Program - Université de Pau et des Pays de l'Adour

2007 y 01 , 

Fourth 

http://www.univ-pau.fr/~cgout/mata2007 

Deadline for abstract submission : Jan. 31, 2007 

Inverse problems, Level set methods... 

In Numerical Algorithms (Springer) 

Scientific Committee: G. Albrecht, D. Apprato, 

C. Brezinski, C. de Boor, C. Gout, A. Kunoth, 

T. Lyche, M.-L. Mazure, C. Rabut, R. Schaback. 

Mira Bozzini 

Christian Gout 

Christophe Rabut 

Leonardo Traversoni 

Lay out: Marjory Lemière (Pau, Honolulu) 

Instituto Politécnico Nacional (CIC) 

INSA de Toulouse 

Università di Milano 

PROGRAM

Fourth International Conference on 

Multivariate Approximation: 

Theory and Applications 

at Cancun (WESTIN Spa & Resort), Mexico April 26-May 01, 2007. 

Index 

I. Introduction................................................................................................... page 5 

II. Proceedings............................................................................................. page 7 

III. Programme - Schedules 

Table .....................................................................................................page 11 

Talks..................................................................................................... page 13 

IV. Participants .................................................................................................. page 17 

IV. Abstracts ....................................................................................................page 23 

3

Introduction 

You will find in this program all the schedule of the conference, and some other 

informations (proceedings, abstracts…). 

Thank you for your participation in MATA 2007. 

MATA 2007 - Scientific Committee 

Gudrun Albrecht (France), Dominique Apprato (France), Claude Brezinski (France), 

Carl de Boor (USA), Christian Gout (France), Angela Kunoth (Germany), Tom Lyche 

(Oslo), Marie-Laurence Mazure (France), Christophe Rabut (Toulouse), Robert 

Schaback (Germany). 

Organizing Committee: 

Mira Bozzini 

Christian Gout 

(SIAM representative) 

Christophe Rabut 


Dipartimento di Matematica e Applicazioni 

Via R. Cozzi n. 53, 20125 Milano, Italy 

Tel: italy+026448+ 

Université de Valenciennes 

LAMAV-ISTV2 

Mont Houy 

59313 Valenciennes cedex 9, France 

chris_gout@cal.berkeley.edu 

INSA Toulouse 

Dpt Génie Mathématique 

135 av. de Rangueil 

31077 Toulouse cedex 4, France, 

Tel: (+33) 561 559 322 : Fax number: +33 561 559 320 

christophe.rabut@insa-toulouse.fr 


Univ. Autonoma Met. (Iztapalapa), 

Ciencias Basicas e Ingenieria, 

Ap Post 55-532, C.P. 09340 Mexico D.F., Mexico, Tel: (52) 

5 7244646 

ltd@xanum.uam.mx 

In cooperation with 

- Ministère de l'Education Nationale et de la Recherche, France 

- SMAI - AFA (Association Française d'Approximation) 

- SIAM 

- UAM (Iztapalapa) 

- INSA de Toulouse 

- Université de Valenciennes et du Hainaut Cambresis 

- Instituto Politecnico Nacional (CIC) 

- Università di Milano 

5

Previous Conference of the series: 

First International Conference on Scattered Data Fitting 

Cancun, Mexico, Thursday, March 2- Wednesday, March 8, 1995 

Org. : A. Le Méhauté, L. L. Schumaker, L. Traversoni 

Second International Conference on Multivariate Scattered Data Fitting 

Puerto Vallarta, Jalisco, Mexico, April 15--20, 1999 

Org. : P. Gonzalez-Casanova, A. Le Méhauté, L. L. Schumaker, L. Traversoni 

Third International Conference on Multivariate Scattered Data Fitting 

Puerto Vallarta, Jalisco, Mexico, April 23--29, 2003 

Org. : C. Gout, C. Rabut, L. Traversoni 

http://www.univ-pau.fr/~mata2003/ 

6

The proceedings 

of the conference will be edited as a special issue of 

Numerical Algorithms (SPRINGER). 

Some remarks: 

1) No difference will be made in the proceedings between an oral talk (invited or regular) 

and a poster talk. 

2) To appear in the proceedings, it will be necessary to have made the communication in 

Cancun. 

3) The paper submitted to the proceedings must correspond to the talk presented in 

Cancun (though possibly actualized). No page limitation is given… But of course, it's 

important to submit a concise paper! 

4) A peer review will be made. The process will be the same as a classical submission 

in the journal "Numerical Algorithms"... The authors will have to take into account the 

"instructions for authors" of this journal (http://www.springerlink.com/link.asp?id=101751). 

The editors will take their decision following the reports made by the referees. 

5) Deadline to submit a paper : OCTOBER 2007. 

Manuscript Presentation 

1. Manuscripts must be written in English and typed on one side of the page only, with wide 

margins. A font size of 11 point is preferred. Manuscripts must include: 

• A short title which will be used together with author‘s names in the running headline. 

• The authors‘ name and addresses 

• A self-contained abstract, preferably without formulae 

• A list of keywords 

• AMS (MOS) classification 

2. Tables and figures should be numbered, have a self-contained caption, and their positions in 

the text must be clearly indicated. In the case of paper-only submission, figures should be 

on separate sheets. In the case of electronic submission, please supply your figures in 

Encapsulated Postscript form. 

References must be numbered. In the text, they should be indicated by a bracketed number : [1]. 

Figures : All photographs, graphs and diagrams should be referred to as a 'Figure' and they should be numbered 

consecutively (1, 2, etc.). Multi-part figures ought to be labelled with lower case letters (a, b, etc.). Please insert keys 

and scale bars directly in the figures. Relatively small text and great variation in text sizes within figures should be 

avoided as figures are often reduced in size. Figures may be sized to fit approximately within the column(s) of the 

journal. Provide a detailed legend (without abbreviations) to each figure, refer to the figure in the text and note its 

approximate location in the margin. Please place the legends in the manuscript after the references. 

Tables : Each table should be numbered consecutively (1, 2, etc.). In tables, footnotes are preferable to long explanatory 

material in either the heading or body of the table. Such explanatory footnotes, identified by superscript letters, should 

be placed immediately below the table. Please provide a caption (without abbreviations) to each table, refer to the table 

in the text and note its approximate location in the margin. Finally, please place the tables after the figure legends in 

the manuscript. 

References 

1. Journal article 

Hamburger, C.: Quasimonotonicity, regularity and duality for nonlinear systems of partial differential equations. Ann. 

Mat. Pura. Appl. 169, 321–354 (1995) 

7

2. Inclusion of issue number (optional) 

Campbell, S.L., Gear, C.W.: The index of general nonlinear DAES. Numer. Math. 72(2), 173–196 (1995) 

3. Journal issue with issue editor 

Smith, J. (ed.): Rodent genes. Mod. Genomics J. 14(6), 126–233 (1998) 

4. Journal issue with no issue editor 

Rodent genes: Mod. Genomics J. 14(6):126–233 (1998) 

5. Book chapter 

Broy, M.: Software engineering – from auxiliary to key technologies. In: Broy, M., Denert, E. (eds.) Software Pioneers, pp. 

10–13. Springer, Berlin Heidelberg New York (2002) 

6. Book, authored 

Geddes, K.O., Czapor, S.R., Labahn, G.: Algorithms for Computer Algebra. Kluwer, Boston (1992) 

7. Book, edited 

Seymour, R.S. (ed.): Conductive Polymers. Plenum, New York (1981) 

8. Chapter in a book in a series without volume titles 

MacKay, D.M.: Visual stability and voluntary eye movements. In: Jung, R., MacKay, D.M. (eds.) Handbook of Sensory 

Physiology, vol. 3, pp. 307–331. Springer, Berlin Heidelberg New York (1973) 

9. Chapter in a book in a series with volume titles 

Smith, S.E.: Neuromuscular blocking drugs in man. In: Zaimis, E. (ed.) Neuromuscular Junction. Handbook of Experimental 

Pharmacology, vol. 42, pp. 593–660. Springer, Berlin Heidelberg New York (1976) 

10. Proceedings as a book (in a series and sub-series) 

Zowghi, D., et al.: A framework for reasoning about requirements in evolution. In: Foo N., Goebel R. (eds.) Topics in 

Artificial Intelligence, 4th Pacific Rim Conference on Artificial Intelligence, Cairns, August 1996. Lecture Notes in 

Computer Science. Lecture Notes in Artificial Intelligence, vol. 1114, pp. 157–168. Springer, Berlin Heidelberg New York 

(1996) 

11. Proceedings with an editor (without a publisher) 

Aaron, M.: The future of genomics. In: Williams, H. (ed.) Proceedings of the Genomic Researchers, Boston, 1999 

12. Proceedings without an editor (without a publisher) 

Chung, S.-T., Morris, R.L.: Isolation and characterization of plasmid deoxyribonucleic acid from Streptomyces fradiae. 

In: Abstracts of the 3rd International Symposium on the Genetics of Industrial Microorganisms, University of 

Wisconsin, Madison, 4–9 June 1978 

13. Paper presented at a conference 

Chung, S.-T., Morris, R.L.: Isolation and characterization of plasmid deoxyribonucleic acid from Streptomyces fradiae. 

Paper presented at the 3rd international symposium on the genetics of industrial microorganisms, University of 

Wisconsin, Madison, 4–9 June 1978 

14. Patent. 

Name and date of patent are optional 

L.O. Norman: Lightning rods. US Patent 4,379,752, 9 Sept 1998 

15. Dissertation, Ph.D. thesis (either text is acceptable) 

J.W. Trent: Experimental acute renal failure. Dissertation, University of California (1975) 

16. Institutional author (book) 

International Anatomical Nomenclature Committee: Nomina anatomica. (Excerpta Medica, Amsterdam 1966) 

17. Non-English publication cited in an English publication 

Wolf, G.H., Lehman, P.-F.: Atlas der Anatomie, vol. 4/3, 4th edn. Fischer, Berlin (1976) [NB: Use the language of the 

primary document, not that of the reference for "vol" etc.!] 

18. Non-Latin alphabet publication. 

The English translation is optional. 

Marikhin, V.Y., Myasnikova, L.P.: Nadmolekulyarnaya struktura polimerov 

(The supramolecular structure of polymers). Khimiya, Leningrad (1977) 

19. Published and In press articles with or without DOI: 

19.1 In press: 

Wilson, M. et al. In: Wilson M (ed.): Style manual. Springer, Berlin Heidelberg New York (2006) 

19.2. Article by DOI (with page numbers) 

Slifka, M.K., Whitton, J.L.: J. Mol. Med. 78:74–80. DOI 10.1007/s001090000086 (2001) 

19.3. Article by DOI (before issue publication with page numbers) 

Slifka, M.K., Whitton, J.L.:J. Mol. Med. (in press). DOI 10.1007/s001090000086 (2006) 

19.4. Article in electronic journal by DOI (no paginated version) 

Slifka, M.K., Whitton, J.L.: Dig. J. Mol. Med. DOI 10.1007/s801090000086 (2000) 

20. Internet publication/Online document 

Doe, J. In: The dictionary of substances and their effects. Royal Society of Chemistry.Available via DIALOG. 

http://www.rsc.org/dose/title of subordinate document. Cited 15 Jan 1999 (1999) 

8

20.1. Online database 

Healthwise Knowledgebase. US Pharmacopeia, Rockville. Http://www.healthwise.org. Cited 21 Sept 1998 (1998) 

Supplementary material/private homepage 

Doe, J.: Http://www.privatehomepage.com. Cited 22 Feb 2000 (2000) 

University site 

Doe, J.: Http://www.uni-heidelberg.de/mydata.html. Cited 25 Dec 1999 (1999) 

FTP site 

Doe, J.: ftp://ftp.isi.edu/in-notes/rfc2169.txt. Cited 12 Nov 1999 (1999) 

Organization site 

ISSN International Centre: Global ISSN database. http://www.issn.org. Cited 20 Feb 2000 (1999) 

Online Manuscript Submission, 

Review and Tracking System for the journal - Numerical Algorithms 

TO SUBMIT A PAPER, IT WILL BE NECESSARY TO SUBMIT DIRECTLY TO NUM. ALGO 

USING THE JOURNAL WEBSITE. 

C. GOUT will send you more details, in pecular the LOGIN and PASSWORD related TO 

MATA 2007. 

So, do not submit paper before you have these informations (around mid-september). 

Authors: wide range of submission file formats is supported, including: Word, WordPerfect, RTF, TXT, 

TIFF, GIF, JPEG, EPS, LaTeX2E, TeX, Postscript, PICT, Excel, Tar, Zip and Powerpoint. PDF is not an 

acceptable file format. 

9

April 26, THURSDAY April 27, FRIDAY April 28, SATURDAY April 30, MONDAY May 01, TUESDAY 

8.00: Registration 

8.30: Welcome session 

8.40 Larry Schumaker 

A C 1 quadratic trivariate macroelement 

space defined over 

arbitrary tetrahedral partitions 

9.20 Hrushikesh N Mhaskar, 

Diffusion polynomial frames on 

metric measure spaces 

9.50 Steven Damelin 

Numerical integration, Energy and 

Discrepancy on Compact 

Homogenous manifolds 

10.20 Shayne Waldron, 

Tight frames and their symmetries 

11.00 Mexican Coffee Break 

& Registration 

11.40 Carole Le Guyader 

Self-repelling snakes for topologypreserving 

segmentation models. 

12.10 Nicolas Forcadel 

Image segmentation using a 

Generalized Fast Marching 

Method 

12.40 Knut Mørken 

Mathematical and computational 

challenges in some applications of 

medical imaging 

13.20 END 

8.00 Maria Cruz Lopez de Silanes 

An extension of a bound for functions 

in Sobolev spaces and its application to 

(m,s)-spline interpolation and 

smoothing 

8.40 Alexander Grebennikov, 

Rotating projection of image 

reconstruction and application for 

computer tomography 

9.10 Juarez Valencia Lorenzo 

Hector 

Numerical Simulation of a Viscous 

Pump by a High Order Finite Element 

Method 

9.40 Milvia Rossini 

Polyharmonic splines 

10.20 Mike Neamtu 

Splines for Surfaces of Arbitrary 

Topology 


11.40 Frank Sommen 

The Dirac equation for bivector mass 

12.20 Bram de Knock 

A new Hilbert transform in a metric 

dependent Clifford setting 

12.50 Liesbel Van de Voorde 

Some results in discrete Clifford 

analysis 

13.20 Klaus Gürlebeck 

On monogenic approximations of 

solutions of the Lamé system 

14.00 END 

9.00 Rafal Ablamowicz 

Computational algebraic geometry methods for 

curves and surfaces 

9.40 Pedro Gonzalez-Casanova 

A Dynamical Node Adaptive RBF Method to Solve 

PDE Problems. 

10.10 Karina Toscano 

Hand-written character recognition using spline 

wavelets. 

10.40 Gabriele Steidl , 

Diffusion Filters and Wavelets 

11h20 Mexican Coffee Break 

12h 15: END 

POSTER SESSION, Monday 10.40 

- Mira Bozzini, A local method for large and 

unevenly distributed data. 

- Anne Gaëlle Saint-Guirons, Approximation of 

surfaces with fault(s) and/or rapidly varying data, 

using segmentation process, D m spines and finite 

element method. 

- Hermenegildo Cisneros, Isolated character 

recognition by using vector machines with a kernel 

of radial base. 

- Benjamin Cruz, Some formal aspects and 

applications of Cartan - Dieudonné theorem. 

- Xiang Feng Bu, Variational calculus and multigrid 

dynamical programmingon expo-rational surfaces 

and volume deformations: theory and applications 

- Sonia Charleston Forward and Inverse Problems 

for Determining Computer Simulated Respiratory 

Sound Sources 

- Christian Gout , Geodesic Active contour under 

geometrical constraints: Theory and 3D applications 

- Christophe Rabut, How to reduce Runge 

phenomenon 

- Jernej Kozak, Lattices on simplices 

8.00 Olivier Gibaru 

Parametric L 1 polynomial splines 

interpolation 

8.40 Lubomir Dechevsky 

Expo-rational B-splines of a complex 

variable and their applications 

9.10 Victoria Hernandez 

Umbrella based triangulation of regular 

parametric surfaces 

9.40 Karine Dadourian 

Analysis of some bivariate non-linear 

interpolatory subdivision schemes and 

associated multiresolution 

10.10 Hoffman Micklos 

General formulae of B-spline curves with 

shape parameters 


and 

POSTER SESSION 

12.00 Yi Xu 

QWT: Retrospective and New Applications 

12.40 Arne Lakså 

Generalized Expo-Rational B-splines 

13.10 Heriberto Casarrubias 

Using cubic spline for the velocity planning 

in mobile robots. 

13.40 Francesca Pitolli 

Sparse Approximation for Source Separation 

in the Magneto-encephalography Inverse 

Problem 

8.00 Jeremy Levesley 

The order of approximation of transport 

equations by the kinetic equation 

8.40 Leonardo Traversoni 

Wavelets and Movements, different 

approaches 

9.10 Luis Eduardo Falcon 

Radon transform and harmonical 

analysis for 3D motion estimation using 

robot omnidirectional vision 

9.40 Rafael Resendiz 

Building solids using quaternions 


10.40 Wolfgang Sprößig 

Quaternionic holomorphic functions in 

fluid dynamics and elasticity 

11.20 Angela Kunoth 

Fast Multiscale Methods for PDE- 

Constrained Control Problems 

12h00 Organizing Committee 

12h10 END 

14.20 END 

MATA 2007 Cancun, Mexico… April 26-May 01, 2007. Org. : M. Bozzini, C. Gout-C. Rabut-L. Traversoni 

Registration required to appear in the program…

APRIL 26, THURSDAY 

8.00: Registration 

8.30: Welcome session 

8.40 Larry Schumaker 

A C 1 quadratic trivariate macro-element space defined over arbitrary tetrahedral partitions 

9.20 Hrushikesh N Mhaskar, 

Diffusion polynomial frames on metric measure spaces 

9.50 Steven Damelin 

Numerical integration, Energy and Discrepancy on Compact Homogenous manifolds 

10.20 Shayne Waldron, 



& Registration 

11.40 Luminita Vese 

Variational image decomposition models into cartoon and texture 

12.20 Carole Le Guyader 

Self-repelling snakes for topology-preserving segmentation models. 

12.50 Nicolas Forcadel 

Image segmentation using a Generalized Fast Marching Method 

11.40 Knut Mørken 

Mathematical and computational challenges in some applications of medical imaging 

14.00 END 

APRIL 27, FRIDAY 

8.00 Maria Cruz Lopez de Silanes 

An extension of a bound for functions in Sobolev spaces and its application to (m,s)-spline 

interpolation and smoothing 

8.40 Alexander Grebennikov 

Rotating projection of image reconstruction and application for computer tomography 

13

9.10 Juarez Valencia Lorenzo Hector 

Numerical Simulation of a Viscous Pump by a High Order Finite Element Method 

9.40 Milvia Rossini 

Polyharmonic splines 

10.20 Mike Neamtu 

Splines for Surfaces of Arbitrary Topology 


11.40 Frank Sommen 


12.20 Bram de Knock 

A new Hilbert transform in a metric dependent Clifford setting 

12.50 Liesbel Van de Voorde 

Some results in discrete Clifford analysis 

13.20 Klaus Guerlebeck 

TBA 

14.00 END 

APRIL 28, SATURDAY 

9.00 Rafal Ablamowicz 

Computational algebraic geometry methods for curves and surfaces 

9.40 Pedro Gonzalez-Casanova 

A Dynamical Node Adaptive RBF Method to Solve PDE Problems. 

10.10 Karina Toscano 

Hand-written character recognition using spline wavelets. 

10.10 Martin Buhmann 

Radial Basis Function Interpolation with Parameters 

10.40 Gabriele Steidl 


11h20 Mexican Coffee Break 

12h 15: END 

14

APRIL 29, SUNDAY 

Group excursion to Chitzen Itza (or Tulum), 

APRIL 30, MONDAY 

8.00 Olivier Gibaru 

Parametric L 1 polynomial splines interpolation 

8.40 Lubomir Dechevsky 

Expo-rational B-splines of a complex variable and their applications 

9.10 Victoria Hernandez 

Umbrella based triangulation of regular parametric surfaces 

9.40 Karine Dadourian 

Analysis of some bivariate non linear interpolatory subdivision schemes and associated 

multiresolution 

10.10 Hoffman Micklos 

General formulae of B-spline curves with shape parameters 

10.40 Mexican Coffee Break and 

POSTER SESSION 

- Mira Bozzini, A local method for large and unevenly distributed data 

- Anne Gaëlle Saint-Guirons, Surface Approximation of surfaces with fault(s) and/or rapidly 

varying data, using segmentation process, D m spines and finite element method 

- Hermenegildo Cisneros, Isolated character recognition by using vector machines with a kernel 

of radial base 

- Benjamin Cruz, Some formal aspects and applications of Cartan - Dieudonné theorem 

- Xiang Feng Bu, Variational calculus and multigrid dynamical programmingon expo-rational 

surfaces and volume deformations: theory and applications 

- Sonia Charleston Forward and Inverse Problems for Determining Computer Simulated 

Respiratory Sound Sources 

- Christian Gout & Dominique Apprato, Geodesic Active contour under geometrical constraints: 

Theory and 3D applications 

- Christophe Rabut, How to reduce Runge phenomenon 

- Jernej Kozak, Lattices on simplices 

15

12.00 Yi Xu 


12.40 Arne Lakså 


13.10 Heriberto Casarrubias 

Using cubic spline for the velocity planning in mobile robots 

13.40 Francesca Pitolli 

Sparse Approximation for Source Separation in the Magneto-encephalography Inverse Problem 

14.20 END 

May 01, TUESDAY 

8.00 Eduardo Bayro-Corrochano 

Applications of quaternion wavelet transform: for robotic vision 

8.40 Leonardo Traversoni 

Wavelets and Movements, different approaches 

9.10 Luis Eduardo Falcon 

Radon transform and harmonical analysis for 3D motion estimation using robot omnidirectional 

vision 

9.40 Rafael Resendiz 

Building solids using quaternions 

10.10 Wolfgang Sprößig 

Quaternionic holomorphic functions in fluid dynamics and elasticity 


11.30 Angela Kunoth 

Fast Multiscale Methods for PDE-Constrained Control Problems 

12.10 Jeremy Levesley 

The order of approximation of transport equations by the kinetic equation 

12h50 Organizing Committee 

13.00 END 

16

LIST OF PARTICIPANTS 

The 65 registered participants come from 18 different countries. 

Belgium (3): De Knock, Sommen, Van de Voorde China (1) : Xu Cuba (1): Hernandez 

Ethiopia (1) : Gudetta France (8): Apprato, Dadourian, Forcadel, Gibaru, Gout, Le Guyader, 

Rabut, Saint-Guirons Germany (5): Buhmann, Guerlebeck, Kunoth, Sprößig, Steidl Hungary (2): 

Hoffmann, Juhasz Israël (1) : Ziegler Italy (3): Bozzini, Pitolli, Rossini Lybia (3) : 

Mexico (21): Arredondo, BarrÓn, Bayro Corrochano, Casarubbias, 

Cervantes, Charleston, Cisneros, Cruz, FalcÓn, Gloria, Gonzalez-Casanova, Grebennikov, Hector, 

Leon, LÓpez, Martinez, Rabinovitch, Resendiz, Sossa Azuela, Toscano, Traversoni New Zealand 

(1) : Waldron Norway (4) : Bu, Dechevsky, Lakså, Mørken Slovenia (1) : Kozak Spain 

(1): Lopez de Silanes United Kingdom (1): Levesley USA (6): Ablamowicz, Damelin, Mhaskar, 

Neamtu, Schumaker, Vese Russia (1) : Berdyshev 

Ablamowicz Rafal (saturday, 8.30) 

Department of Mathematics, Box 5054, Tennessee Technological University , Cookeville, TN 38505 U.S.A. 

rablamowicz@tntech.edu 

Apprato Dominique (monday, 10.40) 

Université de Pau UFR Sciences et Techniques de la Côte Basque 64600 Anglet. France 

Université de Pau, FR 2952 CNRS - IPRA-Av. de l Université - BP 1155 , 64018 Pau cedex, France 

apprato@univ-pau.fr 

Arredondo Brenda 

Univ. Autonoma Met., Iztapalapa Ciencias Basicas e Ingenieria, 

Ap Post 55-532, C.P. 09340 Mexico, 9340 Mexico D.F. Mexico 

BarrÓn Ricardo 

Centro de Investigación en Computación – IPN, Av. Juan de Dios Bátiz, esq. Miguel Othon de Mendizábal 9, 

Mexico City, 07738. Mexico 

rbarron@cic.ipn.mx 

Bayro Corrochano Eduardo (tuesday, 8.00) 

CINVESTAV Unidad Guadalajara, Apartado Postal 31-438, Plaza la Luna , 44550 Guadalajara, Jalisco. Mexico 

edb@gdl.cinvestav.mx 

Berdyshev Vitalii 

URAL University, Ekaterinbourg, Russia 

bvi@imm.uran.ru 

17

Bozzini Mira (monday, 10.40) 

Università di Milano, Dipartimento di Matematica e Applicazioni ,Via R. Cozzi n. 53, 20125 Milano, Italy 

mira.bozini@unimib.it 

Bu Xiang Feng (monday, 10.40) 

Narvik University College, P.O. Box 385, 2 Lodve Langes St., 8505 Narvik, Norway 

430492@student.hin.no 

Buhmann Martin (saturday, 10.10) 

Justus-Liebig-Universität Giessen, Arbeitsgruppe Numerische Mathematik, Mathematisches Institut, Heinrich- 

Buff-Ring 44, 35392 Giessen , Germany 

buhmann@uni-giessen.de 

Casarubbias Heriberto (monday, 13.10) 



heribertocv@gmail.com 

Cervantes-Cabrera Daniel 

Unidad de Coputo Cientifico DGSCA 

UNAM Ciudad Universitaria, Circuto Exterior, Zona Cultural 44510 Mexico D.F. 

danielcc@rv.unam.mx 

Charleston Sonia (monday, 10.40) 

Electrical Engineering Department, 2 Health Science Department, Universidad Autónoma Metropolitana- 

Iztapalapa, 09340 Mexico City, México 

schv@xanum.uam.mx 

Cisneros Hermenegildo (monday, 10.40) 



hcisneros@sagitario.cic.ipn.mx 

Cruz Benjamín (monday, 10.40) 



benjamincruz@sagitario.cic.ipn.mx 

Cruz Lopez de Silanes Maria (friday, 8.00) 

Departamento de Matemática Aplicada C.P.S., Univ. de Zaragoza, María de Luna, 3 - 50018 Zaragoza, Spain. 

mcruz@posta.unizar.es 

Dadourian Karine (monday, 9.40) 

Université de Provence et Monitrice à l'Université Paul Cézanne. CMI, LATP - IMT/LATP, 

39 rue Frédéric Joliot-Curie, 13453 Marseille cedex 13 

dadouria@cmi.univ-mrs.fr 

Damelin Steven (thursday, 9.50) 

Department of Mathematical Sciences, , Georgia Southern University, Postoffice Box 8093, Statesboro, GA 

30460-8093, U.S.A. 

damelin@georgiasouthern.edu 

Dechevsky Lubomir (monday, 8.40) 


18

Lubomir.T.Dechevsky@hin.no 

FalcÓn Luis Eduardo (tuesday, 9.10) 

CINVESTAV Av. CientifIca 1145, Colonia el Bajio, 45010 Zapopan, Mexico 

lfalcon@gdl.cinvestav.mx 

Forcadel Nicolas (thursday, 12.50) 

Ecole Nationale des Ponts et Chaussées, CERMICS, 6 et 8 avenue Blaise Pascal, Cité Descartes - Champs sur 

Marne, 77455 Marne la Vallée Cedex 2, France 

forcadel@cermics.enpc.fr 

Gibaru Olivier (monday, 8.00) 

ENSAM Lille 8 Bd. Louis XIV, 59046 Lille France 

Olivier.Gibaru@lille.ensam.fr 

Gloria Rafael 

Univ. Autonoma Met., Iztapalapa Ciencias Basicas e Ingenieria 


Gonzalez-Casanova Pedro (saturday, 10.10) 

Unidad de Coputo Cientifico DGSCA 

UNAM Ciudad Universitaria, Circuto Exterior, Zona Cultural 44510 Mexico D.F. 

pedrogc@plecticsdgsca2.unam.mx 

Gout Christian (monday 10.40) 

Université de Valenciernnes, LAMAV-ISTV2, Mont Houy, 59313 Valenciennes cedex 9, France 

chris_gout@cal.berkeley.edu 

Grebennikov Alexander 

Autonomous University of Puebla , Puebla, MÉXICO 

agrebe@fcfm.buap.mx 

Guerlebeck Klaus (friday, 13.20) 

University of Weimar Math. Dpt, Coudraystr.13 99423 Weimar. Germany 

klaus.guerlebeck@bauing.uni-weimar.de 

Le Guyader Carole (thursday, 12.20) 

INSA de Rennes,20 Avenue des Buttes de Coësmes, CS 14315, 35043 Rennes Cedex , France 

carole.le-guyader@insa-rennes.fr 

Hector Juarez Lorenzo Valencia (friday, 9.10) 

Departamento de Matematicas, Universidad Autonoma Metropolitana-AV. San Rafael 186 Col. Vicentina, 9340 

Mexico city, Mexico 

hect@xanum.uam.mx 

Hernandez Victoria (monday, 9.10) 

Instituto de Cibernetica, Matematica y Fisica Calle E Nº 309, esquina a 15 - 10400 Vedado 10400 Vedado-La 

Habana. Cuba 

vicky@cidet.icmf.inf.cu 

19

Hoffmann Micklos (monday, 10.10) 

Institute of Mathematics and Informatics University of Eger H-3300 Eger Hungary 

hofi@gemini.ektf.hu 

Juhasz Imre (monday, 10.10) 

Institute of Mathematics and Informatics University of Eger H-3300 Eger Hungary 

agtji@uni-miskolc.hu 

de Knock Bram (friday, 12.50) 

Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Gent, Belgium 

Bram.DeKnock@UGent.be 

Kozac Jernej (monday, 10.40) 

FMF and IMFM, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenija 

Jernej.Kozak@fmf.uni-lj.si 

Kunoth Angela (tuesday, 11.30) 

Universität Bonn Institut für Angewandte Mathematik , Wegelerstr.6 53115 Bonn. Germany 

kunoth@iam.uni-bonn.de 

Lakså Arne (monday, 12.40) 


Arne.Laksa@hin.no 

Leon Diana 



Levesley Jeremy (tuesday, 12.10) 

Leicester University Department of Mathematics and Computer Science LE1 7RH Leicester. UK 

jl1@mcs.le.ac.uk 

LÓpez Jesús 

Centro de Investigación en Computación – IPN, Unidad Profesional Adolfo Lopez Mateos - Col. Nueva Industrial 

Vallejo - Delegacion Gustavo A. Madero, Mexico City, 07738. Mexico 

jesuslrb06@sagitario.cic.ipn.mx 

Martinez Ileana 



Mhaskar Hrushikesh N. (thursday, 9.20) 

Department of Mathematics, California State University, Los Angeles, CA 90032, U. S. A. 

hmhaska@calstatela.edu 

Mørken Knut (thursday, 13.20) 

University of Oslo, Department of Informatics and Center of Mathematics of Applications, PO Box 1053, 

Blindern, Norway. 

knutm@ifi.uio.no 

Neamtu Mike (friday, 10.20) 

Vanderbilt University, Dpt of Mathematics, 1326 Stevenson Center, Nashville, TN 37240 USA 

neamtu@math.Vanderbilt.Edu 

20

Pitolli Francesca (monday 13.40) 

Univ. "la Sapienza" Roma Via Antonio Scarpa 16, 161 Roma. Italy 

pitolli@dmmm.uniroma1.it 

Rabinovitch Vladimir 

Instituto Politecnico Nacional, Mexico, ESIME Zacatenco, Edif.1, av. IPN, 07738, Mexico 

vladimir.rabinovich@gmail.com 

Rabut Christophe (monday 10.40) 

INSA Toulouse 135 av. de Rangueil, 31077 Toulouse cedex4. France 

rabut@insa-toulouse.fr 

Resendiz Rafael (tuesday, 9.40) 

Universidad Autonoma Metropolitana 

Ahuizotl 23 Col. Ancon Los Reyes la Paz 56410 Edomex, Mexico 

rafael.resendiz@gmail.com 

Rossini Milvia (friday, 9.40) 

Università di Milano, Dipartimento di Matematica e Applicazioni ,Via R. Cozzi n. 53, 20125 Milano, Italy 

rossini@matapp.unimib.it 

Saint-Guirons Anne-Gaëlle (monday 10.40) 

Université de Pau, Lab. Math. UMR 5142 CNRS - IPRA-Av. de l Université - BP 1155 , 64018 Pau cedex, France 

Anne-gaelle.saint-guirons@univ-pau.fr 

Schumaker Larry L. (thursday, 8.00) 

Vanderbilt University, Dpt of Mathematics, 1326 Stevenson Center, Nashville, TN 37240 USA 

larry.schumaker@gmail.com 

Sommen Frank (friday, 11.40) 


fs@cage.rug.ac.be 

Sossa Azuela Juan Humberto 

Centro de Investigación en Computación – IPN, Av. Juan de Dios Bátiz, S/N Col. Nueva Industrial Vallejo, Mexico 

City, 07738. Mexico 

hsossa@cic.ipn.mx 

Sprößig Wolfgang (tuesday, 10.10) 

TU Bergakademie Freiberg, Fakultät für Mathematik und Informatik, Institut für Angewandte Analysis, 09596 

Freiberg, Germany 

sproessig@math.tu-freiberg.de 

Steidl Gabriele (saturday, 10.50) 

Univ. of Mannheim, Dpt of Mathematics and Computer Science, A5-Room B107, 68131 Mannheim, Germany 

steidl@kiwi.math.uni-mannheim.de 

Toscano Karina (saturday 9.40) 



likatome@hotmail.com 

Traversoni Leonardo (tuesday, 8.40) 

21

Univ. Autonoma Met. (Iztapalapa Ciencias Basicas e Ingenieria, Ap Post 55-532, C.P. 09340 Mexico, 9340 Mexico 

D.F. Mexico 


Van de Voorde Liesbet (friday, 12.50) 


Liesbet.VandeVoorde@UGent.be 

Vese Luminita (thursday, 11.40) 

UC Los Angeles, 405, Hilgard Avenue, Los Angeles, CA 90095-1555, U.S.A. 

lvese@ucla.edu 

Waldron Shayne (thursday, 10.20) 

Department of Mathematics (Rm 410), University of Auckland, 28 Invermay Ave, Private Bag 92019, Auckland, 

Sandringham, New Zealand 

waldron@math.auckland.ac.nz 

Xu Yi (monday, 12.00) 

Shanghai Jiaotong University, School of electronic, information and electrical engineering, Institute of Image 

communication and information processing, Room 321-323,No. 5 Buliding, school of electronic, information and 

electrical engineering,Shanghai Jiaotong Univ, Dongchuan Rd. 800, 200240 Shanghai, China 

xuyi@sjtu.edu.cn 

22

Computational algebraic geometry methods 

for curves and surfaces 

Rafal Ablamowicz 

rablamowicz@tntech.edu 

ABSTRACT : 

Finding analytic (exact) formulas for parallel lines (envelopes) to 

parabolas, ellipses, hyperbolas, and other curves such as, for example, the Bézier 

cubic, is of importance in structure engineering and theory of mechanisms. 

Likewise, computation of parallel surfaces to the given surface, such, for 

example, a non degenerate quadric, is also of great importance. Polynomials that 

implicitly define parallel lines or surfaces for the given offset to a curve or a 

surface can be computed by finding Gröbner bases for suitable elimination 

ideals of a suitably defined affine variety. Singularity of the lines parallel to the 

conics is discussed and their singular points are explicitly found as functions of 

the offset and the parameters of the conic. Critical values of the offset are linked 

to the maximum curvature of each conic. This approach can be extended to 

parallel surfaces. 

Keywords: Affine variety, elimination ideal, Gröbner basis, homogeneous 

polynomial, singularity, family of curves, envelope, pitch curve, undercutting, 

cam surface.

Applications of quaternion wavelet transform: 

for robotic vision 

Eduardo Bayro Corrochano 

Eduardo Bayro 

Abstract: The QWT using visual cues is applied for 3D reconstruction and 

space-time motion coding. This coding can be used by humanoids as a 

compresed language for understand and mimic human like behaviours.

A local method for large 

and unevenly distributed data. 

MIRA BOZZINI 



Via R. Cozzi n. 53, 20125 Milano, Italy 

In this poster we present a technique based on a local thin plate spline to 

recover functions, when the density of the data locations presents 

discontinuoties or steep gradients.

Some formal aspects and applications 

of Cartan – Dieudonné theorem 

Benjamín Cruz, Ricardo Barrón, Jesús López & Humberto Sossa 

Centro de Investigación en Computación – IPN 

Av. Juan de Dios Bátiz, esq. Miguel Othon de Mendizábal 9 

Mexico City, 07738. MEXICO 

E-mails: benjamincruz@sagitario.cic.ipn.mx, rbarron@cic.ipn.mx, 

jesuslrb06@sagitario.cic.ipn.mx ,hsossa@cic.ipn.mx 

Abstract 

One of the most important results in the area of Euclidean geometry, about Euclidean 

movements, is Cartan – Dieudonné theorem, Euclidean movements in n -dimensional 

Euclidean space are split in a certain number of reflections from hyperplanes of Euclidean 

space itself. 

This theorem is as follows: Any orthogonal transformation in a lineal space of n - 

dimension can be performed at most by n single reflections. 

There are several proofs of this theorem, however only in some of them we find elements 

that allow us to achieve a practical calculus algorithm. In this work one of these proofs is 

studied and some formal steps that are important to base the proof are added, these steps 

show us the power of Geometric Algebra as for simplicity and range. 

It is important to point out the role this theorem plays in practical applications like 

factorization of operators of movements and interpolation in virtual spaces generation, 

computer vision and graphics. 

In this paper besides formal aspects of the theorem, we show an algorithm and practical 

applications itself.

Variational calculus and multigrid dynamical programming 

on expo-rational surfaces and volume deformations: 

theory and applications 

Lubomir T. Dechevsky, Børre Bang, Xian Feng Bu, Arne Lakså 

Narvik University College, NORWAY 

In [1] and [2] we considered a variety of problems of variational calculus and optimal 

control theory with applications to geometric modelling with parametric curves, 

surfaces, volume deformations and higher dimensional manifolds. As a numerical 

method for approximate computation of the extremals, we proposed a fast multigrid 

dynamical programming algorithm. In [1] we used this method for the fast 

computation of geodesics on parametric surfaces, and Section 7 of [1] contains a 

detailed comparison of this algorithm with current state-of-the-art methods for exact 

and approximate computation of geodesics on triangulated surfaces, as described in 

[4] and the references therein. We pointed out that the algorithm from [1] has 

numerous significant advantages compared to the methods discussed in [4]. In the 

present communication we continue this analysis in the case when the original surface 

is approximated (by interpolation or by fitting) with an expo-rational surface (see [3]). 

We show that when the expo-rational interpolation is of Lagrange type, the algorithm 

from [1] emulates an enhanced version of the methods on triangulated surfaces. In the 

expo-rational setting it is very easy to upgrade the Lagrange interpolation to Hermite 

interpolation, whereby the extremal for the smooth Hermite expo-rational surface can 

be obtained as a small perturbation of the extremal for triangulated surfaces 

corresponding to Lagrange expo-rational approximation. In this presentation we shall 

demonstrate also several other ‘superproperties’ of the expo-rational B-splines in 

relevance to multidimensional variational and optimal-control problems with 

application to CAGD. One of the most important of these ‘superproperties’ is the ease 

of upgrading of the algorithm of [1] to higher dimensional manifolds (e.g., from 

geodesics on surfaces to geodesics in volume deformations and higher-dimensional 

vector-fields), while an analogous upgrade of the methods considered in [4] would be 

very cumbersome and labour-consuming, to the point of being beyond feasibility. 

References: 

[1] L. T. Dechevsky, L. M. Gulliksen. A multigrid dynamical programming algorithm 

for discrete dynamical systems and its applications to numerical computation of 

global geodesics. Int. J. Pure Appl. Math. (ISSN 1311-8080) 33(2) (2006) 257-285. 

[2] L. T. Dechevsky, L. M. Gulliksen. Application of a multigrid dynamical 

programming algorithm to optimal parametrization, and a model solution of an 

industrial problem. Int. J. Pure Appl. Math. (ISSN 1311-8080) 33(3) (2006) 381-406. 

[3] L. T. Dechevsky, A. Lakså, B. Bang. Expo-rational B-splines. Int. J. Pure Appl. 

Math. (ISSN 1311-8080) 27(3) (2006) 319-367. 

[4] V. Surazhsky, T. Surazhsky, D. Kirsanov, S. J. Gortler, H. Hoppe. Fast exact and 

approximate geodesics on meshes. ACM Transactions on Graphics (ISSN 0730-0301) 

4(3) (2005) 553-560.

Radial Basis Function Interpolants with Parameters 

Martin Buhmann, University of Giessen, Germany 

Abstract 

A major tool for multivariate approximations by interpolation and other means is the method 

of radial basis functions. Using the well-known radial basis functions such as Gaussians or multiquadrics 

and its generalisations has many advantages such as well-posedness of the interpolation 

problem in high dimensions and fast convergence, with little or no conditions on the geometry of 

the data points. These radial basis functions are extremely popular with engineering and other 

applications. 

√ 

However, many radial basis functions have parameters, such as the parameter c ∈ Φ(r) = 

r 2 + c 2 or Φ(r) =exp−c 2r2 , and especially in the numerical solution of the interpolation problems, 

conditioning problems arise if the parameters tend to zero or infinity. Changing these 

parameters can be very important in practical applications, and although they will never tend to 

infinity in applications, large parameters may be needed. As for example Fornberg and Larsson 

and collaborators noticed, sensible limits of the interpolating functions as a whole can sometimes 

be identified and computed even in these difficult cases. This is the main theme we wish to discuss 

in this talk and present results with have been found in collaboration with Dinew.

Using cubic spline for the velocity 

planning in mobile robots 

Heriberto Casarrubias, Humberto Sossa, Ricardo Barrón 




E-mails: heribertocv@gmail.com, hsossa@cic.ipn.mx, rbarron@cic.ipn.mx, 

Abstract 

The velocity regulation for a mobile robot is a problem of visual control that allows it to 

travel complex trajectories and evade obstacles and achieve objectives even if they are in 

movement. In this work shows how to use the cubic splines to control the position, 

orientation, and velocity of a differential mobile robot got nodes from an image, being careful 

of the kinematic constraints. 

Also practical applications and results are showed.

Forward and Inverse Problems for Determining Computer Simulated Respiratory 

Sound Sources 

S. Charleston-Villalobos 1 , J. Cruz-García 1 , R. González-Camarena 2 , T. Aljama-Corrales 1 

1 

Electrical Engineering Department, 2 Health Science Department, 

Universidad Autónoma Metropolitana-Iztapalapa, 09340 Mexico City, México 

Pulmonary auscultation represents a common clinical procedure to discover and to track 

respiratory diseases by the analysis of breathing sounds (BS) through physician’s ear. Lung 

structural and functional changes due to pathologies lead up to changes in sounds 

production and transmission, acoustical information that could be useful for clinical 

diagnosis. Regarding these aspects, several efforts have been made to determine the sites of 

production of inspiratory and expiratory BS as well as lung acoustic characteristics using 

mainly experimental protocols, sound speed measurements and acoustic models. The 

findings were compatible with the idea of a central origin for expiratory BS while flow 

turbulence seems to be the mechanism of origin. Possibly both, the expiratory and 

inspiratory sounds, have a multi-centre origin. Concerning BS speed and transmission in 

lung, different works have established that possibly BS travels in part through airways and, 

in part through lung parenchyma. In fact, lung parenchyma constitutes 90% of the total 

volume of the lung and at frequencies below the audible range, the parenchyma could be 

considered as a medium composed of bubbles in a fluid, i.e., a homogeneous mixture 

capable of sound attenuation by thermal losses. 

All previous efforts attempt to solve in certain way the forward and inverse problems 

but just recently the localization of BS sources has been established through the solution of 

a formal inverse problem. However, several factors affect the estimated sources. Among 

them, the numerical algorithm to provide the solution, the transfer function, the number of 

measurements taken into account (or microphones used), the sources depth and, the 

measurement noise. In the present work, we present preliminary results to the solution of 

the BS inverse problem establishing firstly, the forward problem in terms of a priori 

information and secondly, evaluating the performance of three inverse linear solution 

algorithms considering the effect of the some of the mentioned factors, by a systematic 

study in computer simulated scenarios. 

Three different algorithms were analyzed and compared: Overdetermined Least-Square 

(LSIS), Underdetermined Minimum Norm (MNIS) and Focal Underdetermined System 

Solution (FOCUSS). Besides visual evaluation of estimated sources, the algorithms 

performance was numerically assessed by MSE. Computer simulations included the 

estimation of single and multiples sources as well as the study of the influence of source 

depth, number of microphones, and thoracic volume discretisation on the inverse solution. 

In particular, for multiple single and non-single point RS sources. Overall, our results 

suggest that the FOCCUS algorithm performed better than the LSIS and the MNIS, as it 

provided more adequate solutions for shape, number, amplitude and position of sound 

sources, in 3D space. Furthermore, FOCCUS required less number of sensors and it was 

less affected by volume discretisation.

Isolated character recognition by using Support 

Vector Machines with a kernel of Radial Base 

Hermenegildo Cisneros, Ricardo Barrón, Benjamín Cruz 




E-mails: hcisneros@sagitario.cic.ipn.mx, rbarron@cic.ipn.mx, benjamincruz@sagitario.cic.ipn.mx 

Abstract 

In this work, we present a combination of techniques for the recognition of isolated 

hand-written characters. We describe a way to obtain the feature vector that allow us to 

describe an isolated character. Flusser invariant moments and other geometric measures are 

combined as objects descriptors. We implement an off-line system. We use the so-called 

Support Vector Machines Multi-class [SVM] with a kernel of radial base as the main 

classifiers. Its know that radial base functions can interpolate and smooth functions, they can 

outline in an efficient way the boundary between the classes, and precisely by this reason we 

propose to use in this work. Some experiments and results are showed.

An extension of a bound for functions in Sobolev spaces 

and its application to (m, s)–spline interpolation and smoothing 

Arcangéli, Rémi 1 ; López de Silanes, María Cruz 2 ; Torrens, Juan José 3 

1 Route de Barat, 31160 Arbas, France 

2 Depto. de Matemática Aplicada, CPS, Universidad de Zaragoza, 50018 Zaragoza, Spain 

3 Depto. de Ingeniería Matemática e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain 

Let f be a function defined over a bounded open subset Ω of R n with a Lipschitz–continuous boundary. 

In this talk, we present a Sobolev bound involving the values of f at finitely many points of Ω. This 

result extends those obtained by H. Wendland and co–workers (cf. [9, 11]). 

We then apply the Sobolev bound to derive error estimates for interpolating and smoothing (m, s)–splines 

(cf. [3, 1]). For interpolating (m, s)–splines, these results generalize those given by various authors (cf., 

for example, [4, 7, 12, 6, 5, 9]). For smoothing (m, s)–splines, we first consider the case of exact data. 

We pay special attention to the behaviour of these splines according to the smoothing parameter ε and 

the number N of data. These results generalize those given by various authors (cf. [7, 11, 10]). 

Finally, we consider the problem of noisy data. Then, in the stochastic case of data perturbed by a 

noise, under a classic random noise hypothesis, we give error estimates in the sense of mathematical 

expectation, for the smoothing D m –splines (the particular case s = 0), which improve previous results 

by various authors (cf. [10, 8, 2]). 

References 

[1] R. Arcangéli, M. C. López de Silanes, and J. J. Torrens. Multidimensional Minimizing Splines. 

Kluwer Academic Publishers, Dordrecht, 2004. 

[2] R. Arcangéli and B. Ycart. Almost sure convergence of smoothing D m –splines for noisy data. Numer. 

Math., 66:281–294, 1993. 

[3] J. Duchon. Splines minimizing rotation–invariant semi–norms in Sobolev spaces. Lectures Notes 

Math., 571:85–100, 1977. 

[4] J. Duchon. Sur l’erreur d’interpolation des fonctions de plusieurs variables par les D m –splines. 

RAIRO Anal. Numer., 12(4):325–334, 1978. 

[5] M. J. Johnson. An error analysis for radial basis function interpolation. Numer. Math., 98:675–694, 

2004. 

[6] W. Light and H. Wayne. On power functions and error estimates for radial basis function interpolation. 

J. Approx. Theory, 92:245–266, 1998. 

[7] M. C. López de Silanes and R. Arcangéli. Estimations de l’erreur d’approximation par splines 

d’interpolation et d’ajustement d’ordre (m, s). Numer. Math., 56:449–467, 1989. 

[8] M. C. López de Silanes and R. Arcangéli. Sur la convergence des D m –splines d’ajustement pour des 

données exactes ou bruitées. Rev. Mat. Univ. Complut. Madrid, 4(2–3):279–294, 1991. 

[9] F. J. Narcowich, J. D. Ward and H. Wendland. Sobolev bounds on functions with scattered zeros, 

with applications to radial basis fuction surface fitting. Math. Comp., 74:743–763, 2005. 

[10] F. Utreras. Convergence rates for multivariate smoothing spline functions. J. Approx. Theory, 52:1– 

27, 1988. 

[11] H. Wendland and C. Rieger. Approximate interpolation with applications to selecting smoothing 

parameters. Numer. Math., 101:643–662, 2005. 

[12] Z. M Wu and R. Schaback. Local error estimates for radial basis function interpolation to sacattered 

data. IMA J. Numer. Anal., 13:13–27, 1993.

Analysis of some bivariate non-linear interpolatory subdivision 

schemes and associated multiresolution 

K. Dadourian ∗ , J. Liandrat 

February 20, 2007 

For linear subdivision schemes or multiresolution analyses, generalisation from univariate to 

multivariate framework can be performed easily by tensorial product. Indeed, tensorial product 

of univariate limit (or scaling) functions provide the limit (or scaling) function in the multivariate 

framework. 

This work is devoted to non linear subdivision schemes and associated multiresolution transforms 

derived by tensor product of some univariate non linear schemes. More precisely, we consider 

univariate subdivision schemes S NL that can be written as a perturbation of a linear convergent 

scheme S as: 

∀f ∈ l ∞ ,S NL (f) = S(f) + F(δ(f)), 

where the perturbation F is a non linear operator and δ is a linear continuous operator (for example 

δf = d k f where df = (f n+1 − f n )). 

In [1], it has been shown that under natural conditions on F, δ and S, the subdivision scheme 

S NL is convergent and stable as well as the associated multiresolution. 

Here we analyse the bivariate scheme obtained by iterating alternatively on lines and rows the 

non linear univariate scheme. Results on convergence and stability will be shown. Our results are in 

agreements with the performances of non linear schemes like WENO or PPH for image processing 

([3], [2]). 

References 

[1] Amat, S., K. Dadourian and J. Liandrat, Analysis of a class of subdivision schemes and 

associated non-linear multiresolution transforms, submitted. 

[2] Amat, S., R. Donat, J. Liandrat and J. C. Trillo, Analysis of a new non-linear subdivision 

scheme: applications in image processing, Foundations of Computational Mathematics 225 

(2005), 193–225. 

[3] Cohen, A., N. Dyn and B.Matei, Quasi linear subdivision schemes with applications to ENO 

interpolation, Applied and Computational Harmonic Analysis 15 (2003), 89–116.

Numerical integration, Energy and Discrepancy on Compact 

Homogenous manifolds. 

Abstract: 

Dr Steven Damelin 

In this talk, we will review a program of research of the author and his collaborators on 

numerical integration estimates on compact homogenous spaces in terms of energy criteria. 

Even for the sphere, our approach is completely new. This is joint work with Peter Grabner 

(Graz), Levesley (Leicester), Sun (Missouri) and Hinkernell (IIT) 

Dr Steven Damelin 

Work Address: 

Department of Mathematical Sciences, 

Georgia Southern University, Postoffice Box 8093, 

Statesboro, GA 30460-8093, U.S.A.

Expo-rational B-splines of a complex variable and their 

applications 

Lubomir T. Dechevsky 

Narvik University College 

I begin by introducing the several possible definitions of multivariate ERBS (exporational 

B-splines, as introduced in [1]) and then concentrate on the bivariate case and 

its translation into complex-variable language. I consider several applications of the 

ERBS of a complex variable. In particular, if the spectrum of a compact operator 

acting on a Banach space be triangulated (as scattered point set in the plane), then a 

minimally supported infinitely smooth ERBS-based partition of unity can be 

constructed. This allows explicit computation of the Cauchy-Riesz-Dunford integral 

representation of analytic functions of this compact operator. One application of this 

is for deriving sharp estimates of the norms of operator resolvents. Another 

application is the possibility to parametrize the auto-diffeomorphisms on the unit ball 

in the N-dimensional Euclidean space. When N is 2 or 3, this has important 

applications to registration problems in brain imaging and other applications relevant 

to computer tomography. 

Reference: 


Math. (ISSN 1311-8080) 27(3) (2006) 319-367.

Radon transform and harmonical analysis for 3D 

motion estimation using robot omnidirectional vision 

Luis Eduardo Falcon 

and 

Eduardo Bayro-Corrochano 

edb@gdl.cinvestav.mx 

Abstract: 

Since images taken by omnidirectional sensors can be mapped to the sphere, the 

problem to estimate the 3D rotation of a camera can be treated as a problem of 

estimating rotations between spherical images. In recent years harmonic 

analysis has been used in computer vision to obtain 3D rotations with the Radon 

and Hough transform, evaluating the Fourier transform on the unit sphere S2 

and on the rotation group SO(3). In particular, we used the Radon transform on 

the space of lines of the omninidrectional images, to recover the relative 

position between two cameras.

IMAGE SEGMENTATION USING A 

GENERALIZED FAST MARCHING METHOD 

Nicolas Forcadel 1 , Carole Le Guyader 2 and Christian Gout 3,4 

1 

: Ecole Nationale des Ponts et Chaussées 

CERMICS, 6 et 8 avenue Blaise Pascal 

Cité Descartes - Champs sur Marne 

77455 Marne la Vallée Cedex 2, France 

2 

: INSA de Rennes 

20 Avenue des Buttes de Coësmes, CS 14315 

35043 Rennes Cedex , France 

3 : Université de Valenciennes 

LAMAV - ISTV2, Le Mont Houy 


4 : INRIA Futurs MAGIQUE 3D 

Université de Pau, IPRA, UMR Math CNRS 5142 

Av. de l’Universités, BP 1155 

64013 Pau cedex, France 

Abstract : 

In previous work, Carlini et al. (2005& 2006) have introduced a new Fast 

Marching algorithm for a non-convex eikonal equation modeling front evolutions in 

the normal direction. The algorithm corresponds to extension of the Fast Marching 

developed by Sethian (1996). 

After a presentation of this new method, we propose to develop & integrate it 

for some applications in image segmentation. 

We give the modelling and some theoretical results of this approach, its 

discretization is given, and we present numerical examples to show the potentiality of 

this novel method. 

References : 

E. Carlini, M. Falcone, N. Forcadel, R. Monneau, Convergence of a Generalized Fast 

Marching Method for a non-convex eikonal equation, 2006. 

E. Carlini, E. Cristiani, N. Forcadel, A non-monotone Fast Marching scheme for a Hamilton- 

Jacobi equation modeling dislocation dynamics, Numerical Mathematics and Advanced 

Applications Proceedings of ENUMATH 2005, Santiago de Compostela, Spain, July 2005 

Bermdez de Castro, A.; Gmez, D.; Quintela, P.; Salgado, P. (Eds.). 

J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in 

Computational Geometry, Fluid Mechanics, Computer Vision, Computer-Aided Design, 

Optimal Control and Material Sciences , Broché, 2 nd Edition, 1999.

C 1 and C 2 -continuous polynomial parametric L p splines (p ≥ 1) 

P. Auquiert (1) ,O.Gibaru (2) ,E.Nyiri (2) 

(1) (LAMAV) Université de Valenciennes et du Hainaut-Cambrésis 

Le Mont Houy, F-59313 Valenciennes cedex 9 

(2) (L2MA) C.E.R. Ecole Nationale Supérieure d’Arts et Métiers de Lille 

8 Bld Louis XIV, F-59046 Lille Cedex 

We introduce L p (1 ≤ p ≤∞) piecewise polynomial parametric splines of degree 3 or 5 which 

smoothly interpolate data points. The coefficients of these polynomial splines are calculated by 

minimizing the L p norm of the second derivative. We demonstrate that the C 1 -continuous cubic 

(resp. G 1 -continuous cubic and C 2 -continuous quintic) L p polynomial splines are unique for 

1

A Dynamical Node Adaptive RBF Method 

to Solve PDE Problems. 

Pedro González-Casanova 1 ; José Antonio Muñoz Gómez 2 and Gustavo Rodríguez 

Gómez 2 . 

The numerical approximation of PDE problems whose solution presents sharp 

gradients is a difficult task. This is particularly true when large real life problems are 

considered. Some algorithms have appeared in the radial basis functions literature 

to deal with this problem. In this work, we purpose a new RBF algorithm -based on 

Kansa’s radial basis functions unsymmetric collocation method-, which combines a 

local node adaptive technique with a domain decomposition method. Specifically, a 

node refinement technique based on local error interpolation estimates and a 

quad-tree type algorithm is formulated. This method is then further combined with a 

multiplicative Schwarz domain decomposition technique with overlapping regions. 

The refinement strategy is designed to keep the load balance in each subdomain. 

This objective is achieved by dynamically adapting the number of subdomains in 

accordance with the knot insertion-remotion technique. Numerical results applied 

to stationary partial differential equations in two dimensions, shows that the 

dynamic partition of data, combined with the domain decomposition method, 

reduce the computational cost and can be used to solve problems with a large 

number of nodes. 

1 Unidad de Investigación En Computo Aplicado, DGSCA, Universidad Nacional 

Autónoma de México, D. F, México. 

2 Ciencias Computacionales, Instituto Nacional de Óptica y Electrónica (INAOE), Puebla 

Tonanzintla, México.

Geodesic active contour under geometrical conditions: 

Theory and 3D Applications 

Christian Gout §,$ and Carole Le Guyader ♥ . 

§ :INSAdeRouen 

Laboratoire de Mathématiques de l’INSA 

Place Emile Blondel, BP 08 

76 131 Mont-Saint-Aignan cedex, France. 

chris gout@cal.berkeley.edu. 

$ :Université de Valenciennes - LAMAV-ISTV2 

Mont Houy, 59313 Valenciennes cedex 9, France 

♥ : INSA de Rennes 

Centre de Mathématiques de l’INSA 

20, Avenue des Buttes de Coësmes, CS 14315 

35043 Rennes cedex, France 

carole.le-guyader@insa-rennes.fr 

Abstract: 

We will first give a survey of some methods we have developed for image segmentation 

under geometrical constraints. 

We then propose a new scheme for both detection of boundaries and fitting of geometrical 

data based on a geometric partial differential equation, which allows for a rigorous 

mathematical analysis. The model is a geodesic-active-contour-based model, in which we 

are trying to determine a curve that best approaches the given geometrical conditions (for 

instance a set of points or curves to approach) while detecting the object under consideration. 

Formal results concerning existence, uniqueness (viscosity solution) and stability are 

presented as well. We give the discretization of the method using an Additive Operator 

Splitting scheme which is very efficient for this kind of problem. We also give numerical 

examples on 3D real data sets. 

Key words: Geodesic active contours, Level set method, viscosity solution, approximation 

of points. 

AMS classification: 35XX, 49L25, 74G65, 65D18, 53A10, 68U10.

ROTATING PROJECTION ALGORITHM OF IMAGE RECONSTRUCTION 

AND APPLICATION FOR COMPUTER TOMOGRAPHY 

Alexander Grebennikov , J. G. Vázquez Luna, T. Valencia Perez, M. Najera Enriquez 

Autonomous University of Puebla , Puebla, MÉXICO 

agrebe@fcfm.buap.mx 

ABSTRACT 

We consider the problem of the image reconstruction of the structure, consisting of the component 

with different characteristics, under the influence of the known external physical field. The image of the 

distribution of these characteristics is described inside the space domain Ω with by some 

function g( x, y, z ) that must be reconstructed based on indirect boundary observations. This problem can 

be resolved by tomography that corresponds to the used external field. There are some tomographies such 

as electrical, infrared, acoustic, Roentgen (radiography), etc. [1], [2]. The traditional approach requires 

scanning the object and calculating the inversion of the Radon transform. It seems necessary for difficult 

structures and can be realized in sufficiently fast manner. But sometimes the investigating object has the 

simple structure, so its reconstruction consists only in localization of some elements with different 

characteristic inside of the homogeneous (or quasi homogeneous) region. In [3] it was proposed the 

simplified variant of scanning algorithm without application of the inverse Radon transform. We call it as 

“Rotating Projection” algorithm. In [3] this algorithm was applied for electric tomography. Here we use 

this idea and develop the approach for abstract computer tomography. 

We explain the idea of this algorithm in the simplest plane case for one element to be localized. 

Suppose that we know values of function ν ( p, ϕ ) (projections) which characterizes the intensiveness of 

passed through the object rays for some fixed angles ϕ and all linear coordinates p of scanning. By 

another words, we know for some fixed angles ϕ i , i = 1,..., n corresponding number of one dimensional 

images (projections) as functions of one variable p. 

We need the next steps. 

1. Prolongation on a plane of calculated projections ν ( p, ϕ i ) for every p along the direction, 

corresponding to angleϕ i , to obtain the extended two dimensional image. 

2. Rotation, i.e., changing number i (scanning for different ϕ ) of prolonged projections and localization 

of the areas of intersection of projections with the same values. 

If we do not interesting in the exact geometrical form of the element and want to localize it as a 

rectangle, it is sufficient [4] to use n=2. 

We developed the explained scheme for the space (three dimensional) case to localize some point 

wise elements and continuous objects, constructed corresponding algorithm and computer programs in 

MATLAB system and justified these programs by a lot of numerical model experiments. 

One of the important applications of constructed algorithm and programs was realized by authors for 

the medicine diagnostics of cancer of the female bosom. In this case it is important to reduce the time of 

the radiation treatment in the tomography process, by other words – to reduce the number of angles of 

scanning. The special algorithms of data pre-processing were developed also for reducing real data for 

the simplified form considered above. This part of investigations was realized in the frame of the Project 

No SALUD 2004-01-021, supported by SSA/IMSS/ISSSTE-CONACYT of Mexico. 

REFERENCES 

1. R.A. Williams and M.S. Beck, Process Tomography: Principles, Techniques and Applications. 

Butterworth-Heinemann, Oxford, 1995. 

2. M.S. Beck and B.H. Brown, Process tomography: a European innovation and its application. 

Measurement Science and Technology, 7, 215–224 (1996). 

3. A. Grebennikov, J. G. Vázquez Luna, M. A. Cruz Gama, Fast Linear Algorithms for Image 

Processing in Electric Tomography. III Taller Internacional sobre Física Aplicada. La Habana, 

Noviembre 30 – Diciembre 3, 2004. 

4. G. Germen, Reconstruction of images using projections. Moscow, Mir, 1983.

On monogenic approximations of solutions of the Lamé system 

K.Gürlebeck 

Abstract: The talk begins with a short introduction to the function theory of quaternion-valued 

monogenic functions. Basic ideas of the construction of polynomial basis systems will be 

briefly explained. We will then focus to the derivatives and primitives of such basis functions. 

To approximate also solutions with singularities like cracks we will define systems of rational 

monogenic functions. All this will then be applied to construct spatial generalizations of the 

well-known Kolossov-Muschelishvili formulas from the plane case of linear elasticity.

Self-repelling snakes for topology-preserving 

segmentation models. 

by 

Carole Le Guyader (INSA de Rennes, France) 

and 

Luminita Vese. (UC Los Angeles, USA) 

The implicit framework of the level set method has proved to be very powerful in 

tracking propagating fronts. Indeed, the evolving contour is embedded in a higherdimensional 

level set function and its evolution can be phrased in terms of an Eulerian 

formulation. 

The ability of this modelling to handle topological changes, its being parameter-free 

and intrinsic make it useful in a wide range of fields and particularly in image segmentation. 

Nevertheless, in some applications, this topological flexibility turns out to be undesirable, for 

instance when the shape to be detected has a known topology, when the resulting shape must 

be homeomorphic to the initial one. 

Thus we propose in this talk a segmentation model which preserves topology and 

which is based on an implicit formulation and on the geodesic active contours. Experimental 

results on various synthetic and real images in 2 or 3 dimensions will be given.

Numerical Simulation of a Viscous Pump by a High Order Finite 

Element Method 

L. Héctor Juárez V. 

e-mail: hect@xanum.uam.mx 

Departamento de Matemáticas 

Universidad Autónoma Metropolitana-I 

México 

Flow generated by a circular rotating cylinder eccentrically placed between parallel plates in a two 

dimensional channel is studied numerically. This simulation describe a pumping device appropriate 

form microelectromechanical systems (MEMS). The device is shown to be capable of generating a net 

flow due to the differential viscous resistance between small and large gaps. The numerical simulation 

is carried out by a high–order Lagrange Finite Element Method in combination with a second order 

time–integration scheme.

Umbrella based triangulation 

of regular parametric surfaces 

Victoria Hernández, Instituto de Cibernética, Matemática y Física, ICIMAF, 

La Habana, Cuba, e-mail: vicky@icmf.inf.cu 

Pedro L. del Angel, Centro de Investigaciones Matemáticas, CIMAT, 

Guanajuato, México, e-mail: luis@cimat.mx 

Jorge Estrada, Instituto de Cibernética, Matemática y Física, ICIMAF, 

La Habana, Cuba, e-mail: jestrada@icmf.inf.cu 

Abstract 

Parametric surfaces are the standard in Computer Aided Design and Manufacturing 

(CAD/CAM) systems. Therefore, a lot of work has been done to obtain good discretizations 

of parametric surfaces, suitable for visualization as well as for numerical solution of partial 

differential equations using finite elements method (FEM) or boundary elements method 

(BEM). Triangulation of parametric surfaces is also an essential problem in computer 

graphics and computational geometry. The most common discretization of a surface is a 

triangular mesh, due to its flexibility for representing complex geometries and also because 

current graphics hardware and software are tuned to handle triangular meshes. The simplest 

way of constructing a triangular mesh on a parametric surface consists in generating a (nice) 

triangulation in parameter space and lifting it to the surface by means of the parametrization. 

Unfortunately, this triangular mesh could be very distorted. 

In this talk we propose an advancing front method for generating an isotropic triangular mesh 

on a regular parametric surface. Starting from a point on the surface, the method computes a 

set of points in the intersection curve between the surface and the sphere centered at that point 

with a prescribed radius. These points are the vertices of a cell composed by triangles 

approximately equilateral. The mesh grows repeating the described computation with 

boundary vertices of the cell as starting points. The resulting isotropic mesh can be refined in 

order to get a better approximation of the surface, making the method also useful for 

visualization of the surface. Finally, we obtain a sufficient condition ensuring that a surface 

triangulation is of Delaunay type. As a particular case, we prove that any isotropic 

triangulation (with vertices on a surface) which provides a good approximation of a surface 

is a Delaunay triangulation. 

Keywords: Parametric surfaces, triangular meshes

GENERAL FORMULAE OF B-SPLINE CURVES WITH SHAPE PARAMETERS 

MIKL OS HOFFMANN AND IMRE JUH ASZ 

In spite of the recently developed curve design methods, B-spline curve still plays essential role in 

computer aided geometric design. The generalization of this curve, however, is also in the forefront of 

research due to its importance in applications. Several authors attempted to extend the approximation 

abilities of the B-spline curve by some shape parameter which allows the user to improve the shape 

of the approximating curve. Obviously, the most known result of this attempt is the NURBS curve 

(c.f. [3]), but this curve has rational coecient functions, thus alternative methods tried to incorporate 

shape parameters into the original, polynomial basis functions. One of the earliest methods in this way 

is -spline curve with two global parameters ([6], [7]). Further methods have been provided by direct 

generalization of B-spline curves as B-splines in [2], [4] and recently as GB-splines in [1] or as SPB-splines 

in [5]. Some alternative spline curves with shape parameters can be found in [8] and [9]. 

The various generalizations of the B-spline curve dier from each other in the modied basis functions 

of the curve. Unfortunately, most of the papers provide the denition of the modied basis functions with 

no further explanations. The aim of this paper is to give a general approach and formulae of B-spline 

curves with shape parameters, including all the previously mentioned methods and give a chance for a 

comparative study. 

It will be shown, that all the curves provided by these methods can be considered as a linear blending 

3X 

3X 

of the classical B-spline curve b (t) = B i (t) p i and another curve g (t) = G i (t) g i . Thus, the 

i=0 

modied curve with shape parameter is of the form 

c (; t) = q () b (t) + (1 q ()) g (t) ; q () 2 [0; 1] : 

In most of the cases g (t) turns to be the parameterized line segment p 1 p 2 , in other cases g (t) is a 

4 th order curve. In this content we give a general framework of spline curves with shape parameters, 

provide a comparative study of the previous methods and dene a curve which, in some sense, the best 

approximation method by generalizing the classical B-spline curve. Using this framework multivariate 

shape control with several shape parameters as well as non-uniform parameterization and continuity of 

the subsequent arcs are also discussed in the paper. 

i=0 

References 

[1] Guo, Q., 2005. Cubic GB-spline curves. Journal of Information and Computational Science 3, 465{471. 

[2] Loe, K.F., 1996. B-spline: a linear singular blending B-spline. The Visual Computer 12, 18{25. 

[3] Piegl, L., Tiller, W., 1995. The NURBS book. Springer Verlag. 

[4] Tai, C.L., Wang, G.J., 2004. Interpolation with slackness and continuity control and convexity-prservation using singular 

blending. Journal of Computational and Applied Mathematics 172, 337{361. 

[5] Yin, M., Wang, Q., 2005. B-spline curve of order 4 with two shape parameters. Journal of Computational Information 

Systems 1, 609{614. 

[6] Barsky, B.A., Beatty, J.C., 1983. Local control of bias and tension in splines. ACM Transactions on Graphics, 2, 

109-134. 

[7] Barsky, B.A.,1988. Computer graphics and geometric modeling using splines. Springer-Verlag, Berlin. 

[8] Habib Z., Sakai M. and Sarfraz M. 2004. Interactive Shape Control with Rational Cubic Splines, International Journal 

of Computer-Aided Design & Applications, 1, 709-718. 

[9] Habib Z., Sarfraz M. and Sakai M., 2005. Rational cubic spline interpolation with shape control, Computers & Graphics 

29, 594-605. 

E-mail address: hofi@ektf.hu 

E-mail address: JuhaszI@abrg.uni-miskolc.hu 

Institute of Mathematics and Computer Science, Karoly Eszterhazy University, Eger, Hungary 

Department of Descriptive Geometry, University of Miskolc, Miskolc, Hungary

A new Hilbert transform in a metric dependent Clifford setting 

F. Brackx, B. De Knock, H.DeSchepper 

Department of Mathematical Analysis, Faculty of Engineering, Ghent University 

Galglaan 2, 9000 Gent, Belgium 

e-mail: Bram.DeKnock@UGent.be 

In one—dimensional signal processing, the Hilbert transform has become an indispensable tool in signal 

analysis: it underlies the concept of analytic signal (see e.g. [1]), which is crucial in the construction of so— 

called quadrature filters, used for obtaining selective spectral information. Mathematically, if f ∈ L 2 (R) 

is a real valued signal of finite energy, and H[f] denotes its Hilbert transform, i.e. the convolution with 

1 

the principal value kernel 

π 

Pv 1 x , then the corresponding analytic signal is the function 1 2 f + i 2 H[f], 

which belongs to the Hardy space H 2 (R),andarisesastheL 2 non-tangential boundary value for y → 0+ 

of the holomorphic Cauchy transform of f in the upper half of the complex plane. 

The Hilbert transform has been generalized to higher dimension (see e.g. [2]) and higher dimensional 

applications have already been addressed as well, for instance in [3], where the concept of analytic signal 

was generalized to two dimensions in order to design appropriate quadrature filters for pattern recognition, 

using a two—dimensional Hilbert transform, also referred to as Riesz transform. Several of the 

higher dimensional generalizations were established in Clifford analysis, a comprehensive theory offering 

an elegant and powerful generalization to higher dimension of the theory of holomorphic functions in 

the complex plane. In its most simple yet still useful setting, flat (m + 1)—dimensional Euclidean space, 

Clifford analysis focusses on monogenic functions, i.e. null solutions of the Clifford-vector valued Dirac 

operator ∂ = P m 

j=0 e j∂ xj where (e 0 ,... ,e m ) forms an orthogonal basis for the quadratic space R m+1 

underlying the construction of the Clifford algebra R 0,m+1 (see e.g. [4],[5]). 

The above described form of Clifford analysis may be referred to as isotropic, since the metric in the 

underlying space is the standard Euclidean one. In our talk however, we will adopt the idea of a metric 

dependent, also called anisotropic or metrodynamical, Clifford setting (see e.g. [6]), which offers the 

possibility of adjusting the co-ordinate system to preferential, and not necessarily mutually orthogonal, 

directions in the m—dimensional signal to be analyzed. This leads to the introduction of a metric dependent 

generalized Hilbert transform in R m ⊂ R m+1 , a special case of which was already introduced and 

used for two—dimensional image processing in [3]. 

References 

[1] A.D. Poularikas (ed.), The transforms and applications handbook, CRC Press, Boca Raton, 1996. 

[2] J. Gilbert and M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge 

University Press, Cambridge, 1991. 

[3] M. Felsberg, Low-Level Image Processing with the Structure Multivector, PhD-thesis, Christian- 

Albrechts-Universität, Kiel, 2002. 

[4] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Pitman Publishers, Boston-London- 

Melbourne, 1982. 

[5] R. Delanghe, F. Sommen and V. Souček, Clifford Algebra and Spinor-Valued Functions, Kluwer 

Academic Publishers, Dordrecht, 1992. 

[6] F. Brackx, N. De Schepper and F. Sommen, Metric Dependent Clifford Analysis with Applications to 

Wavelet Analysis. D. Alpay (ed.), Wavelets, Multiscale Systems and Hypercomplex Analysis, Series 

”Operator Theory: Advances and Applications”, 167, Birkhäuser Verlag, Basel, 2006, 17-67.

Lattices on simplices 

Gašper Jaklič*,JernejKozak,MarjetaKrajnc, 

Vito Vitrih, Emil Žagar 

FMF and IMFM, University of Ljubljana, Slovenia 

In contrast to the univariate case, uniqueness of the solution of a multivariate 

Lagrange polynomial interpolation problem depends not only on the fact that 

interpolation points should be distinct but also on their geometry. Lattices are 

perhaps the most often used configurations of prescribed interpolation points. 

In this talk, three-pencil lattices on triangulations will be considered. The 

explicit representation of a lattice, based upon barycentric coordinates, will be 

presented. This enables us to construct lattice points in a simple and numerically 

stable way and carries over to triangulations in a natural way. 

A generalization to 3D simplex partitions will be presented.

Fast Multiscale Methods for PDE-Constrained Control Problems 

Angela Kunoth 

Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany, 

kunoth@iam.uni-bonn.de www.iam.uni-bonn.de/∼kunoth 

The fast numerical solution of PDE-constrained control problems provides a formidable 

challenge, in particular, for problems constrained by time-dependent PDEs. For a single 

elliptic PDE which may be considered as the core system to be solved in any case, there 

exist by now three classes of preconditioners, all of which are of multiscale structure 

(multigrid, BPX and wavelet preconditioners), which assure uniformly bounded spectral 

condition numbers of the system matrix independent of the discretization. 

In my talk I wish to address fast multiscale solvers for control problems constrained 

by elliptic PDEs with distributed as well as Dirichlet boundary control. I will present 

the main ingredients for obtaining theoretical estimates which guarantee optimality of 

the multilevel preconditioners. Together with employing iterative solvers and a nested 

iteration scheme, I will show that they therefore provide the solution ingredients of the 

control problem (state, costate and control) up to discretization error accuracy in optimal 

linear complexity. Corresponding numerical results confirming these estimates will be 

shown. 

Finally, I would like to address PDE-constrained control problems with additional 

inequality constraints on the control which pose substantial difficulties for fast numerical 

solution schemes.


Lubomir T. Dechevsky, Arne Lakså, Børre Bang 

Narvik University College, NORWAY 

We formalize the fundamental structure and properties of expo-rational B-splines, as 

defined and derived in [1], and identify the maximal class of possible generalized B- 

splines which share the same fundamental structure and properties. We consider this 

maximal class for parametric curves, tensor-product surfaces, triangulated surfaces 

and simplectified multivariate manifolds. We show that besides the true expo-rational 

B-splines, there are also some classical piecewise-polynomial B-splines which satisfy 

the above formal constraints and form closed subspaces in usual classical piecewisepolynomial 

B-spline spaces of higher degree and/or dimension of the knot-vector. We 

consider in detail the simplest case of piecewise-polynomial generalized expo-rational 

B-spline which can be computed explicitly (including in the case of triangulated 

surfaces and simplectified multi-dimensional manifolds). Visualization and a 

discussion on applications are also included in the talk. 

Reference: 


Math. (ISSN 1311-8080) 27(3) (2006) 319-367.

The order of approximation of tranport 

equations by the kinetic equation 

Jeremy Levesley 

Leicester University 

Department of Mathematics and Computer Science 

LE1 7RH Leicester. UK 

jl1@mcs.le.ac.uk 

The Lattice Boltzmann method is a technique for simulating material transport 

via a dynamical system generated by a kinetic equation in phase space. The 

macroscopic quantities of interest (e.g. density or momentum) are moments of 

the phase space distribution. We will outline how Navier-Stokes' equation can be 

viewed as free flight in phase space, followed by an equilibration process. We 

will describe a novel numerical method for simulating fluid flow which 

implements the above free flight and equilibration at high Reynolds' number and 

discuss how well this method approximates the Navier-Stokes' equations. We 

will also look at methods of stabilisation of this process based around various 

sorts of filter.

Diffusion polynomial frames on metric measure spaces 

We construct a multiscale tight frame based on an arbitrary orthonormal 

basis for the L 2 space of an arbitrary sigma finite measure space. The approximation 

properties of the resulting multiscale are studied in the context of 

Besov approximation spaces, which are characterized both in terms of suitable 

K—functionals and the frame transforms. The only major condition required is 

the uniform boundedness of a summabilility operator. We give sufficient conditions 

for this to hold in the context of a very general class of metric measure 

spaces, where the assumption of finite speed of wave propagation might not 

hold. The theory is illustrated using the approximation of characteristic functions 

of caps on a dumbell manifold, and applied to the problem of recognition of 

hand—written digits. Our methods outperforms comparable methods for semi— 

supervised learning. This is joint work with Professor Mauro Maggioni, Duke 

University. 

H. N. Mhaskar 

Department of Mathematics 

California State University 

Los Angeles, CA 90032 

U. S. A.

Mathematical and computational challenges in some 

applications of medical imaging 

Knut Mørken 

knutm@ifi.uio.no 

University of Oslo 

CMA, P.O. Box 1053, Blindern, 0316 Oslo, Norway 

The ability to capture images of the interior of the human body has revolutionized 

both medical diagnosis and therapy. Today, a number of different imaging 

technologies are available, such as CT (Computed Tomography), MRI (Magnetic 

Resonance Imaging), PET (Positron Emission Tomography), ultrasound, X-ray, photo 

and video. The first three techniques provide 3D samples in uniform grids; ultrasound 

can give both 2D and 3D images, while X-ray gives 2D images, but is also the basis 

for CT imaging. Although these technologies have already radically changed many 

aspects of medicine, with for instance the emergence of minimally invasive surgery, 

new surgery techniques based on imaging are likely to further transform both 

diagnosis and therapy in even more profound ways. 

A fundamental challenge in medical imaging is to combine data from different 

imaging techniques to produce a synthetic image that provides more information than 

any of the individual images. This is the image fusion problem, which in turn is 

dependent on the image registration problem, i.e., the problem of aligning two images 

(2D or 3D) in a common coordinate system. In general, these are extremely 

challenging problems, both mathematically and computationally. 

This talk will give a brief introduction to medical imaging and the image registration 

and image fusion problems on a fairly nontechnical level. In addition, we will 

consider some specific applications of medical imaging such as temperature 

monitoring via MRI during heating of tissue, and monitoring of the frozen area during 

freezing of tissue.

Splines for Surfaces of Arbitrary Topology 

Mike Neamtu, Vanderbilt University 

Abstract: 

There is a well-developed theory of bivariate splines on planar triangulations, 

which allows one to construct splines of arbitrary smoothness. However, such 

splines can only be used to obtain parametric surfaces that are topologically 

equivalent to a planar region. This theory of bivariate splines fails completely 

when one wants to define smooth surfaces of arbitrary topological genus. In this 

talk we shall address the causes of this serious gap in the spline theory and 

discuss a possible remedy.

Sparse Approximation for Source Separation 

in the Magnetoencephalography Inverse Problem 

Francesca Pitolli 

Dip. MeMoMat, Università diRoma”La Sapienza”, Italy. 

The aim of magnetoencephalography (MEG) is the analysis of brain functionality through the 

measurements of the tiny magnetic field generated by neuronal currents (see, for instance, [2] and 

references therein). As a matter of fact, since neuron cells functioning is mediated by electric 

currents, to understand brain functionality it is important to gain knowledge about the current 

distribution within the head. Therefore, looking from the physics side, the final goal of MEG is 

to accurately determine the current density flowing within the volume of the head in the working 

human brain. 

The neuromagnetic field is only in the order of 10 −13 Tesla in magnitude, that is much smaller 

than the environmental noise. To successfully resolve the current density flowing within the brain it 

is mandatory to use low noise superconducting magnetometers as well as sophisticated signal processing. 

In addition, the magnetic field measured externally the head has a poor spatial resolution 

- at present MEG devices have at most a few hundred sensors - and can be generated by several 

possible current density configurations. Hence, the identification of a specific current source configuration 

from the measured magnetic fields is an ill-conditioned inverse problem, and is one of the 

challenging aspects of this technology. The main goal of this paper is to give an efficient and stable 

numerical scheme to compute the solution of the MEG inverse problem assuming that the currents 

flowing in the brain satisfy a sparsity constraint [1,3,4]. 

This is a joint work with M. Fornasier and V. Pizzella. 

[1] I. Daubechies, M. Defrise, and C. De Mol, An iterative thresholding algorithm for linear inverse 

problems with a sparsity constraint, Commun. Pure Appl. Math., 57 (2004) 1413–1457. 

[2] C. Del Gratta, V. Pizzella, F. Tecchio, and G. L. Romani, Magnetoencephalography - a noninvasive 

brain imaging method with 1ms time resolution, Rep. Prog. Phys., 64 (2001) 1759–1814. 

[3] M. Fornasier, and H. Rauhut, Recovery algorithms for vector valued data with joint sparsity 

constraints, Report 2006-27, Johann Radon Institute for Computational and Applied Mathematics 

(RICAM), 

http://www.ricam.oeaw.ac.at/publications/reports/06/rep06-27.pdf. 

[4] M. Fornasier, F. Pitolli and V. Pizzella, Blind Source Separation with Sparsity Constraints for 

Magnetoencephalography, Proc. of SIMAI 2006... 

Francesca Pitolli 

Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate 

Università diRoma“LaSapienza” 

ViaA.Scarpa16 

I-00161 Roma, Italy 

email: pitolli@dmmm.uniroma1.it 

WWW: http://www.dmmm.uniroma1.it/∼pitolli

How to reduce Runge Phenomenon 

Christophe Rabut,Toulouse University, (INSA), France 

Two ways are presented to reduce Runge phenomenon (strong oscillations 

for polynomial interpolation). The first one is done for equidistant interpolation 

points and gives some kind of flexibility to the interpolation values. The 

second one is for any point repartition and consists in giving some more freedom 

by a degree elevation and using the so-obtained freedom for minimizing 

a norm significative of the oscillation of the polynomial. 

First method: The points to be interpolated being ( i , y n i) i=0:n , we first 

use the fact that the Bezier polynomial p 0 (x) = ∑ n 

( ) i=0 y i B i (x) (where B i (x) = 

n 

i x i (1 − x) n−i is the i th degree n Bernstein polynomial) oscilates no more 

than the data points (variation diminishing propety), and so we start from 

this polynomial. However we would like a polynomial closer the data. To 

get it, we construct, for any k in IN by induction the family (Bi k ) i=0:n 

by the following : Bi 

k = Bi 

k−1 + ∑ n 

j=0 (δ ij − Bi 

k−1 (x j )) , Bj 

k−1 = 2 Bi 

k−1 − 

∑ nj=0 

Bi 

k−1 (x j )) , Bj 

k−1 . Then the polynomial p k (x) = ∑ n 

i=0 y i Bi k (x) is proved 

to converge to the interpolating polynomial of the points ( i , y n i) i=0:n and 

presents far less oscillations of the interpolation polynomial, even when values 

at x = i are very close to y n i. 

Second method: For any k ≥ n,we define the polynomial p k as being 

the degree k polynomial interpolating the data points (x i , y i ) i=0:n and minimizing, 

over all polynomials p interpolating the data points, the quantity 

∫ xn 

x 0 

(p ′′ (x)) 2 dx. We so use the degree of freedom given by the degree of the 

polynomial to cut down the oscillations. Numerical experiments give the 

evidence that p k converges (over [x 0 ..x n ]) to the interpolating natural cubic 

spline of the data points, which shows that the oscillations are effectively cut 

down. 

Graphs of both the methods will be presented.

BUILDING SOLIDS USING QUATERNIONS 

Universidad Autónoma Metropolitana 

Rafael Reséndiz 

rafael.resendiz@gmail.com 

We show a way to make solid modelling taking the solid surface like a 

trajectory of a particle; it is well known that a trajectory is seen as a 

rational Bézier curve, so we make a dual quaternionic parametrization 

of mentioned trajectory. Then the main idea is to pick a surface given 

by either scattered data or regularly distributed and be able to 

reproduce it carving it in some material.

Polyharmonic B-Splines 

Milvia Rossini 

Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca 

Polyharmonic splines are considered very often in the literature, because of 

their good properties. 

The aim of this talk is to present the recent results obtained jointly by the 

Milano-Bicocca group and Christophe Rabut on some classes of polyharmonic 

B-spline having attractive properties for the applications. 

In particular after giving their construction and presenting their properties, 

we discuss the results related to the construction of a multiresolution analysis 

of L 2 (R 2 ). Examples of applications to different problems will be also shown. 

Milvia Rossini 


Università di Milano-Bicocca 

Via Cozzi 53 

I-20125 Milano, Italy 

email: milvia.rossini@unimib.it

Surface Approximation of surfaces with fault(s) and/or rapidly varying 

data, using segmentation process, D m spines and finite element method. 

C. Gout 1,2 , C. Le Guyader 3 , H. Huru 4 , L. Romani 5 , *A.G. Saint-Guirons 6 

1 Université de Valenciennes 

LAMAV - ISTV2, Le Mont Houy 


2 INSA de Rouen - LMI 

Place Emile Blondel, BP 08 

76 131 Mont-Saint-Aignan cedex, France 

3 INSA de Rennes 

20 Avenue des Buttes de Coësmes, CS 14315 

35043 Rennes Cedex , France 

4 University of Tromsoe 

Department of Mathematics and Statistics, 

University of Tromsoe, N-9037 Tromsoe, Norway,. 

5 Università di Milano, 

Dipartimento di Matematica e Applicazioni ,Via R. 

Cozzi n. 53, 20125 Milano, Italy. 

6 Université de Pau 

LMA UMR 5142, Avenue de l’Université, 

64018 Pau cedex, France 

In many problems of geophysical interest, one has to deal with data that exhibit complex fault structures or at 

least large variations (rapidly varying data). This occurs for instance when describing the topography of 

seafloor surfaces, mountain ranges, volcanoes, islands, or the shape of geological entities, that can present 

large and rapid variations due for instance to the presence of faults in the structure. 

The usual approximation methods use to lead to instability phenomena or undesirable oscillations. The key 

point to get a good approximant consists in precisely define the locations of the large variations and the 

faults. To do that, we propose a method developed by Le Guyader and Gout (2005) to "segment" the data. 

Then, having the knowledge about the location of the discontinuities of the surface, we can generate a mesh 

using triangles (which takes into account the set of discontinuities) and a specific spline approximant is then 

computed. We discuss the convergence of the method when the number of data tends to infinity. We show 

the efficiency of this technique applying it to real datasets in Oceanography and in the Geosciences. This last 

part is crucial because our propose method is generated 

Key words: Level set methods, finite element methods, spline, surfaces having faults, spline functions... 

Meshing constraints 

Without information about the location of the large variations, the usual approximation methods lead to 

instability phenomenae or undesirable oscillations that can locally and even globally hinder the 

approximation. Here is the Tonga trench after the segmentation process on the dataset (we get 3 patches). 

We then apply a spline approximant in order to 

get a surface whithout oscillations, and where the 

large gradients are correctly given. This spline is 

discretized on a finite element space , the 

meshing taking into account the patches obtained 

after the segmentation process.

A C 1 quadratic trivariate macro-element space 

defined over arbitrary tetrahedral partitions 

Larry Schumaker 

larry.schumaker@gmail.com 

Abstract: 

In 1988 Worsey and Piper constructed a trivariate macro-element based on C 1 

quadratic splines defined over the Powell-Sabin splitof a tetrahedron. 

Unfortunately, their element can only be used with tetrahedral partitions that 

satisfy some very restrictive geometric constraints. We show that by further 

refining their split, it is possible to construct a macro-element (based on the 

same spline space) that can be used with arbitrary tetrahedral partitions. The 

resulting macro-element space is stable and provides full approximation power. 

It is of dimension 4V + 2E + 4F, where V,E,F are the numbers of vertices, edges, 

and faces in the partition.


by Frank Sommen 

The classical Klein Gordon equation may be factorized by means of the Dirac operator 

using the Clifford algebra R 3,2 with generators e 1 , e 2 , e 3 , w = -i e 4 and v = -i e 5 whereby 

the elements e j satisfy the standard defining relations e j e k + e k e j = -2 δ jk . The Dirac 

operator that factorizes the Klein-Gordon operator is then given by ∂ x = ∂ x1 e 1 + ∂ x2 e 2 

+ ∂ x3 e 3 + w ∂ t + v m , whereby m is the scalar mass. Within the same algebra one may 

also consider the bivectorial operator v( ∂ x1 e 1 + ∂ x2 e 2 + ∂ x3 e 3 + w ∂ t ) + M whereby M 

is a bivector in the space-time Clifford basis e 1 , e 2 , e 3 , w. There is no reason why the 

vectorial Dirac operator has physical relevance while this bivectorial operator wouldn't; it 

is also Lorentz invariant and the first part corresponds to the usual space-time Dirac 

operator for massless particals. It is in other words an alternative extension of this 

operator leading to an alternative notion of mass. In our presentation we study this 

operator from the function theoretic point of view as well as the corresponding Klein- 

Gordon operator which has the form ∆+ m 0 + e 123 w M 0 + S whereby ∆ is the Laplacian, 

m 0 the standard mass, M 0 the "Klein mass" and S the mass-momentum interaction 

operator.

Hyperholomorphic functions in elasticity and fluid dynamics 

Wolfgang Sproessig 

Institute of Applied Analysis 

Freiberg University of Mining and Technology 

09596 Freiberg, Germany/Saxony 

Abstract: During the last twenty years Clifford analysis has become increasingly important 

tool also for the analysis of (initial) boundary value problems of partial differential 

equations and their applications to problems of mathematical physics and engineering. In 

particular we will apply real and complex quaternions. The Clifford calculus covers a 

branch of mathematics which uses computational techniques for solving problems from a 

wide variety of physical systems which mathematically modeled in 3,4 and more dimensions. 

In my talk I will try to reflect some main ideas in the field of applications. Above 

all I intent to discuss a quaternion operator calculus and its application to boundary and 

initial value problems including lines of a suitable numerical analysis. Examples in the 

applications area will be chosen from problems in elasticity theory and hydrodynamics . 

We start with an algebraic introduction. Then a brief discourse in function theory in R n 

will follow. We define versions of the generalized Cauchy-Riemann operator, the Teodorescu 

transform and the so-called Cauchy-Fueter operator. Formulas of Borel-Pompeiu 

type complete the function theoretic part. Moreover, we introduce a singular integral 

operator (Bitzadse operator) over the boundary of a sufficient smooth bounded domain in 

R n , corresponding Plemelj projections will be described. A Hodge decomposition of the 

quaternionic Hilbert space follows. One of the associated orthoprojections is a generalized 

version of the well-known Bergman projection. Now we enter in the part with applications 

and start with problems in linear elasticity. Neuber-Papkovic statements are included. We 

present in this case a quaternionic multipole method. Furthermore, we consider methods 

for the solution of initial-boundary problems in fluid dynamics and forecasting equations 

on the sphere. The we will explain a corresponding associated discretization concept and 

show the interplay between the discrete and continuous model. We finish our lecture with 

a brief sketch of further interesting fields. Basic results are included in [1], [2],[3],[4], [5],[6] 

and [7]. 

References 

[1] Brackx F., Delanghe R. and Sommen F.(1982) Clifford analysis. Pitman Research Notes in 

Math., Boston, London, Melbourne. 

[2] K. Gürlebeck and Sprössig W.(1990) Quaternionic Analysis and Boundary Value Problems, 

Birkhäuser Verlag, Basel. 

[3] K. Gürlebeck and Sprössig W.(1997) Quaternionic and Clifford calculus for physicists and 

engineers, John Wiley, Chichester. 

[4] K. Gürlebeck and Sprössig W.(2002) Representation theory for classes of initial value problems 

with quaternionic analysis, Mathematical Methods in the Applied Sciences; 25; 1371 — 1382. 

[5] K. Gürlebeck, K. Habetha and Sprössig, W.(2006) Function theory in the plane and in the 

space, (German), Birkhäuser, Basel. (English version in preparation, August 2007) 

[6] Kravchenko, V.V. and Shapiro,M.V.(1996) Integral representations for spatial models of mathematical 

physics, Pitman research notes Mathematics Series 351. 

[7] Mitrea, M.(1994) Clifford wavelets, singular integrals and Hardy Spaces, Lecture Notes in 

Mathematics 1575, Springer, New York.


Gabriele Steidl, 

University of Mannheim, 

Faculty of Mathematics and Computer Science 

Abstract 

Nonlinear diffusion filtering and wavelet shrinkage are two methods 

that can be applied for discontinuity-preserving denoising. In this talk we 

give a survey on relations between both paradigms when space discrete 

or fully discrete versions of nonlinear diffusion filters are considered. We 

study a class of numerical schemes for nonlinear diffusion filtering that 

offers insights on the design of novel wavelet shrinkage rules for isotropic 

and anisotropic image enhancement. These schemes utilise analytical or 

semi-analytical solutions to dynamical systems that result from spacediscrete 

nonlinear diffusion filtering on minimalistic images with 2 × 2 

pixels. They are applicable to singular nonlinear diffusion filters such as 

TV flow, to bounded nonlinear diffusion filters of Perona—Malik type, and 

to tensor-driven anisotropic methods such as edge-enhancing or coherenceenhancing 

diffusion filtering. The fact that these schemes use processes 

within 2 × 2-pixel blocks allows to connect them to shift-invariant Haar 

wavelet shrinkage. This interpretation leads to novel shrinkage rules for 

two- and higher-dimensional images that are scalar- or vector-valued. Unlike 

classical shrinkage strategies they employ a diffusion-inspired coupling 

of the wavelet channels that guarantees an approximation with an excellent 

degree of rotation invariance. Conversely, previousely used shrinkage 

functions can be identified as corresponding to well—known diffusifities. 

ThetalkisbasedonjointworkwithJ.Weickert,M.Welk,P.Mrázek 

from the Saarland University.

Hand-written character recognition using spline wavelets 

Karina Toscano, Ricardo Barrón, Humberto Sossa 




E-mails: likatome@hotmail.com, rbarron@cic.ipn.mx, hsossa@cic.ipn.mx 

Abstract 

Hand-written character recognition is important in multiple artificial intelligence 

applications. Many of the existent solutions use soft-computing models like neural networks, 

support vector machines, and so on. 

Lately it has been propose to use models base on splines, to capture the trace essence to 

adjust nodes to the character an be able to make comparisons. 

In this work is generalized this idea but using spline wavelets to capture the characteristic 

of trace in different scales and be able to make discrimination more robust. 

Finally experiments are made to compare results and show advantages of the proposed 

method.

Hand-written character recognition using spline wavelets 

Karina Toscano, Ricardo Barrón, Humberto Sossa 




E-mails: likatome@hotmail.com, rbarron@cic.ipn.mx, hsossa@cic.ipn.mx 

Abstract 

Hand-written character recognition is important in multiple artificial intelligence 

applications. Many of the existent solutions use soft-computing models like neural networks, 

support vector machines, and so on. 

Lately it has been propose to use models base on splines, to capture the trace essence to 

adjust nodes to the character an be able to make comparisons. 

In this work is generalized this idea but using spline wavelets to capture the characteristic 

of trace in different scales and be able to make discrimination more robust. 

Finally experiments are made to compare results and show advantages of the proposed 

method.

WAVELETS AND MOVEMENTS 

DIFFERENT APPROACHES 


Univ. Autonoma Met. (Iztapalapa) 

Ciencias Basicas e Ingenieria, 



We present several approaches on movement detection and recognition using 

wavelets as well as the classical speed tuned wavelets. As an alternative we propose 

to use a mother wavelet that actually describes a movement, a simple one, in order 

to build any other given by sparse data. We build such wavelet using stereoscopic 

vision and we show that in order to do that there are also several different 

approaches with advantages and disadvantages

Some results in discrete Clifford analysis 

H. De Schepper, F. Sommen and L. Van de Voorde 

Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University 

Let (e 1 ,... ,e m ) be the usual Euclidean basis, taken as a set of generators underlying the construction 

of the Clifford algebra R 0,m , where the geometric multiplication is governed through the defining relations 

e 2 j = −1, j =1,... ,m and e je k + e k e j =0,j, k =1,... ,m. Standard (or orthogonal) Clifford analysis 

then usually focusses on the study of so—called monogenic functions, i.e. R 0,m valued null solutions of 

the Dirac operator ∂ = P m 

j=1 e j∂ xj . In this way, Clifford analysis is often seen as a direct and elegant 

generalization to higher dimensions of the theory of holomorphic functions in the complex plane, since the 

Dirac operator generalizes in a natural way the well—known Cauchy—Riemann operator. Crucial in the 

development of the function theory is the factorization of the Laplacian by the Dirac operator: ∂ 2 = −∆, 

a basic feature underlying the proofs of the most important properties an d results. It is at the same 

time the reason why Cliffordanalysisisalsoconsideredasarefinement of classical harmonic analysis, 

since monogenic functions refine the properties of harmonic functions. 

More recently, discretization has emerged as a new topic in Clifford analysis, focussing on discrete 

counterparts of the concepts and results of the continuous function theory, in view of the eventual numerical 

treatment of boundary value problems and problems from potential theory. In this presentation, 

we aim at contributing to the further development of a theoretical framework for these discrete versions 

of Clifford analysis. We use the factorization of the Laplacian, mentioned above for the continuous case, 

asastartingpointfordefining, on specific geometric structures, discrete Dirac operators which factorize 

a given discrete Laplacian. A first example is the so—called “cross Laplacian” on Z m : 

Lf(x) = 

mX 

[f(x + e j ) − f(x − e j )] − 2mf(x) 

j=1 

In this case, factorizing Dirac operators has already been established by K. Guerlebeck and A. Hommel 

[1, 2], as well as by U. Kaehler and N. Faustino [3]. More generally, one may consider discrete Laplacians 

with weight coefficients on a 3 m -cube centered around a point in Z m . However, in that case, no first 

order factorizations can be found within the usual context of the Clifford algebra R 0,m . A new axiomatic 

framework is needed, built on the notions of two new Clifford bases, e + j and e− j , j =1,... ,m, corresponding 

to the observation that also the partial derivative ∂ xj shows two discretized versions: f(x + e j ) − f(x) 

and f(x) − f(x − e j ). Moreover, so—called curvature vectors have to be introduced. Based on these ideas, 

we develop a theory of discrete Dirac operators on rectangular and triangular grids in the two and three 

dimensional case, the techniques used leading to a factorization of any discrete Laplacian in a cube or 

hexagon. Naturally, we also aim at discrete versions of the function theoretic properties of the Dirac 

operator, in particular a discrete Borel-Pompeiu theorem as well as Cauchy’s integral forms, making use 

of a discrete notion of boundary. In this presentation, we will present our first results. 

References 

[1]K.Guerlebeck,A.Hommel: Onfinite difference potentials and their applications in a discrete 

function theory, Mathematical Methods in the Applied Sciences 25 (2002), 1563—1576. 

[2] K. Guerlebeck, A. Hommel: On finite difference Dirac operators and their fundamental solutions, 

Advances in Applied Clifford Algebras 11 (2003), 89—106. 

[3] N. Faustino, U. Kaehler: Fischer decomposition for difference Dirac operators, Advances in Applied 

Clifford Algebras 17 (2007), 37—58.

Variational image decomposition models into cartoon and texture 

Luminita VESE 

Department of Mathematics, UCLA 

405, Hilgard Avenue 

Los Angeles, CA 90095-1555, U.S.A 

http://www.math.ucla.edu/~lvese 

In 2001, in an American Mathematical Society lecture series entitled 

"Oscillating Patterns in Image Processing and Nonlinear Evolution Equations", Yves 

Meyer has analyzed and questioned the total variation minimization model (Rudin, 

Osher, Fatemi) for separating cartoon from texture. He therefore proposed several 

refined variants of the TV model, by substituting the L 2 norm to the square of the 

fidelity term v=f-u by 

weaker norms of generalized functions. In particular, he proposed to model the texture 

component v in a u+v model by one of the spaces of generalized functions 

G=div(Linfinity), F=div(BMO) and E=Laplacian(B 1 _{infinity,infinity}). 

David Mumford and Basilis Gidas also show that images can be seen as samples 

from probability distributions of random variables supported on spaces of generalized 

functions, not on spaces of functions. 

However, it is not easy to solve such models in practice. In this talk, I will 

review Meyer's models and present some computational approaches for image 

restoration and image decomposition into cartoon and texture.


Shayne Waldron 

Department of Mathematics (Rm 410) 

University of Auckland 

Private Bag 92019, Auckland 

New Zealand 

Frame decompositions are useful because they are technically similar to orthogonal expansions 

(they simply have more terms) and can be constructed to have desirable properties that may be 

impossible for an orthogonal basis, e.g., in the case of wavelets certain smoothness and small 

support properties. 

Here we show that frames are of interest for finite dimensional spaces where the desirable 

properties are symmetries of the underlying space (which an orthogonal basis cannot express). 

We present a number of (hopefully compelling) examples of tight frames. One of these is for the 

orthogonal polynomials of several variables for a weight which has some symmetries. Another, 

which is important in quantum measurement theory, is the so called Heisenberg frames of d 2 

vectors in C d which have minimum cross correlation. We introduce the class of harmonic frames 

of n vectors in C d , which are generated by an abelian subgroup of symmetries. For each n and d 

there are a finite number of harmonic frames, and we discuss how they can be obtained by using 

a symbolic algebra package such as MAGMA.


Author: Yi Xu 

Email: xuyi@sjtu.edu.cn 

Abstract: Quaternion wavelet transform (QWT) achieves much attention in recent years as a new 

multiscale analysis tool for geometric image structures. It is an extension of the real wavelet 

transform and complex wavelet transform (CWT) by using the quaternion algebra and the 2-D 

Hilbert transform of filter theory. It brings exciting properties to many image applications as color 

image processing, object recognition, optical flow estimation and stereo matching. To give an 

overview of the development of QWT and investigate its potential applications, this paper is 

organized as four parts. 

Section I provides a brief introduction of QWT. Section II presents a retrospective of the 

evolution of QWT and its dominant applications. Compared with discrete wavelet transform (DWT) 

and CWT, the development of QWT is reviewed. The principles of quaternion wavelet construction 

are explored for achieving the important properties in multiscale analysis of geometric image 

structures, including the linear-phase property, shift-invariance property and compact support 

property with reduced computational complexity. Then the dominant applications of QWT in the 

existing research works are summarized. It indicates that the mechanism of adaptive scale 

representation of geometry features is important for such multiscale image analysis tasks. The 

improvement of the robustness and accuracy in two application instances of uncalibrated stereo 

matching and optical flow estimation stresses the importance of the mechanism. 

Section III switches its focus on the potential of QWT in new applications such as image fusion 

and registration. As for image fusion application, the reversibility of QWT is discussed. And then a 

new fusion scheme is advanced, incorporating Contourlet transform (CT) into the quaternion 

wavelet decomposition scheme to enhance the direction selectivity of QWT. Because edges and 

textures are the intrinsic information in image representation, it is important to enhance them and 

thus improve the spatial resolution. CT involves the basis functions that are oriented at any 

power-of-two number of directions with flexible aspect ratios. Therefore the introduction of 

Contourlet into QWT scheme brings great probabilities to get a better representation of edges and 

textures than the original quaternion wavelet decomposition scheme. The potential of modified 

QWT scheme in the new application of image fusion is then supported by the encouraging 

experimental results. As for image registration application, the 2-D phase concept of QWT is 

utilized to give an invariant feature detector in scale space. 2-D phase congruency model is 

discussed and then a dimensionless feature measure is proposed. The accordingly extracted features 

are important for the robust estimation of the image affine transform in registration task. We 

provide several registration experiments with quantitative assessments to testify the proposed 

registration method. 

Brief conclusion on the QWT development experience is given in Section IV. In all, this paper 

aims to give a useful reference for the future use of QWT. 

KeyWords: Quaternion wavelet transform (QWT), multiscale image analysis

Program - Université de Pau et des Pays de l'Adour

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