation based on information content
into the set of program meanings. In order to define a
correspondence between the ways that programs are
constructed, and the sets of meanings of programs, Scott

developed an elegant lattice theory for the meanings of
programs [16].
Scientists have data with undefined values, although
their sources are numerical problems and failures of
observing instruments rather than non-terminating
computations. An undefined value for pixels in satellite
images contains less information than valid pixel
radiances and thus creates an order relation between data
values. Data are often accompanied by metadata [18]
that describe their accuracy, for example as error bars,
and these accuracy estimates also create order relations
between data values based on information content (i.e.,
precision). Finally, array data objects are often
approximations to functions, as for example a satellite
image is a finite approximation (i.e., a finite sampling in
both space and radiance) to a continuous radiance field,
and such arrays may be ordered based on the resolution
with which they sample functions.
In general scientists use computer data objects as
finite approximations to the objects of their mathematical
models, which contain infinite precision numbers and
functions with infinite ranges. Thus metadata for
missing data indicators, numerical accuracy and function
sampling are really central to the meaning of scientific
data and should play an important role in a data model.
We define a data model U as a lattice of data objects,
ordered by how precisely they approximate mathematical
objects. To say that U is a lattice [2] means that there is
a partial order on U (i.e., a binary relation such that, for
all u1 , u2 , u3 ∈ U, u1 ≤ u1 , u1 ≤ u2 & u2 ≤ u1 ⇒ u1 = u2 and u1 ≤ u2 & u2 ≤ u3 ⇒ u1 ≤ u3 ) and that any pair
u1 , u2 ∈ U have a least upper bound (denoted by u1 ∨ u2 )
and a greatest lower bound (denoted by u1 ∧ u2 ).
The notion of precision of approximation also
applies to displays. They have finite resolutions in space,
color and time (i.e., animation). 2-D images and 3-D
volume renderings are composed of finite numbers of
pixels and voxels and are finite approximations to
idealized mathematical displays. Thus we will assume
that our display model V is a lattice and that displays are
ordered according to their information content (i.e.,
precision of approximation to ideal displays). In Sections
4 and 5 we will present examples of scientific data and
display lattices.
We assume that U and V are complete lattices, so
that they contain the mathematical objects and ideal
displays that are limits of sets of data objects and real
displays (a lattice is complete if any subset has a least
upper bound and a greatest lower bound). Just as we
study functions of rational numbers in the context of
functions of real numbers (the completion of the rational
numbers), we will study visualization functions between
the complete lattices U and V, recognizing that data
objects and real displays are restricted to countable
subsets of U and V.
3 Conditions on visualization functions
The lattice structures of U and V provide a way to
quantize information content and thus to define
conditions on functions of the form D: U → V. In order
to define these conditions we draw on the work of
Mackinlay [10]. He studied the problem of automatically
generating displays of relational information and defined
expressiveness conditions on the mapping from
relational data to displays. His conditions specify that a
display expresses a set of facts (i.e., an instance of a set
of relations) if the display encodes all the facts in the set,
and encodes only those facts.
In order to interpret the expressiveness conditions
we define a fact about data objects as a logical predicate
applied to U (i.e., a function of the form P: U → {false,
true}). However, since data objects are approximations
to mathematical objects, we should avoid predicates such
that providing more precise information about a
mathematical object (i.e., going from u1 to u2 where
u1 ≤ u2 ) changes the truth value of the predicate (e.g.,
P(u1 ) = true but P(u2 ) = false). Thus we consider
predicates that take values in {undefined, false, true}
(where undefined < false and undefined < true), and we
require predicates to preserve information ordering (that
is, if u1 ≤ u2 then P(u1 ) ≤ P(u2 ); functions that preserve
order are called monotone). We also observe that a
predicate of the form P: U → {undefined, false, true} can
be expressed in terms of two predicates of the form
P: U → {undefined, true}, so we will limit facts about
data objects to monotone predicates of the form
P: U → {undefined, true}.
The first part of the expressiveness conditions says
that every fact about data objects is encoded by a fact
about their displays. We interpret this as follows:
Condition 1. For every monotone predicate
P: U → {undefined, true}, there is a monotone predicate
Q: V → {undefined, true} such that P(u) = Q(D(u)) for
each u ∈ U.
This requires that D be injective (if u 1 ≠ u 2 then
there are P such that P(u 1 ) ≠ P(u 2 ), but if D(u 1 ) = D(u 2 )
then Q(D(u 1 )) = Q(D(u 2 )) for all Q, so we must have
D(u 1 ) ≠ D(u 2 )).
The second part of the expressiveness conditions
says that every fact about displays encodes a fact about
data objects. We interpret this as follows:

Condition 2. For every monotone predicate
Q: V → {undefined, true}, there is a monotone predicate
P: U → {undefined, true} such that Q(v) = P(D -1 (v)) for
each v ∈ V.
This requires that D -1 be a function from V to U, and
hence that D be bijective (i.e., one-to-one and onto).
However, it is too strong to require that a data model
realize every possible display. Since U is a complete
lattice it contains a maximal data object X (the least
upper bound of all members of U). Then D(X) is the
display of X and the notation ↓D(X) represents the
complete lattice of all displays less than D(X). We
modify Condition 2 as follows:
Condition 2'. For every monotone predicate
Q: ↓D(X) → {undefined, true}, there is a monotone
predicate P: U → {undefined, true} such that Q(v) =
P(D -1 (v)) for each v ∈ ↓D(X).
These conditions quantify the relation between the
information content of data objects and the information
content of their displays. We use them to define a class
of functions:
Definition. A function D: U → V is a display
function if it satisfies Conditions 1 and 2'.
In [7] we prove the following result about display
functions:
Proposition 1. A function D: U → V is a display
function if and only if it is a lattice isomorphism from U
onto ↓D(X) [i.e., for all u 1 , u 2 ∈ U, D(u 1 ∨ u 2 ) =
D(u 1 ) ∨ D(u 2 ) and D(u 1 ∧ u 2 ) = D(u 1 ) ∧ D(u 2 )].
This result may be applied to any complete lattice
models of data and displays. In the next three sections
denote the least element of a lattice), and also includes
all closed real intervals. We interpret the closed real
interval [x, y] as an approximation to an actual value that
lies between x and y. In our lattice structure, these
intervals are ordered by the inverse of set containment,
since a smaller interval provides more precise
information than a containing interval. Figure 1
illustrates the order relation on a continuous scalar type.
Of course, an actual implementation can only include a
countable number of closed real intervals (such as the set
of rational intervals).
[0.0, 0.0] [0.01, 0.01] [0.5, 0.5]
[0.0, 0.01]
[0.0, 0.1]
[0.0, 1.0]
⊥
[0.945, 0.945]
[0.93, 0.95]
[0.9, 1.0]
[0.94, 0.97]
Figure 1. The order relations among a few values of a
continuous scalar.
We also define discrete scalars to represent integer
and string variables, such as year, frequency_count and
satellite_name. If s is discrete then I s includes ⊥ and a
countable set of incomparable values (no integer is more
precise than any other integer). Figure 2 illustrates the
order relation on a discrete scalar type.
. . .
-3
-2
-1
0 1 2 3 . . .
⊥
Figure 2. The order relations among a few values of a
discrete scalar.
we will explore its consequences in one setting. Complex data types are constructed from scalar data
4 A Scientific data model
We will develop a scientific data model that
integrates metadata for missing data indicators,
numerical accuracy and function sampling. We will
develop this data model in terms of a set of data types,
starting with scalar types used to represent the primitive
variables of mathematical models. Given a scalar type s,
let I s denote the set of possible values of a data object of
the type s. First we define continuous scalars to
represent real variables, such as time, temperature and
latitude. If s is continuous then I s includes the undefined
value, which we denote by the symbol ⊥ (usually used to
types as arrays and tuples. An array data type represents
a function between mathematical variables. For
example, a function from time to temperature is
approximated by data objects of the type (array [time] of
temperature;). We say that time is the domain type of
this array, and temperature is its range type. Values of
an array type are sets of 2-tuples that are (domain, range)
pairs. The set {([1.1, 1.6], [3.1, 3.4]), ([3.6, 4.1], [5.0,
5.2]), ([6.1, 6.4], [6.2, 6.5])} is an array data object that
contains three samples of a function from time to
temperature. The domain value of a sample lies in the
first interval of a pair and the range value lies in the
second interval of a pair, as illustrated in Figure 3.
Adding more samples, or increasing the precision of

samples, will create a more precise approximation to the
function. Figure 4 illustrates the order relation on an
array data type. The domain of an array must be a scalar
type, but its range may be any scalar or complex type (its
definition may not include the array's domain type).
[6.2, 6.5]
[5.0,5.2]
[3.1, 3.4]
[1.1, 1.6] [3.6, 4.1] [6.1, 6.4]
Figure 3. An array samples a real function as a set of
pairs of intervals.
{([1.33, 1.40], [3.21, 3.24]),
([3.72, 3.73], [5.09, 5.12]),
([6.21, 6.23], [6.31, 6.35])}
{([1.1, 1.6], [3.1, 3.4]),
([3.6, 4.1], [5.0, 5.2]),
([6.1, 6.4], [6.2, 6.5])}
{([1.1, 1.6], ⊥ ),
([3.6, 4.1], [5.0, 5.2]),
([6.1, 6.4], ⊥ )}
{([1.1, 1.6], [3.1, 3.4]),
([3.6, 4.1], [5.0, 5.2]),
([6.1, 6.4], [6.2, 6.5]),
([7.3, 7.5], [8.1, 8.4])}
φ (the empty set)
Figure 4. The order relations among a few arrays.
Tuple data types represent tuples of mathematical
objects. For example, a 2-tuple of values for temperature
and pressure is represented by data objects of the type
struct{temperature; pressure;}. Data objects of this type
are 2-tuples (temp, pres) where temp ∈ I temperature
and pres ∈ I pressure . We say that temperature and
pressure are element types of the tuple. The elements of
a tuple type may be any complex types (they must be
defined from disjoint sets of scalars). A tuple data object
x is less than or equal to a tuple data object y if every
element of x is less than or equal to the corresponding
element of y, as illustrated in Figure 5.
([0.3, 0.4], [2.3, 2.4])
([0.0, 0.9], [2.3, 2.4]) ([0.3, 0.4], [2.0, 2.9])
(⊥, [2.3, 2.4]) ([0.0, 0.9], [2.0, 2.9]) ([0.3, 0.4], ⊥)
(⊥, [2.0, 2.9])
(⊥, ⊥)
([0.0, 0.9], ⊥)
Figure 5. The order relations among a few tuples.
This data model is applied to a particular application
by defining a finite set S of scalar types (these would
represent the primitive variables of the application), and
defining T as the set of all types that can be constructed
as arrays and tuples from the scalar types in S. For each
type t ∈ T we can define a countable set H t of data
objects of type t (these correspond to the data objects that
are realized by an implementation).
In order to apply our lattice theory to this data
model, we must define a single lattice U and embed each
H t in U. First define X = X{I s | s ∈ S} as the cross
product of the value sets of the scalars in S. Its members
are tuples with one value from each scalar in S, ordered
as illustrated in Figure 5. Now we would like to define U
as the power set of X (i.e., the set of all subsets of X).
However, power sets have been studied for the semantics
of parallel languages and there is a well known problem
with constructing order relations on power sets [14]. We
expect this order relation to be consistent with the order
relation on X and also consistent with set containment.
For example, if a, b ∈ X and a < b, we would expect that
{a} < {b}. Thus we might define an order relation
between subsets of X by:
(1) ∀A, B ⊆ X. (A ≤ B ⇔ ∀a ∈ A. ∃b ∈ B. a ≤ b)
However, given a < b, (1) implies that {b} ≤ {a, b} and
{a, b} ≤ {b} are both true, which contradicts {b} ≠
{a, b}. This problem can be resolved by restricting the
lattice U to sets of tuples such every tuple is maximal in
the set. That is, a set A ⊆ X belongs to the lattice U if

a < b is not true for any pair a, b ∈ A. The members of
U are ordered by (1), as illustrated in Fig. 6, and form a
complete lattice (see [7] for more details).
{(Α, Β, ⊥)}
{(Α, ⊥, ⊥), (⊥, Β, ⊥)}
{(Α, ⊥, ⊥)} {(⊥, Β, ⊥)}
{(⊥, ⊥, ⊥)}
φ
(the empty set)
Figure 6. The order relations among a few members
of a data lattice U defined by three scalars.
(temp1, pres1) {( ⊥,
temp1, pres1)}
object of a
tuple type
set of one tuple with
time value = ⊥
Figure 7. An embedding of a tuple type into a lattice.
{(time1, temp1),
(time2, temp2),
(time3, temp3),
. . .
(timeN, tempN)}
array of temperature
values indexed by
time values
{(time1, temp1, ⊥ ),
(time2, temp2, ⊥ ),
(time3, temp3, ⊥ ),
. . .
(timeN, tempN, ⊥ )}
set of tuples with
pressure values =
Figure 8. An embedding of an array type into a lattice.
To see how the data objects in Ht are embedded in
U, consider a data lattice U defined from the three scalars
time, temperature and pressure. Objects in the lattice U
are sets of tuple of the form (time, temperature,
pressure). We can define a tuple data type
struct{temperature; pressure;}. A data object of this type
is a tuple of the form (temp, pres) and can be mapped to
a set of tuples (actually, it is a set consisting of one tuple)
in U with the form {(⊥, temp, pres)}. This embeds the
tuple data type in the lattice U, as illustrated in Figure 7.
Similarly, we can embed array data types in the data
lattice. For example, consider an array data type (array
⊥
[time] of temperature;). A data object of this type
consists of a set of pairs of (time, temp). This array data
object can be embedded in U as a set of tuples of the form
(time, temp, ⊥). Figure 8 illustrates this embedding.
The basic ideas presented in Figs. 7 and 8 can be
combined to embed complex data types, defined as
hierarchies of tuples and arrays, in data lattices (see [7]
for details).
5 A scientific display model
For our scientific display model, we start with
Bertin's analysis of static 2-D displays [1]. He modeled
displays as sets of graphical marks, where each mark was
described by an 8-tuple of graphical primitive values
(i.e., two screen coordinates, size, value, texture, color,
orientation and shape). The idea of a display as a set of
tuple values is quite similar to the way we constructed the
data lattice U. Thus we define a finite set DS of display
scalars to represent graphical primitives, we define Y =
X{I d | d ∈ DS} as the cross product of the value sets of
the display scalars in DS, and we define V as the
complete lattice of all subsets A of Y such that every tuple
is maximal in A.
set of animation steps:
tuple of display
scalar values
for a graphical
mark
interval that mark
persists during
animation
(time, x, y, z, red, green, blue)
red green blue
ranges of values
of mark's color
components
location and size
of mark in volume
x
Figure 9. The roles of display scalars in an animated
3-D display model.
We can define a specific lattice V to model animated
3-D displays in terms of a set of seven continuous display
scalars: (x, y, z, red, green, blue, time}. A tuple of
values of these display scalars represents a graphical
mark. The interval values of x, y and z represent the
z
y

locations and sizes of graphical marks in the volume, the
interval values of red, green and blue represent the
ranges of colors of marks, and the interval values of time
represent the place and duration of persistence of marks
in an animation sequence. This is illustrated in Figure 9.
A display in V is a set of tuples, representing a set of
graphical marks.
Display scalars can be defined for a wide variety of
attributes of graphical marks, and need not be limited to
simple values. For example, a discrete display scalar
may be an index into a set of complex shapes (i.e., icons).
6 Scalar mapping functions
Proposition 1 said that a function of the form
D: U → V satisfies the expressiveness conditions (i.e., is
a display function) if and only if D is a lattice
isomorphism from U onto ↓D(X), a sublattice of V. We
can now apply this to the scientific data and display
lattices described in Section 4 and 5.
The scalar and display scalar types play a special
role in characterizing display functions in the context of
our scientific models. Given a scalar type s ∈ S, define
U s ⊆ U as the set of embeddings of objects of type s in U.
That is, U s consists of sets of tuples of the form
{(⊥,...,b,...,⊥)} (this notation indicates that all
components of the tuple are ⊥ except the s component,
which is b). Similarly, given a display scalar type
d ∈ DS, define V d ⊆ V as the set of embeddings of
objects of type d in V. In [7] we prove the following
result:
Proposition 2. If D: U → V is a display function,
then we can define a mapping MAP D : S → POWER(DS)
(this is the power set of DS) such that for all scalars s ∈ S
and all for a ∈ U s , there is d ∈ MAP D (s) such that
D(a) ∈ V d . The values of D on all of U are determined
of designing displays as an assumption, but that it is a
consequence of a more fundamental set of expressiveness
conditions. Figure 10 provides examples of mappings
from scalars to display scalars (lat_lon is a real2d scalar,
as described in Section 7).
a
n
i
m
a
t
i
o
n
type image_sequence =
s
t
e
p
s
array [time] of array [lat_lon] of structure {ir; vis;}
x
z
y
red green blue
Figure 10. Mappings from scalars to display scalars.
In [7] we present a precise definition (the details are
complex) of scalar mapping functions and show that
D: U → V is a display function if and only if it is a scalar
mapping function. Here we will just describe the
behavior of display functions on continuous scalars. If s
is a continuous scalar and MAP D (s) = d, then D maps U s
to V d . This can be interpreted by a pair of functions
g s :R × R → R and h s :R × R → R (where R denotes the
real numbers) such that for all {(⊥,...,[x, y],...,⊥)} in U s ,
D({(⊥,...,[x, y],...,⊥)}) = {(⊥,...,[g s (x, y), h s (x, y)],...,⊥)},
which is a member of V d . Define functions g' s :R → R
and h' s :R → R by g' s (z) = g s (z, z) and h' s (z) = h s (z, z).
Then the functions g s and h s can be defined in terms of
g' s and h' s as follows:
by its values on the scalar embeddings U s . Furthermore, (2) g s (x, y) = min{g' s (z) | x ≤ z ≤ y} and
(3) h s (x, y) = max{h' s (z) | x ≤ z ≤ y}.
(a) If s is discrete and d ∈ MAP D (s) then d is
(b)
discrete,
If s is continuous then MAPD (s) contains a
single continuous display scalar.
(c) If s ≠ s' then MAPD (s) ∩ MAPD (s') = φ.
This tells us that display functions map scalars,
which represent primitive variables like time and
temperature, to display scalars, which represent
graphical primitives like screen axes and color
components. Most displays are already designed in this
way, as, for example, a time series of temperatures may
be displayed by mapping time to one axis and
temperature to another. The remarkable thing is that
Proposition 2 tells us that we don't have to take this way
These functions must satisfy the conditions illustrated in
Figure 11.
Although the complete lattices U and V include
members containing infinite numbers of tuples (these are
mathematical objects and ideal displays) in [7] we prove
the following:
Proposition 3. Given a display function D: U → V,
a data type t ∈ T and an embedding of a data object from
H t to a ∈ U, then a contains a finite number of tuples
and D(a) ∈ V contains a finite number of tuples.

h' s and g' s
determine
mapping to
interval in a
continuous
display
scalar
no lower bound
no upper bound
h' s
interval in a
continuous scalar
h' s above g' s
g' s
h' s and g' s both continuous
and increasing (could
both be decreasing)
Figure 11. The behavior of a display function D on a
continuous scalar interpreted in terms of the
behavior of functions h' s and g' s .
7 Implementation
The data and display models described in Sections 4
and 5, and the scalar mapping functions described in
Section 6, are implemented in our VIS-AD system [6, 8].
This system is intended to help scientists experiment
with their algorithms and steer their computations. It
includes a programming language that allows users to
define scalar and complex data types and to express
scientific algorithms. The scalars in this language are
classified as real (i.e., continuous), integer (discrete),
string (discrete), real2d and real3d. The real2d and
real3d scalars have no analog in the data model
presented in Section 4, but are very useful as the domains
of arrays that have non-Cartesian sampling in two and
three dimensions. Users control how data are displayed
by defining a set of mappings from scalar types (that they
declare in their programs) to display scalar types. By
defining a set of mappings a user defines a display
function D: U → V that may be applied to display data
objects of any type.
The VIS-AD display model includes the seven
display scalars described for animated 3-D displays in
Section 5, and also includes display scalars named
contour and selector. Multiple copies of each of these
may exist in a display lattice (the numbers of copies are
determined by the user's mappings). Scalars mapped to
contour are depicted by drawing isolevel curves and
surfaces through the field defined by the contour values
in graphical marks. For each selector display scalar, the
user selects a set of values and only those graphical
marks whose selector values that overlap this set are
displayed. Contour is a real display scalar and selector
display scalars take the type of the scalar mapped to
them. We plan to add real display scalars for
transparency and reflectivity to the system (to be
interpreted by complex volume rendering of graphical
marks), as well as a real3d display scalar for vector (to
be interpreted by flow rendering techniques).
VIS-AD is available by anonymous ftp from
iris.ssec.wisc.edu (144.92.108.63) in the pub/visad
directory. Get the README file for complete
installation instructions.
8 Recursively defined data types
The data model in Section 4 is adequate for scientific
data, but is inadequate for complex information
processing which involves recursively defined data types
[14]. For example, binary trees may be defined by the
type bintree = struct{bintree; bintree; value;} (a leaf
node is indicated when both bintree elements of the tuple
are undefined). Several techniques have been developed
to model such data using lattices. In the current context,
the most promising is called universal domains [3, 17].
Just as we embedded data objects of many different types
in the domain U in Section 4, data objects of many
different recursively defined data types are embedded in a
universal domain (which we also denote by U).
However, these embeddings have been defined in order to
study programming language semantics, and have a
serious problem in the visualization context. Data
objects of many different types are mapped to the same
member of U. For example, an integer and a function
from the integers to the integers may be mapped to the
same member of U, and thus any display function of the
form D: U → V will generate the same display for these
two data objects. Thus, in order to extend our lattice
theory of visualization to recursively defined data types,
other embeddings into universal domains must be
developed.
A suitable display lattice V must also be developed
such that there exist lattice isomorphisms from a
universal domain U into V. Displays involving diagrams
and hypertext links are analogous to the pointers usually
used to implement recursively defined data types. Thus
the interpretation of V as a set of actual displays may
involve these graphical techniques. However, since a
large class of recursively defined data types can be
embedded in U, and since V is isomorphic to U, these
graphical techniques must be applied in a very abstract
manner to define a suitable lattice V.
9 Conclusions
It is easy to think of metadata as secondary when we
are focused on the task of making visualizations of data.
However, it is central to the meaning of scientific data
that they are approximations to mathematical objects,
and lattices provide a way to integrate metadata about
precision of approximation into a data model. By

inging the approximate nature of data and displays into
central focus, lattices provide a foundation for
understanding the visualization process and an analytic
approach to defining the mapping from data to displays.
While Proposition 2 just confirms standard practice in
designing displays, it is remarkable that this practice can
be deduced from the expressiveness conditions.
Although we have not derived any new rendering
techniques by using lattices, the high level of abstraction
of scalar mapping functions do provide a very flexible
user interface for controlling how data are displayed.
There will be considerable technical difficulties in
extending this work to recursively defined data types, but
we are confident that the results will be interesting.
Acknowledgments
This work was support by NASA grant NAG8-828,
and by the National Science Foundation and the Defense
Advanced Research Projects Agency under Cooperative
Agreement NCR-8919038 with the Corporation for
National Research Initiatives.
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An Object Oriented Design for the Visualization of
Multi-Variable Data Objects
Abstract
This paper presents an object-oriented system design
supporting the composition of scientific data visualization
techniques based on the definition of hierarchies of typed
data objects and tools. Traditional visualization systems
focus on creating graphical objects which often cannot be
re-used for further processing. Our approach provides
objects of different topological dimension to offer a
natural way of describing the results of visualization
mappings. Serial composition of data extraction tools is
allowed, while each intermediate visualization object
shares a common description and behavior. Visualization
objects can be re-used, facilitating the data exploration
process by expanding the available analysis and correlation
functions provided. This design offers an open-ended
architecture for the development of new visualization
techniques. It promotes data and software re-use,
eliminates the need for writing special purpose software
and reduces processing requirements during interactive
visualization sessions.
1. Introduction
1.1. Visualization Objects and Processing
Modules
We examined several well-known visualization
systems (AVS, Explorer, FAST, Khoros, ...) and the
techniques they use to interconnect independent data
processing modules. A typing system is usually provided
to define process interfaces, and connections are only
allowed between I/O ports accepting the same type of data.
For this paper, the word type is used in the
programming language sense for input and output abstract
data structures. Graphical output is usually the endproduct
of a visualization session. Thus, data of various
types are passed between modules - each module possibly
creating data of a new type - until displayable geometry is
output for a final rendering. Data-flow architectures already
support some form of data sharing. However, the data
produced for the graphical display stage of a visualization
Jean M. Favre and James Hahn
EE&CS Department
The George Washington University
Washington, DC 20052
are collections of graphics primitives, using a systemdefined
geometry type. Rendering/display modules become
in effect a bottleneck, taking geometric data in, without
allowing their re-use by non-graphical processing tools.
Common visualization tools have too often focused on
directly producing geometric data (for example, sets of
polygons or line segments) ready for the fast hardware
rendering workstations of the 1990's. Thus, individual
polygons or lines are generated with normal information
and a color index or an RGB value associated with each
vertex. Rendering such objects can often be done in real
time but unfortunately, rendering is the only operation
that can be applied to such objects.
1.2. Previous work
Foley and Lane [2] have presented multi-valued
visualization techniques. They assume the definition of a
geometric object D (which can be formed of several
disjoint surfaces). D is a user-defined surface or an isosurface
and is used as a value probe to examine volume
data at the surface of the object. Color Blended Contours,
Projected Surface Graphs, Contour Curves on a Projected
Surface Graph, Iso-Surface and Hyper-Surface Projection
Graphs are the tools they use to compose volumetric
rendering techniques. However, their work is restricted to
the use of surfaces for geometric support, and their
analysis of data is limited to rendering operations.
The definition and use of object-oriented abstract data
types for scientific visualization has been documented by
several authors [3,6,8,11]. However, they do not deal
with composition techniques and a system design to
support these techniques. The data types provided do not
foster the re-use of data objects for composition purposes.
In a Visualization '91 Workshop Report [1], the notion of
"Functions of Several Variables" (FOSV) is discussed as
the most important abstract data type relevant to scientific
computations. The workshop participants propose to use
this data class as the starting point for a reference model,
and consider Visualization mappings as operations
between FOSV instances. Lucas et al. [8] use this
paradigm and present a high-level overview of Data
Explorer. Our system design differs from their approach

y putting more emphasis on data integrity and
requirements for Functional Composition.
1.3. Motivation and Design Goals
To promote serial composition of visualization
techniques, we must augment the inter-connectivity of
modules with carefully designed typed output and allow
several visualization primitives to form a composite
visualization object. We give visualization objects a
broader functionality than pure graphics primitives. An
object-oriented design is used to encapsulate both
geometry and field data so that objects can be freely
exchanged between data operators. Each data class is
endowed with data extraction and rendering operators. Our
emphasis is on combining visualization techniques to
create composite visualization designs and providing an
abstraction which favors data and code re-use. It is
particularly useful for multi-variate field data, where more
than one scalar field is considered at each grid point, and
where we can alternate between different data field views,
independently of the underlying geometry.
In Section 2, we briefly introduce the object-oriented
paradigm. Section 3 shows how most sets of scientific
data can be described by the generic unstructured data grids
in 3-D space and gives details about the general purpose
sets of line-, surface- and volume-elements of points and
the functions associated with them. Section 4 shows how
data visualization tools are re-defined and combined to
achieve Data Selection and Functional Composition.
Section 5 describes several examples. We offer some
discussion in section 6 and conclude in section 7.
2. The Object-Oriented Paradigm
In most conventional programming languages, every
name (identifier) has a type associated with it. This type
determines what operations can be applied to the name.
For example, integer and floating point types come with
the pre-defined +, -, *, / operators. The programming
paradigm provided by object-oriented languages favors a
similar process of defining types and associated operators.
Abstraction is defined as the process of extracting
essential properties of a concept. Data structures allow the
abstraction of the structural aspects of the data
organization. Procedures and functions allow the
abstraction of behavioral aspects. The C++ programmer
can combine these two user-defined abstractions to create
data classes, defining new types for which access to data is
restricted to a specific set of access functions. The data
structure thus embedded in a class definition can be
initialized, accessed and operated upon by ways of these
access functions. These functions are shared by all
instances of the class and common behavior among them
is thus insured.
Sub-classes can be derived from a parent class by
sharing data structures and operations. Inheritance is the
technique which allows sub-classes of a parent class to reuse
(inherit) the parent's functions. Using the objectoriented
paradigm, our aim is to define high-level classes
for each set of data common to the visualization field.
Associated with these definitions are display and
processing tools needed to operate on such sets of data.
3. Data Classes based on the Spatial
Domain
Gridded data are a common occurrence in scientific
computations. Data may come in regular, rectilinear,
curvilinear or arbitrary grids. As Butler et al. have
remarked [1], visualization mappings generally use and
produce sets of points in space, with associated data
values. Furthermore, common data extraction tools are
generally mappings from n-D space to (n-k)-D space and
most sets of data can be described by generic data
structures in such spaces.
To manipulate sets of points, several user-defined
types are used. We use a Point class for points in R 3 ,
with associated data values. The elementary linear segment
joining two Points is defined by the class LineCell.
Likewise, the class SurfaceCell is represented by subclasses
of the basic surface elements (triangle, quad, etc.),
and the class VolumeCell, with sub-classes of
elementary volume elements (tetrahedron, prism,
hexahedron, etc.). The C++ classes PointSet,
LineSet, SurfaceSet and VolumeSet are then used to
organize sets of elementary objects of the respective types.
A class hierarchy exists to combine cells of identical
topological dimension, and the sets only manipulate
pointers to the 1-D, 2-D or 3-D classes of cells. This
allows the combinations of cells of different sub-classes
often found in Finite Element Analysis (grids of mixed
elements) and the data processing is then achieved with a
look-up of the appropriate functions of each sub-class.
3.1. Data Fields
A general dataset will consist of points and associated
data values in the form of scalar, vector or tensor fields. A
Field class provides a conceptual definition which
encompasses the PointSet, LineSet, SurfaceSet and
VolumeSet sub-classes. An instance of class Field is a
set of data with a given number of points, elementary
cells, and field variables. Figure 1 shows part of our class
hierarchy, with some of the basic entities we have defined:
The following functions are provided with the
definition of Field. The sub-classes LineSet,
SurfaceSet and VolumeSet take advantage of method
inheritance and can re-use all of these:
• Evaluate the type of the cells and their
connectivity.
• Create a Postscript description file for hardcopy
presentation.

VolumeSets
Hexahedron Tetrahedron Prism
Fields
SurfaceSets
Triangle Quad
LineSets
Sets of ... Sets of ...
Sets of ...
• Display the set as an opaque volume, a wire
frame, a cloud of pseudo-colored points.
• Display the set as deformed geometry.
• Clip against geometric objects and display as
above.
• Clip against another instance of Field.
• Copy, scale, reshape and orient a data glyph at
selected points (See [6]).
• Derive gradient, curl, divergence or Laplacian of a
field variable.
• Clip the set based on data values or based on
spatial coordinates.
• Use the set of points as Point Locators (to
initiate particle tracing for example).
• Use the points' coordinates and/or their associated
data values.
The core of a data visualization process consists of
mappings of the dependent variables to graphical
primitives. These mappings are implemented generically
by the processing functions of each sub-class of Field.
When all visualization tools are designed to use and
output data of type Field (or its sub-types), functional
composition, whose goal is to combine visualization
primitives into a composite visualization technique, can
be readily implemented (see section 4). Instances of the
sub-classes of Field can either use the functions defined
above, or use specialized operations. For example, the
PointSet class has functions to store, access or modify
the coordinates and data values of points and to draw the
Figure 1: Class Hierarchy
Line-Segment
PointSets
points in several ways. The other sub-classes have more
specialized functions.
3.2. LineSets
A LineSet is defined as a set of variables of type
LineCell with associated field values representing a
general 3-D space curve or line. Encapsulated with the
definition of this data structure are a few specialized
operations, available to all instances of the class, such as:
• Display the polyline as a simple curve, a tube,
streamtube, or as a ribbon.
• Compute its length.
3.3. SurfaceSets
The class SurfaceSet represents the data structure of
a grid made of elementary surface elements of type
SurfaceCell. A SurfaceSet is defined as a set of
points with associated field values (a domain in 3-D
space), and an inter-element connectivity function (either
implicit as for regular grids, or explicit as for FEM
meshes) assembling points in the lattice. The set can be
formed of multiple disjoint surface patches. For example,
a 2-D Finite Difference grid is an instance of the class
SurfaceSet. We now list a few common operations for
such sets:
• Compute iso-contour lines for a selected scalar
field (LineSet objects).

• Compute streamline profiles based on a velocity
vector (LineSet objects).
• Decimate or optimize the "mesh" [4, 12, 14].
• Compute the grid's surface area.
• Combine LineSet objects into a SurfaceSet.
• Apply Texture Mapping techniques on the
surface to visualize a data component.
3.4. VolumeSets
The class VolumeSet characterizes the data structure
of a grid made of elementary volume elements like a
tetrahedral mesh for instance. Associated with the volumes
are operators such as:
• Compute iso-contour surfaces for a selected scalar
field (SurfaceSet objects).
• Compute arbitrary cross-sections (SurfaceSet
objects).
• Compute boundary surfaces (SurfaceSet
objects).
• Compute streamlines based on a velocity field
(LineSet objects).
• Compute its volume.
• Volume Render.
At this point, it is easy to recognize that each
visualization technique is defined as an operator for each
class. These functions can be applied on the data values
and are used to create and exchange typed data objects
between each other. Functional Composition derives from
this careful design of typed data. We focus next on the
mappings from VolumeSets, to SurfaceSets, to
LineSets which are compositions of techniques. The reuse
of all the display and processing functions defined for
these data types is at the base of Functional Composition.
Boundary-surface
s
Cross-sections
Iso-surfaces
VolumeSets
Streamlines
SurfaceSets
Iso-contour
Cell-Boundaries Lines
Streamlines
LineSets
Figure 2: Mappings between Field Objects
4. Composition of Field Operators
4.1. Functional Composition
In our system, all functions are defined as mappings from
instances of Fields to other instances of Fields (or their
sub-classes). Figure 2 gives examples of visualization
techniques as mappings between variables of the
subclasses of Field .
Three-dimensional tools such as boundary surface-,
cross-section- and isosurface-extraction, which in current
systems are limited to creating displayable geometry, are
enhanced by creating objects of type Field amenable to
further enhancing and processing. Each instance of a
Field encapsulates an underlying geometry and some field
data. Thus, functional composition of data visualization
tools is made possible by inheriting all the display and
processing methods defined for the grid classes. Multiple
data extractions and geometry color mappings can be
serially composed, each taking a Field object and
producing another Field object.
An example of Functional Composition is to
consider a VolumeSet V with data fields f1 through fn.
An intermediate object S of type SurfaceSet is created
with all of V's data fields stored at its points. Likewise, L
of type LineSet, inherits the data values of S and all
SurfaceSet and LineSet operations remain available
for S and L. For example,
S = Cross-Section(V, F(x, y, z))
/* cross-section surface F(x,y,z) = 0 */
L = Iso-contour-Line(S, fj, nb, min, max)
/* isocontour lines fj(X) = min to fj(X) = max */
Or more succinctly, to highlight the functional
composition taking place:
L = Iso-contour-Line(
Cross-Section(V, F(x, y, z)), fj, nb, min, max)
Figure 3 gives the schematic diagram showing the
functional composition taking place in our example.
Field objects are shown in ellipsoids while operators are
shown in rectangles. We highlight the fact that each
instance of Field can be displayed in several ways or
passed downstream for further data extraction. Defining
and processing the output of all common data
visualization tools as Fields promotes their functional
composition and allow further data processing. This
object-oriented approach of designing high level objects
for input and output of data visualization tools improves
the fan-in and fan-out of processing modules both in the
conventional function-based approach and in the new dataflow
systems. Data and code reuse are highly favored. The
visualization process gains efficiency and practicality by

4D Data
Data
projection
VolumeSet
V
Cross-section
SurfaceSet
S
Iso-contour
Lines
LineSet
L
Bounding
Surface
Display
(Boundary Edges)
Display
(lines)
SurfaceSet
Display
(Bounding Box)
Data Flow
Display Flow
Composite Rendering
Figure 3: System view of Data Objects and Operations for the given example.
abstracting itself from hardware-oriented graphics
primitives. This approach becomes very useful for multivalued
field data sets where we can alternate between
different data field views, independently of the underlying
geometry and without re-processing of the objects.
Computational requirements are thus reduced, promoting
interactivity.
4.2. Fields as Domain Selectors
Functional Composition can be interpreted as a data
filter of the first operand by the first operator, followed by
the application of another technique on the result of the
selection. Thus, Field objects can provide support for a
data extraction operator while remaining partially visible.
Transparency may allow the user to see through an
object, but the accurate display of multiple objects with
various transparency indices is difficult. Field objects can
be used instead without being displayed. Most often, we
will re-use a SurfaceSet (such as a cross-section) as a
data filter. Drawing its outside line-boundary can help
visualize its extent in space, while it is re-used to apply
other data extraction techniques, such as contour line
drawing, or particle tracing confined to the plane. This
functional composition is of great help to the scientist
whose goal is to understand inter-relationships between
data fields. By providing a way to restrict the domain of
application of a data mapping to a sub-set of data of
interest, correlation between data fields is more easily
extracted and analyzed.
Field objects can also be used as input parameters
and their vertices can be used as point locators or seeds for
operations like particle tracing, or iso-surface
computations.
4.3. Other Operations on Fields
When Field instances are regarded as data selectors, it
can be appropriate to apply Boolean operations between
each instance. Union and intersection are very useful
operations that can lead to increased expressiveness.

Figure 4: Data Visualization around a highspeed
train
These operators can be defined for each sub-class of
Field, since the underlying data structure allows multiple
disjoint sub-sets of elementary cells. It is sometimes
necessary to completely remove a part obstructing the
view. Cut-aways, which remove features based on the
spatial coordinates of their vertices, have also been
considered. Since each instance of a Field encapsulates
an underlying geometry, we can perform Boolean
operations on their geometric structures, while conserving
all field data. The union of Fields is quite straightforward
but is only meaningful if the two objects merged carry the
same data information.
Other operations are also available between Fields.
For example, the tiling of a streamsurface (SurfaceSet)
with individual streamlines (LineSets). Ribbon and
streamsurfaces can be constructed thusly. This example
shows that we can reverse the natural order of
visualization processing shown in Figure 2 which
emphasizes going from a higher dimension to a lower
dimension. Of particular interest are grid definition
techniques which allow us to construct computational
grids by scaling, translating and revolving LineSets and
SurfaceSets.
5. Examples
Our library of Field classes and operations has been
implemented in C++ and runs on SGI workstations. The
graphic toolkit Inventor [13] has been used to perform all
display operations. Our examples show compositions of
data extraction capabilities and highlight the polymorphic
display options available to objects of our three most
important classes: VolumeSet , SurfaceSet and
LineSet.
Our first example in Figure 4 consists of a
VolumeSet object of 127,049 tetrahedral elements with
energy, density and velocity (Vx, Vy, Vz) stored at each
Figure 5: Visualization of Velocity Fields by
Polymorphic Rendering
grid point. The object under study is a high-speed train in
a flow field. We focus our attention on the front of the
vehicle and proceed by calculating an isovalue surface of
the x-component of velocity which shows up as a large
bulgy surface on the nose of the train. Cross sections
along the direction of travel and perpendicularly to the
direction of travel are also computed. The addition of the
SurfaceSets created is then used to show a composition
of several data encoding. The cross-sections, the isosurface
and the boundary surface are colored with the pressure field
(lower left image) and combined to restrict the input
domain for the computation of iso-energy lines, shown
with a different colormap encoding the variations of the
cross-velocity field. Note that the iso-surface actually
encodes four scalar variables simultaneously. A pseudocoloring
of the pressure is shown while the iso-lines
which are restricted to its surface are colored with a fourth
data mapping.
Our second example in Figure 5 shows polymorphic
rendering of particle traces computed by a VolumeSet
object. A cylinder in a transversal flow is studied in a
dataset of 246,725 volume cells. The volume object
computes its boundary surface on which we compute an
iso-pressure LineSet. Streamlines are also computed as
instances of LineSets. As such, they can be displayed
in a multitude of ways, without requiring any specialpurpose
coding. A rake of streamlines is computed and
displayed as simple pseudo-colored space curves; another
LineSet is displayed with velocity vector icons placed at
regular intervals, while another one is displayed as a tube
and one is displayed as a ribbon, allowing additional color
encodings (In this example, the streamline LineSets are
colored with Pressure). Mappings to pseudo-colored
geometric objects can be activated interactively on the
various LineSets since the data and the geometry are
encapsulated in the same data-structure.
The performance of our library of tools provides for
interactive inquiries. The train dataset was processed for
isosurface and cross-sections in a few seconds, including
triangulation and connectivity computations for more than

16,000 triangles. The surfaces were then joined, isocontoured
and shaded in a few more seconds on an SGI
Crimson. Similarly, the outside surface of the cylinder
dataset was extracted and iso-contoured at interactive speed.
We have a limited user interface which allows a mix of
keyboard inputs and direct manipulation via the Inventor
toolkit. Since all objects in the graphics scene contain
geometry and data values, computational queries can be
readily answered. Each common grid type has
"constructor" methods available to read in and store sets of
data points and no programming is required. However, the
end-user would need to add an additional member function
for the given data type for data encoded in a new format.
6. Discussion
Since our visualization algorithms rely on connected
components of elementary cells, their implementation
requires more work. For example, functions like Marching
Cubes iso-surface extraction [7] or iso-contour lines are
cell-based by nature and their output is traditionally cast
into sets of disjoint geometric entities. Here, to take
advantage of the grid data structures and guarantee surface
continuity and a consistent right-hand rule ordering of the
vertices, surfaces are more easily constructed as a moving
front intersecting the volume [15]. Likewise, iso-contour
lines must also comply to our design and be fully
connected lines, instead of the concatenation of line
segments at rendering times. (This is curve sequence
contouring versus grid sequence contouring [10]). Note
that by construction, streamlines offer naturally connected
paths and don't require new implementations. We reap an
important benefit from this redesign. Consider iso-level
data mappings. There is a well known ambiguity which
arises when a cell spans a saddle in the data (Two opposite
corners above and two below the threshold value in a 2-D
cell). This may lead to spurious holes or surface segments
at rendering times, unless additional efforts are invested to
carefully handle these cases (See [9] for a very good
survey). We have implemented these functions in an
advancing front fashion to avoid this ambiguity. An
active set of cells is maintained at the boundary of the
isosurface and edge and orientation information is passed
to the new candidate cells. Because isocontour lines or
isosurfaces are constructed incrementally, we obtain local
coherency in the numbering scheme of points and
elementary cells. When re-used, the Field objects are
more efficient and they do not require additional
connectivity mapping or renumbering.
7. Conclusion
The data extraction operations we show in our
examples are not new, but our contribution is to provide
the environment where they can be easily combined. We
defined elementary cells of various topological dimensions
endowed with display and data extraction operators. They
are assembled as Field objects to represent grids of data.
Field objects encapsulate data fields with an underlying
geometry and offer a rich functionality. They avoid the
strong type restrictions typical in other systems by
combining display and data manipulation operations.
They can be re-used at different stages of the visualization
process, thus increasing the fan-in and fan-out of
processing modules and enabling better composition of
data mappings. We could not achieve similar composite
visualization in the other systems we are familiar with,
because of their use of pure graphical primitives.
Our data objects can be joined or act as filters to
promote the serial composition of visualization
techniques. This composition helps understand interrelationships
between data fields and facilitates the
analysis of data correlation. Our system design opens new
ways to scientific exploration and provides an open-ended
architecture for implementing new visual representations
whose effectiveness should be evaluated by using
principles of visual perception [5].
The structured definitions also allow quantitative
analysis to take place. For example, in medical imaging,
doctors may want to compute a volume contained between
iso-radiation surfaces. In fluid flow, flux computations
can be computed through surfaces of interest. Because the
objects we manipulate are all piece-wise approximations
of well-behaved volumes, surfaces and lines, we can
estimate lengths, surface areas, volumes and other
numerical values.
We have limited our discussion to grids of linear
cells. We intend to expand our library of tools to handle
cells with curved boundaries, often used in FEA. We will
also focus our effort to data sets in 3D space + time.
Constructing streamsurfaces from streamlines or isovalue
surfaces from contour lines on successive cross-sections
also shows that visualization processes are not limited to
mappings to lower dimensions. We plan to research other
examples of mappings to higher dimensions.
Acknowledgments
We would like to thank Larry Gritz, Randy Rohrer
and Daria Bergen for their valuable help in editing our
manuscript, Larry Cannon for his encouragement, and the
Department of EE&CS at The George Washington
University for its financial support.
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Abstract
XmdvTool� Integrating Multiple Methods for Visualizing
Multivariate Data
Much of the attention in visualization research has
focussed on data rooted in physical phenomena� which
is generally limited to three or four dimensions. How�
ever� many sources of data do not share this dimen�
sional restriction. A critical problem in the analysis of
such data is providing researchers with tools to gain in�
sights into characteristics of the data� such as anoma�
lies and patterns. Several visualization methods have
been developed to address this problem� and each has
its strengths and weaknesses. This paper describes a
system named XmdvTool which integrates several of
the most common methods for projecting multivari�
ate data onto a two�dimensional screen. This inte�
gration allows users to explore their data in a variety
of formats with ease. A view enhancement mecha�
nism called an N�dimensional brush is also described.
The brush allows users to gain insights into spatial
relationships over N dimensions by highlighting data
which falls within a user�speci�ed subspace.
1 Introduction
The major objectives of data analysis are to sum�
marize and interpret a data set� describing the con�
tents and exposing important features �6�. Visualiza�
tion can play an important role in each of these objec�
tives� both in qualitative evaluation of the data and
in conjunction with focussed quantitative analysis. A
given visualization technique is generally applicable to
data of certain characteristics. This paper describes
a system which has been developed for the display of
multivariate data.
Multivariate data can be de�ned as a set of enti�
ties E� where the i th element e i consists of a vector
with n observations� �x i1� x i2� ���� x in�. Each observa�
tion �variable� may be independent of or interdepen�
dent with one or more of the other observations. Vari�
Matthew O. Ward
Computer Science Department
Worcester Polytechnic Institute
Worcester� MA 01609
ables may be discrete or continuous in nature� or take
on symbolic �nominal� values. Variables also have a
scale associated with them� where scales are de�ned
according to the existence or lack of an ordering rela�
tionship� a distance �interval� metric� and an absolute
zero �origin�.
When visualizing multivariate data� each variable
may map to some graphical entity or attribute. In
doing so� the type �discrete� continuous� nominal� or
scale may be changed to facilitate display. In such sit�
uations� care must be taken� as a graphical variable
with a perceived characteristic �type or scale� which
is mapped to a data variable with a di�erent charac�
teristic can lead to misinterpretation.
Many criteria can be used to gauge the e�ective�
ness of a visualization technique for multivariate data.
Some of these are directly measurable� such as the
number of variables or data points which can be dis�
played. Others require subjective evaluation and are
thus di�cult to quantify. The list below summarizes
some of these criteria. In our studies of the vari�
ous projection techniques we are examining these and
other issues.
Constraints on dimensions� all of the projection
techniques surveyed degrade in usefulness when
the number of dimensions or variables exceeds a
certain size.
Constraints on data set size� each data projec�
tion method allocates a certain amount of screen
space for each data sample. As screen space is
�nite� there exists a limit for e�ective visualiza�
tion.
The e�ect of data distribution� sparse data sets
may lead to poor screen utilization� while highly
clustered data may make it di�cult to identify
individual samples.
Occlusion� in many instances� di�erent data points

will map to the same location on the screen. It
is important that the viewer be aware of these
overlaps and perhaps have a strategy to obtain a
view which avoids a given overlap.
Perceptibility� the goal of visualizing data is to try
to understand the structure of the data or de�
tect some data characteristics� such as anomalies�
extrema� and patterns. These features may be
more readily apparent in some projection tech�
niques than others.
User interactions� visualization is incomplete with�
out interaction. Each projection technique has a
logical set of interactive capabilities for view mod�
i�cation and enhancement.
Interpretation guides� users need reference points
such as keys� labels� and grids to help interpret
the data and determine its context.
Use of color� color can be used to convey one or
more variables of the data or to help highlight
or deemphasize subsets of the data. As color per�
ception varies both contextually and between in�
dividuals� it should be used with care.
3�Dimensional cues� other cues commonly found in
3�D graphics� such as shading� translucency� and
motion� can be used to reduce the overall dimen�
sionality� although often at some cost in inter�
pretability.
2 N�Dimensional Data Visualization
Methods
Many techniques for projecting N�dimensional data
onto two dimensions have been proposed and explored
over the years. This section presents an overview of
four classes of techniques� and describes their imple�
mentation within XmdvTool.
2.1 Scatterplots
Scatterplots are one of the oldest and most com�
monly used methods to project high dimensional data
to 2�dimensions. In this method� N � �N � 1��2 pair�
wise parallel projections are generated� each giving the
viewer a general impression regarding relationships
within the data between pairs of dimensions. The
projections are generally arranged in a grid structure
to help the user remember the dimensions associated
with each projection. Many variations on the scatter�
plot have been developed to increase the information
content of the image as well as provide tools to facili�
tate data exploration. Some of these include rotating
the data cloud �12�� using di�erent symbols to distin�
guish classes of data and occurrences of overlapping
points� and using color or shading to provide a third
dimension within each projection.
The procedure for generating scatterplots within
XmdvTool is quite straightforward. The display win�
dow is divided into an N by N grid� and each data
point results in N 2 points being drawn� using only two
dimensions per view. Columns and rows in the grid
are labeled according to the dimension they represent.
Figure 1 presents a seven dimensional data set
using scatterplots. Note that plotting each dimen�
sion against itself along the diagonal provides dis�
tribution information on the individual dimensions.
The data set contains statistics regarding crime in
Detroit between 1961 and 1973� and consists of
13 data points. The data set was obtained via
anonymous ftp from unix.hensa.ac.uk in the directory
�pub�statlib�datasets. Some dimensions of the origi�
nal set have been eliminated to facilitate display us�
ing scatterplots. The dimensions and their ranges are
given in Table 1. Linear structures within several of
the projections indicate some correlation between the
two dimensions involved in the projections. Thus� for
example� there is a correlation between the number
of full�time police� the number of homicides� and the
number of government workers �with a corresponding
negative correlation in the percent of cleared homi�
cides�.
One major limitation of scatterplots is that they
are most e�ective with small numbers of dimensions�
as increasing the dimensionality results in decreasing
the screen space provided for each projection. Strate�
gies for addressing this limitation include using three
dimensions per plot or providing panning or zooming
mechanisms. Other limitations include being gener�
ally restricted to orthogonal views and di�culties in
discovering relationships which span more than two
dimensions. Advantages of scatterplots include ease
of interpretation and relative insensitivity to the size
of the data set.
2.2 Glyphs
The de�nition of a glyph covers a large number of
techniques which map data values to various geometric
and color attributes of graphical primitives or sym�
bols �10�. Some of the many glyph representations
proposed over the years include the following�

Dimension Minimum Maximum
Full�time police per 100�000 population 255. 400.
Unemployment rate 0. 12.
Number of manufacturing workers in thousands 450. 620.
Number of handgun licenses per 100�000 100. 1200.
Number of government workers in thousands 120. 250.
Percent homicides cleared by arrests 50. 100.
Number of homicides per 100�000 0. 60.
� Faces� where attributes such as location� shape�
and size of features such as eyes� mouth� and ears
are controlled by di�erent data dimensions �5�.
� Andrews glyphs� which map data to functions
�e.g. trigonometric� of N variables �1�.
� Stars or circle diagrams� where each glyph con�
sists of N lines emanating from a point at uni�
formly separated angles with lengths determined
by the values of each dimension� with the end�
points connected to form a polygon �13�.
� Stick �gure icons� where the length� orientation�
and color of N elements of a stick �gure are con�
trolled by the dimensional values �7�.
� Shape coding� where each data point is repre�
sented by a rectangle which has been decomposed
into N cells and the dimensional value controls the
color of each cell �4�.
In XmdvTool� we use the star glyph pattern �13�.
The user can choose between either uniformly spaced
glyphs or using two of the dimensions to determine
the location of the glyph within the window. Each
ray of the glyph has a minimum and maximum length�
determined either by the user �for glyphs with data�
driven locations� or by the size of the view area �for
uniformly spaced glyphs�. A key for interpreting the
dimensions is included in a separate window.
Figure 2 shows an example of glyphs in XmdvTool
using the same data set as in Figure 1. The evolu�
tion of the shape over time indicates both trends and
anomalies. For example� the clear protrusion in the
direction associated with cleared homicides �257 de�
grees� found in the earlier shapes evolves into a con�
cavity over time.
Glyph techniques are generally limited in the num�
ber of data elements which can be displayed simulta�
neously� as each may require a signi�cant amount of
screen space to be viewed. The density and size con�
straints of the elements� however� depend on the level
Table 1� Dimensions of the Detroit data set.
of perceptual accuracy required. Also� it can be di��
cult to compare glyphs which are separated in space�
although if data dimensions are not being used to de�
termine glyph locations� the glyphs can be sorted or
interactively clustered on the screen to help highlight
similarities and di�erences. Most of the glyph tech�
niques are fairly �exible as to the number of dimen�
sions which can be handled� though discriminability
may be a�ected for large values of N �greater than 20
or so�.
2.3 Parallel Coordinates
Parallel coordinates is a technique pioneered in the
1970�s which has been applied to a diverse set of mul�
tidimensional problems �8�. In this method� each di�
mension corresponds to an axis� and the N axes are
organized as uniformly spaced vertical lines. A data
element in N�dimensional space manifests itself as a
connected set of points� one on each axis. Points lying
on a common line or plane create readily perceived
structures in the image.
In generating the display of parallel coordinates in
XmdvTool� the view area is divided into N vertical
slices of equal width. At the center of each slice an
axis is drawn� along with a label at the top end. Data
points are generated as polylines across the N axes.
Figure 3 shows an example of the Parallel Coordi�
nates technique using the same data set as in Figure 1.
Clustering is evident among some of the lines� indicat�
ing a degree of correlation. For example� the X�shaped
structure between the axes for cleared cases and homi�
cides indicates an inverse correlation� and the nearly
parallel lines betwen the axes for manufacturing work�
ers and handgun licenses suggests a relatively constant
increase in the rate of handgun ownership as manufac�
turing jobs increase �some exceptions exist� however�.
The major limitation of the Parallel Coordinates
technique is that large data sets can cause di�culty in
interpretation� as each point generates a line� lots of
points can lead to rapid clutter. Also� relationships

etween adjacent dimensions are easier to perceive
than between non�adjacent dimensions. The number
of dimensions which can be visualized is fairly large�
limited by the horizontal resolution of the screen� al�
though as the axes get closer to each other it becomes
more di�cult to perceive structure or clusters.
2.4 Hierarchical Techniques
Several recent techniques have emerged which in�
volve projecting high dimensional data by embedding
dimensions within other dimensions. In the 1�D case
�11�� one starts by discretizing the ranges of each di�
mension and assigning an ordering to the dimensions
�dimensions are said to have unique �speeds��. A
background color is also associated with each speed.
The next step is to divide the screen into C0 vertical
strips� where C0 is the cardinality of the dimension
with the slowest speed. The strips are colored accord�
ing to that speed. Each of these strips is then divided
into C1 strips and colored accordingly. This is re�
peated until all dimensions have been embedded and
the data value associated with each cell can be plotted
on the vertical axis.
In 2�D �9�� an analogous technique called Dimen�
sional Stacking involves recursively embedding images
de�ned by a pair of dimensions within pixels of a
higher�level image. Unlike the previous system� how�
ever� data is not restricted to functions� thus mak�
ing this technique amenable to a wider range of data
types. In Worlds within Worlds �2�� each location in a
3�D space may in turn contain a 3�D space which the
user may investigate in a hierarchical fashion. The
most detailed level may contain surfaces� solids� or
point data.
XmdvTool requires three types of information to
project data using dimensional stacking. The �rst is
the cardinality �number of buckets� for each dimen�
sion. The range of values for each dimension is then
decomposed into that many equal sized subranges.
The second type of information needed is the order�
ing for the dimensions� from outer�most �slowest� to
inner�most �fastest�. Dimensions are assumed to al�
ternate in orientation. The last piece of information
used is the minimum size for the plotted data item
�the system will increase this value if the entire image
can �t within the view area�. Each data point then
maps into a unique bucket� which in turn maps to a
unique location in the resulting image. If the image
generated exceeds the size of the view area� scroll bars
are automatically generated to allow panning. A key
is provided in a separate window to help users under�
stand the order of embedding� and grid lines of varying
intensity provide assistance in interpreting transitions
between buckets at di�erent levels in the hierarchy.
The sparseness of the data set of Figure 1 makes
uncovering relationships di�cult using Dimensional
Stacking. Figure 4 shows a denser set consisting of 3�
D drill hole data with a fourth dimension representing
the ore grade found at the location �more than 8000
data points�. Longitude and latitude are mapped to
the outer dimensions� each with cardinality 10. Depth
and ore grade map to the inner dimensions �ore grade
is the vertical orientation�� with cardinality 10 and 5�
respectively. There is a clear region in which the ore
grade improves with depth� and other places where
digging had stopped prior to the ore grade falling sig�
ni�cantly. By adjusting the cardinalities and ranges
for the various dimensions� a more detailed view of
the data may be obtained �16�.
The hierarchical techniques are best suited for fairly
dense data sets and do rather poorly with sparse data.
This is due to the fact that each possible data point
is allocated a speci�c screen location �with overlaps
avoidable by careful discretization of dimensions�� and
as the dimension of the data increases� the screen space
needed expands rapidly. In contrast� the techniques
described earlier generally do well with sparse data
over high numbers of dimensions� though scatterplots
are constrained somewhat it the maximum manage�
able dimension. The major problem with hierarchical
methods is the di�culty in determining spatial rela�
tionships between points in non�adjacent dimensions.
Two points which in fact are quite close in N�space
may project to screen locations which are quite far
apart. This is somewhat alleviated by providing users
with the ability to rapidly change the nesting charac�
teristics and discretization of the dimensions.
3 N�Dimensional Brushing
Another useful capability of XmdvTool is N�
dimensional brushing �15�. Brushing is a process in
which a user can highlight� select� or delete a subset
of elements being graphically displayed by pointing at
the elements with a mouse or other suitable input de�
vice. In situations where multiple views of the data
are being shown simultaneously �e.g. scatterplots��
brushing is often associated with a process known as
Linking� in which brushing elements in one view af�
fects the same data in all other views. Brushing has
been employed as a method for assisting data analysis
for many years. One of the �rst brushing techniques
was applied to high dimensional scatterplots �3�. In
this system� the user speci�ed a rectangular region in

one of the 2�D scatterplot projections� and based on
the mode of operation� points in other views corre�
sponding to those falling within the brush were high�
lighted� deleted� or labeled. Brushing has also been
used to help users select data points for which they
desire further information. Smith et. al. �14� used
brushing of images generated by stick �gure icons to
obtain higher dimensional information through soni��
cation for the selected data points.
In XmdvTool� the notion of brushing has been
extended to permit brushes to have dimensionality
greater than two. The goal is to allow the user to gain
some understanding of spatial relationships in N�space
by highlighting all data points which fall within a user�
de�ned� relocatable subspace. N�D brushes have the
following characteristics�
Brush Shape� In XmdvTool� the shape of the brush
is that of an N�D hyperbox. Other generic shapes�
such as hyperellipses� will be added in the future�
as well as customized shapes� which can consist
of any connected arbitrary N�D subspace.
Brush Size� For generic shapes the user simply
needs to specify N brush dimensions. The mech�
anism used by XmdvTool to perform this� albeit
primitive� is to use N slider bars.
Brush Boundary� In XmdvTool� the boundary of a
brush is a step edge. Another possibility would
be a ramp� with many possibilities for the shape
of the ramp. Another interesting enhancement
could be achieved by coloring data points accord�
ing to the degree of brush coverage �where it falls
along the ramp�.
Brush Positioning� Brushes have a position which
the user must be able to easily and intuitively
control. In the general case� the user needs to
specify N values to uniquely position the brush.
This is done in XmdvTool via the same sliders
employed in size speci�cation.
Brush Motion� Although XmdvTool currently sup�
ports only manual brush motion� we hope to im�
plement several forms of brush path speci�cation
in the future.
Brush Display� N�dimensional space is usually
quite sparse� thus it is useful at times to display
the subspace covered by the brush on the data
display. The location can be indicated either by
the brush�s boundary or a shaded region showing
the area of coverage. In XmdvTool� brushes are
displayed as shaded blue�grey regions� with data
points which fall within the brush highlighted in
red.
Brush size and position are currently speci�ed in a
rather simplistic manner. The user selects the dimen�
sion to be adjusted� and then changes the brush size or
position via a slider. There are many opportunities for
allowing the user to directly manipulate the brush in
the display area� although each procedure would need
to be customized based on the projection method in
use. For example� the user could move or resize one
dimension of the brush by dragging the edge or cen�
ter of the brush along one of the axes of the Parallel
Coordinate display� or set the location of the brush
by selecting one of the glyphs. Direct manipulation of
the brush will be one of the features incorporated into
future releases of XmdvTool.
4 Summary and Conclusions
This paper has presented an overview of the �eld
of multivariate data visualization and has introduced
a software package� named XmdvTool� to help users
experiment with di�erent N�dimensional projection
techniques. Each of the techniques has its strengths
and weaknesses in regards to the types of data sets
for which it is most appropriate. The number of di�
mensions in the data as well as the range of values
and distribution within each dimension all play im�
portant roles. The goal of the user in examining the
data� whether it be for patterns� anomalies� or depen�
dencies� is also important in gauging the relative use�
fulness of a technique. One of the long�term goals of
this research e�ort is to create a benchmark for evalu�
ating multivariate data visualization tools using data
sets with a diversity of characteristics. The evalua�
tion criteria listed in Section 2 �as well as other crite�
ria� will be employed in the assessment process� along
with studying the performance of human subjects in
locating structure in data sets using di�erent tools.
Future development of XmdvTool will include
both generic view�enhancement techniques �additional
brush capabilities� panning� zooming� clipping� and
methods related to the speci�c projection techniques.
Some of these will include experimenting with di�er�
ent glyph structures� interactively changing the order
of dimensions� and dynamic control over the binning
for dimensional stacking. Some of these capabilities
have already been implemented into N�Land �16�� a
software package for exploring the capabilities of di�
mensional stacking developed by the author and his

colleagues� and thus should be relatively easy to in�
corporate.
XmdvTool is written in C using X11R5� Athena
Widgets� and the Widget Creation Library �Wcl�
2.5�� and will be made available on anonymous ftp
�wpi.wpi.edu� in the near future. Interested par�
ties should contact the author at matt�cs.wpi.edu or
check in the �contrib�Xstu� directory at the above
mentioned site for the �le named XmdvTool.tar.Z.
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Figure 1� The Detroit data using Scatterplots. Correlation between pairs of dimensions manifest themselves as
linear structures.
Figure 2� The Detroit data using the Star glyph representation. Key provides associations of dimensions with
line orientation.

Figure 3� The Detroit data using the Parallel Coordinates representation. Inverse correlations can be seen
between the number of government workers versus the percent of cleared homicides� as well as between the
percent of cleared homicides versus the number of homicides.
Figure 4� Four�dimensional data set using dimensional stacking. The data consists of ore grades with three spatial
dimensions. Inner dimensions show ore grade and depth. Outer dimensions show longitude and latitude. Highest
levels of ore grade are seen in the third to �fth sections horizontally and the sixth to eighth sections vertically.

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Case Study: Visualization and Data Analysis in Space and Atmospheric
Science
Abstract
In this paper we show how SAVS, a tool for visualization
and data analysis in space and atmospheric science,
can be used to quickly and easily address problems
that would previously have been far more laborious to
solve. Based on the popular AVS package, SAVS presents
the user with an environment tailored specifically
for the physical scientist. Thus there is minimal
"startup" time, and the scientist can immediately concentrate
on his science problem. The SAVS concept readily
generalizes to many other fields of science and engineering.
Overview
Space and atmospheric scientists have long been
faced with huge, continually growing repositories of data
from decades of spacecraft missions. The quantities of
information involved are so enormous that traditional
methods for data analysis and assimilation would quickly
be overwhelmed. This situation is made all the more
critical by unprecedented volumes of new data from
modern, high-bandwidth experiments and by the
widespread use of empirical and large-scale computational
methods for the synthesis of understanding across
datasets and discipline boundaries. A new approach is
clearly necessary.
We are attempting to address this problem with
SAVS (a Space and Atmospheric Visualization Science
system). SAVS is a unique "pushbutton" environment
A. Mankofsky, E.P. Szuszczewicz, and P. Blanchard
Science Applications International Corporation
McLean, Virginia
C. Goodrich, D. McNabb, and R. Kulkarni
University of Maryland
College Park, Maryland
D. Kamins
Advanced Visual Systems
Waltham, Massachusetts
that mimics the logical scientific process in data acquisition,
reduction, and analysis without requiring a detailed
understanding of the underlying software components; in
effect, SAVS gives the physical scientist access to highlevel
functionality, tailored to his particular needs, without
requiring him to become a computer scientist as well
[1]. SAVS provides (1) a customizable framework for
accessing a powerful set of visualization tools, based on
the popular AVS package, with hooks to PV-Wave and
Khoros; (2) a set of mathematical and statistical analysis
tools; (3) an extensible library of discipline-specific
functions and models (providing, for example, satellite
tracks and ionospheric parameters); and (4) capabilities
for local and remote database access, including search,
browse, and acquire functions.
We begin this paper with a brief description of the
SAVS architecture, its components, and its implementation;
the focus is on the integration of visualization with
mathematical and statistical models, advanced database
techniques, and user interfaces for enabling enhanced
acquisition, analysis, comparison, and understanding of
both observational and computational data. Through the
use of examples taken from current research in space
science, we then demonstrate the applicability of the
SAVS concept, with emphasis on the benefits of using an
integrated visualization/data analysis tool. Finally, we
touch upon future directions for SAVS, paying particular
attention to the extension of the SAVS paradigm beyond
space and atmospheric science to other fields of science
and engineering.

Visualization and methods
The SAVS architecture recognizes that scientists and
mission design engineers work routinely with data,
theoretical models, and orbits (including orbital parameters,
payload configuration, instrument fields-of-view,
coordinated ground stations, etc.). At any given time
they may work exclusively with one item, or with combinations
of all three. The most likely scenario involves
an interactive, iterative combination of the three, filtered
through a mathematical interpreter and displayed through
a broad set of visualization tools.
SAVS is a "pushbutton" environment that mimics
the thought processes of the scientist in data acquisition,
reduction, and analysis. As such, SAVS has three entry
concourses for daily activities: data, models, and experiment
envelope. By selecting one of these entry points in
the SAVS front end, the scientist begins a procedure
utilizing SAVS-guided functionality to exercise an
unlimited number of loops in any given concourse or
across concourses. The scientist can treat and overlay
multiple model runs or datasets, and visualize the results
in any number of formats.
SAVS is based on the AVS visualization system, and
also makes use of numerous other modules and packages,
both custom-written and public domain, to enable its
distributed database access, physical modeling,
mathematical, and statistical functionality. It supports a
wide variety of standard data formats (HDF, CDF,
netCDF, etc.), and provides a well-documented set of
"hooks" so that users can easily include readers for other
formats. The key point, for purposes of this paper, is that
the familiar AVS user interface (most notably the
network editor and its associated module libraries) is
usually hidden from the average SAVS user. Instead, the
SAVS interface gives the user "pushbutton" entries into
the three concourses described above, providing instant
access to the wide variety of database, modeling,
analysis, and visualization functions supported by the
system. No knowledge of the methods, modules, and
networks that actually execute these functions is
required. Thus SAVS is immediately accessible and useful
to the scientist.
Examples and discussion
Mission planning
We demonstrate the interactivity and extensibility of
the SAVS concept with an application scenario that can
be considered generic to any mission planning exercise,
or indeed to any activity involving data reduction and
analysis that requires (1) a number of models; (2)
ground-based and spaceborne data; (3) in situ and remote
sensing techniques; (4) perspectives on fluxtube-coupled
domains; and (5) co-registration of satellite-borne and
ground-based diagnostics.
A typical mission scenario is presented in Figure 1.
This scenario has counterparts in several current NASA
missions (e.g., CRRES, TIMED, ISTP) requiring the
coordination of satellite passes with ground-based optical,
ionosonde, and radar diagnostics. The SAVS models
concourse was used to generate the object in Figure 1a
by overlaying (1) thermospheric winds (blue vectors) and
oxygen densities (global color-coded surface) at 350 km
using the WIND and MSIS library models; (2) auroral
oval boundaries from the Feldstein library model; (3)
altitude cut-planes of ionospheric densities from the IRI
library model, in the latitudinal and longitudinal planes
intersecting the coordinates of the Arecibo Observatory;
and (4) geomagnetic field lines (in white) from the IGRF
library model. The satellite trajectory was generated in
the experiment envelope concourse using the CADRE-3
library model.
With button and dial controls, the user can (1) adjust
the auroral oval for any magnetic activity index or
Universal Time; (2) change the heights of the thermospheric
wind and density calculations; (3) vary the activity
index A p , the solar F10.7 flux, and the day-of-year
input drivers for WIND and MSIS; (4) alter the year for
the IGRF; (5) tune the month and sunspot number for the
IRI; and (6) change the coordinates for the intersecting
cut-planes (with interest, for example, in high latitude
passes in conjunction with Sondrestrom radar operations
in Greenland). The cut-planes could also be in the
magnetic E-W and N-S directions instead of being
aligned with the geographic contours of latitude and longitude
at the Arecibo Observatory.
From the CADRE-3 orbit specifications the user can
develop stack plots of any number of relevant parameters
as a function of UT, latitude, longitude, MET, or LT.
These parameters include SZA, MLAT, MLONG, MLT,
L-shell value, point of conjugacy, spacecraft velocity
vector relative to the ambient magnetic field, height of
the terminator relative to the satellite track, magnetic
field values or values of its E-W or N-S components, and
so on.
The user can also call up a 3D-to-1D module to
interpolate model results along any segment of the satellite
track. Through the data concourse, he can then
compare the results with local or remote databases of
along-track measurements.
For co-registration with ground-based diagnostics,
Figure 1b presents a zoomed-in perspective on the
Arecibo site, including the projection of the groundbased
HF radar beam up to an altitude of 400 km. That
projection is developed and displayed with the SAVS
"field-of-view" module, controllable by the user to

epresent any ground-based remote sensing diagnostics.
The user need only input the coordinates of the ground
site, the azimuth and elevation of the instrument's lineof-sight,
and its beam width (or acceptance) half angles
in the N-S and E-W directions.
Inspection of Figure 1b shows that the satellite (the
black sphere) at the given UT in its trajectory in not on a
field line that connects to the region diagnosed by the
ground-based radar. With animation and pause controls,
the user can determine the exact time of the field line
coupling and begin a number of SAVS-controlled
operations. For example, the 3D-to-1D module can be
used to interpolate ionospheric and thermospheric model
densities, temperatures, and composition onto the
connecting field line, and the model data used as input to
calculate fluxtube-integrated conductivities from the
satellite to the region covered by the ground-based
diagnostics. The module could also be used to calculate
integrated densities or emissions along the field-of-view
of the ground-based sensor. The field-of-view module
can in turn be "attached" to the spacecraft to project and
visualize the field-of-view of an on-board remote sensing
device. In this case the animation and visualization
products could focus on the determination of the time of
intersection of the fields-of-view of the satellite and
ground-based sensors.
The scenarios are unlimited. Models can be tuned
and retuned, and results visualized in simple stack plots
of along-track correlations for comparisons with data.
Visualizations can include constant altitude surfaces of
any geophysical parameters; alternately, the user can
choose altitude cut-planes, isosurfaces, or isocontours.
The "proper" visualization product is only determined by
the perceptive and inquisitive scientific mind, with the
freedom of the "pushbutton" operation of the SAVS system
making the choices immediately available. One can
only imagine how difficult this process would be without
visualization tools (for example, requiring a mathematical
analysis to determine the intersection of a particular
flux tube with the field-of-view of a ground-based
sensor), or if the scientist was required to become an
AVS expert before he could analyze his mission.
Experimental analysis
As another example of the importance to the scientist
of easily-accessible visualization and analysis tools,
we consider a situation where the visualization products
are not representations of data and model runs. Rather,
they provide tests of model or algorithmic integrity and
time saving views of an experimental scenario that facilitate
determination of the shortest path in the process of
data reduction and analysis. This is illustrated in a
SAVS application to a rocket-borne chemical release
experiment in the NASA CRRES program that involved
in situ and ground-based systems and aspects of coupled
phenomena along geomagnetic field lines.
Figure 2a shows a rocket trajectory along with an
array of magnetic field vectors determined by the IGRF
library model. The obvious discontinuity in the magnetic
field representation immediately revealed a subtle
problem in an interpolation module that might have gone
undetected without this simple visual aid.
Figure 2b illustrates the application of SAVS
visualization tools to help understand the actual
configuration of the experiment. In the simplest
description of the experiment scenario, a Ba/Li mixture
in a sealed canister was separated from the rocket's diagnostics
payload, and the vaporized gases were released
on the upleg portion of the trajectory at an altitude near
180 km. The colored disk in the panel is a color-coded
model depiction of the cloud's ion density in the plane
perpendicular to the cloud's bulk velocity at a very early
phase of the expansion process. The red line is the suborbital
trajectory, and the long and short vectors are the
magnetic field and payload velocities at the point of the
release. The cone-shaped object is the projection of a
ground-based HF heater beam intended to intercept the
chemical cloud at its ionospheric altitude, excite the
electrons, and induce enhanced optical emissions and
plasma instability processes.
The initial process of understanding the actual configuration
of the experiment involved the following
questions: (1) What were the magnitude and directions of
the cloud's bulk velocity relative to the geomagnetic
field? (2) How well was the chemical release coupled
along the magnetic field to the diagnostic payload? (3)
How effectively did the heater beam intercept the cloud?
(4) Was it possible to observe expanding cloud ions on
the downleg portion of the trajectory?
The first three questions were answered using the
rotation and zooming tools in SAVS, with some of the
image products presented in Figures 2c and 2d. The last
question was answered with yet another rotation that
projected an image perpendicular to the plane of the
trajectory, with the downleg portion of the trajectory in
the foreground. That image, presented in Figure 2c,
shows that the magnetic field at the release site did
indeed intersect the downleg trajectory. A simple hand
scaling of altitude and range showed that the flux tube
passing through the release missed the downleg
trajectory by 13 km, however, a sufficiently great
distance to minimize interest in searching the data for
possible late-time ion signatures of the expansion process
on the downleg trajectory.
The entire exercise of answering the questions listed
above required less than five minutes with SAVS.
Without the visualization tools, the answers would have

equired the development of a number of mathematical
algorithms and a sequence of stack plots. That exercise
could have taken several days. Once again, we see that
visualization is a time saving and insightful element in
the process of data reduction and analysis.
The future
The SAVS concept is designed for broad appeal and
functionality, since its architecture accommodates a general
scientific approach to data access, manipulation,
analysis, and visualization. Every scientific discipline
deals with data, either in scalar, vector, or image format.
Every discipline needs to develop, test, and apply empirical
and first principle model descriptions of cause-effect
relationships. Every science needs to deal with the spatial
and temporal coordinates of datasets (generally
represented by a satellite's orbital elements, payload
configuration, and instrument fields-of-view in a space
science application). And finally, every scientific
discipline requires a suite of mathematical and statistical
tools to handle data and prepare it for visualization. This
is the generic system capability of SAVS, making it
applicable to virtually any scientific discipline.
For a system such as SAVS to reach its full potential
there is the intrinsic need for scientists to open their
minds to the broad spectrum of capabilities embedded in
the visualization tools available today. Unfortunately,
many scientists have been conditioned to regard data and
model outputs as entities with inviolate properties that
dictate the use of traditional analysis techniques. This
attitude must be abandoned for the full potential of visualization
to be realized. Visualization is a rapidly developing
field that holds great promise. But like any
resource, it must be effectively mined and made accessible
to its potential constituency.
Acknowledgment
This work was supported by the NASA Applied
Information Systems Research Program (AISRP) under
contract NAS5-32337.
References
[1] Szuszczewicz, E.P., A. Mankofsky, P. Blanchard, C.
Goodrich, D. McNabb, and D. Kamins, "SAVS: A Space and
Atmospheric Visualization Science System," in Visualization
Techniques in Space and Atmospheric Sciences, E.P.
Szuszczewicz and J. Bredekamp, eds., NASA Headquarters
Publication, 1993 (and references therein).

A Case Study on Visualization for Boundary Value Problems
G�abor Domokos Randy Pa�enroth
Technical University of Budapest University of Maryland
Department of Strength of Materials Advanced Visualization Laboratory
H�1521 Hungary College Park� MD� 20742
tel� 011�36�1�1813798 tel� 301�405�4865
fax� 011�36�1�1813798 fax� 301�314�9363
e�mail� domokos�botond.me.bme.hu e�mail� redrod�avl.umd.edu
Abstract
In this paper we present a method� and a software
based on this method� making highly inter�active visu�
alization possible for computational results on nonlin�
ear BVPs associated with ODEs. The program PCR
relies partly on computer graphics tools and partly on
real�time computations� the combination of which not
only helps the understanding of complex problems� it
also permits the reduction of stored data by orders of
magnitude. The method has been implemented on PCs
�running on DOS� and on the Application Visualiza�
tion System �AVS� for UNIX machines� this paper
provides a brief introduction to the latter version be�
sides describing the mathematical background of the
method.
1 Introduction
This paper is intended to describe a method� and
a software based on this method� which may be very
helpful when presenting� storing or trying to under�
stand computational results on nonlinear boundary�
value problems �BVPs� associated with ordinary dif�
ferential equations �ODEs�. Such BVPs occur in var�
ious �elds of applied mathematics� in mechanics� in
particular in rod theory and also in structural engi�
neering� also in robotics� �nding optimal paths for a
device. There are standard codes available �for exam�
ple AUTO�Doedel 1986�� solving such BVPs� however�
the problem is often that the huge data �les created
by such a software are hard to store and even much
harder to understand. The best tool to visualize such
data are bifurcation diagrams which serve as �charts��
interpreted in some parameter space. Such bifurcation
diagrams can be extremely complicated� and relating
them to the physical behaviour of the system is a very
hard task. Our method is based on the construction
of a very special bifurcation diagram� which enables
the user to identify instantaneously the physical con�
�gurations corresponding to the points of the diagram.
In our software PCR this is realized simply by click�
ing with the mouse on the desired point� in the adja�
cent window the corresponding physical con�guration
would show up immediately. This kind of special di�
agram also helps to shrink the size of the stored data
radically� in some cases by several orders of magnitude.
We will sketch the history of our research program
in section 2� then outline brie�y the underlying deep�
but well�known and non�technical ideas from the the�
ory of ODEs and dynamical systems in Section 3.
Athough these ideas might seem simple� it is non�
trivial to apply them to particular problems and even
less so to build a code which realizes them in an inter�
active way. The code PCR has been implemented
on Personal Computers running DOS� using Microsoft
QuickBasic 4.5 and on UNIX workstations� using the
Application Visualization System �AVS��1992�.
2 Motivation and background
This research was motivated by a project of Li and
Maddocks �1994�� They constructed a mathematical
model for DNA molecules �a homogeneous� elastic ring
with circular cross section� and did extensive analyt�
ical and numerical studies on this model. The nu�
merical investigations were carried out with AUTO
�Doedel� 1986�� a continuation code for BVPs associ�
ated with ODEs. The data produced by AUTO ex�
ceeded 1.5 Gbytes� also� the produced bifurcation di�
agrams were extremely di�cult to relate to the phys�
ical behavior of the ring. There were several conjec�
tures related to the computation results� however� it
was nearly impossible to check them �visually�� due
to the large amount of stored data and also because
points on the bifurcation diagram could not be iden�
ti�ed with phyisical con�gurations. At this stage we
started to build a visualization tool which would per�

mit to reduce the size of the stored data and also link
the bifurcation diagram to the physical con�gurations
in an inter�active way. Our goal was to build a gen�
eral visualization tool� adaptable to any BVP associ�
ated with ODEs. We used ideas published earlier by
Domokos �1994�. As a result� the package PCR was
produced� running both on Personal Computers and
also on UNIX workstations. PCR helped to reduce
the data size by a factor over 1000� it proved to be
also very helpful in analyzing the dataset. We will
indicate the application of PCR to the problem of Li
and Maddocks �1994� in section 5.
3 Mathematical background� formula�
tion of results
Results in bifurcation problems are usually pre�
sented in the form of bifurcation diagrams. We are
not aware of any general de�nition for �bifurcation di�
agram�� in fact� depending on what purpose we have
in mind� we may design rather di�erent kinds of bifur�
cation diagrams. As examples we mention the widely
applied method of using Fourier coe�cients as coor�
dinates. Our present goal when designing bifurcation
diagrams is threefold�
1.� To create diagrams which are topologically cor�
rect� i.e. one point of the diagram corresponds to one
solution of the BVP and vice versa� also that the map�
ping is continuous.
2.� To create diagrams from which the physical con�
�gurations of the BVP can be easily reconstructed.
3.� To minimize the amount of stored data.
All three of the above criteria are met by the set of
coordinates ��global coordinates�� to be described be�
low.
Global coordinates can be created for explicit ODEs
satisfying the Lifschitz�condition� i.e. for ODEs which
meet the requirements of the Uniqueness Theorem
�Bieberbach� 1923�. Nearly all applications in mechan�
ics are �well�behaved� and fall into this category.� For
a rare exceptions see �Domokos 1993�.� For such well�
behaved ODEs a semi�in�nite trajectory is completely
and uniquely determined by the initial conditions. For
example� consider a linearly elastic cantilever beam�
clamped at the left end and free at the right end sub�
jected to the compressive force P �Fig.1�.
The ODE describing the shape of the beam in terms
of the slope � as a function of the arclength s was �rst
described by Euler�
� 00 � P sin��� � 0� �1�
The trajectories of this equation are uniquely de�
termined by the three scalars ��0�� � 0 �0� and P �the
s=0
deformed shape
α( s)
s=1
trivial shape
Figure 1� The cantilever beam
P
α ’(0)
Figure 2� Segment of the global bifurcation diagram
for the cantilever beam.
former being �true� initial conditions� the latter one a
parameter�. However� not all trajectories are of inter�
est for us� only those which meet the boundary condi�
tions
a� ��0� � 0
b� � 0 �1� � 0
P
�2�
Condition �2�a� eliminates one of the �variable� ini�
tial conditions as a constant� so all trajectories which
might meet the boundary conditions can be uniquely
represented in the �� 0 �0�� P � plane. This plane has
property 1�� i.e. one point of the plane corresponds to
one trajectory and vice versa. The scalars � 0 �0� and
P are the global coordinates for this BVP� the plane
�space� spanned by them will be called the global rep�
resentation space of the BVP. A �nite segment of the
bifurcation diagram for the BVP �1�2� is illustrated in
Fig.2 in the global representation space.
Based on the example of the cantilever we can now
de�ne the global coordinates of a two�point BVP as
those �Cauchy�type� initial conditions and parameters
which are not �xed by the boundary conditions at the
origin. The global representation space of the BVP
is spanned by the global coordinates. �Domokos� 1994�
The bifurcation diagram in the global representation
space not only has the advantage that it is topologi�
cally correct �Property 1��� it also makes the recon�

M 0
M 0
"PARAMETER CURVE"
CORRESPONDING
PHYSICAL SHAPE
P 0
P
POINT IDENTIFIED
BY GRAPHICS TOOL
P 0
"REAL CURVE"
α’(0)
Figure 3� The double�window system of PCR
struction of physical shapes extremely simple �Prop�
erty 2��. Take any point from the bifurcation diagram
and add the remaining constant initial condition �in
out cantilever example� ��0� � 0� to obtain a com�
plete set of initial conditions. Running an IVP solver
with these conditions regenerates the physical shape
instantly . This feature of the global bifurcation dia�
gram dictates the basic idea of PCR� Have constantly
two windows open on the screen. One window shows
the bifurcation diagram �or a projection of it� and by
some device we can locate points on this diagram. The
other window instantly shows the corresponding phys�
ical shape. Such a con�guration is shown in Fig.3�
This combination of graphics with real�time com�
puting is the key idea of PCR.
If the dimension of the global representation space
is more than 2 then we seemingly loose the topological
correctness since projection to the 2D s creen may
result in spurious intersection points. However� such
points may be eliminated� as pointed out in the next
section. We remark that the bifurcation diagram in
the global representation space requires minimal data
storage� thus has property 3.
4 Software implementation
The software PCR was developed based on the
above described method. PCR has two implementa�
tions� a PC�DOS version� requiring 386DX�33 proces�
sor and VGA graphic screen �or up�. A more sophisti�
cated version is running on the Application Visualiza�
tion System �AVS� on UNIX workstations� this latter
version o�ers better graphics and more �exibility. In
this section we describe the main features character�
istic for both versions.
Our software is called PCR� which stands for�
P arameter � Curve � Real� The input for the soft�
ware is an n�m matrix� n �number of rows� denoting
the number of data points in the global representation
space� m �number of columns� denoting the dimen�
sions of this space. As described in section 2� each
of the rows contains the set of initial values which
are not set constant by the boundary�conditions at
the origin. The integers m and n can be arbitrary.
PCR also needs an IVP solver routine for the speci�c
BVP� there is an empty �slot� in the AVS network
for that routine. After reading in the data �le� PCR
opens up two graphic windows�according to the setup
in Figure 2. A point on the parameter curve �bifur�
cation diagram� can be selected by clicking with the
mouse on the point. Given an n dimensional bifur�
cation diagram� PCR allows the user to choose any 3
dimensions and display these. In higher dimensional
diagrams spurious intersection points may arise. PCR
o�ers several tools to get rid of these points� In 4D
the user may de�ne a �nonlinear� color map according
to the 4th dimension. The user can also de�ne a 2D
subspace of the m�dimensional global representation
space in which a rotation can be performed� removing
the spurious intersection points.
5 Summary
In this paper we presented the theoretical basis and
description of the software PCR� intended for inter�
active visualization of results on BVPs. This usage
of the �global coordinates� de�ned in section 2� of�
fers several advantages� It minimizes the size of the
stored data� it provides a topologically correct bifur�
cation diagram and makes the real�time reproduction
of the physical con�gurations possible. PCR o�ers a
variety of inter�active tools to deal with the dataset
in the global representation space. Its main feature is
the simultaneous viewing of the bifurcation diagram
and of the physical con�guration corresponding to a
user�de�ned point on this diagram. PCR also o�ers
several tools to understand higher�dimensional bifur�
cation diagrams� including colorization and rotations
in the higher dimensional space.
The study of bifurcation diagrams with a tool like
PCR might very often help revealing otherwise hidden
features� such as connectivity of branches in the bifur�
cation diagram� or� realtionship between symmetries
in the physical� con�guration space and the global rep�
resentation space� leading to deeper understanding of

the problem.
In Fig. 5 we give an interesting example of appli�
cation� torsional buckling of a uniform� circular rod.
PCR was running on data obtained by Li and Mad�
docks �1994�. The global representation space is 4 di�
mensional in this problem and is spanned by the initial
moments M1� M2� M3 and the force F . Fig 4 shows the
M1� M2� M3 projection of this diagram and one phys�
ical con�guration corresponding to the point identi�
�ed on the bifurcation diagram. Without explaining
it further� we would like to point out an interesting
relationship between the bifurcation diagram and the
physical con�guration. The latter one is in unstressed
state a circular ring� which has a continuous rotation
symmetry group �called SO�2��. On the other hand�
the bifurcation diagram reveals an 8th order discrete
symmetry group generated by the three re�ections to
the coordinate planes �even apparent from this single
picture�. How this two groups relate can be explored
if one is able to investigate the phyisical con�gurations
corresponding to points which are mapped into each
other by this discrete group. �This is called a group
orbit.� PCR is optimally suited for such an investi�
gation and� in fact� helped to explain the relationship
between the two mentioned symmetry groups. We also
remark that in this problem PCR helped to reduce the
size of the data �le by a factor of 100. We feel that
problems of this complexity might be very hard to un�
derstand without a tool like PCR.
Acknowledgement
The authors are happy to thank Professor John H.
Maddocks for continuous encouragement and support
during their work. This research was supported by the
Dr Imre Kor�anyi Fellowship and the Hungarian Na�
tional Science Foundation grant No. F7690 �G.D.� and
the ASSERT award from the Air Force O�ce of Scien�
ti�c Research �R.C.P.�. The software PCR has been
developed on hardware supplied from Digital Equip�
ment Corporation under a grant from the Scienti�c
Innovators Program.
References
AVS� Inc. �1992�� AVS User�s Guide Waltham�
Massechusetts
Bieberbach� L. �1923�� Di�erentialgleichungen
Springer� Berlin
Doedel� E. �1986�� AUTO 86 user manual Cal�
tech�Dept. Applied Mathematics� Pasadena� Cal�
ifornia
Domokos� G.� �1993�� Can strings
buckle� AMD �Vol 167 �Recent development in
stability� vibration and control of structural sys�
tems� ASME 1993� Applied Mechanics Division�
ISBN 0�7918�1146�8� pp 167�174
Domokos� G.� �1994�� Global Description of Elas�
tic Bars ZAMM 74 �4� T289�T291
Li� Y and Maddocks� J.H. �1994�� On
The Computation of Spatial Equili bria of Elastic
Rods Including E�ects of Self�Contact submit�
ted for publication
Figure 4� Bifuircation diagram and one physical con�
�guration of ring under torsion

Case Study: Severe Rainfall Events in Northwestern Peru
(Visualization of Scattered Meteorological Data)
Lloyd A. Treinish
IBM Thomas J. Watson Research Center, Yorktown Heights, NY
lloydt@watson.ibm.com
Abstract
The ordinarily arid climate of coastal Peru is disturbed
every few years by a phenomenon called El Niño, characterized
by a warming in the Pacific Ocean. Severe rainstorms
are one of the consequences of El Niño, which
cause great damage. An examination of daily data from
66 rainfall stations in the Chiura-Piura region of northwestern
Peru from late 1982 through mid-1983 (associated
with an El Niño episode) yields information on the mesoscale
structure of these storms. These observational
data are typical of a class that are scattered at irregular locations
in two dimensions. The use of continuous realization
techniques for qualitative visualization (e.g., surface
deformation or contouring) requires an intermediate
step to define a topological relationship between the locations
of data to form a mesh structure. Several common
methods are considered, and the results of their application
to the study of the rainfall events are analyzed.
Introduction
The climate of coastal Peru and southwestern Ecuador
is mainly controlled by the Humboldt current, a cold
ocean current which travels northward along the coastline
of Chile and Peru before dispersing near the equator. The
current helps cause dry conditions to persist continuously
along the Peruvian littoral, making the land strip between
the Andes and the Pacific Ocean one of the most arid
deserts in the world. Every few years this condition is disturbed
by a phenomenon called El Niño, characterized by
an ocean warming which appears off the coastlines of
northwestern Peru and southwestern Ecuador. This warming
modifies the Humboldt current destroying the persistent
high pressure zone normally induced by the
Humboldt on the west side of the Andes, which in turn
generates major changes in the local meteorology and
climate [13]. Excessive and severe rainstorms are the
most disastrous consequences of El Niño, and such storms
can cause great damage to human life, property, crops, and
animal life. The rainfall from such episodes causes the
flooding of existing rivers, huaycos (mudslides), and the
sudden creation of new rivers and lakes.
The 1982-1983 El Niño has received wide attention
for its severity [10]. In Peru alone, it was responsible for
much loss of life, damage affecting over 80% of the
highway system, railroad washouts, and material loss estimated
in the billions of dollars. The heating of the ocean
off the Peruvian coast during El Niño periods has also
caused the loss of much marine life. For example, the El
Niño of 1972 virtually destroyed the Peruvian anchovy
fishing industry, which at that time represented a significant
percentage of the world's protein supply with a catch
of about 12 million tons per year [11]. Such destruction
emphasizes the need to better understand the meteorological
forces unleashed by this powerful ocean-air interaction.
Goldberg et al [6] have investigated the mesoscale
structure of severe rainfall events during the 1982-1983
period by examining daily data from 66 rainfall stations in
the Chiura-Piura region of northwestern Peru. Figure 1
shows the location of this region, which was selected because
it was most severely affected by the 1982-1983 El
Niño and because the data were highly reliable and complete.
Figure 1. Location of Peruvian Rainfall Stations.
These data support the study of rainfall characteristics over
this localized region during El Niño and non-El Niño periods,
as a function of elevation, geographic location, and
time of year.
Treatment of scattered data
The data from the rainfall stations are typical of observational
data that are scattered at irregular locations in
two or three dimensions (i.e., data with no notion of connectivity
or topology). Figure 2 is representative of a
straightforward discrete realization of such data as a scatter
plot to show the spatial distibution. Figure 3 illustrates
the temporal distribution for a single station.

Figure 2. Spatial Distribution of Peruvian Rainfall
Measurements on January 26, 1983.
Although such techniques preserve the fidelity of the
data, they fail to impart qualitative information about the
spatial characteristics of the measurements or the phenomena
of which they represent discrete samples. Thus, the
application of continuous realization techniques (e.g., surface
deformation or contouring for two-dimensional data,
volume rendering or surface extraction for three-dimensional
data) is necessary. An intermediate step of defining
a topological relationship between the locations of data to
form a mesh structure is required. Conventional continuous
realization techniques can then be applied to such a
mesh. There is a long history of mathematical methods
to create such meshes. Each method does change the data
and their artifacts must be understood because they will
carry through to the actual visualization. This discussion
is only meant as a very brief introduction to the topic.
Nielson [8] summarizes many of the methods in use today
and their relative advantages and disadvantages.
Figure 3. Time History Plot of Rainfall Measurements at Chulucanas, Peru with January 26, 1983
Highlighted.
The simplest and quickest approach is to create a regular
grid from the point data by nearest neighbor meshing -find
the nearest point to each cell in the resultant grid and
assign that cell the point's value as illustrated in Figure 4.
Such a technique is valuable because it preserves the original
data values and distribution of a grid after a coordinate
transformation may have taken place on a collection of
points. Although computationally inexpensive, the results
may not be very suitable for qualitative display because of
the preservation of the discrete spatial structure.
Ungridded
Points
Gridded
Figure 4. Nearest Neighbor Gridding.

An alternate approach that preserves the original data
values involves imposing an unstructured grid dependent on
the distribution of the scattered points. In two dimensions,
this would be a method for triangulating a set of scattered
points in a plane [2]. This technique first requires the
Voronoi tesselation of the plane with a polygonal tile surrounding
each of the scattered points. These tiles are such
that the locus of all points within a particular tile are closer
to the scattered point associated with that tile than they are
to any other points in the set. A triangulation can then be
constructed which is the dual of the Voronoi tesselation
(i.e., connecting a line between every pair of points whose
tiles share edges). This is known as Delauney triangulation
and is illustrated in Figure 5 as applied to the rainfall
stations.
Figure 5. Delauney Triangulation of Rainfall
Stations.
For a relatively random distribution of a small number
of points such as these rainfall data, the application of continuous
realization techniques to the triangulated mesh does
not yield useful qualitative results. Consider Figure 6, in
which the mesh from Figure 5 is pseudo-colored by
amount of rainfall. The rendering process applies bilinear
interpolation to the value at each node to determine the
color of each pixel in the image. Although the original
data are preserved, the sparseness of the points results in a
pseudo-colored distribution that is difficult to interpret.
A potentially more appropriate method, and certainly
one that is more accurate than nearest neighbor meshing,
uses weighted averaging as illustrated in Figure 7. For any
given cell in a grid, the weighted average of the n nearest
values in the original data distribution spatially nearest to
that cell is chosen. A weighting factor, w i = f(d i), where d i
is the distance between the cell and the ith (i = 1 , …, m)
point in the original distribution, is applied to each of the
n values. Figure 7 illustrates the case where n = 3. A
common weight is w = d -2. These are variants of Shepard's
method [15]. For example, Renka [14] modified this approach
with local adaptive surface fitting. Collectively,
these methods are typically O[nlog(n)] in cost.
Intermediate in quality and computational expense would be
using linear instead of weighted averaging.
Ungridded
Points
Interpolator
Gridded
Figure 7. Weighted Average Gridding
All methods in the aforementioned class do introduce
aliasing or smoothing of the data to achieve a gridded structure.
The form of the interpolation may also impart artifacts
on the results depending on the relative spatial variability
of the original data vs. how close the interpolant
function may be able to model that structure. Given a goal
of qualitative visualization, such artifacts may be acceptable.
Figure 8 shows a regular mesh with spacing of 0.04
degree (of latitude and longitude), onto which the rainfall
data for January 26, 1983 has been gridded using d-2 weighting for n = 5. The mesh and the data locations have
been similarly pseudo-colored. There is good, but NOT
perfect correspondence between the original data values and
that of the interpolated grid, sufficient for qualitative realization.
From this grid isocontour lines of constant rainfall
every 25 mm and a pseudo-colored image are created as
shown in Figures 9 and 10, respectively.
Figure 9. Isocontour Lines of Rainfall from
Weighted Average Gridding of Stations.
Implementation
The techniques described herein have been developed
with the IBM Visualization Data Explorer (DX), a generalpurpose
software package for scientific data visualization
and analysis. It employs a data-flow-driven client-server
execution model and is currently available on Unix workstations
from Sun, Silicon Graphics, Hewlett-Packard,
IBM, DEC and Data General [7]. DX provides tools for
operating on both scattered and gridded data. The DX
Connect module performs the Delauney triangulation used
in Figures 5 and 6 while the Regrid module performs the
weighted average interpolation used in Figures 8, 9 and 10.

The Regrid module provides independent control of the exponent
of the weighting factor, the size of n and the radius
of influence from each node of the grid within which to
consider data points. In this case a radius of 0.36 degree (of
latitude and longitude) was used. It should be noted that for
each day of data, not all stations have rainfall measurements.
This is NOT the same as a station reporting no
rain. DX supports a notion of data invalidity. Hence, for
any given day, only those stations having measurement are
considered by both the Connect and Regrid modules in creating
gridded versions of the data. The choice of modules
that support continuous realization is independent of the
use of Connect or Regrid even though they result in different
mesh structures because these DX operations are polymorphic.
These tools were used to create a DX application
that supports the study of the rainfall data, which is shown
in Figure 11.
Results
The detailed results of these analyses are presented by
Goldberg et al [6]. The data have shown that enhanced
rainfall during El Niño can be divided into two categories;
first, a general increase in background levels over non-El
Niño periods and second, sporadic bursts of intense rainfall
superimposed over the enhanced background levels (cf.,
Figure 3). It is these sporadic bursts which were most responsible
for the great damage during the 1982-1983 El
Niño episode.
The data further show that the severe storms (or bursts)
often originate near the Andean foothills and may be induced
by the interaction of rainbands moving inland from
the coast with mountain downslope winds. This is best
shown with a time sequence during such an event. Figures
12 and 13 illustrate the distribution of rainfall for January
24-27, and May 19-22, 1983, respectively, when a severe
series of storms took place. The gridding process described
earlier is used to create a pseudo-colored mesh independently
for each day. The mesh is deformed by the altitude
at each node, which is determined from the same gridding
process applied to the altitude of each station. This deformed
surface gives a reasonable approximation of the topography
in northwestern Peru, especially given the
paucity of high-resolution elevation data for this region.
Conclusions and future work
The characteristic topography near regions such as
Chulucanas (roughly in the center, cf., Figures 3, 11, 12
and 13), where such storms were observed to occur on a
frequent basis, is ideal for the aforementioned interaction
between the rainbands and the Andean foothills. The origin
of the east-west rainbands near the north Peruvian coast is
less clear but may be caused by low altitude wind surges,
which are driven northward along the coast of Peru by a
large and quasi-permanent high in the southeastern Pacific.
Investigation of the applicability of other methods of
gridding scattered data will be considered for these and similar,
but larger sets of meteorological data [1], [3], [4], [5],
[9], [12], [16] and [17].
Acknowledgments
The data are available courtesy of NASA/Goddard
Space Flight Center, Greenbelt, Maryland.
References
[1] Alfeld, P. "Scattered Data Interpolation in Three or
More Variables". Mathematical Methods in Computer
Aided Geometric Design, Academic Press, pp. 1-33,
1989.
[2] Agishtein and Migdal. "Smooth surface reconstruction
from scattered data points". Computer Graphics
(UK), 15, n. 1, pp. 29-39, 1991.
[3] Akima, H. "A Method of Bivariate Interpolation and
Smooth Surface Fitting for Irregularly Distributed Data
Points". ACM Transactions on Mathematical
Software, 4, 2, pp. 148-159, June 1978.
[4] Hardy, R. L. "Multiquadric Equations of Topography
and Other Irregular Surfaces". Journal of Geophysical
Research, 76, n. 8, pp. 1905-1915, March 10, 1971.
[5] Gmelig-Meyling, R. H. J. and P. R. Pfluger. "Smooth
Interpolation to scattered data by bivariate piecewise polynomials
of odd degree". Computer Aided Geometric
Design, 7, pp. 439-458, 1990.
[6] Goldberg, R.A., G. Tisnado M. and R. A. Scofield.
"Characteristics of Extreme Rainfall Events in Northwestern
Peru during the 1982-1983 El Niño Period". Journal of
Geophysical Research, 92, C13, pp. 14225-14241,
1987.
[7] Lucas, B., G. D. Abram, N. S. Collins, D. A. Epstein,
D. L. Gresh and K. P. McAuliffe. "An Architecture for a
Scientific Visualization System". Proceedings IEEE
Visualization '92, pp. 107-113, October 1992.
[8] Nielson, G. M. "Scattered Data Modeling". IEEE
Computer Graphics and Applications, 13, n. 1. pp.
60-70, January 1993.
[9] Perillo, G.M.E. and M. C. Piccolo. "An interpolation
method for estuarine and oceanographic data". Computers
and Geosciences, 17, n. 6, pp. 813-820, 1991.
[10] Philander, S. G. H. "Anomalous El Niño of 1982-
1983". Nature, 305, p. 16, 1983.
[11] Quinn, W. H., D. O. Zopf, K. S. Short and R. T. W.
Kuo-Yang. "Historical Trends and Statistics of the Southern
Oscillation, El Niño, and Indonesian Droughts". Fishery
Bulletin, 76, p. 663, 1978.
[12] Ramstein, G. "An Interpolation Method for
Stochastic Models". Proceedings Eurographics '90,
pp. 353-365, 1990.
[13] Rasmusson, E. M. "El Niño and Variations in
Climate". American Scientist, 73, 168, 1985.
[14] Renka, R. J. "Multivariate interpolation of large
sets of scattered data". ACM Transactions on
Mathematical Software, 14, 2, pp. 139-148, June 1988.
[15] Shepard, D. "A Two-dimensional interpolation function
for irregularly-spaced data". Proceedings 23rd
National ACM Conference, pp. 517-524, 1968.
[16] Smith, W. F. and P. Wessel. "Gridding with continuous
curvature splines in tension". Geophysics, 3, pp. 293-
305, 1990.
[17] Yue-sheng, L. and G. Lü-tai. "Bivariate Polynomial
Natural Spline Interpolation to Scattered Data". Journal of
Computational Mathematics, 8, n.2, pp. 135-146,
1990.

Figure 6. Pseudo-Colored Rainfall Distribution
from Delauney Triangulation.
Figure 10. Pseudo-Colored Rainfall from
Weighted Average Gridding of Stations.
Figure 12. Rainfall Distribution in Northwestern
Peru on January 24-27, 1983.
Figure 8. Pseudo-Colored Mesh from Weighted
Averaging and Rainfall Stations.
Figure 11. User Interface of a DX-based
Application for Studying Peruvian Rainfall Data.
Figure 13. Rainfall Distribution in Northwestern
Peru on May 19-22, 1983.

Appendix. Alphabetical Listing of Rainfall Stations and Coordinates.
Name Number Latitude Longitude Altitude
(°S) (°W) (m)
ALTAMIZA 29 5.07 79.73 2600
ANIA 17 4.85 79.48 2450
ARANZA 16 4.85 79.58 1300
ARDILLA 40 4.52 80.43 150
ARENALES 8 4.92 79.85 3010
ARRENDAMIENTOS 57 4.83 79.90 3010
AUL 42 4.55 79.70 640
AYABACA 2 4.63 79.72 2700
BARRIOS 33 5.28 79.70 310
BERNAL 36 5.47 80.73 32
BIGOTE 34 5.33 79.78 200
CANCHAQUE 35 5.37 79.60 1200
CHALACO 25 5.03 79.80 2250
CHIGNIA 60 5.60 79.70 360
CHILACO 3 4.70 80.50 90
CHULUCANAS 9 5.10 80.17 95
CHUSIS 37 5.52 80.82 12
CIRUELO 64 4.30 80.15 202
CORPAC 15 5.20 80.62 49
ESPINDOLA 49 4.63 79.50 2300
FRIAS 20 4.93 79.93 1700
HUANCABAMBA 68 5.23 79.43 1052
HUAR HUAR 62 5.08 79.47 3150
HUARA DE VERAS 47 4.58 79.57 1680
HUARMACA 14 5.57 79.52 2100
JILILI 46 4.58 79.80 1330
LA ESPERANZA 7 4.92 81.07 12
LA TINA 1 4.40 79.95 427
LAGARTERA 54 4.73 80.07 307
LAGUNA RAMON 59 5.55 80.67 9
LAGUNA SECA 19 4.88 79.48 2450
LANCONES 45 4.57 80.47 110
LAS LOMAS 66 4.65 80.25 265
Name Number Latitude Longitude Altitude
(°S) (°W) (m)
LOS ALISOS 21 4.97 79.53 2150
MALLARES 6 4.85 80.73 45
MIRAFLORES 11 5.17 80.62 30
MONTEGRANDE 13 5.35 80.72 27
MONTERO 48 4.63 79.83 1070
MORROPON 10 5.18 79.98 140
NANGAY DE MATALACAS 18 4.87 79.77 2100
OLLEROS 53 4.70 79.65 1360
PACAYPAMPA 23 5.00 79.67 1960
PAITA 67 5.08 81.13 6
PALOBLANCO 28 5.05 79.63 2800
PALTASHACO 30 5.12 79.87 900
PANANGA 43 4.55 80.88 450
PARAJE GRANDE 65 4.63 79.92 1500
PASAPAMPA 31 5.12 79.60 2410
PICO DEL ORO 41 4.53 79.87 1325
PIRGA 61 5.67 79.62 1510
PUENTE INTERNACIONAL 63 4.38 79.95 408
SAN JOAQUIN 32 5.13 80.35 100
SAN MIGUEL 12 5.23 80.68 29
SAN PEDRO 27 5.08 80.03 254
SANTO DOMINGO 24 5.03 79.87 1475
SAPILLICA 56 4.78 79.98 1446
SAUSAL DE CULUCAN 4 4.75 79.77 980
SICCHEZ 44 4.57 79.77 1435
SUYO 39 4.50 80.00 250
TACALPO 50 4.65 79.60 2010
TALANEO 26 5.05 79.55 3430
TAPAL 55 4.77 79.55 1890
TEJEDORES 5 4.75 80.25 230
TIPULCO 52 4.70 79.57 2600
VADO GRANDE 38 4.45 79.60 900
VIRREY 58 5.53 79.98 230

Please reference the following animation located in the MPG directory:
RAINFALL.MPG 242-frame animation (November 1, 1982 through
June 30, 1983) of the daily rainfall distribution in
northwestern Peru (MPEG-1).
Copyright © 1994 by IBM Thomas J. Watson Research Center

Case Study: Visualization of Mesoscale Flow Features in Ocean Basins
Andreas Johannsen Robert Moorhead
andreas@erc.msstate.edu rjm@erc.msstate.edu
NSF Engineering Research Center for Computational Field Simulation
P.O. Box 6176, Mississippi State University, Mississippi State, MS 39762
Abstract
Environmental issues such as global warming are an
active area of international research and concern today.
This case study describes various visualization paradigms
that have been developed and applied in an attempt to elucidate
the information provided by environmental models and
observations. The ultimate goal is to accurately measure the
existence of any long term climatological change. The global
ocean is the starting point, since it is a major source and
sink of heat within our global environment.
Introduction
The NSF Engineering Research Center (ERC) for
Computational Field Simulation at Mississippi State University
was established in 1990 with the mission of mounting
a concerted research program to provide U.S. industry
with the capability for the computational simulation of
large–scale, geometrically–complex physical field problems.
For the Scientific Visualization Thrust of the ERC,
oceanographic visualization has been a major area of research
and development. An interactive acoustic visualization
tool has been developed for the Naval Oceanographic
Office [4,6], which allows an analyst to visualize 3D ocean
sound speed fields volumetrically to determine the location
of subsurface oceanographic features. An eddy detection,
extraction, tracking, and animation capability has been developed
with the MSU Center for Air–Sea Technology
[5,10,11].
The present visualization focus on ocean data is part of
the Acoustic Monitoring of Global Ocean Climate (AM-
GOC) Program, which involves many of the major oceanographic
researchers and centers in the United States, as well
as many abroad, in an effort to measure ocean temperature
using acoustic thermometry to provide direct evidence of
the rate of global climate change. The 30–month pilot project
is primarily focused in the Pacific Ocean.
1
Since oceans cover about three–quarters of the earth
and are vast reservoirs of heat and carbon dioxide build–up
from our industrialized world, an accurate measure of ocean
temperature on a global scale can provide direct evidence of
the rate of global climate change caused. In fact, the global
ocean has been called the climatological flywheel.
Within the AMGOC program, global climate models
are being studied and refined. Data from these models aid
in the prediction of global warming trends and the prediction
of acoustic travel time trends, which are tested against measurements.
Both long–range acoustic propagation and
ocean climate models provide a better understanding of the
effects of seasonal and other natural variability of ocean
temperatures.
Sound speed in the ocean is a function of temperature,
pressure, and salinity. Global ocean warming of as little as
3–5 millidegrees Celsius produces a sound speed increase of
0.02 m/s. Over global paths (16,000 km) this results in a
travel time reduction of 0.1–0.2 seconds. Thus even a miniscule
global warming signal can be detected by the decreased
propagation time. However, this decrease in travel time
must be detected against a background of travel time fluctuations
caused by oceanographic variability arising from
sources other than warming. Such fluctuations are largely
due to mesoscale, seasonal, and longer term variability. It
is these oceanographic “noise” sources that the work described
in this case study attempts to visualize effectively.
As the AMGOC program evolves, the visualization system
will allow the visualization of fluid flow and acoustic propagation
within the same scene. This comprehensive visualization
system will allow detection of computational and experimental
problems, discovery of unexpected physical
phenomena and confirmation of the expected, and will provide
an effective means of visual communication among
project participants and to the scientific and lay communities
in general.
The visualization system will include representations
of ocean temperature fields, ocean current fields, ocean surface
height, sound speed data, acoustic source and receiver

locations and structure, and acoustic propagation paths
showing propagation losses, boundary interactions, and
acoustic arrival patterns. Effective representation of mesoscale
eddies, internal waves, and other relevant ocean characteristics
will be developed.
This data visualization capability will include both experimental
and computational model data. Effective comparison
of forward predictive modeling with resulting assimilated
data will be included. The visualization system
will be interactive, providing a global view and high resolution
sectional views under user control of both location and
viewpoint. It will be possible to move about the targeted section
for alternate views or to move around geometrical features.
The key elements of the visualization system have been
the GUI, the data management, and the flow visualization algorithms.
The value of the work is determined by the degree
to which it allows the users to explore their data and extract
information. The ability to see contextual information ––
bathymetry, coastlines, location of acoustic sources and receivers,
physical extent labelling (latitude, longitude, and
depth) –– is crucial.
Dataset
The primary data set that has been visualized is the result
of an ongoing simulation of the Pacific Ocean [2], the
so–called NRL model. It covers the area from 109.125�E
to 282�E and from 20�S to 62�N with a fixed latitude/longitude
sampling of 0.125� x 0.17578125�, which results in a
resolution of 989 x 657. The model has 6 isopycnal
(constant density) horizontal layers. Although the internal
timestep of the model is 20 minutes, data is saved only every
3.05 model days, which yields 120 samples per model year.
The outputs of the simulation are layer thickness deviations,
h, and the horizontal current, [u,v] T . This produces 3 GB of
data per model year.
Visualization System Issues
For the design and implementation of the ocean model
visualization software, an object–oriented approach has
been adopted. Advantages of object–oriented programming
are well–documented, e.g., [3]; in particular it facilitates
stable extensions to large software systems. Extensibility is
a major concern for the ocean model visualization software,
as it is used by a wide variety of institutions within the AM-
GOC program. Furthermore, the visualization paradigms
found useful within the context of ocean modeling should be
applicable to other fields. For example, the techniques developed
for oceanographic flow visualization should also be
useful for hydrodynamic, meteorological, and aerodynamic
flow visualization, necessitating extensions to computational
grids other than the ones used for the AMGOC ocean
models.
2
The inherent size of ocean models makes data handling
an important part of the visualization system. For an initial
implementation, data handling has been restricted to structured,
four–dimensional grids. This provides enough flexibility
for the model, while allowing efficient data access.
Access to raw model data is supplied by an IOStructuredGrid4D
object. This object administers all data files, it
chooses appropriate readers for specific file formats and, in
compliance with main memory size, makes an attempt to
buffer data.
Results from one ocean model experiment are typically
archived in several files to keep file sizes manageable. For
example, the data for the Pacific Ocean from the NRL model
is split into files which contain all the output data for a quarter
year. However, this partitioning should not be visible
during the analysis of the data. Therefore, the visualization
system provides the means to describe a Logical Grid; the
current implementation uses an ASCII file, since this is permanent,
data–related information. Figure 1 shows part of the
Logical Grid File for the Pacific Ocean data set.
\gridname history_1981–1993
\dimension 0..1439 # Time i
0..5 # Layer j
0..656 # y k
0..988 # x l
\variable h float ‘Thickness anomaly’
\variable u float ‘Transport velocity u’
\variable v float ‘Transport velocity v’
\filetype NRL/History/Hydro
\pathname /leo7/hisdat/
\filename h124y151.da i= 0.. 29 c=h,u,v
\filename h124y151.db i= 30.. 59 c=h,u,v
\filename h124y151.dc i= 60.. 89 c=h,u,v
\filename h124y151.dd i= 90..119 c=h,u,v
\pathname /leo1/hisdat/
\filename h124y152.da i=120..149 c=h,u,v
\filename h124y152.db i=150..179 c=h,u,v
\filename h124y152.dc i=180..209 c=h,u,v
# ...
Figure 1. Extract from a Logical Grid File
The \dimension command provides a description of index
ranges for the entire logical grid. The data itself is described
by \variable commands, each specifying the name
of a data variable, its data type and its meaning (as a comment
string). With this information available, each \filename
command specifies the index subrange and the data
variables stored in one particular file. Any continuous index
subrange and any combination of variables is allowed; the
example given in Figure 1 shows only subranges in i and for
all files variables h, u, and v. The \pathname command is
provided for convenience, all following filenames are ap-

pended to the current pathname. The actual access to data
files is determined by their \filetype, which the IOStructuredGrid4D
uses to choose specific readers.
The IOStructuredGrid4D object provides data access
through its method GetVertex (see Figure 2). In each of the
four dimensions, an arbitrary index subrange (or the special
symbol fullRange) can be specified for any variable, allowing
flexible random access of any zero– to four–dimensional
data subset. This includes obvious choices like ‘‘one layer
of h at a particular time’’ and less obvious, but potentially
useful ones like ‘‘one vertical line of h for all timesteps.’’
Providing an easy–to–use but very general data interface
like GetVertex encourages localized data access, leading to
improved system performance through a reduced number of
actual disk file accesses.
IOStructuredGrid4D_GetVertex(
Object *ptrSender,
long iStart, iEnd,
long jStart, jEnd,
long kStart, kEnd,
long lStart, lEnd,
char *ptrVariableName,
void **ptrptrData )
Figure 2. Basic data access method
The current implementation of IOStructuredGrid4D
buffers rectangular subsets of layers, so–called tiles. The
number of tile buffers can be tailored to the available main
memory by using the method SetNoOfBuffers. Individual
tiles are loaded with filetype–specific readers, which receive
a filename, file–relative indices, and a variable name. This
establishes a compact and clean interface and makes readers
for new filetypes easy to add.
The size of ocean model datasets usually prohibits local
storage on a visualization computer. Data is stored on a file
server and accessed through a network, which typically
presents a serious performance bottleneck. In order to address
this problem, one filetype/reader combination in the
visualization system uses a wavelet compression method
[8,9]. Data files are compressed off–line, transmitted when
accessed, and decompressed on the fly.
Flow Visualization
For visualization purposes, graphical representations
with suitable visual attributes have to be derived from the
ocean model data. The structure of layered ocean models
suggests layer interfaces for graphical representation. At
any fixed time, each of the layer interfaces is defined by a
height field, with height values (or depth values, rather) provided
for each of the model’s latitude/longitude pairs.
The geometrical shape of a layer interface can be visualized
by an appropriate surface (interpolated or approxi-
3
mated), with visual clues provided by a lighting model. Figure
3 shows the layer interfaces for the six–layer NRL
model. To avoid aliasing artifacts along the shoreline and
for increased drawing performance, the land data has been
masked for all but the lowest layer interface.
At any fixed time, the horizontal current in one layer is
characterized by vectors [u,v] T on a latitude/longitude rectilinear
grid. Visualizations of steady (instantaneous) and unsteady
(time dependent) flow traditionally use tufts or arrows
to indicate flow directions at fixed and/or moving
points, e.g., [7]. Figure 4 shows a 10x10 degree subsection
of the uppermost two layers in the Pacific Ocean dataset,
with an arrow anchored at each grid point. This is a useful
technique and it has been used extensively, but for larger datasets,
i.e., larger regions and/or finer grids, pictures tend to
be overcrowded with arrows, making flow features practically
undetectable.
For the visualization of horizontal flow within the layers
of a layered ocean model, a technique termed colorwheel
has been developed. The basic idea is to abandon any auxiliary
graphical objects like arrows, which might overcrowd
or clutter the picture, and instead use color for representing
flow direction. A circular color lookup table provides an individual
color for all directions; given a vector [u,v] T , it is
straightforward to compute its direction in terms of an angle
�, which is used to access the corresponding color in the
table.
Figure 5 shows the colorwheel technique applied to the
uppermost layer in the Pacific Ocean model. In order to relate
colors and directions, the colorwheel itself is reproduced
on a planar surface with a fixed position relative to the
layer; it follows all transformations (e.g., rotations) of the
layer in 3–space. For ease of computation and understanding,
the colorwheel has been designed using the HSV color
model, with the flow direction corresponding to Hue. In
Figure 5, the other two channels, S and V, are set to a fixed
value of 1.0, i.e., full saturation and value (lightness).
For the detection of flow features, it is desirable to visualize
not only vector directions, but also vector lengths. Given
a vector [u,v] T , this can be achieved by letting its direction
determine the color and ‘‘scaling’’ that color according to
the vector’s length. In the HSV color model, for example,
the vector’s length can scale the saturation (S) and/or lightness
(V) channels. Figures 6 and 7 illustrate logarithmic
scaling of S and V. Prominent flow features include the
equatorial (east–to–west) and equatorial counter (west–to–
east) currents, the Kuroshio Current off Japan, and eddies,
which, as circular flow, appear as a rotation of the colorwheel
configuration. Clockwise and counter–clockwise eddies
can be easily distinguished. A rotated view, Figure 8,
clearly shows the rising of the uppermost layer interface between
the equatorial and counter–equatorial currents. For a
detailed analysis of flow features, the visual representation

can be probed by pointing with the mouse and reporting the
flow vector on the colorwheel legend.
These initial experiments with the colorwheel technique
have shown its usefulness in visualizing ocean model
flow data. However, there is ample room for improvements
and extensions. The HSV color model has been chosen for
convenience of computation; other color models are being
investigated. To facilitate the qualitative interpretation of
visualizations, changes of perceived colors on the wheel
should correspond directly to (angular) changes of vector
direction. In addition, various configurations on the wheel
seem to be useful, for example, concentrating the color gamut
in a wedge of the colorwheel to increase the resolution of
particular flow directions.
Summary
A large part of the task has been and continues to be
gaining an understanding of the oceanographers’ and acousticians’
knowledge, nomenclature, questions, and interests.
The basic task of visualizing ocean models and acoustic
paths is not likely to change over the next few years. The focus
will be on developing and integrating more tools (other
vector/flow visualization techniques, tensor visualization
for acoustic propagation, etc.) and extending the flexibility
of the system (e.g., handling data on other grids structures,
etc.). For example, coastal modelers with whom we also collaborate
are interested in applying the colorwheel technique
to visualize flow in their problem domain, e.g., flow in tributaries
and bays [1]
Acknowledgements
The work described herein is part of a larger project involving
many people. Kelly Parmley Gaither developed the
vector visualization routines which produced Figure 4, Scott
Nations wrote much of the data management routines and
GUI code, and Bernd Hamann provided much information
on surface reconstruction and vector visualization paradigms.
Their contributions are gratefully acknowledged.
This work has been supported in part by ARPA and the
Strategic Environmental Research and Development Program
(SERDP).
4
References
1 Lee Butler, Coastal Engineering Research Center, US Army
Corps of Engineers, private communications, March 29,
1994.
2 H.E. Hurlburt, Alan J. Wallcraft, Ziv Sirkes and E. Joseph
Metzger, “Modelling of the Global and Pacific Oceans: On
the Path to Eddy–Resolving Ocean Prediction,” Oceanography,
Vol. 5, No. 1, 1992, pp. 9–18.
3 B. Meyer, Object Oriented Software Construction, Prentice–
Hall, London, 1988.
4 R.J. Moorhead, B. Hamann, and J. Lever, ”OVIRT – Oceanographic
Visualization Interactive Research Tool,” Navy
Scientific Visualization and Virtual Reality Seminar, Bethesda,
MD, June 1993.
5 R.J. Moorhead and Z. Zhu, “Feature Extraction for Oceanographic
Data Using A 3D Edge Operator,” IEEE Visualization
’93, San Jose, CA, Oct. 1993.
6 R.J. Moorhead, B. Hamann, C. Everitt, S. Jones, J. McAllister,
and J. Barlow, “Oceanographic Visualization Interactive
Research Tool (OVIRT),” SPIE Proc. 2178, SPIE/IS&T Electronic
Imaging, San Jose, CA, February 6–10, 1994.
7 Frits H. Post and Theo van Walsum, “Fluid Flow Visualization,”
Focus on Scientific Visualization, H. Hagen, H. Müller,
and G.M. Nielson (Eds.), Springer–Verlag, 1993.
8 Hai Tao and Robert Moorhead, “Lossless Progressive Transmission
of Scientific Data Using Biorthogonal Wavelet
Transform,” IEEE International Conference on Image Processing,
Austin, TX, Nov. 1994.
9 Hai Tao and Robert Moorhead, “Progressive Transmission of
Scientific Data Using Biorthogonal Wavelet Transform,”
IEEE Visualization ’94, Washington, D.C., Oct. 1994.
10 Z. Zhu, R.J. Moorhead, H. Anand, and L.R. Raju, “Feature
Extraction and Tracking in Oceanographic Visualization,”
SPIE Proc. 2178, SPIE/IS&T Electronic Imaging, San Jose,
CA, February 6–10, 1994.
11 Z. Zhu and R.J. Moorhead, “Exploring Feature Detection
Techniques for Time–Varying Volumetric Data,” IEEE Workshop
on Visualization and Machine Vision, Seattle, WA, June
24, 1994.

Fig. 3. Layer interfaces of the six-layer NRL
ocean model
Fig. 5. Flow visualization with the colorwheel on the
uppermost layer interface
Fig. 7. As Figure 6, top view
Fig. 4. Flow visualization with arrows on the
uppermost two layer interfaces
Fig. 6. Flow visualization with the colorwheel, color
intensity and saturation scaled by flow vector lengths
Fig. 8. Flow visualization with the colorwheel, rotated view
of the uppermost layer interface, height exaggerated

Case Study: Integrating Spatial Data Display with Virtual Reconstruction
Philip Peterson Brian Hayden F. David Fracchia
School of Computing Science Dept. of Archaeology School of Computing Science
Simon Fraser University, Burnaby, B.C. V5A 1S6 Canada
Abstract
In the process of archaeological excavation, a vast amount
of data, much of it three-dimensional in nature, is
recorded. In recent years, computer graphics techniques
have been applied to the task of visualizing such data. In
particular, data visualization has been used to accomplish
the virtual reconstruction of site architecture and to enable
the display of spatial data distributions using threedimensional
models of site terrain. In the case we present
here, these two approaches are integrated in the modeling
of a prehistoric pithouse. In order to better visualize
artifact distributions in the context of site architecture,
surface data is displayed as a layer in a virtual
reconstruction viewable at interactive rates. This
integration of data display with the architectural model
has proven valuable in identifying correlations between
distributions of different artifact categories and their
spatial proximity to significant architectural features.
1. Introduction
In studying the remains of prehistoric architecture, it is
generally of interest to identify how different areas of a
structure were utilized. The answer to this question can be
seen as an important step in the development of a theory of
the social organization of the site’s former inhabitants.
Under the assumption that certain artifacts have a
strong association with a specific activity, by examining
the spatial distribution within the site architecture of
different artifact types, it is possible to formulate a
hypothesis which identifies areas according to usage.
However, in order to form a more complete understanding
of dwelling organization, one must question why one area
is associated with a particular usage pattern and not
another.
In the case presented here, we describe how, using
computer graphics techniques, it is possible to incorporate
the display of artifactual data distributions into the
structural and environmental context of site architecture (in
this case, a prehistoric pithouse). We have found that
visualizing excavation data in this manner can, in addition
to being a valuable aid in the identification of correlations
between the spatial distribution of artifacts and significant
site features, be a useful tool in identifying the implications
of architectural constraints on usage areas.
1.1 Visualization in archaeology
Computer graphics techniques are increasingly being
used to visualize complex data in archaeological
investigation. In recent years, several projects involving
the creation of detailed virtual reconstructions of
archaeological sites have been undertaken. In these
applications, a three-dimensional computer graphics model
of the site is constructed and viewed using standard
modeling, rendering, and animation techniques.
Some of the most well-known examples of virtual
reconstruction have involved the application of solid
modeling techniques (CSG) in the recreation of historical
architecture. Initially applied to the modeling of the temple
precinct of Roman Bath, CSG techniques were later
refined and applied to the more ambitious modeling of the
Saxon Minster of Winchester using the WINSOM solid
modeller [13]. A later project, the modeling of the Furness
Abbey [5], expanded on the techniques pioneered in the
earlier efforts and offered interactive viewing capabilities,
although at a fairly low degree of realism.
Recent efforts in virtual reconstruction have produced
increasingly photorealistic results. A detailed model of the
Dresden Frauenkirche, destroyed during the second world
war, has been used to create a high resolution computer
generated film [4]. Unlike most archaeological sites where
the only source of data is the site itself, original plans and
photographs of the architecture existed on which to base
the model. One of the outstanding features of this
reconstruction is the attention paid to the recreation of
interior lighting and surface detail, evoking in the viewer a
sense of the architectural space. Attention to detail and
accurate surface characteristics are also evident in the
virtual reconstruction of the Visir tomb in Egypt [11]. In
both of these cases, the visualization is not the ultimate
goal, but is being used as part of the process leading to the
eventual physical reconstruction of the site.
Computer graphics techniques have also been applied
to the analysis of archaeological sites. From site survey
data, a three-dimensional model of terrain can be

constructed [7]. Digital terrain models (DTM) can then be
used as a context in which to display spatial data such as
artifact distributions [2]. This approach is fundamental to
geographic information systems (GIS), and their
applicability to archaeology has also become evident,
particularly when considering data at the site or intra-site
level [9,10]. A related approach considers the problem of
visualizing spatial data throughout several noncontemporaneous
layers [1]. In this early system, a data
layer can be displayed in a plan view, or a cross-sectional
view can be used to slice through multiple layers.
The ability to visually relate the distribution of data to
surface characteristics has proven useful in the
identification of spatial correlations. This technique can be
extended to include less tangible features such as
viewsheds and solar paths, While less physical than
surface characteristics, these environmental factors are
nonetheless, sufficiently important. Analyses of these
features have been applied in architecture [6,12], and have
potential application in archaeology [8].
2. Visualization
The Keatley Creek site in British Columbia consists of
over 100 cultural depressions, several of which have been
fully excavated since 1986. The majority of these
depressions are the remains of residential dwellings that
were occupied over 2000 years ago. The case we present
here describes the virtual reconstruction of one of the
largest of these dwellings.
Figure 1. Pithouse architecture.
The type of dwellings associated with these cultural
depressions are referred to as pithouses. Pithouse
construction consists of an approximately circular
subterranean floor with a raised earthen rim and a roughly
conical roof structure. The roof is constructed of log beams
pinned to the rim and sloping inward towards the center of
the pit. The inward sloping beams, supported by upright
log posts are interlaced with joists and covered with wood
thatching and dirt. At the apex of this roof structure is a
central smoke-hole, which, in addition to serving as the
source of light and ventilation, also serves as the pithouse
entrance. Ingress and egress from the smoke-hole is
accomplished using a log ladder. The major structural
components are illustrated in Figure 1.
2.1 Virtual reconstruction
In constructing the model, the focus was on creating a
visual representation that conveyed the major structures,
yet was simple enough to navigate at interactive rates on
available hardware. Moreover, structural features that were
not necessary to convey the basic architectural shape were
intentionally left out.
The floor was reconstructed from survey data by
fitting a non-uniform b-spline surface that interpolated
known surface points. Beams and posts were modeled
using cylindrical primitives, and the roof covering was
constructed from simple polygonal patches (Plates 1 and
2). A simple shading model with minimal texture mapping
is used.
In addition to the major structural components of the
pithouse, simple representations of significant interior
features are also included. The locations of hearths are
indicated by the position of red spheres on the pithouse
floor. Similarly, the positions of storage pits are denoted by
black disks (Plate 3)
The user can move in, out and around the model, in
real time, by steering a virtual camera using a mouse or
SpaceBall. The standard display is in the first-person, and
the interaction method used does not require the user to
change focus away from the model. In addition, an
orthographic overview of the floor can be displayed
simultaneously. In this view, an arrow glyph is used to
indicate position and orientation in the main view (Plate 5).
2.2 Spatial data display
The spatial distributions of a variety of artifacts can be
displayed within the model. Distribution plots are layered
on top of the pithouse floor using a model in which the
greater the artifact density, the more the surface colour is
modified (Plate 4). Multiple plots can be layered at once,
using different colors, transparency or vertical
displacements if desired.
Most artifact data at this site was recorded as discrete
counts within square regions. Therefore, spatial data is

plotted at this resolution, and no colour interpolation is
used.
2.3 Light availability
A characteristic feature of pithouse architecture is the
central smoke-hole. This opening at the apex of the roof
structure is the only source of natural light in the dwelling.
Because of the limited availability of light, it is
hypothesized that the usage of different areas of the house
floor was subject, at least partially, to the availability of
working light.
In order to approximate the directly lit areas of the
house interior, a light source approximating the emission
area of the smoke-hole is modeled. This light source
illuminates the floor model so that areas of direct light can
be easily seen. The solar path of the sun is calculated for
the site, and can be specified to correspond to a given date.
The user can then interactively manipulate the light source
through its apparent daily motion to see its relation to data
distributions layered onto the floor surface (Plate 6). This
method, while not a correct model of the actual sunlight
distribution, is a useful approximation for the purpose of
data exploration.
4. Results
In the case presented here, we have used threedimensional
computer graphics techniques to integrate the
display of spatial data into a model of a prehistoric
pithouse. This has provided us with some valuable insight
into the implications that the architectural structure of a
pithouse can have on usage areas within the dwelling. By
viewing artifact data in an approximation to its original
context, we were able to more readily identify potential
relationships between data distributions and the pithouse
structure.
Artifact data from the Keatley Creek excavation is
probably the most detailed and comprehensive ever
collected from a similar site [14]. Various hypotheses exist
relating the spatial distribution of certain artifact types to
areas of the pithouse floor. In order to visualize how
particular data distributions relate to the site, distributions
were layered onto the surface model of the floor (Plate 4).
Distribution plots of this type are commonly displayed
on two-dimensional site maps. Placing the data in the
three-dimensional model has the advantage of making
obvious any relation to surface slope. As can be seen in
Plate 4, it is readily apparent when a high density of
artifacts, in this case, debitage, are on a wall slope
(possibly indicating a storage area).
While the analysis of spatial relationships between
data and floor position has proved useful, we have found
the most insightful information from this visualization
comes as a result of considering that the structure imposes
constraints on usage areas.
In the orthographic projection shown in Plate 5, two
different artifact distributions are displayed. One of the
data sets is clearly more concentrated in the periphery than
the other. The data set that is more central is likely related
to its proximity to the hearths as it is commonly thought to
be associated with food preparation. The outer data set,
however, is associated with personal possessions. One
possible reason for this peripheral distribution might be
that sleeping areas were on the edge of the structure. While
not identifiable in the two-dimensional plot, it is readily
seen in Plates 3 and 6 that the ceiling height at the edge is
quite low, and therefore, the edge could not have been
useful for much else other than sleeping or storage.
Another form of architectural constraint inherent in the
pithouse structure is the availability of working light. Since
the only source of natural light is the smoke-hole, we have
approximated how much of the interior would be lit at a
given time of day. In Plate 5, the distribution of the data set
shown in blue (heavily-retouched scrapers), is more
concentrated towards one end of the floor. By examining
the same plot in Plate 6, it is readily apparent that this
artifact is more dominant in the area lit by the midday sun.
This would indicate that this artifact type is associated with
work requiring some visual acuity.
5. Future work
While the use of computer graphics as a visualization
technique in archaeology is becoming more common, there
are still many possibilities for further research. In
particular, the case presented here can be expanded on in
several ways. Our research up to this point has
concentrated on creating a model of major structural
features only. We have taken this approach for two
reasons: First, it was necessary to minimize the complexity
of the model in order to maintain interactivity on available
hardware. Second, the structural factors that we were most
interested in testing the implications of (direct light
availability and ceiling height), did not require significant
structural complexity. With access to faster hardware, more
detailed features could be incorporated without sacrificing
display speed, and as a result interactivity.
While the simple lighting model implemented thus far
has proven useful, the development of a more accurate
model that accounts for light from hearths, in addition to
natural light, would provide a better basis for work area
theories.
A further enhancement to the model would be to
implement it in a virtual reality (VR) environment. In
addition to providing the user with a sense of the

architectural space, VR techniques have potential for
integrating the display of data in the context of a model [3].
Finally, it would be of interest to integrate the
visualization model with the underlying data such that it is
no longer a passive display device only, but can be used to
query data and perform statistical functions.
References
[1] Norman I. Badler and Virginia R. Badler. Interaction with a
color computer graphics system for archaeological sites. In
Computer Graphics (Proceedings of SIGGRAPH ’78),
pages 217–221, Atlanta, GA, August 1978.
[2] W. A. Boismier and Paul Reilly. Expanding the role of
computer graphics in the analysis of survey data. In C. L.
N. Ruggles and S. P. Q. Rahtz, editors, Computer and
Quantitative Methods in Archaeology 1987, BAR
International Series 393, pages 221–225, Oxford, 1988.
British Archaeological Reports.
[3] Steve Bryson and Creon Levit. The virtual windtunnel: An
environment for the exploration of three-dimensional
unsteady flows. In Proceedings of Visualization ’91, San
Diego, CA, October 1991. IEEE.
[4] Brian Collins. From ruins to reality—the Dresden
Frauenkirche. IEEE Computer Graphics and Applications,
pages 13–15, November 1993.
[5] Ken Delooze and Jason Wood. Furness abbey survey
project—the application of computer graphics and data
visualisation to reconstruction modelling of an historic
monument. In Kris Lockyear and Sebastian Rahtz, editors,
Computer and Quantitative Methods in Archaeology 1990,
BAR International Series, pages 141–148, Oxford, 1990.
British Archaeological Reports.
[6] Stephen M. Ervin. Visualizing n-dimensional implications
of two-dimensional design decisions. In Proceedings of
Visualization ’92, Boston, MA, October 1992. IEEE.
[7] T. M. Harris and G. R. Lock. Digital terrain modelling and
three-dimensional surface graphics for landscape and site
analysis in archaeology and regional planning. In C. L. N.
Ruggles and S. P. Q. Rahtz, editors, Computer and
Quantitative Methods in Archaeology 1987, BAR
International Series 393, pages 161–172, Oxford, 1988.
British Archaeological Reports.
[8] Martin Kokonya. Data exploration in archaeology: New
possibilities and challenges. In Paul Reilly and Sebastian
Rahtz, editors, Communication in Archaeology: a global
view of the impact of information technology, Volume
One: Data Visualisation, pages 49–64. 1990.
[9] Kenneth L. Kvamme. Geographic information systems and
archaeology. In Gary Lock and Jonathan Moffett, editors,
Computer and Quantitative Methods in Archaeology 1991,
BAR International Series S577, pages 77–84, Oxford,
1991. British Archaeological Reports.
[10] Gary Lock and Trevor Harris. Visualizing spatial data: the
importance of geographic information systems. In Paul
Reilly and Sebastian Rahtz, editors, Archaeology and the
Information Age: a global perspective, chapter 9, pages 81–
96. Routledge, 1992.
[11] P. Palamidese, M. Betro, and G. Muccioli. The virtual
restoration of the Visir tomb. In Proceedings of
Visualization ’93, pages 420–423, San Jose, CA, October
1993. IEEE.
[12] Jon H. Pittman and Donald P. Greenberg. An interactive
environment for architectural energy simulation. In
Computer Graphics (Proceedings of SIGGRAPH ’82),
pages 233–242, Boston, MA, July 1982.
[13] P. Reilly. Data visualization in archaeology. IBM Systems
Journal, 28(4):569–579, 1989.
[14] James G. Spafford. Artifact distributions on housepit floors
and social organization in housepits at Keatley Creek.
Master’s thesis, Simon Fraser University, Department of
Archaeology, 1991.
Acknowledgements
This work was supported in part by a Natural Sciences
and Engineering Research Council (NSERC) post-graduate
scholarship. Equipment, resources, and encouragement
were provided by the faculty, statff, and students of the
Graphics and MultiMedia Research Lab at Simon Fraser
University.

Plate 1. Pithouse exterior
Case Study: Integrating Spatial Data with a Virtual Reconstruction
Philip Peterson Brian Hayden F. David Fracchia
VIS ’94

Plate 2. Pithouse interior.
Case Study: Integrating Spatial Data with a Virtual Reconstruction
Philip Peterson Brian Hayden F. David Fracchia
VIS ’94

Plate 3. Pithouse interior showing hearths,
strorage pits and smoke−hole.
Case Study: Integrating Spatial Data with a Virtual Reconstruction
Philip Peterson Brian Hayden F. David Fracchia
VIS ’94

Plate 4. Spatial data layering.
Case Study: Integrating Spatial Data with a Virtual Reconstruction
Philip Peterson Brian Hayden F. David Fracchia
VIS ’94

Plate 5. Plan view of pithouse floor.
Case Study: Integrating Spatial Data with a Virtual Reconstruction
Philip Peterson Brian Hayden F. David Fracchia
VIS ’94

Plate 6. Spatial data layer with
midday lighting model
Case Study: Integrating Spatial Data with a Virtual Reconstruction
Philip Peterson Brian Hayden F. David Fracchia
VIS ’94

Case Study: Observing a Volume Rendered Fetus within a Pregnant Patient
Andrei State, David T. Chen, Chris Tector, Andrew Brandt, Hong Chen,
Ryutarou Ohbuchi, Mike Bajura and Henry Fuchs
Abstract
Augmented reality systems with see-through headmounted
displays have been used primarily for
applications that are possible with today's computational
capabilities. We explore possibilities for a particular
application—in-place, real-time 3D ultrasound
visualization—without concern for such limitations. The
question is not “How well could we currently visualize the
fetus in real time,” but “How well could we see the fetus
if we had sufficient compute power?”
Our video sequence shows a 3D fetus within a
pregnant woman's abdomen—the way this would look to a
HMD user. Technical problems in making the sequence
are discussed. This experience exposed limitations of
current augmented reality systems; it may help define the
capabilities of future systems needed for applications as
demanding as real-time medical visualization.
1 Introduction
Interpreting 3D radiological data is difficult for nonexperts
because understanding spatial relationships
between patient and data requires mental fusion of the
two. Volume rendering has been useful for visualization
but has typically been viewed separately from the patient.
Ideally one would like to see directly inside a patient.
Ultrasound echography allows dynamic live scanning and
patient-doctor interaction; an augmented reality system
displaying live ultrasound data in real time and properly
registered in 3D space within a scanned subject would be
a powerful and intuitive tool; it could be used for needleguided
biopsies, obstetrics, cardiology, etc.
2 Previous work
Many researchers have attempted visualization of 3D
echography data [1, 2, 3, 4]; some have volume visualized
data sets that were acquired as a series of hand-guided 2D
echography slices with 6 DOF [5, 6]. Compared to
University of North Carolina
echography imaging by current state-of-the-art 2D
scanners, such volume visualizations promise to reduce
the difficulty of mentally combining 2D echography slices
into a coherent 3D volume.
However, almost all of these systems have used
conventional stationary video monitors for presentation,
so that a user must still mentally fuse 3D volume images
on the monitor with the 3D volume of the patient.
One system [7] tried to visualize live 2D echography
images in-place within the patient using a see-through
HMD system. While that system demonstrated the initial
concept of “augmented reality,” it could show only a few
image slices (no 3D volume) at a relatively low frame rate
Those few ultrasound images appeared to be pasted in
front of the patient’s body rather than fixed within it.
3 Near-real-time visualization system
In January 1993 we attempted to improve upon [7]
with a system designed to perform real-time, in-place
volume visualization of a live human subject. It contained
two major real-time features: a continuously updated and
rendered volume, and an image compositor. The volume
was updated from a series of 2D echography images
acquired by a tracked ultrasound transducer. The image
compositor combined each frame of the volume rendering
with a live HMD video image of the patient (Plate 1).
Unfortunately, due to the requirements of real-time
operation, the performance was seriously inadequate. The
system's major shortcomings were:
• the ultrasound acquisition rate was only 3 ultrasound
frames per second, causing both temporal and spatial
undersampling of the volume,
• the reconstructed volume was crudely sampled (to
100 x 100 x 100), and could not be updated at more
than 1 ultrasound slice per second,
• the volume was rendered at low resolution (65 x 81 rays
cast) and interpolated to the display resolution of
(512 x 640) to achieve 10 fps,
• the tracking system resolution and accuracy were poor,
with significant lag and noise in the data.
As a result of these problems, the (interactive) volume

enderings displayed by this system were unrecognizable.
The scans of a nearly full-term fetus did not reveal
human-like forms to any member of the research team
other than the M.D. ultrasonographer.
4 Hybrid real-time / off-line system
To answer the question “How much better would the
visualization be if the real-time computational demands
were met by the available resources,” this system was
designed so that the most computationally expensive tasks
(volume reconstruction and rendering) are done off-line.
Figure 1 shows the steps involved in generating a
video sequence that combines volume rendered
echography data with HMD camera visuals. The figure
shows the dependency of various tasks on one another.
The tasks can be grouped roughly into calibration tasks
performed prior to scanning a subject, data acquisition
tasks performed during the scan, and image generation
tasks which are a post process.
4.1 Calibration
Calibration procedures compute
several sets of parameters. One such set
relates echography pixels to the
transducer tracker origin; together with
the transducer tracker position, it
determines the location of the
echography pixels in 3D world space.
Similarly, the camera position and
orientation relative to the HMD tracker
origin must be determined; the optical
distortion of the camera lens must be
modeled so that the CG imagery can be
made to match the camera images.
Echography pixel to tracker
calibration. This function (which is not
necessarily linear in pixel space) is
measured by imaging a point target (a
4 mm bead obtained from GE medical
systems and suspended at the tip of a pin
in a water tank) at a known location
relative to the transducer. Plate 2 shows
the calibration setup for the ultrasound
transducer. The transducer is attached to
a precision translation stage which
moves under computer control to chart
out the point spread functions of the
echography pixels (this function grows
with distance from the transducer tip).
The 2D transducer slice was measured to
be rotated 2.3 degrees from the
transducer's axis.
OFF-LINE LIVE PRE-SCAN
SCAN
U/S scanner
Echography pixel
calibration
U/S calibration
parameters
�������
Sweep abdomen
�������
�����
�����
Digitizing stylus
�����
tracking data
�����
�����
Define pit
geometry
�����
Pit geometry data
�����
���
���
Data streams
Transducer
tracking data
(w/ time code)
Low-pass filter
(fc=1Hz)
Filtered transducer
tracking data
Tasks
���Data and data sources
Failed subprocess
Camera calibration. Position and orientation of the
HMD camera relative to the HMD tracking origin are
determined by an iterative semi-automatic method. The
optical distortion of the lens is determined by imaging a
grid pattern (Plate 2, inset). A circularly symmetric
model based on a 5th degree polynomial is used.
Our optical tracking system is described in [8]; the
calibration methods used for it are described in [9].
4.2 Real-time acquisition
Unburdened by visualization processing needs, this
phase takes place in true real-time. Both the ultrasound
images and the HMD camera images are recorded at
30 fps on Sony D2 digital tape recorders. At the same
time, tracking data for the ultrasound transducer and for
the HMD is saved on a UNIX workstation which
controls the D2 recorders. With each tracker record the
time code of the corresponding video frame is stored for
later synchronization, thus establishing correspondence
between the tracking and video data streams.
To create an illusion of the visualized volume
“Emergency measures”
Calibrated tracker
Acquire echography data stream
Reconstruct
Reconstructed
volume
Manual editing
Edited volume
Pregnant patient
Echography
slices
(digital video)
Render volume,
pit, w/ distortion
CG Element
(digital video)
HMD
tracking data
(w/ time code)
Low-pass filter
(fc=6Hz)
Composite
Composited footage
HMD with video camera
Pos. / orientation
calibration;
Optical distortion
correction
Acquire video data stream
Filtered HMD
tracking data
Define pit
geometry
Pit geometry data
Video calibration
parameters
HMD camera
images
(digital video)
Figure 1: Flow diagram for hybrid experiment combining
real-time acquisition with off-line visualization. The top
row shows the “basic ingredients” of our system.

esiding inside the abdomen, a (polygonal) model of a
“pit” must be created; the pit must conform to the shape
of the abdomen and be placed at the correct position. To
achieve this, the geometry of the abdomen is acquired by
making a zig-zag sweep of the abdomen. The tip of the
tracked transducer is used as a 3D digitizing stylus.
4.3 Off-line image generation
In this phase volume visualized echography images
and the images captured by the HMD-mounted camera are
combined into composite HMD viewpoint imagery. The
major steps in generating the composite are:
Tracking noise filtering. The tracking data we
acquire fluctuates even if the tracked target remains
stationary. Such noise causes misregistration between
video and CG imagery. To reduce this effect, a noncausal
low-pass filter without phase shift is used [10],
with cut-off frequencies of 1 Hz for the transducer tracker
and 6 Hz for the HMD tracker.
Reconstruction. The echography pixels are
positioned and resampled into a regularly gridded volume,
using a simple approximation algorithm based on a linear
combination of Gaussian weighting functions which are
translated and scaled to minimize artifacts [6, 11]. Size
and shape of an echography pixel in world (or tracker)
space are approximated by a point spread function which
falls off away from the world space position as a nonspherical
Gaussian. Since an image frame and its tracking
information are related by the frame's time code,
digitization of echography frames from video tape and
volume reconstruction can take place automatically on a
workstation under program control.
Visualization. A volume renderer vol2 [12] running
on a graphics multicomputer Pixel-Planes 5 is used to
render the reconstructed volume(s) from viewpoints
matching those of the HMD-mounted camera. By
modulating the direction of rays, the images are distorted
based on the model described above; the polygonal pit,
built using the data from the abdomen geometry sweep, is
included in the rendering. The images are recorded in
single-frame mode onto digital video tape.
Compositing. Camera and CG images are mixed by
chroma-keying on a Sony video mixer. The mixer
replaces blue background in the CG frames by HMD
frames; the time codes recorded during HMD acquisition
are used to ensure synchronization of the 2 elements.
5 Live subject experiment and results
In January 1994 we scanned two pregnant subjects
with the hybrid system. Since the current tracking setup
allows only one target at a time, the abdomen sweep data,
the ultrasound data (Plate 3, left) and the head camera
data (Plate 4) had to be recorded in 3 successive passes.
The patients had to remain motionless throughout the
acquisition phase.
From the acquired ultrasound imagery, we selected
and digitized a short sequence of about 15 seconds, during
which the ultrasonographer had made a continuous sweep
(455 slices) of the fetus from the middle of the skull down
to the bottom of the hip (Plate 3, right). The slices were
reconstructed into a 165 x 165 x 150 volume with a
resolution of 8.2 voxels/cm (or a voxel size of .122 cm 3 ),
which is better than the highest resolution of the
ultrasound machine/transducer combination.
After reconstruction, objects such as uterus and
placenta were edited out manually using an editing mask
with Gaussian fall-off to avoid introducing artifacts in the
volume. Finally a small 3D Gaussian filter (standard
deviation 2 voxels) was applied to the volume.
The abdomen geometry sweep had failed due to a
minor technical problem during the live scan; we derived
abdomen geometry data by triangulating a number of
small structures visible on the skin of the abdomen in
frames videographed from different viewpoints and
extracted from the HMD video sequence.
The reconstructed and edited volume (Plate 5) and
the pit were rendered by vol2 with optical distortion (Plate
5, inset); the CG sequence was combined with the HMD
camera sequence (Plate 6).
6 Conclusion and future directions
Many aspects of our experiment suffered from lack of
immediate, real-time 3D feedback. During HMD video
acquisition, the HMD wearer could not really see inside
the patient; thus we unfortunately ended up looking at the
patient from the “wrong” side and thus viewing the fetus
from behind in the resulting composite video sequence.
During echography acquisition, we were unable to gather
enough echography slices to reconstruct a complete fetus
due to lack of real-time 3D feedback on the geometry of
the scanned anatomy. Still, in separately generated
images from viewpoints chosen more advantageously than
those of the HMD camera during the HMD video
acquisition, one can recognize more anatomical features
(Plate 5).
What resources would be required to present an online
visualization of comparable quality in an augmentedreality
system? We expect advances in volume rendering
software and hardware to soon provide high-speed
stereoscopic rendering of volumetric data sets (see for
example [13]). As for reconstruction, our volume
contained nearly 4 times as many voxels as the one used
in the 1993 experiment. Since the latter was being
reconstructed at a rate of about 1 Hz, we need an increase
of 2 orders of magnitude in computational speed for the

econstruction subsystem.
We learned from the January 1993 experiment and
others that, besides the image generation frame rate, short
lag in both volume reconstruction and visualization is
very important. Our hybrid system has avoided this
problem through off-line processing. For an on-line realtime
system, however, we need to design and implement
hardware and algorithms that provide not only high
throughput but also short lag. In addition, we need fast,
minimal-lag, high-precision tracking.
In the area of our application—visualizing ultrasound
as a “flashlight” into the body—we conclude that a step
forward has been achieved. The sequence showing a fetus
registered inside the pregnant subject provides a “brass
standard” (if not a gold one) as a target for our next
real-time efforts. One way to acquire the high amounts of
computing resources needed for such efforts is through
the current research on high-bandwidth links between
powerful computing stations, which hints at
computational capabilities that might be available in the
next few years and are within the desired range of power.
In general, complex visualizations presented within
augmented vision systems make greater application
demands than either virtual environments or scientific
visualization individually. Any closed virtual
environment or scientific visualization system lacks the
error-emphasizing cues that a combined system provides.
An augmented reality application provides sufficient
information to enable the user to easily notice registration
errors, tracker lag, computational errors, calibration errors
and real time delays which destroy the attempted illusion.
Augmented systems or any virtual reality system dealing
directly with the real world will not be easy to create.
Simple applications with little computational demand,
such as overlays for wiring guides or informational
pointers, will be able to get by, but applications with
complex visualization goals will be heavily burdened by
the demands. Researchers should be sensitive to these
issues and attempt to evaluate carefully their impact in the
intended application.
7 Acknowledgments
Jack Goldfeather provided mathematical models for
the echography calibration procedure. Gary Bishop wrote
filtering software for the tracking data. Nancy Chescheir,
M.D. and Vern Katz, M.D. were our ultrasonographers.
Kathryn Y. Tesh and Eddie Saxe provided experimental
assistance. Scott Shauf created an early version of figure
1. We thank our anonymous subject for her patience.
Funding was provided by ARPA (ISTO DABT
63-92-C-0048 and ISTO DAEA 18-90-C-0044) and by
CNRI / ARPA / NSF / BellSouth / GTE (NCR-8919038).
8 References
1. McCann, H.A., Sharp, J.S., Kinter, T.M., McEwan, C.N.,
Barillot, C., and Greenleaf, J.F. “Multidimensional
Ultrasonic Imaging for Cardiology.” Proc. IEEE 76.9
(1988): 1063-1073.
2. Lalouche, R.C., Bickmore, D., Tessler, F., Mankovich, H.
K., and Kangaraloo, H. “Three-dimensional reconstruction
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1989. 59-66.
3. Pini, R., Monnini, E., Masotti, L., Novins, K. L.,
Greenberg, D. P., Greppi, B., Cerofolini, M., and Devereux,
R. B. “Echocardiographic Three-Dimensional
Visualization of the Heart.” 3D Imaging in Medicine. Ed.
Fuchs, H. Höhne, K. H., and Pizer, S. M. NATO ASI
Series. Travemünde, Germany: Springer-Verlag, 1990. F
60: 263-274.
4. Tomographic Technologies, GMBH. 4D Tomographic
Ultrasound, A clinical study. 1993.
5. Ganapathy, U., and Kaufman, A. “3D acquisition and
visualization of ultrasound data.” Visualization in
Biomedical Computing 1992. Chapel Hill, NC: SPIE, 1992.
1808: 535-545.
6. Ohbuchi, R., Chen, D., and Fuchs, H. “Incremental volume
reconstruction and rendering for 3D ultrasound imaging.”
Visualization in Biomedical Computing 1992 . Chapel Hill,
NC: SPIE, 1992. 1808: 312-323.
7. Bajura, M., Fuchs, H., and Ohbuchi, R. “Merging Virtual
Objects with the Real World: Seeing Ultrasound Imagery
within the Patient.” Computer Graphics (Proceedings of
SIGGRAPH’92) 26.2 (1992): 203-210.
8. Wang,, J., Chi, V., and Fuchs, H. “A Real-Time Optical
3D Tracker for Head-Mounted Display System.” Computer
Graphics (Proceedings of 1990 Symposium on Interactive
3D Graphics) 24.2 (1990): 205-215.
9. Gottschalk, S. “Autocalibration for Virtual Environment
Tracking Hardware.” Computer Graphics (Proceedings of
SIGGRAPH’93) 27 (1993): 65-72.
10. “Digital Low-Pass Filter Without Phase Shift.” NASA Tech
Briefs KSC-11471. John F. Kennedy Space Center, Florida.
11. Ohbuchi, R. “Incremental Acquisition and Visualization of
3D Ultrasound Images.” Doctoral dissertation. University
of North Carolina at Chapel Hill, Computer Science
Department, 1994.
12. Neumann, U., State, A., Chen, H., Fuchs, H., Cullip, T. J.,
Fang, Q., Lavoie, M., and Rhoades, J. “Interactive
Multimodal Volume Visualization for a Distributed
Radiation-Treatment Planning Simulator.” Technical
Report TR94-040. University of North Carolina at Chapel
Hill, Computer Science Department, 1994.
13. Cullip, T. J., and Neumann, U. “Accelerating Volume
Reconstruction With 3D Texture Hardware.” Technical
Report TR93-027. University of North Carolina at Chapel
Hill, Computer Science Department, 1994.

Please reference the following QuickTime movie located in the MOV
directory:
STATE1.MOV ( Macintosh only)
Copyright © 1994 by University of North Carolina
QuickTime is a trademark of Apple Computer, Inc.

New Techniques in the Design of Healthcare Facilities
Tarek Alameldin Mardelle Shepley
The Visualization Laboratory Department of Architecture
Texas A�M University Texas A�M University
College Station� TX 77843�3137 College Station� TX 77843�3137
Abstract
The recent advent of computer graphics techniques
has helped to bridge the gap between architectural con�
cepts and actual buildings. Closing this gap is espe�
cially critical in healthcare facilities. In this paper� we
present new techniques to support the design decision
process and apply them to the design of a neonatal in�
tensive care unit. Two issues are addressed� ergono�
metric accessibility and visual supervision of spaces.
These two issues can be investigated utilizing new tech�
nologies that demonstrate that computers are more
than a medium of communication in the �eld of ar�
chitecture� the computer can make a signi�cant con�
tribution as a proactive design tool.
1 Overview
This paper describes an interdisciplinary project
between faculty of the Visualization Laboratory and
the Department of Architecture at Texas A�M Uni�
versity.
The need to design appropriate environments is
most keenly felt in the area of healthcare facility ar�
chitecture. The built environment can in�uence health
outcomes by mitigating the impact of stressful envi�
ronments �16� and supporting the processes required
to provide a high level of patient care. In some cases�
good facility design and appropriate space adjacencies
may enhance the ability of medical sta� to save lives.
The key to good design of healthcare facilities is
accurate and thorough communication with the client
during the design process. The traditional techniques
utilized by design professionals to enhance interaction
have included interviews and questionnaires� post�
occupancy evaluations� behavioral observation� gam�
ing �tools that enable design team members partici�
pate directly in the design process by manipulating
game pieces that represent the elements of the envi�
ronment� and full�scale mockups. Of these techniques�
full�scale mockups provide the most veridical informa�
tion about the nature of an environment� although
they are expensive and time�consuming to build� and
inappropriate for large components of the design.
Unfortunately� none of these techniques� including
mockups� provide su�cient information to enable ar�
chitects to account for all signi�cant implications of
a proposed design. With the advent of virtual real�
ity technology and computer graphics� however� a less
cumbersome and more accurate alternative to these
techniques is available. Using such technology� the
healthcare client is empowered to participate directly
in the design process and the architect is provided the
opportunity to fully understand the implications of
his�her design. A designer can test concepts about a
space� a component of a building�s interior�exterior�
or the use of a space by representing� analyzing� and
re�ning the design before it is adopted. Visualization
makes it possible for designers to preview the results
of their theories via pictures or animations.
In this paper� which utilized drawings of a proposed
intensive care nursery by The Design Partnership� Ar�
chitects� the visualization construct demonstrates that
the project team can experience the medical setting
before it is built� nurses and physicians can watch a
sta� member in action� working within the environ�
ment. Such opportunities allow administrators� sta�
and designers to communicate more e�ciently during
the design process. Additionally� by allowing the de�
sign team to pre�experience the space� controversial
innovations can be tested without fear of failure.
While the application in this project focused on
interior space� the technology is also intended to il�
lustrate the implications of entire buildings and sites.
Support during the design process can be given at a
variety of environmental scales ranging from master
planning to furniture and equipment design. Other as�
pects of a potential project that can be studied include
the adequacy of barrier�free design� the implications of
spatial volumes and the e�ciency of way��nding �the
ability to understand the layout of a building and ar�
rive at a destination�. With regard to way��nding�

hospitals are notoriously confusing to negotiate� and a
whole specialty area in the design profession has been
developed to enhance the cognitive clarity of a build�
ing �journey�.
Additionally� the technology can be used to evalu�
ate code compliance. This is often neglected in the
early stages of the design process� but increasingly
complex laws and professional liabilities reinforce the
importance of considering these factors throughout
the development of a project. Our innovative tech�
niques will enable architects to think about these fac�
tors at the appropriate time.
The architect can focus on particular aspects of a
design by using separate programs that perform calcu�
lations on speci�c properties �14� 15�. In this paper� we
used human �gure models within the 3D CAD mod�
els in order to experience the spaces and the ergono�
metric accessibility of di�erent objects in the environ�
ment. The problem of accessibility cannot be easily
understood by using traditional representations.It is
also di�cult to imagine what a person can see in a
space by using traditional drawings and CAD system
abstractions. For the neonatal intensive care unit de�
sign� we were able to address ergonometric accessibil�
ity and visual supervision issues by integrating human
�gure models into our geometric CAD models prior to
construction. Our goal is to provide the design team
with an understanding of the completed architecture
design before it is built.
2 Issues and Results
In this section� two problems associated with the
design of physical environment and the results of our
study are discussed.
2.1 The Ergonometric Accessibility Issue
In recent years� the architectural community has
placed an emphasis on ergonometric compatibility in
workspace design. In response to this approach� we
developed techniques utilizing redundant articulated
chain physiology modeling �2� 3� 4� 1�. An articulated
chain is a series of links connected with either revolute
or prismatic joints such as a robotic manipulator or a
human limb.
The reachable workspace of an articulated chain is
the volume or space encompassing all points that a
reference point P on the hand �or the end e�ector�
traces as all the joints move through their respective
ranges of motion �11�. Realistic human �gure models
are based on anthropometric characteristics such as
link �segment� dimensions and joint limits. Di�erent
sized �percentile� human �gures are modeled as pa�
rameterized anthropometric articulated chains. An ef�
�cient system for 3D workspace visualization of redun�
dant articulated chains is utilized in simulating task�
oriented activities of di�erent size�percentile agents.
Three�dimensional workspace visualization has ap�
plications in a variety of �elds such as computer�
aided design� human �gure modeling� ergonomic stud�
ies� computer graphics and arti�cial intelligence� and
robotics. In computer�aided design� the three dimen�
sional workspace of a human limb or robotic manip�
ulator can be used in the interior design of build�
ings. Computer graphics human �gure modeling has
been used in cockpit and car occupant studies� space
station design� product safety studies� and mainte�
nance assessment �7� 5� 9� 10�. An interactive system
which assists the designer or a human factors engi�
neer to graphically simulate task�oriented activities of
di�erent human agents has been under development
�6� 12� 13�. In the ergonomic experiments� this sys�
tem was used to study the e�ect of limb lengths and
joint limits on the reach capabilities of di�erent size�
percentile people. In addition� this system can also
be used to test design solutions for compliance with
American Disabilities Act �ADA� experiments by cre�
ating human models limited by speci�c physical dis�
abilities. Finally� this system can be used in visualiz�
ing sequences of human movements in order to attain
the optimal working environment.
2.2 The Visual Supervision Issue
Traditionally� the design process has emphasized
the two�dimensional relationship between spaces.
Even with the support of models and perspective
drawings� it is di�cult to precreate the visual experi�
ence of moving through a building. Utilizing the tech�
nology developed in this paper� however� the architect
and the client can visualize and experience what a hu�
man can see inside the building during the early stages
of the design process by attaching a camera to the
eyes of the simulated human �gure. A selected �eld of
view is displayed as a translucent pair of view cones�
one for each eye �8�. Viewable objects are shadowed
by the translucent cones which move with the eyes� as
illustrated in Figure 3.
2.3 Result Summary
The information contained in this spatial visualiza�
tion enables architects and clients to validate design
decisions. In this application� access to outlets and
equipment as well as quality of visual supervision of
the intensive care nursery and monitors was examined.
Figure 1 illustrates an overview of a nursery bay
looking toward an exterior window. This scene en�
ables sta� to visualize the impact of windows on the

nursery environment. A view of baby station showing
cabinet and isolette location is shown in Figure 2. The
greater detail provided here allows medical sta� to un�
derstand the scale of the workspace. Finally� Figure 3
illustrates the view from the infant�s vantage point in�
side an isolette. Addressing the needs of both patients
and sta� is essential for good health facility design.
A video tape has been developed to describe these
new visualization techniques as applied to healthcare
facilities design.
3 Conclusions and Future Work
The recent advent of computer graphics techniques
has helped to bridge the gap between the conceptual
and the actual building. In this paper� we found that
the following steps are necessary to re�ne the technol�
ogy�
� New fast�rendering algorithms are needed to pro�
duce more realistic pictures in a shorter amount
of time.
� A simulation of the acoustical environment should
be integrated into the visualized environment to
enhance the veridicality of the experience.
� New visualization tools such as stereo glasses wll
further enhance the 3D experience.
� Multi�level systems need to be designed that will
allow a trade�o� between time and quality.
While work remains to be done to complete the de�
velopment of a virtual reality technology that can
be readily utilized by the average design professional�
the basic technology is already available. Ultimately�
these computer techniques are likely to revolutionize
the design process.
Acknowledgments
We wish to thank Karl Maples �the Visualization
Laboratory�� Jean�Claude Kalache �the Architecture
Department�� and David Walvoord �the Computer
Science Department� for their e�ort in developing the
video associated with this paper.
This research was supported in part by NSF Grants
DUE�9254357� USE�925 1461� Research Enhancement
Program� O�ce of the Associate Provost for Research
and Graduate Studies� Texas A�M University� and
College of Architecture Research Enchancement Pro�
gram� Texas A� M University. Questions regarding
this article can be directed to tarek�viz.tamu.edu.
References
�1� T. Alameldin. Three Dimensional Workspace
Visualization for Redundant Articulated Chains.
PhD thesis� University of Pennsylvania� 1991.
�2� T. Alameldin� N. Badler� and T. Sobh. An Adap�
tive and E�cient System for Computing the 3�D
Reachable Workspace. In Proceedings of IEEE In�
ternational Conference on Systems Engineering�
1990.
�3� T. Alameldin� M. Palis� S. Rajasekaran� and
N. Badler. On the Complexity of Computing
Reachable Workspaces for Redundant Manipula�
tors. In Proceedings of SPIE Intelligent Robots
and Computer Vision IX� Algorithms and Tech�
niques� 1990.
�4� T. Alameldin and T. Sobh. A Hybrid System for
Computing Reachable Workspaces. In Proceed�
ings of SPIE Machine Vision Systems Integra�
tion� 1990.
�5� N. Badler. Articulated Figure Animation� Guest
Editor�s Introduction. IEEE Computer Graphics
and Applications� June 1987.
�6� N. Badler. Arti�cial Intelligence� Natural Lan�
guage� and Simulation for Human Animation. In
N. Magnenat�Thalmann and D. Thalmann� ed�
itors� State�of�the Art in Computer Animation�
pages 19�31. Springer�Verlag �New York�� 1989.
�7� N. Badler� K. Manoochehri� and G. Walters.
Articulated Figure Positioning by Multiple Con�
straints. IEEE Computer Graphics and Applica�
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�8� N. Badler� C. Phillips� and editors B. Webber.
Simulating Humans� Computer Graphics Anima�
tion and Control. Oxford University Press� 1993.
�9� M. Dooley. Anthropometric Modeling Programs �
A Survey. IEEE Computer Graphics and Appli�
cations� 2�9��17�25� November� 1982.
�10� W. Fetter. A Progression of Human Figures Sim�
ulated By Computer Graphics. IEEE Computer
Graphics and Applications� 2�9��9�13� November�
1982.
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Mechanical Manipulator. ASME Journal of Me�
chanical Design� 103�665�672� July 1981.

�12� P. Lee� S. Wei� J. Zhao� and N. Badler. Strength
Guided Motion. Computer Graphics� pages 253�
262� 1990.
�13� C. Phillips� J. Zhao� and N. Badler. Interactive
Real�Time Articulated Figure Manipulation Us�
ing Multiple Kinematic Constraints. Computer
Graphics� 24�2��245�250� 1990.
�14� G. Schmitt. Computer Graphics in Architecture.
In N. Thalman and D. Thalman� editors� New
Trends in Animation and Visualization� pages
153�163. John Wiley � Sons� 1991.
�15� G. Schmitt. Virtual Reality in Architecture. In
N. Thalman and D. Thalman� editors� Virtual
Worlds and Multimedia� pages 85�97. John Wi�
ley� 1993.
�16� R. Ulrich. E�ects of Interior Design on Wellness�
Theory and Recent Scienti�c Research. Journal
of Health Care Interior Design� 3�97�109� 1991.

Case Study: Visualization of an Electric Power Transmission System
Pramod M. Mahadev Richard D. Christie
pramod@u.washington.edu christie@ee.washington.edu
Abstract
Visualization techniques are applied to an electric
power system transmission network to create a graphical
picture of network power flows and voltages. A geographic
data map is used. Apparent power flow is encoded
as the width of an arrow, with direction from real
power flow. Flows are superposed on flow limits. Contour
plots and color coding failed for representing bus voltages.
A two-color thermometer encoding worked well.
The resulting visualization is a significant improvement
over current user interface practice in the power industry.
1. Introduction
Electric power systems are operated by dispatchers located
in a control center equipped with a process control
computer system called an Energy Management System
(EMS). [1] The EMS collects data from locations
throughout the power system, allows the dispatchers to
control power system equipment in remote locations and
executes analysis functions on the collected data. EMSs
have a complex user interface based on CRT displays and
keyboard and pointing device input.
The power system itself is a large electrical network,
with hundreds of buses (nodes) connected by hundreds of
transmission lines and transformers (branches) operating
in sinusoidal steady state. Existing CRT displays can
show a few of the many collected and computed voltages
and power flows in numerical form placed on a small
portion of the complete circuit diagram.
Because of the size of the power system, dispatchers
can suffer severely from information overload in normal
operation, and even more so, and more critically, in
casualties. The EMS user interface is therefore a logical
candidate for the application of visualization techniques.
2. The Visualization
Department of Electrical Engineering, FT-10
University of Washington
Seattle, WA 98195
(206) 543-9689 FAX: (206) 543-3842
Principles for effective visualization [2], guidance
found in Tufte [3,4] and from scientific visualization [5]
are used to create an improved representation for power
system operating state for use in the routine monitoring
task, specifically the line power flows and bus voltage
magnitudes of a power system. The objective of this representation
is to give a global view of the line flows and
bus voltages, to communicate information about proximity
to limits, and to call attention to limit violations.
A first principle is to show as much of the data as possible,
ideally all of it. A geographic data map of the entire
power system is chosen as the most appropriate representation.
There are two commonly used graphical representations
for the transmission system found in engineering
drawings: geographic and orthogonal. The geographic
representation shows the actual routes of the lines between
substations, while the orthogonal representation is
more like a circuit diagram, with the lines laid out with
right angle bends. Even in the orthogonal representation,
the substations are arranged approximately by geography.
Many dispatchers tend to discuss system state problems in
conjunction with a geographic map. The major advantage
of a geographic representation appears to be that it engages
the natural human ability to remember and to reason
about geographic data. The direction of arrival of
transmission lines at substations may be an important
visual cue. For this reason, a representation is used that
places the substations in their approximate locations, and
connects them with straight line segments representing
the transmission lines. The strict geographic layout is
modified to show transformers as lines of some minimum
length. They are an important part of the overall system
flow picture and of flow limitations, but would be invisible
in a strict geographic representation.
2.1 Line Flow Encoding
Graphical encodings work best when they are intuitive.
It is easy to recognize such natural encodings, but hard to
devise them. For flow, Minard successfully encoded the

flow of wine exports as the width of paths. [3] The flow of
power may be similarly encoded. The encoding of flow as
width appears to be a useful natural encoding related to
the effect of water flow in a stream. As the amount of
flow increases, the stream gets deeper and wider. When
too much flow exists, it overflows its banks. This suggests
encoding power flow as line width, and power flow limits
also as a width. Limits are colored gray, to set them further
into the background, while a subdued brown color is
used for flow.
Because line flows and limits are commonly discussed
in terms of apparent power (MVA), apparent power is
encoded. However, apparent power has no direction, and
direction of power flow is important to the dispatcher.
Direction, encoded with an arrowhead, is taken from real
power (MW) (Fig. 1). While it would be ideal to also see
reactive power flow (MVAR), an effective encoding for
presenting both real and reactive power on one display
has not yet been developed. The nature of the monitoring
task places a higher importance on the MVA and MW
direction.
Line flow and limits are encoded as actual MVA, not as
a percentage of limits, using the same scale for every line.
The human eye can easily extract the approximate percentage
loading from the representation of the actual values,
while the actual flow permits comparison with other
overload problems in the system.
Because an overload is a qualitatively different state,
the encoding should undergo a qualitative change. This is
done by changing the color of the arrow. Line flow limits
are displayed as a hollow box inside the arrow. These
visual cues, in addition to the width of the line flow arrow,
call attention to overloads (Fig. 2).
Relatively few variations of line flow encoding were
tried. Most involved getting rid of spontaneous chartjunk,
such as arrows with heads a different color than the body.
A few variations were significant. Shaded areas for limits
proved easier to comprehend than the original idea of
using an outline box. (Fig. 3)
In theory, all of the information about line flow and its
limits is encoded in one half of the line. The other side is
just redundant. Based on this reasoning, it was proposed
to split the line in half lengthwise and display only one
half. This way the same information would be displayed
in half the space, improving information density. (Fig. 4)
This reasoning proved entirely specious, when the encoding
was applied to the global display. (Fig. 5) Human
pattern recognition apparently prefers symmetry and has
trouble with asymmetrical encoding in random orientations.
Red was the first color chosen to indicate a limit violation.
This would be a natural encoding for the general
public, but power system dispatchers have a different redgreen
encoding that can create confusion. In the power
system world, closed switches and circuit breakers show a
red light, to indicate that the equipment is energized, and
therefore dangerous, while open devices show a green
light to indicate safety. However, in normal operation,
most devices are closed, so to a dispatcher, red means OK
and green means trouble! This crossed encoding was
avoided by adopting a bright yellow color, which unambiguously
means abnormality or caution, to indicate
overloads. This illustrates how special characteristics of a
specific user group can influence the visualization.
2.2 Bus Voltage Encoding
A workable representation of bus voltage magnitude
was more difficult to obtain. Bus voltages are not flows,
but can be thought of as height measurements at various
points in the power system. This led naturally to the idea
of a contour plot, similar to a contour map of a mountainous
region. (Fig. 7) Contour plots using contour lines, and
the use of colors to identify contour regions, were tried.
Neither representation seemed to provide a natural encoding.
It was difficult to distinguish between high points
and low points, and difficult to identify limit violations.
Areas of high or low voltage did not seem to have much
physical meaning because they were only vaguely associated
with the buses. A three dimensional perspective view
of the voltage contour surface, as used in [6], does make it
easy to distinguish between highs and lows, but moves the
voltage information even farther from the associated bus,
and has unsatisfactory loss of detail for large systems.
Furthermore, a clear display of all values in a realistically
sized system will require some dynamic element, such as
"nodding" of the entire display. This was felt to be unacceptable
in an operational environment. Contour mapping
was discarded as an encoding technique for bus voltages.
Perhaps the fundamental reason why contour plots of
voltage fail is that they are not fiducial representations of
the relationships of the encoded data. In a power system,
voltage only exists at the bus. The contour plot graphically
gives voltage values for points between buses, where
no voltage exists, and so misleads the user.
The next iteration was to identify voltages at each bus
in some way local to the bus. The progression of colors
adopted was the progression from the natural spectrum,
Red, Orange, Yellow, etc. It was thought that most dispatchers
would be familiar with this progression. Unfortunately,
they are familiar with the acronym, ROYGBIV,
and not with the colors themselves. It was not intuitively
obvious to a user that yellow was a higher voltage than
red. Even with a key, this was not a useful encoding. Gray

scale was also tried, with disappointing results. Colorbased
encoding of voltages was abandoned. In retrospect,
an encoding based on visual temperature (red-yellowwhite)
might have been more successful.
Next, a thermometer approach was tried. The first implementation
used a pointer on a vertical scale. (Fig. 6a)
The pointer was effective at communicating individual
bus voltages, but did not provide a global perspective.
Users had to focus on each bus in turn to see the voltages.
A colored bar, however, proved more visible. The bus
symbol used was a hollow square. The bottom of the box
represented the low voltage limit, and the top of the box
the high voltage limit, typically 0.95 and 1.05 per unit.
The box was filled with a distinctive color from the bottom
up, with the amount of fill proportional to the bus
voltage. (Fig. 6b) This allowed easy identification of patterns
of voltage in the power system by looking for concentrations
of the fill color on the representation. However,
only high voltage patterns could be seen. Low voltages
which are usually operationally more important,
were not as visible.
To improve task specificity, the box was then filled
from the top down. (Fig. 6c) This made low voltages
more visible, at the expense of high ones, but it was also
counter-intuitive. Humans seem to expect things to fill
from the bottom up. In frustration, starting the bar at the
midpoint of the outline box was tried. (Fig. 6d) This was
a total failure, since high and low voltages had the same
amount of color, so no global patterns could be seen,
while extracting individual voltages remained confusing.
The encoding which finally worked combines two approaches,
filling the box from the bottom up with one
color, and from the top down with a contrasting hue. The
bus voltage value is shown by the position of the dividing
line between the two colors. (Fig. 6e) As bus voltage increases,
the line moves up in the box until it exceeds the
limit, where the color outside the limit changes to the
violation color and the box is outlined to show that limits
are being exceeded. (Fig. 6f) The colors chosen are blue
and green, since they contrast with each other are not
used elsewhere on the display.
The combination of line flow and bus voltage representations
has been implemented for the IEEE 118 bus test
case (Fig. 8). The displays were implemented in X-
Windows with X-Toolkit and Athena widgets. They have
been run on both Sun Sparc and HP workstations. A Visual
C++ implementation has also been made.
3. Assessment
The developed visualization permits users to “see” the
power system in a way that is completely new. Power
system state can be taken in at a glance, rather than laboriously
extracted from inspection of many different displays,
after much user interface manipulation. Rapid
identification of problems should permit corrective actions
to be taken promptly enough to avert certain types of
power system collapses. Fig. 9 shows multiple overloads
and bus violations due to a significant load increase. All
of the problems, and their relationships, are clear at a
glance.
The visualization also makes power system operating
conditions more accessible to non-technical persons such
as managers and regulators, and should improve the integration
of technical considerations into financial and
regulatory decisions.
Potential future research directions include usability
testing to obtain objective measures of the benefits of
visualization, integration of visualization into the complex
EMS user interface, application of visualization to
additional dispatcher tasks, and attention to the planning
problem, which may include visualization of power system
dynamics.
4. Acknowledgments
This work has been supported in part by the National
Science Foundation under award number ECS-9058060.
5. References
[1] "Graphics Put Control Room in the Picture," Modern
Power Systems, Vol. 12, No. 4, pp. 43-45, April
1992.
[2] Electric Power Research Institute, Visualizing Power
System Data, IEEE Transactions on Power Systems,
Report TR-102984, EPRI, Palo Alto, California,
April 1994.
[3] E.R. Tufte, The Visual Display of Quantitative Information,
Graphics Press, Cheshire, Connecticut,
1983.
[4] E.R. Tufte, Envisioning Information, Graphics Press,
Cheshire, Connecticut, 1989.
[5] G.M. Nielson and B. Shriver, Visualization in Scientific
Computing, IEEE Computer Society Press,
Los Alamitos, CA, 1990.
[6] Electric Power Research Institute, Advanced Graphics
for Power System Operation, Project Report
RP4000-13, EPRI, Palo Alto, California, August
1993.

Figure 1 - Width Encoding of Line Flow
Figure 2 - Overloaded Line Flow Encoding
Figure 3 - Outline Box Encoding of Line Limits
Figure 4 - Single Sided Line Flow Encoding
Figure 5 - System Display With Single Side Flow
a b c d e f
Figure 6 - Bus Voltage Representations
Figure 7 - Voltage Contour Map
Figure 8 - IEEE 118 Bus Test Case
Figure 9 - Multiple Violations

Case Study: Volume Rendering of Pool Fire Data
H. E. Rushmeier and A. Hamins M.-Y. Choi
CAML and BFRL School of Mech. Engr.
NIST Univ. of Illinois at Chicago
Gaithersburg, MD 20899-0001 Chicago, IL 60607-7022
Abstract
We describe how techniques from computer graphics are
used to visualize pool fire data and compute radiative effects
from pool fires. The basic tools are ray casting and accurate
line integration using the RADCAL program. Example
images in the visible and infrared band are shown which
give qualitative insights about the fire data. Examples
are given of irradiation calculations and novel methods to
visualize the results of irradiation calculations.
1 Overview
In a pool fire, a puddle or pool of liquid fuel is ignited
and burns in the atmosphere. Understanding pool fires is
important to devising methods to control the impact of hazardous
situations resulting from spilled fuels. In this paper
we consider techniques for visualizing the data measured
in pool fires, and for computing the radiative transfer from
pool fires. Combustion is a challenging example for development
of visualization techniques. Fires are turbulent,
non-steady, and multi-wavelength in emission. Fire data is
an example of a class of visualization problems for which
it is important to consider the radiative simulation used to
generate visualizations.
We see objects as a result of the visible light they emit,
transmit or reflect. All rendering methods in some way
simulate the radiative transfer of light [4]. When visualizing
abstract representations of objects – such as molecular
models – simple, heuristic rendering methods are useful.
This is particularly true in volume visualization, in which
a complete physical simulation of light propagation can be
extremely computationally demanding [5]. However, in
some volumetric problems, radiative transfer is a critical
part of the phenomenon being visualized. For these problems,
using radiometrically accurate techniques to generate
images of the data can provide useful insights.
In the case of fire data, radiative transfer plays two important
roles. First, data is obtained from fires by detecting
radiation. Data is collected with probes inserted in the fire to
measure radiative transfer over short distances. Qualitative
information about the fire is obtained by simply observing
the luminosity of the flame. Second, a major quantity of interest
is the radiation from the fire that reaches surrounding
surfaces. For example, the flame can ignite adjacent pools
of fuel by radiative heat transfer. Using the techniques to
flame
pan of fuel
z
grid of
measurement
locations
Figure 1: Typical locations for taking pool fire data.
accurately image fire data, efficient calculations can also be
made about the impact of the fire on its surroundings.
In the particular experiments we consider in this paper,
data were taken for a 10 cm diameter heptane pool fire,
using a pair of opposed, water-cooled, nitrogen-purged,
intrusive probes separated by 12 mm, which defined the
optical path. Figure 1 diagrams the fire and typical measurement
locations. Because the time averaged flame is
cylindrically symmetric, data were taken only for varying
radii r from the center of the pool and for height z above
the pool. Local instantaneous temperature, soot volume
fraction and CO2 concentrations were determined from
radiation measurement in three wavelength regions (900,
1000 and 4350 nm.) These data were used to estimate the
concentrations of H2O and CO. Typical data locations are
shown in two-dimensional plots in Figure 2. Details of the
experimental procedure are given in [1]. These data were
taken to examine various features of radiative transfer from
the fire.
In this paper we consider how visualization and graphical
techniques can be used along with this data to answer
the following questions:
� How well do the measurements represent the actual
data? The two-dimensional plots shown in Figure 2
are difficult to relate to one another, and to the luminous
flame observed in the laboratory.
r

z
r
T, K
1450
300
CO2, kPa
3.6
0
soot
(volume fraction)
4.8x10−7 Figure 2: Typical temperature, CO2 partial pressure, and
soot volume fraction data for a pool fire. The data are
shown in two-dimensional, gray-scaled plots.
� What is the importance of the radiation from the gases
relative to the radiation from the soot? Usually in the
past only soot has been considered.
� What is the radiative feedback to the fuel surface?
How much heat does the flame contribute to the pool
surface for vaporization?
� What is the irradiation at points in the space around
the fire? At what distance from the fire would other
materials be ignited?
2 Visualization
The basic tool for imaging and computing radiative
transfer is ray casting. Figure 3 diagrams casting a ray
through the volume of data. Because data were only taken
at varying r and z, the volume of data is a stack of cylindrical
rings. Rays are simply followed from the observer (or
in the case of irradiation calculations from the collector position)
through the cylinder. For each ring encountered, the
next pierce point is found. If the wall of the ring is pierced,
the r index is incremented (or decremented) to find the
next ring encountered, else the z index is incremented or
decremented.
Once the complete traversal of the ray is found, a list is
constructed of the ring segments that the ray passes through.
For each segment distance, temperature, soot volume fraction
and gas partial pressure are collected. These data are
passed to the routine RADCAL [2] for performing the line
integration which gives intensity i�l� (energy per unit projected
area, time and solid angle, referred to as radiance in
the lighting and graphics literature) at the end of the ray
path l. RADCAL accurately calculates the radiation taking
into account the detailed spectral properties of CO2, H2O,
CH4, CO, N2, O2 and soot. In [2], RADCAL is presented
as an independent FORTRAN program (code is included
in the publication), however it can readily be adapted to
a callable subroutine. RADCAL evaluates the following
equations to obtain i�l�:
0
eye
z
Figure 3: Using the measured data, the fire is numerically
modelled as a stack of cylindrical rings, with uniform properties
in each ring. To generate images or calculate incident
irradiation, rays are cast through the numerical model and
the points where the ray pierces each cylindrical ring are
found.
Z �� �l�
0
Z �2
i�l� � i��l�d� (1)
�1
i��l� � i��we ��� �l� �
ib���l � �exp�����l� � ���l � ���d���l � �
�� �
Z l
0
a��l � �dl �
In the above equations � is wavelength, and �1 and �2
are the limits of the wavelength band of interest for the
calculation. The band can be the entire infrared spectrum
for irradiationcalculations, or narrow bands for the purpose
of examining radiation in the visible, or other bands of
interest. �� is the optical thickness of the medium, which
by definition is a function of the absorption coefficient a�
along the path. i��w is the spectral intensityat the beginning
of the ray path. For the pool fire problem, this is taken to be
the black body intensity at normal room temperature (about
300 K). ib���l � � is the blackbody intensity along the path.
RADCAL considers only absorption and emission along
the path – not scattering, since the effect of scattering is
negligible for the problem of radiation in combustion products.
RADCAL assumes local thermodynamic equilibrium
(LTE), which is an accurate assumption for the radiative
transfer of interest in thermal problems. It should be noted
that non-LTE radiation, luminescence which occurs by excitation
other than thermal agitation, is responsible for a
significant amount of visible radiation in some types of
flames. However visible radiation in most fires, such as the
heptane pool fire considered in this paper, are dominated
by LTE radiation from soot particles.
r

Figure 4: See color plate at end.
One way to evaluate the measurements is to use a camera
model which is sensitive to wavelengths throughoutthe
infrared. Another way is to view the measured data in the
same manner as the fire is physically observed. The luminous
flame is imaged by computing radiances in the visible
band and performing appropriate perceptual transformations
to produce a true color image. This would be done
most accurately by using RADCAL to calculate spectral intensities
in the range of 400 nm to 700 nm, convolvingthese
intensities with the CIE X���, Y ��� and Z��� functions
for the human visual system, and converting from XY Z
coordinates to RGB using measured monitor characteristics
(e.g. see [3]). Because the radiated spectrum from the
soot is relatively smooth, to generate images rapidly, we
sampled the spectrum at the peaks of the X� Y� Z curves
only to estimate the ratios X : Y : Z. The sample spectral
intensities are in units of watts per meter squared-steradian.
These intensities must be scaled to CRT display values in
the range 0-255. Again, to generate images rapidly, the
results were displayed by simply scaling the average luminance
value to 128 in the final image.
3 Discussion
Figure 4 shows time-averaged images of a pool fire. For
purposes of comparison, Fig. 4a is the average of 30 images
grabbed from video tape of the actual pool fire. An
uncalibrated video camera was used, and the flame colors
in the video are quite desaturated compared to the colors
observed directly in the laboratory. Figure 4b shows a synthetic
image generated from measured data. Two thousand
measurements were taken at each spatial location. Figure
4b was generated using the average of all 2000 values
at each location. As expected, the synthetic image shows
that the spacing of the measurements doesn’t represent the
geometric detail of the flame. However, the measurements
are adequate to capture the necking in of the flame near the
fuel surface, and broadening further above the surface.
Figures 4c and d illustrate the difference between radiation
from time averaged measurements, and time averaged
images from a time series of measurements. The image in
Fig. 4c as generated using the average of 30 measurements
at each location. The image in Fig. 4d was generated by
averaging 30 images, each generated by using just one of
the 30 measurements at each location. Because intensity
values must be clipped for each image, the image in 4d
is desaturated in a similar manner as the averaged video
image in 4a. A comparison of Figs. 4c and d shows that
there is a substantial difference between the radiation from
averaged data and the average radiation. This indicates
that just as spatially detailed data is needed, rather than
data averaged over the volume, temporally detailed rather
than time-averaged data is needed to accurately estimate
radiation transfer from the flame.
Note that in all of the synthetic images in Fig. 4 no
numerical legend is shown. This is because these are visual
simulations, not pseudo-colored displays.
Intensity,
W/m 2 str
3600
0
gas + soot soot only
Figure 5: Infrared images of radiation from soot and gas,
versus soot alone.
Insight into the relative importance of soot versus gas
radiation can be gained by imaging the fire using a synthetic
camera which includes wavelengths through the entire infrared
range. Figure 5 shows the resulting image when temperatures,
soot volume fraction. and gas partial pressure
data are given to RADCAL (on the right), versus the image
which results when temperatures and soot volume fraction
data only are used. A comparison of the figures shows that
the gases are responsible for a significant amount of radiation.
A numerical legend is shown on these images, since
the gray shades are assigned based on the total intensity
visible at each pixel, not on perceptual principles.
The same ray casting and line integration techniques
used to generate images can be used to compute irradiance
on surfaces around the fire. For example, the radiative
feedback per unit area, q00�r� to a point on the pool surface
at radius r is given by:
q00�r� �
Z 2�
0
Z ��2
i��� ��cos���d�d� (2)
0
The angles � and � are respectively the polar and azimuthal
angles of a coordinate system based on the pool
surface. The intensity i��� �� is the incident intensity from
direction ��� ��. i��� �� is found by casting in a ray in the
direction ��� �� and computing i�l� for that direction, as in
Eq. 1. The integral can be evaluated using a Monte Carlo
method, summing up the results of casting rays in a large
number of randomly chosen directions. Figure 6 shows
typical results for the fire data given in Figure 2. Figure 6
also shows small, wide angle images (136 degree field of
view) generated from each of the points for which q00�r�
was calculated. These images demonstrate that the fall off
in irradiation is primarily due to the decreasing solid angle
subtended by the hot products of combustion. (Note, these
results are only shown as examples of the calculations. Details
of numerical results for total radiative feed back to the
pool fire can be found in [1]).
The incident flux q00 at other surfaces around the fire is
also of interest, to study the potential for ignition of other

3000
Irradiation , W/m2
1500
0.00 0.05
radius, m
Figure 6: Irradiation of the pool surface by the hot combustion
products. The small images show a wide angle view
image of the view of the combustion products at the various
radial positions looking upward from the pool surface.
materials. The incident flux is a function of position in the
volume around the fire, and orientation relative to the fire.
To estimate the flux in the space around the fire, Eq. 2 was
evaluated for a grid of points in the space around the fire,
with a surface normal vector oriented towards the center of
the fire to estimate the maximum flux. Because the fire data
is rotationally symmetric, the results can be diagrammed in
two dimensions as shown in Figure 7. However, this does
not give a good sense of the meaning of the data to a casual
observer – it doesn’t communicate that the irradiation is
spread through space and that it applies to surfaces oriented
towards the fire. In Figure 8, a volumetric rendition of
the irradiation data is shown. A collection of spheres are
colored according to the maximum incident flux at that
location. A simple lighting heuristic is used to highlight
the sphere in the direction of the fire, to emphasize the role
of orientation in the flux calculation.
4 Summary and Future Work
We have shown how techniques from computer graphics
can be used to gain insight into data from pool fires. Visible
images generated using measured data demonstrated
how well the measurements represented the fire. Images
generated over the entire infrared spectrum demonstrated
the relative importance of gas versus soot emission. The
same ray casting that was used to generate images was used
to calculate irradiation from the fire. Graphical techniques
were further used to illustrate the results of the irradiation
calculations.
The data in the examples given in this paper were obtained
using an immersive probe. Such a probe has the
disadvantage that the entire temperature and emissivity distributioncannot
be obtained simultaneously. An alternative
approach for obtaining fire data is imaging, and applying
techniques from tomography. A potential future role for
synthetic images of fire data may be in the design of such
imaging systems.
0.5
z,m
0.4
0.3
0.2
0.1
1500
1200
900
600 W/m2
0.1 0.2 0.3 0.4 0.5r,m
Figure 7: Two-dimensional plot of contours of constant
irradiation for points around the pool fire.
References
Figure 8: See color plate at end.
[1] M.Y. Choi, A. Hamins, H. Rushmeier and T. Kashiwagi.
Simultaneous optical measurement of soot volume
fraction, temperature and CO2 in heptane pool
fire. To appear in The Proceedings of the 25 th Symposium
(Int’l) on Combustion .
[2] W.L. Grosshandler. RADCAL: A narrow-band model
for radiation calculations in a combustion environment.
NIST Technical Note 1402. National Institute
of Standards and Technology, April 1993.
[3] R. Hall. Illumination and Color in Computer Generated
Imagery. Springer-Verlag, 1989.
[4] J.T. Kajiya. The rendering equation. In Computer
Graphics (SIGGRAPH ’86 Proceedings), pages 143-
150. ACM SIGGRAPH, 1986.
[5] H.E. Rushmeier and K.E. Torrance. The zonal method
for calculating light intensities in the presence of a
participating medium. In Computer Graphics (SIG-
GRAPH ’87 Proceedings), pages 293-302. ACM SIG-
GRAPH, 1987.

0.5m
volume of
measured data
0.5 m
W/m2
3200
500
Figure 4:
a. (upper left) Average of
30 video frames of pool
fire.
b. (upper right) Synthetic
image generated using
data averaged from 2000
time samples.
c. (lower left) Synthetic
image generated using data
averaged from 30 time
samples.
d. (lower right) Synthetic
image generated by averaging
images of 30 time samples.
Figure 8:
Three dimensional
representation of irradiation
in the volume of space
surrounding the pool fire.
The color of each sphere
is determined by the
maximum irradiation at
that point. The shading
of each sphere is determined
by the direction to the
center of the fire.

Case Study� Visualization of Volcanic Ash Clouds
Mitchell Roth Rick Guritz
Arctic Region Supercomputing Center Alaska Synthetic Aperture Radar Facility
University of Alaska University of Alaska
Fairbanks� AK 99775�6020 Fairbanks� AK 99775
Abstract
Ash clouds resulting from volcanic eruptions are a
serious hazard to aviation safety. In Alaska alone�
there are over 40 active volcanoes whose eruptions
may a�ect more than 40�000 �ights using the great cir�
cle polar routes each year. The clouds are especially
problematic because they are invisible to radar and
nearly impossible to distinquish from weather clouds.
The Arctic Region Supercomputing Center and the
Alaska Volcano Observatory have collaborated to de�
velop a system for predicting and visualizing the move�
ment of volcanic ash clouds when an eruption occurs.
The output from the model is combined with a dig�
tal elevation model to produce a realistic view of the
ash cloud which may be examined interactively from
any desired point of view at any time during the pre�
diction period. This paper describes the visualization
techniques employed in the system and includes a video
animation of the 1989 Mount Redoubt eruption which
caused complete engine failure on a 747 passenger jet.
1 Introduction
Alaska is situated on the northern boundary of the
Paci�c Rim. Home to the highest mountains in North
America� the mountain ranges of Alaska contain over
40 active volcanoes� shown in red in Figure 1. In the
past 200 years most of Alaska�s volcanoes have erupted
at least once. Alaska is a polar crossroads where air�
craft traverse the great circle airways between Asia�
Europe and North America� as shown in white in Fig�
ure 1. Volcanic eruptions in Alaska and the resulting
airborne ash clouds pose a signi�cant hazard to more
than 40�000 transpolar �ights each year.
The ash clouds created by volcanic eruptions are
invisible to radar and are often concealed by weather
clouds. This paper describes a system developed by
the Alaska Volcano Observatory and the Arctic Region
Supercomputing Center for predicting the movement
of ash clouds. Using meteorological and geophysical
data from volcanic eruptions� a supercomputer model
provides predictions of ash cloud movements for up
to 72 hours. The AVS visualization system is used to
control the execution of the ash cloud model and to
display the model output in three dimensional form
showing the location of the ash cloud over a digital
terrain model.
Figure 1� Active volcanoes and air routes in Alaska.
Eruptions of Mount Redoubt on the morning of De�
cember 15� 1989� sent ash particles more than 40�000
feet into the atmosphere. On the same day� a Boeing
747 passenger jet experienced complete engine failure
when it penetrated the ash cloud. The ash cloud pre�
diction system was used to simulate this eruption and
to produce an animated �yby of Mount Redoubt dur�
ing a 12 hour period of the December 15 eruptions
including the encounter of the jetliner with the ash
cloud. The animation combines the motion of the
viewer with the time evolution of the ash cloud above
a digital terrain model.
The visualization of the aircraft encounter with the
ash cloud can be used to test the accuracy of the ash
plume model. The aircraft position was known ac�
curately in three dimensions at the time of the en�
counter and can be compared with the position of the
ash cloud produced by the model. Satellite imagery is
also used to assess the accuracy of the model by com�

paring the observed horizontal extent of the ash cloud
with the visualization produced from the model data.
2 Ash Plume Model
The ash cloud visualization is based on the out�
put of a model developed by Hiroshi Tanaka of the
Geophysical Institute of the University of Alaska and
Tsukuba University� Japan. Using meteorological
data and eruption parameters for input� the model
predicts the density of volcanic ash particles in the at�
mosphere as a function of time. The three dimensional
Lagrangian form of the di�usion equation is employed
to model particle di�usion� taking into account the
size distribution of the ash particles and gravitational
settling described by Stokes� law. Details of the model
are given in �2� 3�.
The meteorological data required are winds in the
upper atmosphere. These are obtained from UCAR
Unidata in NetCDF format. Unidata winds are inter�
polated to observed conditions on 12 hour intervals.
Global circulation models are used to provide up to
72 hour predictions at 6 hour intervals.
The eruption parameters for the model include the
geographical location of the volcano� the time and du�
ration of the event� altitude of the plume� particle den�
sity� and particle density distribution.
The raw output from the model for each time step
consists of a list of particles with an �x�y�z� coordinate
for each particle. An AVS module reads the particle
data and increments the particle counts for the cells
formed by an array indexed over �x�y�z�. We chose a
resolution of 150x150x50 for the particle density array�
which equals 1.1 million data points at each solution
point in time. For the video animation� we ran the
model with a time step of 5 minutes. For 13 hours
of simulated time� the model produced 162 plumes�
amounting to approximately 730 MB of integer valued
volume data.
3 Ash Cloud Visualization
The ash cloud is rendered as an isosurface with a
brown color approximating volcanic ash. The render�
ing obtained through this technique gives the viewer a
visual e�ect showing the boundaries of the ash cloud.
Details of the cloud shape are highlighted through
lighting e�ects and� when viewed on a computer work�
station� the resulting geometry can be manipulated
interactively to view the ash cloud from any desired
direction.
At any point in time� the particle densities in
the ash cloud are represented by the values in a
150x150x50 element integer array. The limits of the
cloud may be observed using the isosurface module
in the AVS network shown in Figure 2 with the iso�
surface level set equal to 1.
Figure 2� AVS network to generate plume isosurface.
As the cloud disperses� the particle concentrations
in the array decrease and holes and isolated cells be�
gin to appear in the isosurface around the edges of
the plume where the density is between zero and one
particle. These e�ects are especially noticeable in a
time animation of the plume evolution. To create a
more uniform cloud for the video animation� without
increasing the overall particle counts� the density ar�
ray was low pass �ltered by an inverse square kernel
before creating the isosurface. An example of a �ltered
plume created by this technique is shown in Figure 3.
Figure 3� Ash cloud visualized as an isosurface.
4 Plume Animation
The plume model must use time steps of 5 minutes
or greater due to limitations of the model. Plumes
that are generated at 5 minute intervals may be dis�
played to create a �ip chart animation of the time
evolution of the cloud. However� the changes in the
plume over a 5 minute interval can be fairly dramatic
and shorter time intervals are required to create the

e�ect of a smoothly evolving cloud. To accomplish
this without generating additional plume volumes we
interpolate between successive plume volumes. Using
the �eld math module� we implemented linear inter�
polation between plume volumes in the network shown
in Figure 4.
Figure 4� AVS plume interpolation network.
The linear interpolation formula is�
P �t� � P i � �P i�1 � P i�t� 0 � t � 1� �1�
where P i is the plume volume at model time step i
and t is time. The di�erence term in �1� is formed in
the upper �eld math module. The lower �eld math
module sums its inputs. Normally� a separate module
would be required to perform the multiplication by t.
However� it is possible to multiply the output port of
a �eld math module by a constant value when the
network is executed from a CLI script and this is the
approach we used to create the video animation of the
eruption. If it is desired to set the interpolation pa�
rameter interactively� it is necessary to insert a third
�eld math module to perform the multiplication on
the output of the upper module. This can be an ex�
tremely e�ective device for producing smooth time an�
imation of discrete data sets in conjunction with the
AVS Animator module.
One additional animation e�ect was introduced to
improve the appearance of the plume at the beginning
of the eruption. The plume model assumes that the
plume reaches the speci�ed eruption height instanta�
neously. Thus� the plume model for the �rst time step
produces a cylindrical isosurface of uniform particle
densities above the site of the eruption. To create the
appearance of a cloud initially rising from the ground�
we de�ned an arti�cial plume for time 0. The time 0
plume isosurface consists of an inverted cone of nega�
tive plume densities centered over the eruption coordi�
nates. The top of the plume volume contains the most
negative density values. When this plume volume is
interpolated with the model plume from time step 1�
the resulting plume rises from the ground and reaches
the full eruption height at t � 1.
5 Terrain Visualization
The geographical region for this visualization study
is an area in southcentral Alaska which lies between
141 � � 160 � west longitude and 60 � � 67 � north lati�
tude. The corners of the region de�ne a Cartesian co�
ordinate system and the extents of the volcano plume
data must be adjusted to obtain the correct registra�
tion of the plume data in relation to the terrain.
The terrain features are based on topographic data
obtained from the US Geological Survey with a grid
spacing of approximately 90 meters. This grid was
much too large to process at the original resolution
and was downsized to a 1426x1051 element array of
terrain elevations� which corresponds to a grid size of
approximately 1�2 mile. As shown in Figure 5� the
terrain data were read in AVS �eld data format and
were converted to a geometry using the �eld to mesh
module.
Figure 5� AVS terrain network.
We also included a downsize module ahead of �eld
to mesh because even the 1426x1051 terrain exceeded
available memory on all but our largest machines. For
prototyping and animation design� we typically down�
sized by factors of 2 to 4 in order to speed up the
terrain rendering.
The colors of the terrain are set in the generate
colormap module according to elevation of the ter�
rain and were chosen to approximate ground cover
during the fall season in Alaska. The vertical scale
of the terrain was exaggerated by a factor of 60 to
better emphasize the topography.

The resulting terrain is shown Figure 6 with labels
that were added using image processing techniques.
Features in the study area include Mount Redoubt�
Mount McKinley� the Alaska Range� Cook Inlet and
the cities of Anchorage� Fairbanks and Valdez.
Figure 6� Terrain visualization of study area.
To create the global zoom sequence in the introduc�
tion to the video animation� this image was used as a
texture map that was overlaid onto a lower resolution
terrain model for the entire state of Alaska. This tech�
nique also allowed the study area to be highlighted in
such a way as to create a smooth transition into the
animation sequence.
6 Flight Path Visualization
The �ight path of the jetliner in the animation was
produced by applying the tube module to a polyline
geometry produced by the read geom module. The
animation of the tube was performed by a simple pro�
gram which takes as its input a time dependent cubic
spline. The program evaluates the spline at speci�ed
points to create a polyline geometry for read geom.
Each new point added to the polyline causes a new
segment of the �ight path to be generated. Four sep�
arate tube modules were employed to allow the �ight
path segments to be colored green� red� yellow� and
green during the engine failure and restart sequence.
The path of the jetliner is based on �ight recorder
data obtained from the Federal Aviation Administra�
tion. The �ight path was modeled using three dimen�
sional time dependent cubic splines. The technique
for deriving and manipulating the spline functions is
so powerful that we created a new module called the
Spline Animator for this purpose. The details of
this module are described in �1�. A similar technique
is used to control the camera motion required for the
�yby in the video animation.
By combining the jetliner �ight path with the ani�
mation of the ash plume described earlier� a simulated
encounter of the jet with the ash cloud can be studied
in an animated sequence. The resulting simulation
provides valuable information about the accuracy of
the plume model. Because ash plumes are invisible to
radar and may be hidden from satellites by weather
clouds� it is often very di�cult to determine the ex�
act position and extent of an ash cloud from direct
observations. However� when a jetliner penetrates an
ash cloud� the e�ects are immediate and unmistakable
and the aircraft position is usually known rather ac�
curately. This was the case during the December 15
encounter.
By comparing the intersection point of the jetliner
�ight path with the plume model to the point of inter�
section with the actual plume� one can determine if the
leading edge of the plume model is in the correct posi�
tion. Both the plume model and the �ight path must
be correctly co�registered to the terrain data in order
to perform such a test. Using standard transforma�
tions between latitude�longitude and x�y coordinates
for the terrain� we calculated the appropriate coor�
dinate transformations for the plume model and jet
�ight path. The �rst time the animation was run we
were quite amazed to observe the �ight path turn red�
denoting engine failure� at precisely the point where
the �ight path encountered the leading edge of the
modeled plume. Apparently this was one of those rare
times when we got everything right. The fact that the
aircraft position is well known at all times� and that it
encounters the ash cloud at the correct time and place
lends strong support for the correctness of the model.
Figure 7 shows a frame from the video animation at
the time when the jetliner landed in Anchorage. The
Figure 7� Flight path of jetliner through ash cloud.

Figure 8� AVHRR image taken at 1�27pm.
ash cloud in this image is drifting from left to right
and away from the viewer.
7 Satellite Image Comparison
Ash clouds can often be detected in AVHRR satel�
lite images. For the December 15 events� only one
image recorded at 1�27pm AST was available. At the
time of this image most of the study area was blan�
keted by clouds. Nevertheless� certain atmospheric
features become visible when the image is subjected
to enhancement� as shown in Figure 8. A north�south
frontal system is moving northeasterly from the left
side of the image. To the left of the front� the sky
is generally clear and surface features are visible. To
the right of the front� the sky is completely overcast
and no surface features are visible. One prominent
cloud feature is a mountain wave created by Mount
McKinley. This shows up as a long plume moving in
a north�northeasterly direction from Mount McKinley
and is consistent with upper altitude winds on this
date.
The satellite image was enhanced in a manner
which causes ash clouds to appear black. There is
clearly a dark plume extending from Mount Redoubt
near the lower edge of the image to the northeast and
ending in the vicinity of Anchorage. The size of this
plume indicates that it is less than an hour old. Thus�
it could not be the source of the plume which the jet
encountered approximately 2 hours before this image
was taken.
There are additional black areas in the upper right
quadrant of the image which are believed to have origi�
nated with the 10�15am eruption. These are the clouds
which the jet is believed to have penetrated approx�
imately 2 hours before this image was taken. The
image has been annotated with the jetliner �ight path
entering in the top center of the image and proceeding
Figure 9� Simulated ash cloud at 1�30pm.
from top to bottom in the center of the image. The
ash cloud encounter occurred at the point where the
�ight path reverses course to the north and east. How�
ever� the satellite image does not show any ash clouds
remaining in the vicinity of the �ight path by the time
of this image.
When the satellite image is compared with the
plume model for the same time period� shown at ap�
proximately the same scale in Figure 9� a di�erence in
the size of the ash cloud is readily apparent. While the
leading edge of the simulated plume stretching to the
northeast is located in approximately the same posi�
tion as the dark clouds in the satellite image� the cloud
from the simulated plume is much longer. The length
of the simulated plume is controlled by the duration
of the eruption� which was 40 minutes.
Two explanations for the di�erences have been pro�
posed. The �rst is that the length of the eruption was
determined from seismic data. Seismicity does not
necessarily imply the emission of ash and therefore the
actual ash emission time may have been less than 40
minutes. The second possibility is that the trailing end
of the ash cloud may be invisible in the satellite image
due to cloud cover. It is worth noting that the ash
cloud signatures in this satellite image are extremely
weak compared to cloudless images. In studies of the
few other eruptions where clear images were available�
the ash clouds are unmistakable in the satellite image
and the model showed excellent agreement with the
satellite data.
8 Conclusions
An ash plume modeling and prediction system has
been developed using AVS for visualization and a Cray
supercomputer for model computations. A simulation
of the December 15 encounter with ash clouds from

Mount Redoubt by a jetliner provides strong support
for the accuracy of the model. Although the satel�
lite data for this event are relatively limited� agree�
ment of the model with satellite data for other events
is very good. The animated visualization of the erup�
tion which was produced using AVS demonstrates that
AVS is an extremely e�ective tool for developing vi�
sualizations and animations. The Spline Animator
module was developed to perform �ybys and may be
used to construct animated curves or �ight paths in
3D.
9 Acknowledgments
The eruption visualization of Mount Redoubt Vol�
cano was produced in a collaborative e�ort by the Uni�
versity of Alaska Geophysical Institute and the Arctic
Region Supercomputing Center. Special thanks are
due Ken Dean of the Alaska Volcano Observatory and
to Mark Astley and Greg Johnson of ARSC.
This project was supported by the Strategic En�
vironmental Research and Development Program
�SERDP� under the sponsorship of the Army Corps
of Engineers Waterways Experiment Station.
References
�1� Astley� M. and M. Roth� �Spline Animator�
Smooth Camera Motion for AVS Animations��
AVS �94 Conference Proceedings� Boston� Mas�
sachusetts� May 1994� pg 142�151.
�2� Tanaka� H.� �Development of a Prediction
Scheme for the Volcanic Ash Fall from Redoubt
Volcano�� First International Symposium on Vol�
canic Ash and Aviation Safety� Seattle� Washing�
ton� July 1991� U. S. Geological Survey Circular
165� 58 pp.
�3� Tanaka� H.� K.G. Dean� and S. Akasofu� �Predic�
tion of the Movement of Volcanic Ash Clouds��
submitted to EOS Transactions� Am. Geophys.
Union� Dec. 1992.

Introduction
Challenges and Opportunities in Visualization for
NASA's EOS Mission to Planet Earth
Visualization will be vital to the success of the NASA
EOS Mission to Planet Earth (MTPE), which will gather,
generate, and distribute an unprecedented volume of data
for the purpose of global change research and
environmental policy decisions. No other planned or past
mission will influenced such a large, diverse scientific
community, consisting of atmospheric scientists,
oceanographers, geologists, ecologists, environmental
scientists, climatologist, computer modelers, and social
scientists. This influence will extend well beyond those
individuals and institutions presently involved in the
mission planning, resulting in a very large potential client
community with a high demand for new and innovative
visualization tools.
The needs for visualization tools within EOS extend
beyond post-processing of generated data, into areas of
quality control and data validation, database query and
browse, mission planning, algorithm development and
testing, processing pipeline control, and data integration.
Requirements for visualization within the EOS mission
will greatly challenge existing and future visualization
technology. In addition, traditional boundaries between
visualization, analysis, and data management will need to
be dissolved. Appropriate advancements in scientific
visualization will require greater interaction between
Mike Botts, Chair
The University of Alabama in Huntsville
Jon D. Dykstra
Intergraph Corporation
Lee S. Elson
Jet Propulsion Laboratory
Steven J. Goodman
NASA Marshall Space Flight Center
Meemong Lee
Jet Propulsion Laboratory
scientists and software developers than there has been in
the past.
This panel will focus on the challenges and opportunities
for visualization with regard to the Mission to Planet
Earth. Directions presently being taken within NASA to
fund and assist development of new tools will also be
discussed. The panel session will be designed to hopefully
spark innovative ideas for meeting present and future
needs within EOS and the general scientific visualization
community. The panel consist of a mix of earth scientists
and computer scientists who have a strong interest in
advancing visualization technology and are involved in
various aspects of EOS.
Panel Statements:
Mike Botts
Visualization within the EOS era will provide significant
challenges to present and future tool development, but will
also provide an abundance of opportunities for innovated
ideas and markets. Many of these challenges stem from
requirements to handle large and abundant data files, to
integrate disparate data sets from many types of sensors,
to incorporate more science intelligence and data-specific
knowledge into tools, to allow expanded analytical
capabilities within visualization tools, and to account for

the movement toward more "softcopy" capabilities rather
than post-processing on pre-generated data sets.
Capabilities for extending general visualization tools to
incorporate science or data-specific needs must continue
to advance beyond that presently available. NASA MTPE
is moving toward efforts to modularize and encapsulate
such expert knowledge into both the data formats and tool
development. Thus, data files will become more complex
but more intelligent than at present and will generally
require EOS-supplied API's to fully understand. In
addition, expert processing and analysis algorithms will
be available from the science community; it will be
important to standardize on the presentation of these
algorithms such that they could easily be incorporated into
visualization and analysis tools. Visualization tools which
not only allows access to EOS data, but which can
maximize on the use of the ancillary information that will
be provided, will be the tools most popular within the very
large community which will use EOS data.
The ability to use lower-level data (i.e. sensor data that
has not been mapped and preprocessed) is an increasingly
important need within the earth science community and is
a requirement not being well met by present tools.
Efforts, such as the Interuse Experiment, are underway
which should allow easier incorporation of advanced
capabilities for geolocating, mapping, integrating, and
processing of lower level data within visualization tools.
These are initial efforts toward a more "softcopy"
approach to data, in which higher level data products are
processed as needed within tools rather than precomputed,
stored, and distributed as inflexible data sets.
Jon Dykstra
Satellites within the national and international EOS
family will be generating a wide spectrum of data set
volumes and geometry. Clearly, in order to promote and
encourage the use of these data by the largest number of
end-users, it will be necessary to: 1) facilitate the process
of data search and acquisition, 2) provide the data in an
appropriate geometry and, 3) format the data in an
optimal manner for viewing and processing.
These challenges, especially the one of format, are
particularly acute when dealing with high volume data
sets, e.g. high spatial resolution data or mosaics of
multiple EOS data sets. The commercial photogrammetry
and image processing industry has developed several
formatting techniques for dealing efficiently with large (>
50 megabytes to over a Gigabyte) images. These
techniques include image tiling, data compression and
extensive use of image overviews. In order to best meet
the EOS customer's needs, the issue of optimum data
format ought to be thought through. The more conducive
the EOS image format is to the end user's application, the
more efficiently and effectively the data will be used.
Lee Elson
Visualization in the EOS era (end of the century and
beyond) will have to satisfy emerging technical, social,
and scientific needs. Perhaps the most common element
in these requirements is data availability and access.
Visualization tools will be forced to accept data from a
wide variety of sources and locations. An obvious result
of this trend is the need for standardization: tools must
converge on community-wide practices which become defacto
standards. This process must proceed by consensus
rather than by externally imposed edicts. Examples of
problems inherent in this process will be discussed.
Scientific research in the coming decades is likely to move
strongly in the direction of making results available to and
understandable by the public and policy makers. Not only
is this a political necessity, but it is beneficial to the
intellectual health of humanity. For this and other
reasons, it is important that visualization tools
accommodate lower end platforms. Specific examples,
such as the use of X for display, can be cited.
Exchange of information between scientists is undergoing
rapid evolution. Visualization tools are playing a central
role in this change and must continue to do so. For
example, the standard print media (e.g. journals) is
loosing ground to desktop multimedia tools. The day may
not be far off when I will be able to "publish" a refereed
paper which contains hyperlinks in the references section,
animations for some of the "figures", costs a few dollars
instead of the $5000 that I paid for my last publication,
and will take a few weeks from submission to distribution
instead of 6 months to a year.
Steve Goodman
The integration of multiple data sets from different
satellite sensors, ground stations, numerical models is
becoming increasingly important for research and data
validation. In our particular case, superposition of
satellite sensor data with cloud volume data and groundbased
data is important for interpreting the space data
which show only the cloud top structure and integrated
radiance information, but no vertical structure. The ability
to show model retrieval data side by side with radar and
in-situ observations would be helpful for validating such

models. However, in addition to simply viewing multiple
data sets together, comparative analysis and data fusion
capabilities within the tools are vital to meeting many
scientific needs. For instance, some algorithms use cloud
models combined with constraints from satellite data to
get vertical profiles of cloud liquid water, ice etc.
The ability to locate data within a time and physical space
domain is critical for the scientist. Working in x-y-z
space, rather than latitude-longitude-altitude space for
instance, is very inhibitive to the the scientist. Many
visualization tools are incapable of tieing data down into
physical space which has meaning to a scientist,
particularly if the data is gridded using spatial transforms,
such as map projections. Accurate and complete
geolocation of data within visualization tools, similar to
that available in many GIS systems, is particularly
important for meeting the research needs of the EOS
scientist. Additionally, the element of time needs to have
more meaning than simply a driver of animation.
Adequate parsing and transformation of time domains is
important, as is the ability to analyze temporal sequences
within visualization and analysis tools. Furthermore,
unlike typical animations which presently assume that all
pixels within a given time slice were sampled at the same
time, the sampling of time in orbital swath data is
continuous with each pixel having it own sampling time.
This is particularly important for our research into highly
transient phenomena, such as thunderstorm growth and
decay, cloud electrification and lightning.
There is a strong need to more closely tie visualization
and analysis within the same tool. Tools must allow the
scientist to work with real values, in floating point or twobyte
integers for example, instead of 8-bit numbers
between 0 and 255. Tools should allow for the
recognition and proper handling of "missing data" and
"bad data" flags. Also, Instead of serving as simply a
viewing tool after processing and analysis is complete,
visualization should serve the function of providing the
scientist with insight into the data and thus interactively
driving much analysis. This requires a simple path back
and forth between the visual and analytical domains, as
well as integration of the more powerful visualization
techniques with the more fundamental visualization
techniques such as data plots and graphs. Finally, a closer
interaction between scientists and tool developers is
needed to allow scientist the ability to take advantage of
the latest computing technology, while assuring that real
science needs are being met by the visualization
community.
Meemong Lee
Visualization is a graphical representation technology
which relies on the highly advanced human vision system
for grasping multi-dimensional complex phenomena from
the visual queues (color, shape, perspective geometry, and
temporal sequence). The ability to transform a large
amount of incomprehensible data or complicated
mathematical models into intuitive graphical forms has
greatly expedited the information processing and has
provided an interactively manipulatible data space (virtual
reality).
Visualization plays an important role in every aspect of
NASA mission cycle: mission planning, instrument
design, operation monitoring, data base access, science
processing and sharing the mission objective with the
public. Though EOS doesn't require visualization
technology unique to itself, it must actively participate in
advancing the visualization technology as a whole so that
it can utilize the technology for advancing the mission
data utilization, science product generation, and sharing
the science with the world.
Technology that will be essential in EOS includes
capabilities for 1. presenting a large volume of data from
various instruments and their derived products in a
comprehensible and inter-usable form, 2. enabling
comprehensive analysis of the observations in connection
with the over-all mission environment including
observation dynamics, payload characteristics, and various
other aspects that are related to the resulting observation.
3. providing an integrated virtual observation platform
where the inter-dependencies between different
observations as well as the temporal and spatial
relationship of an observation can be intuitively
appreciated, and 4. sharing the achieved science with
other discipline scientists and public to ensure meeting the
mission objectives.
Panel Participants
Dr. Mike Botts is formally educated in the fields of
Geology, Planetary Sciences, Image Processing/Remote
Sensing, and Geotechnical Engineering. For the last six
years, Dr. Botts has specialized in the use and
development of scientific visualization tools within the
Earth sciences. As a Senior Research Scientist within the
Earth System Science Laboratory (ESSL) at the University
of Alabama in Huntsville (UAH), he has worked on-site at
NASA Marshall Space Flight Center within the Earth
Science and Applications Division (ESAD), assisting
Earth scientists in meeting their visualization and analysis

needs through a combination of in-house development and
off-the-shelf software. In 1992, for a period of six
months, Dr. Botts served on temporary assignment at
NASA Headquarters, evaluating the state of scientific
visualization for NASA's EOS Mission. His report to
NASA Headquarters entitled "The State of Scientific
Visualization with Regard to the NASA EOS Mission to
Planet Earth" established many of the guidelines for
visualization requirements for NASA EOS. He is
presently PI on an EOS/Pathfinder grant ("The Interuse
Experiment") from NASA Headquarters, designed to
improve the interdisciplinary and multi-sensor integration
of NASA EOS and Pathfinder data sets within
visualization and analysis tools.
Dr. Jon Dykstra received his degrees in Geologic Remote
Sensing and is currently the Executive Manager of
Intergraph's Imaging Systems (a component of
Intergraph's Mapping Sciences Division). In this capacity
he is responsible for the development of Intergraph's
commercial image processing and photogrammetric
application software. Dr. Dykstra initiated the design and
development of Intergraph's commercial image processing
software in 1987 and has managed its continuing
evolution since that time. By design, the imaging
software is fully integrated into Intergraph's digital
mapping and Geographic Information Systems. Dr.
Dykstra has managed the groups responsible for the
integration of digital image processing techniques with
those of classical photogrammetry. The resulting digital
stereo photogrammetric products are being used both with
the government and commercial sector for stereographic
exploitation of a wide variety of digital imagery. Before
coming to Intergraph, Dr. Dykstra served for two years as
the Senior Applications Scientist at the Earth Observation
Satellite Corporation (EOSAT). Prior to EOSAT, he
spent seven years as a satellite imaging specialist for Earth
Satellite Corporation (EarthSat).
Dr. Lee Elson is currently a Research Scientist in the
Earth and Space Sciences Division at the Jet Propulsion
Laboratory. His most recent activities have centered on
the study of ozone depletion in the middle atmosphere
using data from the Upper Atmosphere Research Satellite.
He is a co-investigator on the Microwave Limb Sounder
(MLS) experiment and is an active member of a UARS
Theoretical Investigation Team. He is also involved with
planning efforts for the EOS version of the MLS
scheduled to fly on the Chemistry platform. Other
activities have centered around the development of
visualization tools. In particular, he has been an active
user of the LinkWinds package developed by A. Jacobson
at JPL.
Dr. Steve Goodman is an atmospheric scientist within the
Earth Science and Application Division at NASA
Marshall Space Flight Center (MSFC). His present
scientific research interests are in regional and global
studies of the hydrologic cycle, with special emphasis on
process studies and interannual variability. He serves as
both an instrument investigator and interdisciplinary
scientist supported by the NASA Earth Observing System
(EOS) program. Dr. Goodman is the team leader for
science algorithm development and mission operations for
the OTD and TRMM-LIS sensors. Dr. Goodman is also
interested in methods of intelligent data base searching
through large, distributed environmental data bases,
visualization for satellite and radar meteorology,
algorithm optimization, data fusion, and adaptive
forecasting methods.
Dr. Meemong Lee is the Technical Group Supervisor of
the Image Analysis Systems group at JPL, which is
primarily involved in scientific information processing
and visualization, concurrent and distributed data
processing, and real-time data processing for on-board
spacecraft applications. Dr. Lee received degrees in
electrical engineering and computer science. She has
been actively involved in the development of systems for
analyzing seismic data, speech, image, and multi-spectral
data for 12 years, and has been the PI on various
algorithm development tasks including Automatic
Currency Inspection, Markov Random Field Model based
Texture Classification, Automatic Sea-ice Motion
Tracking, and Multi-resolution Pyramid based Image
Registration. She directed the development of various
parallel data processing software systems including the
SPectral data Analysis Manager (SPAM), the Concurrent
Image Processing Executive (CIPE), the Hubble space
telescope PSF model generation and verification, and a
parallel DEM extraction algorithm for Magellan project
(Shape). Dr. Lee is currently involved in developing a
systolic dataflow system for an end-to-end Instrument
Simulation Subsystem development task for JPL Flight
System Testbed, as well as the development of tools for
integrating EOS data under the Interuse Experiment.

Visualization in Medicine:
VIRTUAL Reality or ACTUAL Reality ?
Christian Roux Co-Chair
Jean Louis Coatrieux Co-Chair
1. Departement Image et Traitement de l'Information, Ecole Nationale Superieure des
Telecommunications de Bretagne. BP 832, 29285 Brest Cedex. France
2. Laboratoire deTraitement du Signal et de l'Image, Universite de Rennes I, Campus de Beaulieu
35042 Rennes Cedex. France
Panelists
Jean-Louis Dillenseger, University of Rennes I, France
Elliot K. Fishman, Johns Hopkins School of Medicine, U.S.A.
Murray Loew, George Washington University, U.S.A.
Hans-Peter Meinzer, German Cancer Center Heidelberg, Germany
Justin D. Pearlman, Harvard Medical School, U.S.A.
Abstract
This panel will discuss and debate the role played by 3D visualization in Medicine as a set of methods and
techniques for displaying 3D spatial information related to the anatomy and the physiology of the human
body.
1. Introduction
3D medical imaging is now more than twenty years old. It is still an extremely active field with numerous
theoretical, technological and clinical issues. One of the most critical issues is visualization. Efficacious
visualization of medical objects is indeed of utmost importance for the physicians who analyze the
anatomical structures, their spatial relationships and their physiology.
3D visualization in the medical field is unique with respect to many other domains because it has to
handle large volume data which may come from multiple imaging systems and also from sources other
than imaging modalities. This is why it is so complex and still needs a lots of work. When reviewing the
literature of the domain [1] one realizes very quickly that they are many ways to "see the unseen" [2].
This panel will present some approaches to 3D visualization in Medicine and debate its effectiveness. The
panelists will be asked to answer the question: Visualization in Medicine, Virtual Reality or Actual
Reality ? To this end, several alternatives will be considered:
• Aesthetics vs realism
• Models vs actual values
• Surface rendering vs volume rendering
• Technology for the future vs already available technology
•Other topics will be addressed in the panel: What is the need for display devices ? In which way do the
practitioners actually use 3D visualization ? How can the medical efficacy of the various ways of
producing 3D displays be evaluated ?
The panelists will finally try to point out the main limitations in this domain and to bring out some new
challenges for the future.

2. Visualization in Medicine: What Can Be Learned from Epilepsy Research ? (Jean-Louis Dillenseger)
The capability to display and manipulate three-dimensional medical images is now widely recognized as
fundamental in clinical practice. Several frameworks have been designed in the last two decades from
surface representations to volume rendering. Some hot debates on the right or the best way to visualize
datasets have taken place where more emphasis has been put recently on realistic and quantitative features
rather than on aesthetic images. Here I would like to address another view : what can be learned from
application-oriented approaches? I will do it by using the epilepsy research field.
First, I will try to present this complex application in simple words to facilitate the understanding of each
stage and to identify the multiple roles that visualization can play. Second, the specific problems to be
solved will be examined in depth. They cover the clinical needs and constraints, as well as the
methodological aspects. Raw and transformed data, multimodal sources (model, signal, images),
point/surface/volume based display, direct and inverse problems related to epileptic focus localization will
be analyzed. Third, according to bottom-up and top-down views, inspired from knowledge based
methods, specialization and generalization cues will be discussed.
3. 3D Clinical Imaging in Radiology: Current Applications, Future Needs. (Elliot K. Fishman)
Radiologic imaging requires far more than the detection of a lesion or description of its potential etiology.
The radiologist in 1990s must present the abnormality detected in a form that referring physician can use
for patient management. Unless information is provided in a form that can effect patient outcome, the
study will have little value in the overall scheme of things.
Recent advances in radiology have resulted in the ability to create highly accurate three-dimensional
volumes for visualization. This is especially true with the recent introduction of Spiral CT scanning
which allows acquisition of large data volume in single breath-holds. The role of three-dimensional
visualization in a wide range of clinical applications including that of oncology, the gastrointestinal tract
and orthopedics will be addressed. Specific advantages of volume visualization in terms of therapeutic
management and understanding of treatment options will be presented. Potential directions for threedimensional
imaging and current deficiencies in system renderings and display will be addressed as well.
4. 3D Visualization in medicine: virtual reality or actual reality? (Murray Loew)
Visualization in medicine must extend to the representation of data that arise from sources other than
imaging modalities. Clinicians need better and more quantitative ways to understand the vast array of
information that confronts them when they are making diagnoses or prescribing treatments.
Our familiarity with the three-dimensional world provides an intuition that can be used to help us
navigate among and understand inclines, ridges, and points of inflection in the data space, and thus to
develop further an understanding of the sensitivities of dependence of certain quantities on others. This
should be especially helpful in medical decision-making (balancing costs against benefits, each with its
respective probability), where sharp optima are unusual and opinion often plays a major role. And in
diagnosis there would be the opportunity, for example, to examine graphically the contributions of various
measurements to an overall estimate of the patient's state.
We discuss some issues in 3-D display for data visualization, and how they relate to the goals of providing
a useful view of even higher dimensionality, and of developing a more-quantitative intuition among
practitioners.

5. 3D Visualization in Medicine (Hans-Peter Meinzer)
Now that virtual reality is very popular in many fields from engineering to entertainment many people
also look into medical applications. There have been attempts of 3D visualization in medicine, most of
them follow the vector based approaches of the VR world. The main sources of medical 3D data are CT
and MR images. They are pixel (or voxel) volume data which are therefore translated into vector based
surface descriptions. This step is highly critical as there are usually no clear surfaces in a body. The better
approach is a voxel based 3D volume visualization algorithm omitting the critical step of surface
identification.
Volume based 3D visualization can show objects like clouds, flames or densities (of any origin). This
allows a view on semitransparent objects that may incorporate further objects. It also can show spatial
textural information, which is often of high interest. Even surfaces are better displayed, if they surface is
not highly polished and completely smooth. Textural information on surfaces (i.e. if the Mandelbrot
dimension of a surface is much higher than 2) is falsely suppressed by 3D surface visualization but
correctly shown by 3D volume renderers, as they can show `real' volumes. In this aspect a massively
textured 2D surface can be considered as a 3D volume object.
The problem of the volume renderers today is that they are too slow. Every image still takes a few
minutes on any non parallel hardware while we would need 10 images per second. The vector based
renderers run already in real time and they are embedded in nice 3D input and output man-machine
interfaces. Unfortunately it shows to be very difficult to integrate the vector based algorithms into the
volume renderers and vice versa. Fast parallel hardware can and will solve the time bottleneck and within
the next very few years VR in medicine will become actual reality if the segmentation and classification
problem can be overcome.
While the step of 3D data visualization is not anymore a research problem but an engineering one, the
problem of object identification remains basic research. Before we can show an organ or a tissue we must
first identify it in the volume data. Here we have the signal-to-symbol gap between the enormous amount
of numeric data and the symbolic description of a (medical) scenario. The keywords are e.g.: AI, neural
nets, topological maps, cognitive textures, active contours and topological morphology. The truth is, that
we do not understand how man interprets images and therefore there is no algorithm. This is the real
drawback on the way to virtual reality in medicine.
6. 3D Visualization in medicine: virtual reality or actual reality? (Justin D. Pearlman)
Advances in data acquisition now routinely collect 3D data sets documenting anatomic and physiologic
status changes inside the body. The primary limitation is no longer the ability to acquire the data but now
is the ability to extract and understand the useful information content. In the real world, we use a dozen
mechanisms to appreciate 3D relationships. In the computer world, 3D image data is commonly presented
using only one or two depth cues, often relying on a flat 2D computer screen to visualize the "3D"
information, by extensive use of models of lighting, shading, surface orientation, texture, reflection,
connectivity and vector interpolation. Although the end results can "look very realistic" and are "easy to
manipulate" they in fact contain very little of the information in the original 3D data set.
Virtual reality supplies our senses with a satisfying simulation that mimics what we might encounter in
the real world, using made-up data. Actual reality supplies to our senses information about the real world,
not made up. To a large extent, derived surface models using vector graphics present a virtual reality,
even if the models are based in part on actual 3D data. Much more of the information bandwidth
presented comes from the modeling than from actual data. For medical decision making, I contend that
"looking realistic" is much less important than conveying useful information. In fact, the vector models

commonly suppress information about data quality, so confidence in the data is difficult to assess except
when there are gross errors such as in 3D surface modeling from CT of the head, showing teeth through
the lips because of the effect of the bone signal on the intensity gradient used to derive the "surface
normal."
There are alternatives that come much closer to presentation of actual reality. CUBE software* provides
exploration and composite rendering of the entire actual 3D data set at interactive speeds on standard
hardware at the full depth of the acquired data. The tool provides 3D object recognition and interactive
editing to remove uninformative and obstructive elements, but it presents actual values, not models, to the
viewer. Applied to 3D MRI of the chest, a 3D map of the heart and of the coronary artery supply to the
heart is produced, non-invasively. Obstructing objects not of interest, such as chest wall and other blood
pools, are removed. The surgery is performed at interactive speed on the computer using a simple pointand-click,
rather than on the patient. The results may be viewed interactively, and they may be exported
to volumetric holograms (VOXEL) to rebuild the structures of interest as a 3D sculpture of light.
*Dr. Pearlman has commercial interest in CUBE
The panelists
Jean-Louis Coatrieux
Jean-Louis Coatrieux received the Electrical Engineering degree from the Grenoble Institute in 1970, and
the Third Cycle and the State Doctorate degrees from the University of Rennes I in 1973 and 1983,
respectively. He previously served as an Assistant Professor at the Technological Institute of Rennes and
he became Director of Research (INSERM) in 1986. In addition, he presently is a lecturer at the Ecole
Nationale Superieure des Telecommunications de Bretagne. His research interests include signal and
image processing, knowledge based-techniques and the fusion of data and models in medical applications.
Jean-Louis Dillenseger
Jean-Louis Dillenseger is Maitre de Conference at the University Institute of Technology of Rennes,
France. He is with the Laboratoire de Traitement du Signal et de l'Image at the University of Rennes I.
His research is mainly devoted to 3D and multivariate medical visualization and 3D positioning in
multimodal imaging. He works on ray tracing and volume rendering techniques on anatomical datasets
(MRI, CT,...) and also on functional signals mapping on morphological data. The application field of his
research includes cardiology and neurology. He received a Mechanical, Electronic Engineering degree in
1988 from E.N.I.B., Brest, France, and a Ph.D. degree in biomedical science from the University of Tours,
France, in 1992.
Elliot K. Fishman
Elliot K. Fishman is currently Professor of Radiology and Oncology at the Johns Hopkins School of
Medicine in Baltimore. He is also Director of Body CT and Abdominal Imaging in the Department of
Radiology. Dr. Fishman's interests include optimizing imaging techniques for diagnosis and therapy
planning as well as applications of computer-based imaging particularly in terms of three-dimensional
visualization for optimizing diagnosis and management. His specific areas of interest include that of
musculoskeletal 3D imaging and oncologic applications.
Murray Loew
Murray Loew received the Ph.D. in electrical engineering from Purdue University in 1972, and then in
industry continued his work in pattern recognition and applications. At GWU since 1978, he has taught
and conducted research in pattern recognition, image processing and computer vision, and medical
engineering. Some recent topics are: probabilistic modeling of x-ray images and determination of their
information-theoretic properties; development and comparison of diffusion-based, and robust chain-codebased
descriptors of shape; and data compression and quality measurement for medical images. He was a
founder (in 1987) and is co-director of the university's Institute for Medical Imaging and Image Analysis.

Hans-Peter Meinzer
Hans-Peter Meinzer is a scientist at the German Cancer Research Center in Heidelberg since 1974. Since
1983 he directs a research team specializing in the 3D visualization of 3D tomographies. The team
developed a ray tracing algorithm for medical 3D data cubes. In this context he works on neural nets, AI,
human perception, cognitive texture analysis, morphology, and parallel computing concepts. His special
interest lies in modeling and simulating tissues and tissue kinetics.
After studying physics and economics at Karlsruhe University and obtaining an MS in 1973, Meinzer
received a doctorate from Heidelberg University in medical computer science (1983) and his habilitation
(1987). He is associate professor at the Heidelberg University on medical computer science and a member
of the Gesellschaft fuer Informatik (GI) and theGesellschaft fuer Medizinische Dokumentation und
Statistik (GMDS). He received the Ernst-Derra award from the German Society of Heart Surgeons (1992)
and the Olympus award from the German Society for Pattern Recognition (1993).
Justin D. Pearlman
Justin D. Pearlman MD ME PhD is an Associate Professor of Medicine at Harvard Medical School with a
joint appointment at M.I.T. in the Health Sciences Technology program. He is board certified in Internal
Medicine and in Cardiology, and holds a full-time appointment in Radiology. He received his Masters
degree studying computer architecture and visualization, and received his PhD in Biomedical
Engineering, by detecting and visualizing chemical changes within the wall of human arteries by
magnetic resonance and multidimensional analysis. He is Director, Magnetic Resonance Imaging
Technologies and Computing at Boston's Beth Israel Hospital, where he performs research on
multidimensional imaging of cardiovascular disease, with a focus on new magnetic resonance imaging
techniques for visualization of early atheromatous disease in the coronary and other arteries.
Christian Roux
Christian Roux is Professor of Image Processing and Pattern Recognition at the Ecole Nationale
Superieure des Telecommunications de Bretagne in Brest, France. He leads the Image and Information
Science department where he is currently conducting research programs on image reconstruction, analysis
and interpretation. He is an Associate Editor of the IEEE Transactions on Medical Imaging and Director
of the Brest University's Institute for Medical Information Processing.
References
[1] G.T. Herman "3D Display: A Survey from Theory to Applications" Proc. IEEE Satellite Symposium
on 3D Advanced Image Processing in Medicine Ch. Roux, G.T. Herman, R. Collorec (Eds), Rennesfrance,
November 1992
[2] B.H. McCormick, T.A. Defanti, M.D. Brown, "Visualization in Scientific Computing", Computer
Graphics, Vol. 21, No. 6, 1987.
[3] T. Todd Elvins, "A Survey of Algorithms for Volume Visualization", Computer Graphics, Vol. 26,
No. 3, 1992.

Visualization and Geographic Information
System Integration:
What are the needs & requirements, if any ?
Chair:
Theresa Marie Rhyne, (Martin Marietta/U.S. EPA Vis. Ctr.)
Panelists:
William Ivey, (SAS Institute Inc.)
Loey Knapp, (IBM & University of Colorado)
Peter Kochevar, (DEC/ San Diego Supercomputer Center)
Tom Mace, (Scientific Computing Branch, U.S. EPA)
Introduction:
Historically, a geographic information system (GIS) has been
defined as a combination of a database management system and a
graphic display system which is tied to the process of spatial
analysis. GIS environments also permit building maps in real time
and examining the impacts of changes to the map interactively.
Visualization embraces both image understanding and image
synthesis. It is a methodology for interpreting image data entered
into a computer and for generating images from multi-dimensional
data sets. Visualization research and development has focused on
issues and techniques pertaining to image rendering and not
necessarily on mapping these capabilities to specific geographical
problem areas. On the other hand, GIS environments have not
readily applied visualization methodologies.
This panel addresses the needs and requirements of integrating
visualization and GIS technologies. There are three levels of
integration methods: rudimentary, operational and functional. The
rudimentary approach uses the minimum amount of data sharing
and exchange between these two technologies. The operational
level attempts to provide consistency of the data while removing
redundancies between the two technologies. The functional form
attempts to provide transparent communication between these

espective software environments. At this level, the user only needs
to request information and the integrated system retrieves or
generates the information depending upon the request. This panel
examines the role and impact of these three levels integration.
Stepping further into the future, the panel also questions the longterm
survival of these separate disciplines.
Position Statements:
Theresa Marie Rhyne, Martin Marietta/U.S. EPA
Visualization and GIS methodologies are frequently used to
examine earth and environmental sciences data. Interestingly
enough, both of these disciplines developed and have often been
implemented in parallel to each other. At the U.S. Environmental
Protection Agency, there are separate organizational units which
provide hardware and software support for these functions.
Visualization is primarily associated with the computational
modeling efforts of the EPA's supercomputer and data obtained from
satellite remote sensing systems. GIS environments have been
installed in EPA Research Laboratory, Program and Regional Offices
to collect, analyze, and display large volumes of spatially referenced
data pertaining to remote sensing,. geographic, cultural, political,
environmental, and statistical arenas.
Often the hardware configurations optimized to support GIS
are not compatible with visualization methodologies. Frequently,
visualization software has been customized to encompass standard
cartographic and spatial display capabilities of GIS environments.
The transfer of visualization technology to EPA regional offices and
State environmental protection agencies has both budgetary and
staff support constraints which could benefit from effective
integration of Visualization and GIS disciplines.
Perhaps Robertson and Abel articulated these concerns in the
IEEE CG&A discussion on Graphics and Environmental Decision
Making: "Why are these fields so slow to exchange ideas ?? There are
plausible reasons. To us, the most likely reason is the relative
immaturity of both fields, particularly in the integration of their

technologies into usable systems that address difficult real-world
problems. " (1).
Effective environmental decision making for regulatory
purposes is a real world problem faced on a daily basis at the global,
national, state and local levels. The integration of visualization and
GIS methodologies can facilitate this decision making process and
minimize economic constraints associated with the purchase,
training, support and maintenance of redundant closed systems.
(1) Robertson, Philip K. and David J. Abel 1993, Graphics and
Environmental Decision Making, IEEE Computer Graphics and
Applications, Vol. 13, No. 2, pp. 25 - 27.
William Ivey: SAS Institute, Inc.
Of the levels of GIS integration, the functional level holds the
most promise for effectively applying scientific visualization
techniques to geographic data. At this level, the geographic data is
already stored in a format that is native to the visualization system,
thus requiring no extraordinary user or system intervention to
interact with the data when performing visualizations. This is not to
say that there is no data transformation taking place between the
GIS and visualization environment, there is. The data structures
that support GIS and visualization are designed to be most efficient
for those purposes so some transformation is necessary.
This level of integration still offers many advantages. One is
the ability to generate visualizations much faster since the data does
not have to be converted from a non-native format. This is
especially important for animations where a smooth transition
between time steps yields a more effective visualization. Another
advantage is the link to the GIS data base. With this link, spatial or
logical queries can to be made to the GIS allowing for subsets or
different spatial attributes to be visualized easier. A third advantage
is the link to other system functionalities. In the case of the SAS
system, statistical procedures could be used to perform analysis on
attributes and attribute combinations. Visualizing the results of
these analyses could be the most effective way to understand

complex attribute relationships. A fourth advantage is that attribute
or spatial resolution need not be compromised. With a direct link to
the GIS database, no filtering is necessary so the visualization can
take place using all the data available. And this is what visualization
excels at, making sense out of extremely large amounts of
information.
Examples of visualizations that make use of these advantages
are numerous. In the case of faster is better, consider the following
scenario. Suppose that landuse changes are being discussed at a
town meeting. Historical landuse patterns in digital format for each
of the past five years is available. A model, built into the GIS, that
predicts future landuse patterns based on trends, zoning regulations,
and demographics is also available. Preparing an animation
showing landuse changes using the historical data is a simple task
and is prepared beforehand. The model, however, adds a new
wrinkle. Town commissioners want to use the model to look into the
future and see if they are making the right decisions. Functional
integration makes this possible by allowing modeling scenarios to be
executed in the GIS and the time steps to be animated using the
visualization system.
Loey Knapp: IBM & University of Colorado
Environmental applications have three notable characteristics
which must influence the direction of supporting software tools.
First, the amount of data is increasing dramatically, posing
performance and analysis problems. Second, the data are complex
and heterogeneous representing multiple formats, scales,
resolutions, and dimensions. Finally, a wide range of personnel
must interact with the data as solutions to environmental problems
increasingly involve interdisciplinary teams.
The current software infrastructure for such applications does
not adequately address these three characteristics, making
enhancements and extensions to current products mandatory. One
extension which is gathering force and interest is the integration of
geographic information systems (GIS) and scientific visualization
systems (SVS).

GISs are now used extensively in the analysis of
environmental data due to their capability to manage, manipulate,
and display spatial data but there are several significant problems
with GISs in the environment described above. First, GISs handle
only two-dimensional data; second, displays are generally limited to
spatial views of the data; and third, the capability to support user
interaction where the data is negligible. SVSs present a different set
of strengths and weaknesses. Where GISs provide strong support
for analytical functions, SVSs provide the capabilities to visually
interact with the data using a variety of sophisticated techniques.
These systems can support n-dimensional data, providing the
infrastructure to generate two and three-dimensional spatial views,
animations of time series, and graphical or statistical plots.
However, the lack of analytical operations such as neighborhood
analysis in three dimensions or queries across time series limit the
capabilities of these tools. Bringing the analytical strengths of GISs
together with the visualization strengths of SVSs would provide a
more robust set of software tools for environmental applications.
Three levels of GIS-SVS integration have been identified:
rudimentary, operational, and functional. At a rudimentary level
data can be passed from a GIS to a SVS or vice versa, often saving
extensive reformatting time. The main issues with this level of
integration is redundant data, sequential processing, dual system
interface, and the lack of new functionality relative to analysis of ndimensional
data. An operational level of integration, which allows
real-time data passing removes the data redundancy issue but
introduces a potential performance problem, depending on the size
of the data set and available compute power. Integration at a
functional level holds the most potential for iterative and interactive
analysis and visualization. Such a system would extend the visual
programming concepts of SVSs to include GIS functions, extend GIS
analytical operations to three and four dimensions, and provide a
single interface to environmental scientists.
While a functional level of integration should clearly be the
ultimate goal, there are significant barriers to implementation.
Software remains proprietary and a tight level of integration implies
levels of cooperation difficult to achieve across companies. The cost

of multiple software packages can also be exorbitant, requiring a
cooperative cost agreement between the vendors. These factors can
be overcome but will require users in the environmental community
to play a strong role in demanding and helping design tools which
provide the requisite infrastructure for their applications.
References:
Buttenfield, Barbara P. and John H. Ganter, 1990.
Visualization and GIS: What Should We See? What Might We Miss?
Proceedings of the 4th International Symposium on Data Handling,
July 1990, Zurich, Switzerland.
Campbell, Craig S., and Stephen L. Egbert, 1990. Animated
Cartography, Thirty Years of Scratching the Surface,
Cartographica, Vol. 27, No. 2, pp. 64-81.
DiBiase, David, et al, in press. Animation and the Role of Map
Design in Scientific Visualization, submitted to Cartography and
GIS.
DiBiase, David, 1990. Visualization in the Earth Sciences,
Earth and Mineral Sciences, Vol. 59, No. 2, pp. 13-18.
MacEachren, Alan M. and David DiBiase, 1991. Animated
Maps of Aggregate Data: Conceptual and Practical Problems,
Cartography and Geographic Information Systems, Vol. 28, No. 4,
pp. 221-229.
Monmonier, Mark, 1992. Summary Graphics for Integrated
Visualization in Dynamic Cartography, Cartography and
Geographic Information Systems, Vol. 19, No. 1, pp. 23-36.
Treinish, Lloyd A., 1990. The Visualization Software Needs of
Scientists, NCGA'90 Conference Proceedings, Vol. 1, pp. 6-15.
Treinish, Lloyd A., and C. Goettsche, 1991. Correlative
Visualization Techniques for Multidimensional Data, IBM Journal of
Research and Development, Vol. 25, No. 1/2, pp. 184-304.

Weibel, Robert, and Barbara P. Buttenfield, 1992.
Improvements of GIS Graphics for Analysis and Decision-Making,
International Journal of Geographic Information Systems, Vol. 6,
No. 3 pp. 223-245.
Peter Kochevar: Seqouia2000-San Diego Supercomputer Center
Wherefore GIS ??
The development of data visualization systems is progressing
in a way that makes the whole issue of integration of visualization
and GIS a moot point. Future systems will be based on a very broad
definition of what data visualization is, a means to help transform
data into human understanding. With this broader definition, data
visualization will no longer be looked upon as simply an act of
information presentation but rather as a bi-directional process that
takes into account interaction with end-users.
Furthermore, future visualization systems will deal with
"data" in the most general sense as any digital encoding of
representations of things, measurements, concepts, and relations,
that may be real or imagined. These data may be of the relational
variety that traditionally has not been within the realm of
visualization systems. Relational data is tabular in nature and may
represent such information as sensor measurements, meta-data
(data describing data), stock prices, airline schedules, telephone
directory listings, etc. Alternately, data may be non-relational in
character where geometric and topological relationships exist
between data elements as is frequently the case with scientific data.
Non-relational data may consist of images, digital maps, polygonal
terrain models, gridded output from simulation programs,
CAD/CAM models, etc. Furthermore, any kind of data may vary in
time thus forming data streams of which two or more may be
synchronized during visualization as would be the case, say, with
video data.
Effective data visualization requires close coordination with
data sources whether they be scientific instruments, sensors,
computer simulations, or database management systems (DBMSs).

DBMSs are of particular interest because with them, data can be
stored in a way that makes their subsequent use not only possible but
efficient. They offer the capability to search for data based on
content rather than solely by name, something that is crucial to
enhancing understanding. This act of browsing is an important
interactive visualization function and as such, data visualization
systems can be thought of as portals into databases. They are the far
more sophisticated analog of the presentation subsystems that list
the tuples resulting from queries in relational DBMSs. The results of
such queries are data that must be effectively visualized just like
scientific data is now.
So, wherefore GIS? GIS will cease to be a field unto its own-it
will be completely subsumed by general, interactive data
visualization systems. GIS data is simply that in which data
components are arrayed in latitude and longitude. When structuring
an appropriate visualization, this fact should be utilized just as
knowledge of chemistry is exploited when structuring molecular
visualizations, say. As with any visualization, the structure of the
data and its meaning must be taken into account when fashioning a
salient visualization. In this regard, there is nothing special about
GIS.
Nonetheless, in building next-generation data visualization
systems, much can be learned from the GIS community. There is a
long tradition of handling spatial, non-relational data that can be
exploited. To support the manipulation of such data, the GIS
community has developed sophisticated data models, e.g. SAIF, that
should be studied by the visualization community. Finally, the GIS
community has more experience in using DBMSs than practitioners
of visualization and that experience needs to be tapped.
Dr. Thomas H. Mace: USEPA National Data Processing Division
Visualization and GIS are not necessarily mutually exclusive.
The difference may be more one of emphasis than substance. GIS
functions include data formatting, management, analysis, and
display (or visualization). The power of GIS lies in its ability to assist
in spatial analysis. The other tools exist to support that function.

The map, whether in paper form or more ephemeral, video, form is a
visualization of some aspect of data, real or imagined, which
conforms to the mapmakers model of the topology of the subject.
Spatial analysis is not a new discipline. In the past, spatial
analysis was performed using visualizations of spatial relationships
on paper with scales ranging from sub-atomic to cosmic. We now
use imaging systems and algorithms as our scribes. The tool is
essentially the same when one is looking at the geography of
anemone colonies on a rock or the geography of world political
events. We have developed the use of process science to attempt to
describe our world and predict consequences. Process science
requires both temporal and spatial contiguity to determine cause and
effect. (Otherwise, you have action at a distance--magic.) One
needs both historical and geographical tools. GISs are the tools of
geography. Visualization is as necessary to geography as
mathematics is to physics. We humans need to see our logic.
Cartographers have long known that the map is not only a tool for
getting from here to there (still a process), but a way of organizing
knowledge to make it understandable. Some of this is even evident
in pictograms and petroglyphs left centuries ago in our southwestern
desert. Euclidean space is generally used, but not required in all
instances. All that is required for spatial analysis is a space-time
model and a subject. The advent of computers has only changed the
speed at which we can render our analyses.
It seems that since visualization must be an integral part of
spatial analysis, and spatial analysis is the reason for GIS, why all
the fuss? Part of the problem may be vocabulary. The term
"visualization" may mean one thing to geographers and quite
another to computer scientists. I suspect, however, that the "devil is
in the details". Software for contemporary GIS is still based on
Euclidean representations of reality projected onto a flat surface.
The typical map. As far as I know, there are no commercial GIS
packages that possess data management structures that deal with 3,
4, or n-dimensional space. Virtual reality packages and
photogrammetric workstations visualize in 3 dimensions and some
"visualizations" use animations to deal with model output that
varies over time. This seems to be the arena in which integration can
occur nearly immediately.

For the long term, integration needs to happen in all the
aspects of spatial/temporal analysis. CAD/CAM and AM/FM need
to be seamless with statistics, GIS, modeling, remote and in situ
monitoring, visualization, generalization, and other tools that are
now discrete functions of specialized communities. We need to be
able to work in a networked, collaborative, interdisciplinary
community, with common tools to apply the scientific method to the
highly complex and interrelated environmental problems facing us.
We need to see our logic to predict our future. Integration of GIS
and visualization software seems to be an interesting and useful
place to start.
Biographical Info. of Panelists:
Theresa Marie Rhyne: Martin Marietta/U.S. EPA
Theresa Marie Rhyne is currently a Lead Visualization
Researcher for the U.S. EPA's High Performance Computing and
Communications Initiatives and employed by Martin Marietta
Technical Services. From 1990 - 1992, she was the technical leader of
the U.S. EPA Scientific Visualization Center and was responsible for
building the Center since its founding in 1990.
William Ivey: SAS Institute, Inc.
William Ivey has been involved with the development of GIS
applications since 1988. The first applications he developed were
while employed by Computer Sciences Corporation under contract
to the US EPA. These applications used ARC/INFO and were for the
analysis and visualization of modeling results from gridded
atmospheric pollution models. Subsequently, he was employed by
the MCNC/North Carolina Supercomputing Center. There he
developed an interface between ARC/INFO and the Application
Visualization System(AVS). Currently, William is a systems
developer at SAS Institute and is working on the development of the
first SAS GIS product as well as investigating the integration of this
product with Spectraview, the SAS visualization product.

Loey Knapp: IBM & University of Colorado
Loey Knapp is a PhD candidate in Geography at the University
of Colorado and works for IBM as a research analyst in
visualization. For the last two years she has been working with the
U.S. Geological Survey examining the benefits of enhancing GIS
with scientific visualization techniques for hydrologic applications.
She is currently the project manager of the IBM-ESRI effort to
integrate IBM's Data Explorer with ESRI's ARC/INFO.
Peter Kochevar: DEC/San Diego Supercomputer Center
Peter Kochevar received a BS in Mathematics from the
University of Michigan and an MS in Mathematics from the
University of Utah. Peter also holds MS and Phd degrees in
Computer Science from Cornell University where he was a member
of the Program of Computer Graphics. Peter worked for a number
of years for the Boeing Commercial Airplane Company in Seattle,
Washington where he helped develop a computer-aided airplane
design system. Since 1990, Peter has been employed by the Digital
Equipment Corporation. Currently, he is a visiting scientist at the
San Diego Supercomputer Center where he heads up the data
visualization research efforts of the Sequoia 2000 Project.
Dr. Tom H. Mace: USEPA Scientific Computing Branch
Tom Mace leads EPA's National Data Processing Divsion
efforts to develop and support infrastructure for the use of
advanced information science and technology. He is the Agency's
representative to interagency panels on Landsat and advanced
sensor development, liaison to the NASA EOS Program, the Lead
Contact to the Interagency Working Group on Data Management
for Global Change and Chair of its Access Subgroup, Project Officer
on cooperative agreements with the Consortium for International
Earth Science Information Network, and technical expert to EPA's
Office of International Activities on the development of monitoring
and information systems in Russia and Eastern Europe.

Visualizing Data: Is Virtual Reality the Key?
Abstract
A visualization goal is to simplify the analysis of
large-quantity, numerical data by rendering the data as
an image that can be intuitively manipulated. The
question this controversial panel addresses is whether or
not virtual reality techniques are the cure-all to the
dilemma of visualizing increasing amounts of data.
This panel determines the usefulness of techniques
available today and in the near future that will ease the
problem of visualizing complex data. In regards to
visualization, the panel members will discuss
characteristics of virtual reality systems, data in threedimensional
environments, augmented reality, and
virtual reality market opportunities.
Position Statement by Tom Erickson:
Virtual Reality as a Medium for Data
Visualization
I approach virtual reality (VR) and visualization
from a design perspective. That is, I focus on
understanding what sort of support people need to
accomplish a given task, and on the ways in which
technologies can provide that support. Thus I will talk
Linda M. Stone, Chair
LORAL Space & Range Systems
lstone@srs.loral.com
Thomas Erickson
Apple Computer, Inc.
thomas@apple.com
Benjamin B. Bederson
University of New Mexico
bederson@cs.unm.edu
Peter Rothman
Avatar Partners
avatarp@well.sf.ca.us
Raymond Muzzy
LORAL Rolm Computer Systems
phone: 408-423-ROLM
about both the charac-teristics of VR systems, and the
nature of visualization.
1. Virtual Reality
I see VRs as having three important characteristics:
First, VRs exhibit high interactivity - there is a tight
coupling between the user's actions and the feedback
generated by those actions. Second, they support
embodiment: some sort of representation of the user is in
the same spatial framework as the data. Third, the VR
representation is spatial in nature; virtual objects are
situated in a spatial framework. Note that my definition
makes no mention of the technologies such as the gloves
and head-mounted displays with which VR has become
popularly associated. It also embraces MUDs (AKA
multi-user text-based virtual realities), systems which use
text to represent users, objects, and the spatial
environment in a purely metaphorical manner.
2 . Some Characteristics of Visualization
The aim of visualization is to represent a data set in
a way that makes it perceptible, and thus able to engage
our vaunted ability to recognize patterns. Thus, in my

view, 'visualization' is multi-sensory: it also includes
sonic and haptic representations of data. Second,
visualization is entwined with manipulation. We don't
just want to look at data, we want to move around it,
twist it, shake it, change its color mapping, expand it,
rotate it, shrink it, or slice and dice it. Finally,
visualization is often a collaborative activity. It may be
employed in group problem solving, or in a teaching
situation; thus, the degree to which a visualization
environment can support human-human interaction may
be critical.
3. Supporting Visualization
Visualization is more of a technique than a task.
That is, people don't usually visualize data just for the
hell of it. They may be trying to solve a problem, test a
theory, understand a data set, or communicate their
understanding to others. These differing goals have an
impact on how visualization should be supported, and
make it difficult to discuss the problem at a general
level. If you're designing a knife, it makes a difference
whether the users are surgeons or butchers; simply
designing something with a sharp edge able to cut flesh
will not result in a tool that is equal to all tasks.
Nevertheless, I will at least take a few stabs at ways in
which VR is well suited to support visualization in
general.
I will explore two directions. First, all three
characteristics of VR -- high interactivity, spatiality, and
embodiment -- support very natural ways of
manipulating the data, so that attention can be focused
on the data rather than on the interaction. For example,
the ability to 'grab' a data set at two points (rather than
the single point manipulation supported by most mouse
based systems), provides natural ways of stretching,
shrinking and rotating a data set. Second, because VR
systems represent both data and users within a spatial
framework, they provide support for the collaborative
component of visualization. The on-going use of MUDs
provide striking illustrations of the ability of even
metaphorical spatial environments to allow groups to
save state, communicate, and otherwise structure their
interactions.
Position Statement for Peter Rothman:
Virtual Reality for Multi-Dimensional
Environments
Historically, data visualization has relied on two
dimensional and three dimensional perspective
techniques to enable researchers to visualize complex
multi-dimensional data. Virtual reality technology
presents a revolutionary ability to visualize data in a true
three dimensional environment.
Using virtual reality technology, users are able
to immerse themselves in their data. This data can
dynamically change in real-time or be derived from
previously recorded data sets. Data with
dimensionalities higher than three can be visualized by
using multiple characteristics of the virtual objects.
Examples include:
Size (e.g. large, medium, small)
Shape (e.g. triangle, square, pentagon)
Color (e.g. red, purple, blue)
Texture (photographic images can be "painted"
onto objects)
Position (X, Y, and Z)
Orientation (roll, pitch, yaw)
Behavior (spinning, bouncing, blinking,
breathing, etc.)
Sound (chiming, singing, buzzing, etc.)
By making use of the inherent abilities of
people to visualize and navigate three dimensional
spaces, virtual reality technology enables researchers to
gain new insights into the structure of complex data
streams.
As an example of the power of virtual reality
technologies for data visualization, the presenter will
demonstrate an example of the use of virtual reality for
data visualization in the financial market.
Position Statement for Ben Bederson: Audio
Augmented Reality
For many types of information retrieval, social
interaction is critical to the experience. For instance,
why do people go to live music concerts instead of
listening to the CD? Partly because of the different kind
of sound, but I believe it is also largely due to the social
experience of being in an auditorium with the rest of the
audience. This identifies one area where I do not think
Virtual Reality will be successful. I believe that one of
the biggest challenges for today's technical designers is
to develop ways that computers can improve
communication and interaction among people.
There are also times when it is appropriate to
provide information without replacing the world for
practical reasons. For instance, when giving traffic
directions while driving, a typical approach is to put
video on the dashboard with a map application, or to
supply a printed piece of paper with directions. Both of
these situations are potentially dangerous because they

overload the visual channel by replacing it, at least
temporarily.
One approach to these challenges is to use
"Augmented Reality" - this is, to superimpose computer
generated data on top of the real world, as the person
moves within it. A number of groups are beginning to
experiment with this notion, notably Steve Feiner at
Columbia. My work is different in that it replaces the
video augmentation that Feiner uses with audio. As you
interact with the world, you are provided with relevant
audio annotations which give you extra information
about the current situation without interrupting what you
are doing. This works by using a different and largely
unused communication channel.
For a practical application based on this idea, it
may be necessary for the entire computer system to be
carried around, and for the computer system to know
where it is in the environment. To generalize this
research question, I have begun to ask "What happens
when a computer that you carry knows where it is?"
We have been applying these concepts by
creating an automated tour guide for museums. The
system will automatically describe pieces in the museum
as they are approached. One advantage of this method
over traditional taped tours is that people will have the
choice to choose what they want to see, in what order,
and for how long. We currently have a prototype
demonstrating this technology. This concept could also
be applied to the above mentioned traffic direction
problem so that the driver automatically gets verbal
traffic commands such as "turn right at the light" in situ.
I believe that Virtual Reality may be appropriate
for certain kinds of information visualization, such as
architectural walkthroughs - which can not be done any
other way. However, it may not be able to replace other
kinds of information retrieval, such as museum tours. I
believe that this is due to the social aspect of the
information retrieval process. Virtual Reality takes
people away from people. It replaces the natural
environment with a virtual one - and the society inherent
in the natural one. Social situations in our physical
world are important in our educational, entertaining, and
creative experiences. It is one thing to replace social
situations with technology, but the real challenge is to
use technology to enhance rather than "replace" them.
Position Statement for Raymond Muzzy:
A Marketing Perspective
Extensive customer interest already exists in VR
and expectations are high. It has attained considerable
"hype" in the technical and business press. In this
regard, there exists a similarity to AI in it's early stages
of development. In fact, the analogy to AI is very good
since it can provide some insights into possible outcomes
for VR in terms of marketplace expectations and future
realities. AI has taken time to develop its market niches
and products that meet customer needs and a similar
process will also occur for VR. Leveraging the AI
experience should make the VR transition more efficient.
The "real data visualization needs" of an
application will be the critical drivers that determine the
degree to which VR will realize its full potential. VR
can add a new dimension to interpretation of information
and provide insights that would otherwise be missed such
as scientific interpretation of data. It also provides a
method of presenting results so it can be more readily
understood by others such as in engineering design and
manufacturing. VR can add utility to applications like
training through its interactive attributes which would
improve the learning experience. All these areas
represent "value added" contributions that VR could
make to visualizing data.
The "value added" contribution needs to be
separated from the other aspects of VR hype which only
result in increased costs of doing business and also
contribute to a potential negative view of the "real value"
of VR. This is where the analogy to AI is very important
since the excessive initial AI "hype" did impact the time
required to realize its full potential.
A large number of conventional techniques and
products are capable of satisfying the bulk of the users
"real needs". But there still exists a select and growing
market where VR will be the key because it provides
"value added" functionality. "Focus" will therefore be
the key to exploring market opportunities for VR.
Current technology drivers for the market will be
discussed.
========================================
Biographies of the Panelists
Linda M. Stone is a computing systems researcher
for the Advanced Technology Program at LORAL Space &
Range Systems in Sunnyvale, CA. She researches upcoming
technologies for potential infusion into the Air Force Satellite
Control Network. Her interests include virtual reality,
computer graphics, and software engineering techniques.
Earlier, she supported remote satellite tracking stations as a
human-machine interface software developer. Linda earned
her B.S. in electronic engineering from California Polytechnic
State University in 1990, and will be receiving her M.S. in
engineering management/computer engineering from Santa
Clara University in 1995. She is a member of ACM and IEEE.
Thomas Erickson is an interaction analyst and
designer in Apple Computer's User Experience Architect's
Office in Cupertino, CA. His background is in cognitive
psychology. He has experience as a programmer and writer,
and today practices a mixture of design and (rough)
ethnography. His responsibilities at Apple include designing
interfaces for future technologies and applications, and

developing strategies for future products. Among his current
research interests are understanding what makes real world
environments rich and inviting places (or impoverished and
forbidding places), and applying that understanding to the
design of human-computer interfaces, intelligent devices, and
to physical environments with embedded computational
technology. Tom has published a variety of papers describing
particular design projects, and discussing design process. He
has a bachelor's degree and master's degree in psychology from
University of California-San Diego.
Peter Rothman is the managing partner and one of
the founders of Avatar Partners in Boulder Creek, CA. His
areas of expertise include computer graphics, object-oriented
programming, multiple target tracking, artificial neural
networks, and pattern recognition. Peter is currently
performing research on object oriented programming for
computer graphics applications and advanced pattern
recognition systems. Peter has lectured at numerous
universities, conferences, and seminars. He is a frequent
contributor to AI Expert magazine, and is in the process of
publishing three books. He is the host of the WELL's virtual
reality conference, and influential electronic conference on
virtual reality technology. He has a bachelor's degree in
mathematics from University of California-Santa Cruz, and a
master's degree in computer engineering from University of
Southern California.
Benjamin B. Bederson is currently an Assistant
Professor of Computer Science at the University of New
Mexico in Albuquerque. Before that, he was a research
scientist at Bellcore in Morristown, NJ. There, he developed
the Pad++ system for exploring zooming graphical user
interfaces. Ben is also a visiting research scientist at the New
York University's Media Research Lab. This is where he
developed the Audio Augmented Reality system for automated
tour guides. Ben has published a variety of articles,
conference papers, technical reports, and patents. He earned
his B.S. in computer science from Rensselaer Polytechnic
Institute in 1986, and his M.S. and Ph.D. in computer science
fro New York University in 1989 and 1992, respectively. Ben
is a member of ACM and IEEE.
Raymond Muzzy is Vice President of Business
Development at LORAL Rolm Computer Systems (LRCS) in
San Jose, CA. He is responsible for future business
development plans and marketing at LRCS, which is
expanding its business focus to produce powerful design
software and integrated systems for military digital signal
processing applications. Ray has been active in a number of
industrial groups and led an Electronic Industry Association 10
year forecast of the Synthetic Environments Marketplace. He
has a B.S. in engineering from University of California-
Berkeley, a M.S. and an engineer's degree in
aeronautics/astronautics from Stanford, and a M.B.A. from
Santa Clara University.
Selected References
[1] Bederson, B. B., G. Davenport, A. Druin and P.
Maes. Immersive Environments: A Physical Approach to
Multimedia Experience. Panel at IEEE International
Conference on Multimedia Computing and Systems, 1994.
[2] Bernd, Hamann. Visualization and Modeling
Contours of Trivariate Functions. Ph.D., Thesis, Arizona State
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1 Introduction
Validation� Veri�cation and Evaluation
Chair� Sam Uselton� CSC�NASA Ames
Panelists�
Geo� Dorn� Arco Exploration and Production Technology
Charbel Farhat� Univ. of Colorado
Michael Vannier� Washington University School of Medicine
Kim Esbensen� SINTEF
Al Globus� CSC�NASA Ames
The �How to Lie with Visualization� sessions of the
last couple of years have been interesting and enter�
taining� but it is time to take this discussion further.
Research and development in the �eld of scienti�c vi�
sualization must be concerned with tools scientists can
use to produce good visualizations. The next question
to address in improving visualization as a discipline
might be �How to tell good tools from bad�� However�
a better question is �How to select the most appropri�
ate and useful tools for a particular job�� Scientists
really need support in selecting the �right� tools and
verifying their quality.
Most scienti�c visualizations are exploratory rather
than expository� produced by scientists in the course
of exploring their own data. These visualizations
are usually done by scientists without the support of
graphics or visualization specialists who might be con�
sulted for making expository visualizations. Minor
errors in a visualization are easily overlooked. The
consequences of such errors range from increased time
required to gain the desired understanding to a com�
pletely incorrect understanding. The more innovative
and unusual the application� the less likely such an er�
ror will be detected. This vulnerability is due to the
absence of experience with �how things ought to look�
which is inherent in research.
A �bug� usually refers to software doing something
di�erent than the programmer intended. Comprehen�
sive testing� especially for software intended for use
in innovative environments� is hard. Descriptions and
summaries of the tests we have done are often not
available to the users. A di�erent source of visualiza�
tion errors is software that does something di�erent
than what the scientist thinks it does. The particu�
lar methods used to compute values in the process of
creating visualizations are important to the scientists�
but vendors are understandably reluctant to reveal all
the internals of their products. Is there a workable
compromise�
Another vulnerability of visualization users is in the
choice of a technique which is less e�ective than oth�
ers equally available. Visualization researchers and
developers should give users the information required
to make good decisions about competing visualization
techniques. What information is needed� What will
it take to gather and distribute it� How should it be
tied to visualization software�
This panel is composed of four scientists who use
visualization extensively in their diverse specialties�
and a �fth whose expertise is speci�cally in visual�
ization software. Each of the four application scien�
tists has a unique perspective on current problems and
what could be done better. The �fth member of the
panel has put signi�cant e�ort into designing test data
and test procedures for use in a research environment.
Their opinions make an interesting starting point for
improving our discipline.
2 Position Statement� Geo� Dorn
In oil and gas exploration and production� visu�
alization is becoming increasingly important to solv�
ing problems in data processing� interpretation� and
modeling. Our problems are characterized by large
data volumes �an average processed 3�D seismic vol�
ume may contain 2 billion points�. We interpret this
data to guide drilling a 1 foot diameter well to a depth
of 12�000 feet� where an positional error of 100 to 300
feet laterally can be the di�erence between a good well
and a dry hole. At a cost that ranges from several hun�
dred thousand to about ten million dollars per well�
we need to be accurate and precise.
When we speak of validating� verifying and evalu�
ating a visualization application� we�re talking about
answering a basic set of questions�

1. What does it do�
2. How does it do it�
3. What are its advantages�
3. When is it useful �technically��
4. When is it appropriate �economically��
5. How does it �t with other applications�
With our internal development� answers to what it
does and how it does it are straightforward � we know
the algorithms because we design them. We try to get
the answers to the remaining questions through proto�
typing� involving experienced users as early as possi�
ble� and testing with both �eld and model data. Model
data has the advantage of exact knowledge� �eld data
is necessary because a model is never as complex as
the real world.
What we need and want with outside development
are answers to the same �ve questions. We need to
know what an application does and �to perhaps a
lesser extent� how it does it. We need to know some�
thing about the algorithms that are operating on the
data because we are responsible for properly intepret�
ing that data. Comparisons between various appli�
cations from di�erent developers on a standard set of
data would help us answer the questions about the rel�
ative advantages of various applications� the circum�
stances under which they are useful� and how the ap�
plication �ts with other software tools we use. The
question of economics is probably only answerable by
the user and the user�s management.
Experience with some externally developed visual�
ization applications suggests that too little attention
is placed on the potential suite of other applications
that your clients use in their work. The visualization
application by itself will rarely if ever be the complete
answer to a client�s problem. In order to be used ef�
fectively and e�ciently� you need to be able to easily
pass data between the visualization application and
the rest of the tools being used to solve the problem.
In this same vein� it is important to provide a means
by which the user�developer can tie proprietary anal�
ysis techniques to your visualization application. The
point here is that we need both to visualize and to
use � interpret� interact with � the data. It�s not good
enough just to put up a pretty picture.
What we don�t need is more �ashy demos.
3 Position Statement� Charbel Farhat
In the context of our research� Coupled Field Prob�
lems are continuous mechanical or non�mechanical
systems mathematically modeled by partial di�eren�
tial equations that are coupled at the system in�
terfaces by non�homogeneous boundary conditions.
Some generic examples follow.
�1� A structure is submerged in a gas� �uid or
solid medium. �Speci�c application examples would
be an aircraft� submarine or buried silo.� The prob�
lem components are the structure and the external
�uid medium.
�2� A superconducting medium operates in a
thermal�electromagnetic �eld. The problem compo�
nents are the full�space electromagnetic �eld� the ther�
mal �eld and the superconductor viewed as a material
medium.
�3� Thermomechanical extrusion of a metal or plas�
tic in a fabrication process. The problem components
are the thermal �eld� the moving material� and the
constraining medium.
�4� A structure interacting with an active control
system of mechanical or embedded piezoelectric actu�
ators. The problem components are the �exible struc�
ture� the control system� and the sensor�control de�
vices.
All above examples involve a mechanical system as
one of the problem components� which interacts with
other mechanical� thermal or electromagnetic compo�
nents. The interaction is two�way in the sense that
in principle it is necessary to solve simultaneously for
the state of all components. The interaction domains
may have di�erent dimensionality� a surface in �1��
the mechanical volume in �2�� volume and surfaces in
�3�� and a discrete set �actuator locations� in �4�.
Visualizing the solutions of Coupled Field Problems
is almost as challenging as computing them. Indeed�
commercial visualization software is split between�
a� structured grids
b� unstructured grids
c� CFD �the famous or infamous Q �le�
d� structural mechanics
e� �nite element��nite volume
f� �nite di�erence
g� etc ...
It is out of the question for an engineer to use a
separate visualization package for each �eld because�
a� this is unmanageable
b� it is also unattractive
c� most importantly� the results for each �eld should
be visualized simultaneously in order to understand
the physics of the coupling phenomena.
This illustrates the need for scienti�c visualization
paradigms that are either independent or encapsulate
structure�unstructured concepts� �nite element��nite
di�erence schemes� distinct physical and mathemati�
cal �elds� scalar�vector�and tensor variables.

4 Position Statement� Michael Van�
nier
Visualization is central to diagnostic radiology.
Imaging modalities� such as radiography� computed
tomography �CT�� scintigraphy� magnetic resonance
imaging �MRI�� positron emission tomography �PET�
and others generate data sets which are viewed sub�
jectively by experts to generate text�based reports and
satisfy clinical consultation requests. Diagnostic per�
formance in medical imaging is measured using many
methods� most important among these is Receiver Op�
erating Characteristic or ROC analysis. This method
provides a means to measure the diagnostic process
at di�erent threshold settings in observer�based rated
response experiments.
ROC methods require knowledge of diagnostic
truth. In general� truth is established by independent
means �not the images themselves or simply consensus
of experts�. Application of ROC analysis has shown
that diagnostic performance and user preference for a
certain type of image are not necessarily the same. In
other words� simple visualization of results and sub�
jective impressions can be misleading. The accuracy
and precision of a diagnostic imaging test measured
with ROC methods determine the ability of the test
to discriminate an abnormal individual from a normal
population or to measure changes within an individual
over time.
The veri�cation of diagnostic medical imaging
methods and procedures is done in a staged fash�
ion� beginning with mathematically simulated data�
followed by physical test objects �called phantoms��
cadavers� animals� and ultimately human volunteers.
The interpretation of medical images requires knowl�
edge of norms � usually obtained by study of reference
populations and observation of normal variation. The
de�nition of �normal� is often di�cult� due to limi�
tations in our ability to adequately characterize and
sample su�cient representative cases and account for
their variation.
Evaluation of medical imaging systems and proce�
dures is both formative and summative. Process eval�
uation is done based on cost or safety criteria. Out�
come evaluation is done based on morbidity� mortality
�survival�� and quality of life criteria. Mandatory as�
sessments of medical imaging devices and accessories
including software and pharmaceuticals on the basis
of safety and e�cacy are arbitrated by the FDA.
Overall� medical imaging system visualization soft�
ware validation� veri�cation and evaluation is per�
formed in a well established framework. Presently�
multicenter� multiobserver and multimodality tri�
als of imaging systems are being developed to for�
mally assess imaging technology before adoption for
widespread use or reimbursement by third party pay�
ers.
5 Position Statement� Kim Esbensen
Scientists often have data which has no prede�ned
geometric or spatial interpretation. Visualization is
valuable for these types of data too however.
Exploratory visualization of complex� multivariate
data without an inherent spatial geometry is a speci�c
and important aspect of �standard� multivariate data
analysis. Some form of projective geometry is often
invoked� e.g. one of the powerful �bilinear� projec�
tion methods� principal component analysis� partial
least squares regression etc. With these methods it
is possible to gain valuable insight into the underly�
ing covariance data structures� while simultaneously
screening o� error components in the original data.
All analysis in this regimen has hitherto been carried
out on the basis of the familiar Cartesian coordinate
system. Recently a novel approach based on a set
of parallel coordinate axes in the 2�D plane allows a
completely di�erent visualization avenue� the Parallel
Coordinates �PC� approach.
Amongst other attributes the PC approach allows
for simultaneous display of a virtually unlimited num�
ber of variables without the characteristic projection
bias which is always accompanying bilinear methods.
This approach has been hailed as superior in many
ways to the traditional regimen� especially for process
monitoring and control. This new visualization tech�
nique is speci�cally o�ered not simply as an alterna�
tive way to presenting the same data as the traditional
techniques� but is said to allow completely new types
of insights and to reveal covariance relationships in a
fashion more easily grasped by the uninitiated user.
Illustrating complex process data relationships�
necessarily multivariate� by both the standard multi�
variate data analytical approach and the Parallel Co�
ordinates methodology thus comprises an illustrative
case study on how developers might go about present�
ing evidence on the use and merits of a new visualiza�
tion technique. Because the traditional multivariate
methods are very well known and evaluated� it will
su�ce to evaluate the PC approach on representative
data sets which have already been succesfully anal�
ysed� and understood in this regimen. In order to
familiarize oneself with the new technique� one might
at �rst use model� or simulated� data� some examples
of this avenue will be presented. For more realistic
Veri�cation� Validation and Evaluation however� one
is strongly advised to employ well�chosen real�world

data� process chemical data from a Norwegian py�
rometallurgical smelter plant �more than 25 raw ma�
terials and salient process variables for a complete one
year operation� serve as such a comparison vehicle.
The results from this comparison speaks volumes
of the ACTUAL merits of the new vs. the older tech�
niques in general� and of the speci�c PC attributes
that REALLY represents new types of insights in par�
ticular � and which are indeed nothing but old wine
in new bottles. There would appear to exist a DU�
ALITY between the Cartesian and the Parallel Coor�
dinate approaches. For example� the ordering of vari�
ables in the Parallel Coordinates approach is a critical
issue. The number of permutations of alternative or�
derings of variables is often quite staggering. However�
the ordering issue can be reduced signi�cantly� often
completely eliminated� by using an initial multivari�
ate data analysis. This standard covariance structure
delineation provides important clues for the optimal
parallel coordinate ordering which furthermore can be
obtained simultaneously with outlier�screening.
Neither is the old regimen obsolete� nor is the PC
approach a new panacea ready to substitute the for�
mer. Rather� the scienti�c visualization of complex
non�geometrical data has found a viable new comple�
mentary dual. Generalisations from this intercompar�
ison will be furthered for the speci�c purposes of this
panel.
6 Position Statement� Al Globus
I want to make two points� 1. visualization software
needs rigorous veri�cation in the form of much better
testing and� 2. experiments with human subjects are
essential to scienti�cally validate and evaluate visual�
ization techniques.
Veri�cation � does the software do what the devel�
oper thinks it does�
I�ve found much visualization software to be pretty
buggy. Crashes and mysterious behavior are common.
Only when I�ve used a package for some time and know
it quite well can I get reliable results. What�s a devel�
oper to do�
I�ve tried a few things not speci�c to visualization�
adding a �static test��� to each C�� class� �hand sim�
ulating� by single stepping through the code using a
visual debugger watching all local variables and ob�
ject members update� and building a random widget
tweaker. I�ve also tried developing a test set genera�
tor for unsteady �ows. The �rst two are pretty obvi�
ous and fairly standard� although �apparently� rarely
done. The last two merit some discussion.
A random widget tweaker keeps a list of all of the
user input widgets. In overnight runs� the tweaker re�
peatedly chooses a widget at random and sends it the
tweak message. The tweak message changes the value
of a widget �e.g.� slider position� at random. This tech�
nique simulates a monkey at the keyboard and mouse.
It e�ectively �nds crash and burn bugs.
Building a good test set generator is an interesting
problem. First of all� unsteady data sets can be very
large� so distributing the source to a test set generator
saves a lot of network bandwidth. Output size can be
a parameter to the code so that small data sets can
be generated for debugging and larger sets �that just
�t currently available disk space� generated to inves�
tigate program performance. Test sets should reveal
common and subtle bugs and de�ciencies that gener�
ate incorrect pictures. For example� circular stream�
lines will stress some particle tracing codes.
Validation � does the visualization accurately� and
e�ectively� represent the data�
Many visualization programmers come from the
computer graphics community� as I do. This commu�
nity values pretty pictures� which are not necessarily
correct or informative. In many cases� visualizations
are accepted if they look �more or less right�. Some�
times a user is called in to glance at the visualization
and make a few comments. This is mediocre science�
at best.
We claim that visualization increases human un�
derstanding. This can only be proven by experiments
with human subjects. As far as I know� no such exper�
iments have ever been conducted. Such experiments
are di�cult to design and so require collaboration with
psychologists and�or human factors experts.
Evaluation � is thing A better than thing B�
When is one visualization techniques better than
another� We can �ame or run experiments. For ex�
ample� two groups of subjects are given a data set
and asked to �nd important features. Each group is
given a di�erent visualization tool �e.g.� isosurfaces vs.
scalar mapped cut planes�. Time to completion and
correct results are measured.
7 Panelists�
7.1 Geo�rey Dorn
Geo�rey Dorn received his B.S. in Astrophysics
�1973� and his M.S. in Geophysics �1978� from the
University of New Mexico� and Ph.D. in Exploration
Geophysics �1980� from the University of California�
Berkeley. He has held positions in acquisition and in�
terpretation research in ARCO Oil and Gas Co. since
1980� including four years as director of interactive
interpretation techniques research. He is currently
a Research Advisor at ARCO. His interests include
3�D seismic interpretation� interactive interpretation

techniques and system design� 3D visualization tech�
niques� and reservoir geophysics. Geo� is a member
of the Society of Exploration Geophysicists �SEG� Re�
search Committee. He was general chairman of the
1993 SEG Research Workshop on 3�D Seismology and
has co�chaired SEG workshops on Scienti�c Visualiza�
tion� Emerging Workstation Technology� and Seismic
Stratigraphy.
7.2 Charbel Farhat
Charbel Farhat is Associate Professor of Aerospace
Engineering at the University of Colorado at Boulder.
He holds a Ph.D. in Computational Mechanics from
the University of California� Berkeley �1987�. He is
the recipient of several prestigious awards including
the Presidential Young Investigator Award �National
Science Foundation� 1989�1994�� the Arch T. Colwell
Merit Award �1993�� a TRW fellowship �1989�92�� the
CRAY Research Award �1990�91�� and the PACER
Award �Control Data Corporation� 1987�89�. Profes�
sor Farhat has been an AGARD lecturer on compu�
tational mechanics at several distinguished European
institutions. He is the author of over 50 journal papers
on computational mechanics and parallel processing.
His interest in computer graphics has recently led him
to develop TOP�DOMDEC� an object�oriented inter�
active �nite element pre� and post�processor.
7.3 Michael Vannier
Mike Vannier is a diagnostic radiologist �M.D.� and
researcher in medical imaging. He worked for NASA
as a contractor employee in the 1970�s and completed
a diagnostic radiology residency at the Mallinckrodt
Institute of Radiology� Washington University School
of Medicine in St. Louis� Mo. in 1982. In 1988� he
was visiting scientist in Materials Science and Engi�
neering at Argonne National Laboratory. Presently�
he is professor of radiology at Washington University.
He has extensive experience with the development and
application of 3�D visualization methods in diagnos�
tic imaging� especially for craniofacial deformities. He
serves as Editor�in�Chief of the IEEE Transactions on
Medical Imaging� and on the editorial board of Radiol�
ogy� Investigative Radiology� Diagnostic Imaging� and
others. He is the chair �for 1994�5� of the NIH study
section at the National Library of Medicine concerned
with the initial review of medical informatics projects.
7.4 Kim H. Esbensen
Kim H. Esbensen� Ph.D. �Technical University
of Denmark� Copenhagen� 1979�� is a senior scien�
tist with the Foundation for Scienti�c and Indus�
trial Research �SINTEF�� Oslo� Norway� where he is
in charge of research� development and applications
within chemometrics� multivariate image analysis and
acoustic sensing and imaging. He also holds posi�
tions with the University of Oslo� Telemark Engineer�
ing University� Norway and Universite Laval� Quebec�
Canada. Previously he spent eight years as a research
scientist at the Norwegian Computing Center� Oslo.
Honors include the 1979 University of Copenhagen Sil�
ver Medal� he has held a Royal Danish Academy of Sci�
ence fellowship. He is a member of the Chemometrics
Society� American Geophysical Union� Norwegian Sta�
tistical Society� Meteoritical Society and the Planetary
Society. His involvement with visualization includes
extensive experience in multivariate data analysis and
multivariate image analysis� both in applications and
in development of methodology. He is a co�developer
of the MIA �Multivariate Image Analysis� approach�
now embodied in the ERDAS image analysis system.
His current research interests focus on exploring the
duality between �Cartesian� multivariate data analy�
sis and the new Parallel Coordinates approach.
7.5 Al Globus
Al Globus is a senior computer scientist with Com�
puter Sciences Corporation at NASA Ames Research
Center. His research interests include scienti�c visu�
alization� space colonization� and computer network
enhanced education. Globus received a B.A. degree in
information science from the University of California
at Santa Cruz in 1979 following a previous life as a
musician. He is a member of the IEEE Computer So�
ciety and the American Institute of Aeronautics and
Astronautics.
7.6 Samuel P. Uselton
Sam Uselton is a researcher in visualization and
computer graphics. He is a senior computer scientist
with Computer Sciences Corp. �CSC� working in the
Applied Research Branch of the NAS Systems Divi�
sion at NASA Ames Research Center. He received his
BA in Math and Economics in 1973 from the Univer�
sity of Texas �Austin�� and his MS in 1976 and PhD in
1981� both in Computer Science� from the University
of Texas at Dallas. Sam has been working in com�
puter graphics and scienti�c visualization since 1976.
He has worked with scientists in �elds as diverse as
medicine� oil exploration and production� and compu�
tational �uid dynamics. He at the University of Tulsa
and the University of Houston for a total of ten years.
He is a member of hte IEEE Computer Society� the
ACM� and SIGGRAPH. His current research projects
are in direct volume rendering� realistic image synthe�
sis and uses of parallel and distributed computing for
visualization.

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