Eigen-Based Signal Processing Methods for Ultrasound Color Flow ...

Eigen-Based Signal Processing Methods for Ultrasound 

Color Flow Imaging 

by 

Alfred C.H. Yu 

A thesis submitted in conformity with the requirements 

for the degree of Doctor of Philosophy 

The Edward S. Rogers Sr. Department of Electrical and Computer Engineering 

Collaborative Program with the Institute of Biomaterials & Biomedical 

Engineering 

University of Toronto 

© Copyright by Alfred C.H. Yu 2007

ABSTRACT 

Eigen-Based Signal Processing Methods for Ultrasound 

Color Flow Imaging 

Alfred C.H. Yu, 2007 

Edward S. Rogers. Sr. Department of Electrical and Computer Engineering & 

Institute of Biomaterials and Biomedical Engineering, 

University of Toronto, 

Toronto, ON, Canada 

Purpose: This thesis presents a study on the design of new eigen-based signal processing 

methods for use in color flow imaging. Specifically, the proposed methods are designed to 

address three main problems in color flow signal processing: the lack of abundant Doppler data 

samples, the possible presence of wideband Doppler clutter, and the potential flow signal 

distortions that arise during clutter suppression. 

Theoretical Contributions: A clutter filter (the Hankel-SVD filter) and a flow estimator (the 

Matrix Pencil method) were respectively developed by exploiting the eigen-space principles 

related to two matrix forms known as the Hankel matrix and the matrix pencil. Both techniques 

were then combined to form a new color flow data processor that performs flow detection and 

flow estimation in parallel. All of these methods are adaptive to the Doppler signal contents 

through an SVD of the Hankel matrix created from a Doppler data vector. Moreover, they are 

intended to work with each Doppler ensemble separately (i.e. they do not require Doppler 

ensembles from multiple sample volumes). 

Simulation Assessment: A Doppler signal simulation model was developed to generate Doppler 

signals consisting of non-stationary (phase-modulated) tissue clutter and spectral-broadened 

(amplitude-modulated) flow echoes. The synthesized datasets were used to analyze the Hankel- 

- ii -

Abstract 

iii 

SVD filter’s flow detection performance and the Matrix Pencil’s velocity estimation 

performance. A comparison of the Hankel-SVD filter with an existing type of adaptive filter 

(clutter-downmixing filter) showed that this new filter is more capable of discriminating flow 

echoes from Doppler clutter. Also, it was found that the Matrix Pencil can provide less biased 

flow estimates as compared to other frequency-based flow estimators like the lag-one 

autocorrelator. 

Experimental Assessment: Color flow datasets were obtained respectively from a steady-flow 

phantom and from the carotid arteries of a youth subject. These datasets were used to analyze 

the performance of the proposed single-module processor. For the flow phantom study, the 

single-module processor showed that it can reconstruct velocity maps similar to the theoretical 

flow profile. As well, for the in vivo studies, it gave a better flow detection performance than 

conventional two-module processors especially when there is substantial tissue motion.

ACKNOWLEDGEMENTS 

It was a snowy winter day when I first officially started my doctoral study at the 

University of Toronto. From then to now, this research journey has been rather overwhelming 

with many rough rides in between. In particular, many long school days were spent on the 4 th 

floor of Rosebrugh Building and the 3 rd floor of Mining Building as I tried to decipher various 

abstract concepts from the literature. Throughout the course of study, many important 

individuals have offered advices and support that are invaluable to me, and it is my honor to 

receive their words of wisdom. Prior to the presentation of this dissertation, I would like to give 

big credit to these important individuals. 

First and foremost, I would like to express my sincere gratitude to my thesis advisor, Prof. 

Richard Cobbold, for giving me the freedom to explore and develop my own research directions. 

Dr. Cobbold has always been very supportive of my work and has provided lot of insightful 

comments to help me fine-tune my research results. My co-advisor, Dr. Wayne Johnston, has 

also been an influential figure throughout my study. I have really appreciated his strong 

encouragement along with his great sense of humor. 

During a random occasion, I had the chance to meet with one of our lab’s alumnus, Dr. 

Larry Mo. After our initial discussion on that day, we eventually turned out to become 

collaborators in research. I would like to thank him for sharing his insights on the design aspects 

of ultrasound systems. I am also very grateful for his help in acquiring parts of the experimental 

data used in this thesis. 

When I first started my doctoral study, Prof. Raviraj Adve introduced me to the concept 

of eigen-based signal processing. I really appreciated his advice and insights in this research 

field that eventually evolved as the theme of this thesis. Special thanks also go to Prof. Michael 

Joy and Prof. Berj Bardakjian for their ever challenging questions on the fundamental 

motivations of my studies. Their constructive comments have stimulated many after-thoughts 

following every committee meeting and defense. 

The occasional visits to the 6 th floor of the Research Wing at Sunnybrook & Women’s 

College Health Sciences Centre were always fun and interesting. I would like to thank Prof. 

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Acknowledgements 

v 

Peter Burns for granting me access to the ultrasound scanners in his laboratory. I also wish to 

thank Dr. Milan Banerjee and other members of the Burns research group for helping me resolve 

various problems encountered during the data acquisition process. It is very unfortunate that Dr. 

Banerjee passed away before this thesis was completed, but his smiles and humor will always be 

remembered. 

This research would not have been completed without the help of Dr. Matthew Bruce 

from Philips Ultrasound. I am grateful for his valuable help on fixing the Philips HDI-5000 

machine that I used to collect parts of my experimental data. I would also like to thank Dr. Eric 

Cohen-Solal from Philips Research for offering me a special opportunity to work as a research 

intern and learn about the developmental aspects of medical ultrasound systems. This internship 

experience has widely broadened my knowledge in the ultrasound field. 

School has always been a mix of pain and pleasure. Although the quest for knowledge 

has often been a struggle, the joy of working alongside with the past and present members of the 

Cobbold research group has certainly been a pleasant experience. I will also remember those 

Dilbert-style chats with officemates at RS 411 and classmates from many courses. 

Perhaps the most gracious moment at the end of a long school day was the subway ride 

back to my uptown home. During the ride from Queen’s Park to Finch, I usually collapsed into a 

state of coma, with anything happening around me being considered as white noise. The 

moment when I opened the door of my home was ever pleasing, as those familiar voices were 

often there yapping about my unpunctuality for supper. I would like to thank my parents, 

Frankie and Candy Yu, for moving over to Toronto during my last year of study. With their 

great backline support, I never had to worry about my dietary needs. It is also my honor to 

receive strong encouragements from my former housemates Simon Chan and Ka-Wai Leung. 

They have always put up a platform for me to unload my personal frustrations. As well, I have 

really appreciated the occasional morale boasters from all my great friends: especially the 

entertaining ones from Martin Mak, Jenny Choi, and fellow Yu’s Club members. 

Lastly, I would like to acknowledge the scholarship support from the Natural Sciences 

and Engineering Research Council of Canada as well as the short-term funding from the Walter 

Sumner Foundation and the Fraser Elliot Chair of Vascular Surgery. Their generous support has 

greatly assisted my financing throughout this thesis study.

TABLE OF CONTENTS 

ABSTRACT ................................................................................................................................. ii 

ACKNOWLEDGEMENTS.................................................................................................................. iv 

TABLE OF CONTENTS .................................................................................................................... vi 

LIST OF ABBREVIATIONS AND SYMBOLS ....................................................................................... x 

CHAPTER 1 Introduction to Color Flow Imaging................................................................... 1 

1.1 Clinical Background ............................................................................................................. 1 

1.2 Outline of Thesis Study ........................................................................................................ 2 

1.2.1 Research Motivation and Hypothesis ............................................................................ 2 

1.2.2 Goals and Objectives ..................................................................................................... 2 

1.2.3 Chapter Overview .......................................................................................................... 3 

1.3 Overview of Doppler Ultrasound.......................................................................................... 4 

1.3.1 General Concepts........................................................................................................... 4 

Physical Principles ............................................................................................................. 4 

Doppler Equation................................................................................................................ 4 

Range Localization ............................................................................................................. 5 

1.3.2 Display Modes ............................................................................................................... 6 

Spectrogram........................................................................................................................ 6 

Color Flow Image............................................................................................................... 7 

M-Mode Profile................................................................................................................... 7 

1.4 Principles of Ultrasound Color Flow Imaging...................................................................... 8 

1.4.1 System-Level Overview .................................................................................................. 8 

Data Acquisition ................................................................................................................. 8 

Signal Processing................................................................................................................ 9 

1.4.2 Imaging Considerations............................................................................................... 10 

Transducer Type and Transmit Pulse Characteristics ..................................................... 10 

Image Resolution and Penetration Depth......................................................................... 12 

Velocity Resolution and Aliasing Limit............................................................................. 13 

Pulse Repetition Interval and Doppler Ensemble Size ..................................................... 14 

Parameter Optimization.................................................................................................... 15 

1.4.3 Advanced Flow Imaging Approaches .......................................................................... 15 

Three-Dimensional Mapping............................................................................................ 15 

Vector Flow Imaging ........................................................................................................ 16 

High-Frequency Flow Imaging......................................................................................... 17 

Microbubble Contrast Agents........................................................................................... 17 

Additional Remarks........................................................................................................... 18 

1.5 Signal Processing Challenges in Color Flow Imaging ....................................................... 18 

1.5.1 Background Considerations ........................................................................................ 18 

1.5.2 Challenges in Clutter Suppression............................................................................... 20 

Doppler Ensemble Size Limitations.................................................................................. 20 

Wideband Clutter Problems.............................................................................................. 20 

1.5.3 Challenges in Velocity Estimation............................................................................... 22 

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Table of Contents 

vii 

Carrier Frequency Variations .......................................................................................... 22 

Clutter Filter Distortions .................................................................................................. 23 

1.6 Concluding Remarks........................................................................................................... 24 

CHAPTER 2 Clutter Suppression in Color Flow Imaging: A Hankel-SVD Approach ...... 25 

2.1 Chapter Overview ............................................................................................................... 25 

2.2 Existing Clutter Filter Designs............................................................................................ 26 

2.2.1 Non-Adaptive Filters.................................................................................................... 26 

Background Considerations.............................................................................................. 26 

FIR and IIR Filters............................................................................................................ 27 

Regression Filters ............................................................................................................. 29 

2.2.2 Adaptive Filters............................................................................................................ 30 

Background Considerations.............................................................................................. 30 

Clutter-Downmixing Filters.............................................................................................. 31 

Eigen-Based Filters .......................................................................................................... 32 

2.2.3 Comparison of Clutter Filters...................................................................................... 33 

2.3 The Hankel-SVD Filter....................................................................................................... 35 


Design Motivations ........................................................................................................... 35 

Fundamental Principles.................................................................................................... 36 

2.3.2 Theoretical Formulation.............................................................................................. 36 

Construction of Hankel Data Matrix ................................................................................ 36 

SVD Analysis..................................................................................................................... 37 

Estimation of Clutter Eigen-Space Dimension ................................................................. 38 

Computational Considerations ......................................................................................... 40 

2.4 Simulation Method.............................................................................................................. 41 

2.4.1 Signal Synthesis Model ................................................................................................ 41 

General Principles............................................................................................................ 41 

Synthesis Procedure.......................................................................................................... 42 

Model Parameters............................................................................................................. 44 

2.4.2 Method of Analysis....................................................................................................... 46 

General Overview ............................................................................................................. 46 

Performance Measure....................................................................................................... 47 

2.5 Simulation Results .............................................................................................................. 47 

2.5.1 Characteristics of Principal Hankel Components ....................................................... 47 

Arterial Flow Scenario ..................................................................................................... 47 

Low-Velocity Flow Scenario............................................................................................. 49 

2.5.2 Post-Filter Clutter-to-Blood Signal Ratios.................................................................. 50 

Arterial Flow Scenario ..................................................................................................... 50 

Low-Velocity Flow Scenario............................................................................................. 51 


CHAPTER 3 Velocity Estimation in Color Flow Imaging: A Matrix Pencil Approach .... 54 


3.2 Existing Flow Estimation Strategies................................................................................... 55 

3.2.1 Non-Parametric Estimators......................................................................................... 55 

Fundamental Considerations............................................................................................ 55 

Frequency-Based Methods................................................................................................ 55


viii 

Time-Shift-Based Methods................................................................................................ 56 

3.2.2 Parametric Estimators................................................................................................. 57 

Fundamental Considerations............................................................................................ 57 

Autoregressive Modeling .................................................................................................. 58 

Multiple Signal Classification........................................................................................... 60 

3.3.2 Comparison of Frequency-Based Velocity Estimators................................................ 62 

3.3 The Matrix Pencil Estimator............................................................................................... 63 


Design Motivations ........................................................................................................... 63 


3.3.2 Theoretical Formulation.............................................................................................. 64 

Construction of Data Matrix Pair..................................................................................... 64 

Low-Rank Matrix Reduction............................................................................................. 67 

Numerical Computation.................................................................................................... 68 

Relationship to Eigen-Analysis......................................................................................... 70 

3.3.3 Applications in Color Flow Data Processing.............................................................. 71 

Rank-One Matrix Pencil ................................................................................................... 71 

Rank-Two Matrix Pencil................................................................................................... 72 

Rank-Adaptive Matrix Pencil............................................................................................ 73 

3.4 Simulation Methods............................................................................................................ 75 

3.4.1 Method of Study ........................................................................................................... 75 

General Overview ............................................................................................................. 75 

Performance Measures ..................................................................................................... 75 

3.4.2 Simulation Parameters................................................................................................. 76 

Data Synthesis................................................................................................................... 76 

Data Analysis.................................................................................................................... 77 

3.5 Simulation Results .............................................................................................................. 77 

3.5.1 Flow Scenario with Stationary Clutter ........................................................................ 77 

Performance of Low-Rank Reduction............................................................................... 77 

Comparative Assessment at Different BSNRs................................................................... 78 

Comparative Assessment at Different Blood Velocities.................................................... 81 

3.5.2 Flow Scenario with Non-Stationary Clutter ................................................................ 81 

Efficacy of Adaptive Rank Selection ................................................................................. 81 

Comparative Assessment at Different Blood Velocities.................................................... 83 


Single-Module Approach to Color Flow Data Processing: 

An Experimental Study ...................................................................................... 86 


4.2 The Single-Module Color Flow Data Processor................................................................. 87 


Design Motivation............................................................................................................. 87 


4.2.2 Processor Details......................................................................................................... 88 

Conceptual Overview........................................................................................................ 88 

Description of Processing Module.................................................................................... 89 

4.3 In Vitro Color Flow Imaging Study.................................................................................... 91 

CHAPTER 4


ix 

4.3.1 Experimental Protocol................................................................................................. 91 

Data Acquisition ............................................................................................................... 91 

Signal Processing Procedure............................................................................................ 92 

Data Analysis Method....................................................................................................... 93 

4.3.2 Experimental Results ................................................................................................... 93 

Color Flow Images ........................................................................................................... 93 

Velocity Profiles................................................................................................................ 94 

4.4 In Vivo Color Flow Imaging Studies.................................................................................. 96 

4.4.1 Experimental Protocol................................................................................................. 96 

Data Acquisition ............................................................................................................... 96 

Signal Processing Procedure............................................................................................ 96 

Data Analysis Method....................................................................................................... 97 

4.4.2 Case Study with Vessel Wall Motion ........................................................................... 97 

Overview of Scenario........................................................................................................ 97 

Results: Color Flow Images.............................................................................................. 98 

Results: Regional Power Comparison............................................................................ 100 

4.4.3 Case Study with Translational Tissue Motion ........................................................... 100 

Overview of Scenario...................................................................................................... 100 

Results: Color Flow Images............................................................................................ 102 

Results: Regional Power Comparison............................................................................ 104 

4.5 Concluding Remarks......................................................................................................... 105 

CHAPTER 5 Thesis Summary and Future Directions......................................................... 106 

5.1 Chapter Overview ............................................................................................................. 106 

5.2 Summary of Thesis Study................................................................................................. 106 

5.2.1 Overall Scope............................................................................................................. 106 

5.2.2 Specific Developments ............................................................................................... 107 

Hankel-SVD Filter .......................................................................................................... 107 

Matrix Pencil .................................................................................................................. 108 

Single-Module Color Flow Data Processor ................................................................... 109 

Doppler Signal Synthesis Model..................................................................................... 110 

5.2.3 List of Primary Research Contributions.................................................................... 111 

5.3 Future Research Directions............................................................................................... 111 

5.3.1 Developmental Aspects .............................................................................................. 111 

Real-Time Computation Issues ....................................................................................... 111 

Adaptive Thresholding.................................................................................................... 112 

5.3.2 Experimental Aspects................................................................................................. 112 

Flow Imaging Phantom Design ...................................................................................... 112 

Doppler Clutter Analysis ................................................................................................ 113 

5.3.3 Theoretical Aspects.................................................................................................... 113 

Processing of Non-Uniformly Sampled Color Flow Data.............................................. 113 

Contrast-Enhanced Color Flow Data Processing.......................................................... 114 

Other Single-Snapshot-Based Flow Estimators.............................................................. 114 

Flow Analysis Using Image Processing Tools................................................................ 115 

5.4 Final Remarks ................................................................................................................... 115 

REFERENCES ............................................................................................................................. 117

LIST OF ABBREVIATIONS AND SYMBOLS 

Abbreviations 

2D Two-dimensional 

3D Three-dimensional 

ACR Autocorrelator 

AR Autoregressive 

B-mode Brightness-mode 

BSNR Blood-signal-to-noise ratio 

BV Blood vessel 

CBR Clutter-to-blood signal ratio 

CW Continuous-wave 

ESPRIT Estimation of signal parameters via rotational invariance techniques 

FIR Finite impulse response 

GE Generalized eigenvalue 

IIR Infinite impulse response 

I/Q In-phase/quadrature 

KL Karhunen-Loeve 

M-mode Motion-mode 

MP Matrix Pencil 

MT Moving tissue 

MUSIC Multiple signal classification 

PW Pulsed-wave 

RK-1 MP Rank-one Matrix Pencil 

RK-2 MP Rank-two Matrix Pencil 

RK-A MP Rank-adaptive Matrix Pencil 

RMS Root-mean-squared 

ST Static tissue 

SVD Singular value decomposition 

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List of Symbols and Abbreviations 

xi 

Physical and Imaging Parameters 

α dB Ultrasound attenuation factor (in dB scale) 

Δφ Inter-pulse phase change 

Δt ( =T PRI ) Inter-pulse period (essentially the pulse repetition interval) 

λ o ( = c/f ) Wavelength of ultrasound 

θ Beam-flow angle 

c o Ultrasound propagation speed 

d max Maximum imaging depth of color flow image 

d pen Penetration depth of ultrasound 

d res Depth (or axial) resolution of color flow image 

f D Doppler shift (frequency) 

f D(res) Doppler spectral resolution 

f o Transmitted ultrasound frequency 

f r Received ultrasound frequency 

F Focal length 

I pen(dB) Intensity attenuation (in dB) at a given penetration depth 

l res Beam line (or lateral) resolution of color flow image 

N cyc Number of cycles in transmit pulse 

N D Doppler ensemble size (or number of firings per beam line) 

N L Number of beam lines in color flow image 

t D Inter-pulse arrival time shift 

T F Frame period of color flow image 

T P Temporal length of transmit pulse 

T PRI Pulse repetition interval 

v 

Scatterer velocity 

v alias Velocity aliasing limit of color flow image 

v res Velocity resolution of color flow image 

W Aperture width 

Doppler Signal Processing Parameters 

χ k 

Fitting coefficient of the k th basis vector in a regression filter or principalcomponent 

signal model 

Kronecker delta function that equals to one when n = i+j 

δ n=i+j


xii 

δ n=j–i 

Δf thr 

γ k 

λ 

λ k 

ρ y 

σ k 

φ(n) 

ϕ k (n) 

a(i,j) 

a k (i,j) 

b k 

c k 

e n (i,j) 

f D,k 

f D,kb 

f D,thr 

f D(c) 

f D(est) 

f thr(clut) 

H(f D ) 

K 

K c 

K nom 

M 

P 

R y (0) 

R y (1) 

Kronecker delta function that equals to one when n = j–i 

Clutter bandwidth threshold in the Hankel-SVD filter; also the spectral 

spread threshold in rank-adaptive Matrix Pencil 

Coefficient of the k th basis vector in a KL expansion or Hankel component 

approximation 

Eigenvalue 

Eigenvalue (expected value) of the k th basis vector in a KL expansion 

Post-filter Doppler signal power 

k th singular value in a SVD 

Instantaneous phase of carrier in a clutter-downmixing filter 

n th sample of the k th basis vector in a Hankel-SVD 

(i,j) th entry in the Hankel data matrix 

(i,j) th entry in the k th rank-one Hankel component 

Weight of the k th past input sample in a FIR/IIR filter 

Weight of the k th past output sample in an IIR filter; coefficient of an AR or 

MUSIC characteristic polynomial 

(i,j) th H 

entry of the noise subspace projection matrix E n E n 

Mean Doppler frequency of k th principal basis in a Hankel-SVD or principalcomponent 

signal model 

Principal frequency corresponding to the modal flow component 

Doppler frequency threshold in the Hankel-SVD filter 

Mean Doppler frequency of clutter (used for clutter downmixing) 

Doppler frequency estimate (mean or mode) 

Spectral threshold for checking clutter presence (used in Hankel-SVD filter) 

Filter response at a given Doppler frequency f D 

Order for a filter or eigen-structure rank for a principal-component signal 

model 

Estimated clutter eigen-space dimension in the Hankel-SVD filter 

Nominal eigen-structure rank used by rank-adaptive Matrix Pencil 

Number of snapshots used to estimate Doppler correlation matrix 

Dimension parameter of a Hankel data matrix or matrix pencil 

Autocorrelation function at zero lag 

Autocorrelation function at single sample lag


xiii 

w(n) 

x(n) 

x DM (n) 

X AR (f D ) 

X MUSIC (f D ) 

y(n) 

z k 

Δ 

ϕ k 

Θ 

Φ 

a 0 

a 1 

A 

A 0 

A 1 

Ã 

Ã 0 

Ã 1 

A k 

b 

c 

ĉ 

C 

D 

e k 

E n 

F 

I 

p k 

q 

R x 

n th sample of white noise in an ensemble 

n th sample of Doppler signal in an ensemble 

n th sample of the downmixed Doppler signal 

Signal spectrum for an AR model at a given Doppler frequency f D 

Signal spectrum for a MUSIC model at a given Doppler frequency f D 

n th sample of filtered Doppler signal in an ensemble 

k th characteristic mode in a principal-component signal model 

Noise perturbation matrix in a low-rank matrix reduction 

k th basis vector in a Hankel component approximation 

Diagonal matrix operator for clutter downmixing 

Diagonal matrix of GEs in the decomposition proof of Matrix Pencil 

Data vector for first N D –1 samples in an ensemble 

Data vector for last N D –1 samples in an ensemble 

Hankel data matrix 

Hankel data matrix for the first N D –1 samples in an ensemble 

Hankel data matrix for the last N D –1 samples in an ensemble 

Rank-reduced Hankel data matrix in a low-rank approximation 

First N D –1 rows of a rank-reduced Hankel data matrix 

Last N D –1 rows of a rank-reduced Hankel data matrix 

k th rank-one Hankel component in a SVD 

Blood component of Doppler signal vector 

Clutter component of Doppler signal vector 

Clutter model vector of a regression filter 

Clutter space matrix of a regression filter 

Diagonal matrix of model coefficients in the decomposition of Matrix Pencil 

k th basis vector in a KL expansion 

Noise subspace matrix in a KL expansion 

Filter matrix 

Identity matrix 

k th -order polynomial basis vector in a regression filter 

Eigenvector 

Doppler correlation matrix


xiv 

S Steady-state filter matrix for the state-space form of FIR/IIR filter 

T Transient filter matrix for the state-space form of FIR/IIR filter 

u k k th left singular vector in a SVD 

u 0,k First N D –P samples of the k th left singular vector in a Hankel-SVD 

u 1,k Last N D –P samples of the k th left singular vector in a Hankel-SVD 

v k k th right singular vector in a SVD 

v CS Complex sinusoid vector 

v CS(k) 

Complex sinusoid vector for the k th component in a principal-component 

signal model 

v init Initial state vector for the state-space form of FIR/IIR filter 

w White noise component of Doppler signal vector 

w f White noise component in a filtered Doppler signal vector 

x 

Doppler signal vector 

x m m th Doppler signal vector used to estimate Doppler correlation matrix 

x DM Doppler signal vector after clutter downmixing 

y 

Filtered Doppler signal vector 

Z L Left complex-sinusoid matrix in the decomposition proof of Matrix Pencil 

Z R Right complex-sinusoid matrix in the decomposition proof of Matrix Pencil 

Doppler Signal Modeling Parameters 

β b Bandwidth of the blood signal template 

φ c,max Peak clutter phase deviation in the Doppler signal synthesis model 

κ b Blood magnitude coefficient in the synthesized signal 

κ c Clutter magnitude coefficient in the synthesized signal 

κ w Noise magnitude coefficient in the synthesized signal 

f D(b) Modulating frequency of the blood signal template 

f vib Clutter vibration frequency 

g b (n) n th random sample in the blood scatterer distribution function 

g c Random sample used to model clutter scattering distribution 

h b (n) n th sample in the blood signal template of signal synthesis model 

h c (n) n th sample in the clutter signal template of signal synthesis model 

Temporal width of the blood signal template 

T dur

CHAPTER 1 

Introduction to Color Flow Imaging 

1.1 Clinical Background 

Cardiovascular diseases are health disorders related to the functioning of the heart and the 

blood vessels. According to statistics from the Heart and Stroke Foundation of Canada (2003), 

this class of health problems is the leading cause of death in Canada, affecting over 70,000 

Canadians every year and costing the Canadian economy $18.4 billion annually due to direct 

hospitalization charges and lost productivity. In terms of its causes, vascular stenoses and 

strokes have contributed to 84% of cardiovascular patient cases. As such, detailed diagnoses are 

often performed to study the cardiovascular flow dynamics inside a patient. From the results 

acquired through these diagnoses, clinicians can then deploy suitable therapies and surgeries to 

the patient as necessary. 

Imaging studies of cardiovascular flow dynamics are traditionally performed using X- 

rays. In fact, Nederkoorn et al. (2003) pointed out in their recent review that X-ray angiography, 

which involves the insertion of a catheter device into the patient’s vasculature, still remains as 

the gold standard for such diagnosis. However, as an invasive modality, this imaging approach 

carries the risk of accidentally dislodging fatty tissues and calcified deposits from vessel walls. 

Over the past twenty years, new non-invasive imaging tools such as ultrasound color flow 

imaging and magnetic resonance angiography have been developed in attempt to obtain the same 

diagnostic information without intrusion to the patient’s vasculature. As suggested by Beach et 

al. (1997), these two new imaging tools have similar spatial imaging resolution and appear to be 

complementary methods of each other. Nevertheless, ultrasound color flow imaging has better 

real-time scanning capabilities and is more affordable by most clinics. 

Ever since its development, ultrasound color flow imaging has been used for various 

types of cardiovascular flow studies. As reviewed by Ferrara and DeAngelis (1997), it has been 

used to identify the presence of vascular stenoses, assess complicated flow patterns, diagnose for 

the development of aneurysms and tumors, study the patency of implanted shunts, and visualize 

- 1 -

Chapter 1. Introduction to Ultrasound Color Flow Imaging 2 

blood regurgitations between heart chambers during a cardiac cycle. More recently, this imaging 

tool has found new applications such as microcirculation assessment in the eyes (Williamson and 

Harris 1996), treatment response monitoring during cancer therapies (Lagalla et al. 1998), 

contractility examination of heart muscles (Waggoner and Bierig 2001), and guidance of 

interventional devices (Armstrong et al. 2001). From these clinical applications, it can be seen 

that ultrasound color flow imaging is often used as a high-level vascular visualization tool. 

1.2 Outline of Thesis Study 

1.2.1 Research Motivation and Hypothesis 

Given the widespread use of ultrasound color flow imaging in medical diagnostics, it is 

important for these images to provide accurate visualization of the vascular flow dynamics. As 

suggested in some evaluation studies (e.g. Arbeille et al. 1999, Stewart 2001), failure to provide 

accurate flow information in color flow images may lead to an increased risk of misdiagnosis 

and may cause assessment difficulties during long-term patient monitoring. However, from a 

signal processing perspective, there are several factors that tend to reduce the accuracy of blood 

flow estimates displayed in color flow images. As will be described in Section 1.5, these sources 

of errors can be divided into two categories: 1) those that arise during suppression of clutter in 

the color flow data, and 2) those that arise during estimation of flow velocities. In terms of their 

impact, these error sources may lead to the appearance of spurious or erroneous pixels in the 

color flow images. In turn, they tend to obscure visualization of the actual flow dynamics. 

To address the challenges in color flow signal processing, it is necessary to develop 

suitable methods that can adequately suppress clutter and derive flow estimates with minimal 

bias. We hypothesize that a more effective clutter suppression method can be developed through 

the use of eigen-based approaches like principal component analysis, while a more accurate flow 

estimation strategy can be formulated through parametric spectral analysis along with adaptive 

rank selection. With the advent of improved clutter filters and flow estimators, we can anticipate 

that the clinical reliability of color flow images will be improved. 

1.2.2 Goals and Objectives 

Driven by the research hypothesis, the overall goal of this thesis study is to develop new 

eigen-based signal processing methods for color flow imaging applications. Specifically, it is


our intent to devise a new clutter filter design and a novel flow estimation strategy by exploiting 

the eigen-space properties of two matrix forms known as the Hankel matrix and the matrix 

pencil. We will also consider how these two eigen-based methods can be combined to create a 

new color flow data processor that can perform flow detection and flow estimation in parallel 

within the same processing module. 

To achieve our overall research goal, this thesis study has been broken down into the 

following three specific objectives: 

• Design a new eigen-based clutter filter (referred to as Hankel-SVD filter) based on Hankel 

matrices created from the acquired signals and evaluate its efficacy in clutter suppression; 

• Develop a new eigen-based flow estimator (referred to as Matrix Pencil estimator) from the 

properties of matrix pencil and investigate its theoretical estimation performance; 

• Integrate the above two designs together to form a new color flow signal processor and study 

its performance under various in vitro and in vivo flow scenarios. 

Note that the signal processing strategies listed in the first two objectives can be considered as 

the building blocks for the new signal processor mentioned in the last objective. 

1.2.3 Chapter Overview 

The remainder of this introductory chapter is intended to provide an overview on the 

principles of color flow imaging and describe the signal processing problems being addressed in 

this thesis. In particular, the chapter contents have been organized as follows: 

• Section 1.3 provides a general overview on how ultrasound can be used to study blood flow 

dynamics and describes the display modes in ultrasound-based flow estimation methods; 

• Section 1.4 gives a system-level description of ultrasound color flow imaging, discusses the 

various imaging parameters that need to be considered, and surveys some advanced color 

flow imaging schemes reported in the literature; 

• Section 1.5 discusses the major signal processing challenges in ultrasound color flow 

imaging and reviews existing ways on how these challenges are being addressed; 

• Section 1.6 summarizes the research problem and outlines the organization of the subsequent 

chapters in this thesis. 

Readers who are familiar with color flow imaging principles are encouraged to proceed 

directly to Section 1.5 which describes the sources of errors in color flow signal processing.


1.3 Overview of Doppler Ultrasound 

1.3.1 General Concepts 

Physical Principles 

As recently reviewed by Cobbold (2006, Sec. 9.1 and 10.1), flow velocity estimation 

using ultrasound was first successfully attempted in the late 1950s. This approach involves the 

transmission of ultrasound waves (i.e. acoustic waves with frequencies beyond the human 

audible range of 20 kHz) into the blood vessel and the subsequent reception of echoes returned 

from red blood cell scatterers flowing inside the vessel. In general, there are two excitation 

modes in ultrasound-based flow estimation methods: the continuous-wave (CW) mode and the 

pulsed-wave (PW) mode. As revealed by its name, the CW mode is based on the use of 

continuous ultrasound waves, and it estimates flow velocities by measuring the Doppler 

frequency shifts (i.e. changes in the perceived frequency due to source or target motion) between 

the transmitted wave and the received echoes. Since the Doppler Effect is exploited, this 

excitation mode is often referred to as CW Doppler ultrasound (or simply called CW Doppler). 

On the other hand, the PW mode is based on the use of finite-duration ultrasound pulses, and it 

estimates flow velocities by measuring the time shifts between pulse echoes. This excitation 

mode is historically referred to as PW Doppler ultrasound (or simply called PW Doppler), even 

though it is somewhat of a misnomer † . 

Doppler Equation 

For both CW and PW Doppler ultrasound, the scatterer velocity distribution is one of the 

factors that the shift magnitudes depend on (regardless of whether they are Doppler shifts or time 

shifts). Since vascular flow dynamics in turn govern the scatterer velocity distribution, it is 

possible to gain knowledge about the flow profile from studying the induced shift magnitudes. 

This principle essentially forms the working basis of Doppler ultrasound. In terms of the other 

factors that govern the shift magnitudes, the Doppler frequency shifts seen in the backscattered 

echoes of CW Doppler also depend on the beam-flow angle θ, transmitted ultrasound frequency 

f o , and acoustic propagation speed c o . The equation that relates the Doppler shift f D to the various 

† In early research studies, it was believed that the arrival time shifts are induced by the classical Doppler Effect. 

Hence, the “Doppler” misnomer has evolved. Further discussion on this misnomer can be found in many Doppler 

ultrasound textbooks (e.g. see Jensen 1996, Ch. 4; Evans and McDicken 2000, Ch. 4; Cobbold 2006, Ch. 10).


contributing factors is often referred to as the Doppler equation, and it has the following 

mathematical form (Cobbold 2006, Sec. 9.1): 

f 

D 

2vf 

o 

= f 

o 

− f 

r 

≈ cosθ (for c o >> v), (1-1) 

c 

o 

where v is the scatterer velocity and f r is the received ultrasound frequency. 

In comparison to CW Doppler, the time shifts in PW Doppler ultrasound are also 

dependent on similar factors even though the two excitation modes are based on different 

physical principles. As derived in various ultrasound textbooks (e.g. see Cobbold 2006, Sec. 

10.2), the time shift t D between pulse echoes can be described by the following expression: 

2vTPRI 

t 

D 

= cosθ , (1-2) 

c 

o 

where T PRI is the pulse repetition interval. Note that the time shifts seen in PW Doppler can also 

be interpreted as a frequency-shift mechanism. In particular, given that frequency is equal to the 

phase change over time by definition, the time shift between pulse echoes carries the following 

frequency interpretation: 

Δφ 

f 

ot 

≈ 

Δt 

T 

D 

PRI 

2vf 

= 

c 

o 

o 

cosθ 

, (1-3) 

where Δφ = f o t D indicates the phase change between pulse echoes. Comparing between (1-1) and 

(1-3), it can be seen that the frequency shift in PW Doppler ultrasound is essentially equivalent 

to the classical Doppler shift observed in CW Doppler. Hence, without loss of generality, we 

will interchangeably use the symbol f D and the term “Doppler frequency” throughout this thesis 

to denote the frequency shifts in PW Doppler ultrasound as well. 

Range Localization 

Even though it is more complicated, PW Doppler ultrasound is often more preferred 

because of its range localization capabilities. In particular, by using a range gate to separate the 

echoes originating from outside scatterers, this excitation mode can provide analysis of the flow 

velocity distribution at specific ranges of interest (Cobbold 2006, Sec. 10.3). Such range 

localization is useful to blood flow studies since it facilitates the identification of local changes 

in the flow profile inside the vasculature. Hence, much effort has been devoted to the 

development of PW Doppler ultrasound over the years, but it should be noted that CW Doppler


(b) Velocity Map 

(a) Spectrogram 

(c) Flow Power Map 

Fig. 1-1. Various display modes in Doppler ultrasound: (a) spectrogram (time-varying 

velocity distribution from one sample volume); (b) velocity map (directional flow estimates 

within an image plane in red and blue hues); (c) flow power map (blood signal power within 

an image plane in pale-red hues). These figures were reproduced with permission from 

ZONARE Medical Systems (Mountain View, CA, USA). 

ultrasound still remains of use in studies where range localization is not important. In this thesis, 

we shall focus the attention on PW Doppler ultrasound, and from hereon, it will be implicitly 

assumed that PW excitations are used. 

1.3.2 Display Modes 

Spectrogram 

One commonly used display mode in Doppler ultrasound is a spectrogram (or spectral 

Doppler display) that shows the time-varying flow velocity distribution derived from the 

measured Doppler frequencies. An example of this type of display is shown in Fig. 1-1a. Note 

that the presented flow information is derived from a single sample volume whose size can be 

readily adjusted by the user during a diagnostic procedure. Also, the placement of the sample


volume is guided by a grayscale ultrasound image (i.e. a brightness-mode or B-mode image) that 

reveals the anatomical structure within the ultrasound transducer’s field of view. In terms of the 

spectrogram’s significance, clinicians often use this display mode to gain quantitative insights on 

the vascular flow dynamics within a certain range of interest inside the blood vessel. In 

particular, since vascular stenoses often give rise to disturbed and even turbulent flow conditions, 

clinicians can study the spectrogram to assess the healthiness of a patient’s vasculature. 

Color Flow Image 

Another mode of display for Doppler ultrasound is a color flow image (or velocity map) 

that shows flow velocity estimates of different sample volumes within the transducer’s field of 

view. An example of this type of display is illustrated in Fig. 1-1b. As shown in the figure, the 

velocity estimates are displayed in the form of color pixels, and they are superimposed onto a B- 

mode image of the anatomical structure. This duplex image display format allows clinicians to 

easily identify blood flow within an entire imaging view. In comparison to the spectrogram that 

shows the flow distribution, the color flow image only provides a single velocity estimate 

(usually the mean or modal velocity) for each sample volume. As such, during a diagnostic 

procedure, the color flow image is usually used as a high-level flow visualization tool prior to 

detailed diagnoses at specific parts of the vasculature. 

For the color flow image, it is possible to display flow power estimates instead of flow 

velocities. The resulting duplex images are often referred to as flow power maps (or Doppler 

power maps), and an example of this type of color flow image is given in Fig. 1-1c. Note that, at 

large beam-flow angles, the flow power map has diagnostic advantage over the velocity map 

because the echo power returned from blood scatterers is less dependent of the beam-flow angle 

(while the Doppler shift magnitude is angle-dependent). On the other hand, the velocity map has 

the advantage of being able to provide directional flow information that is clinically important 

for the identification of arteries and veins. 

M-Mode Profile 

A third but less common mode of display for Doppler ultrasound is a motion-mode (or 

M-mode) profile that presents color-coded flow information over time along one ultrasound 

beam line. In connection with the spectrogram and the color flow image, the M-mode profile 

can be considered as a hybrid display mode because it gives time-varying velocity estimates like


Waveform 

Oscillator 

Pulser 

Echo Amplifier 

Mixer 

Transducer 

Lowpass Filter 

Surface 

Doppler 

lateral 

(or time) 

Image View 

Artery 

depth 

Sampling Memory 

(depth res. d res ) 

(Doppler res. T PRI ) 

(image mode: lateral res. l res ) 

(M-mode: time res. N D T PRI ) 

To Flow 

Estimation 

Fig. 1-2. Overview of the data acquisition setup used for ultrasound color flow imaging. 

Note that the sampling memory in the image mode has three dimensions: depth, lateral, and 

Doppler. In M-mode, the lateral dimension becomes the time dimension since the multipulse 

data acquisition process is repeated over time along a single beam line. 

a spectrogram whilst providing multi-depth flow information like a color flow image. The 

advantage of using this display mode is that it allows clinicians to study how the flow profile 

temporally evolves through a particular cross-section of a blood vessel. Note that the velocity 

estimates provided in an M-mode profile are derived using the same methods as the ones used to 

produce color flow images. Also, instead of showing velocity estimates, it is possible to display 

flow power estimates in an M-mode profile. 

1.4 Principles of Ultrasound Color Flow Imaging 

1.4.1 System-Level Overview 

Data Acquisition 

From a system-level perspective, the formation of each color flow image frame or M- 

mode profile can be divided into two main stages: data acquisition and signal processing. As 

illustrated in Fig. 1-2, the data acquisition stage first involves the use of an ultrasound transducer 

(often an array transducer) to transmit pulse firings and collect the returned echoes. In order to


produce an image, the pulse firings are repeated for different beam lines within the field of view; 

in contrast, if an M-mode profile is to be formed, then pulses are only fired along a single beam 

line. Subsequently, the received echoes from the different pulse firings are individually 

downmixed to baseband by mixing the echoes with the carrier frequency of the transmitted pulse 

and applying a lowpass filter to retain only the baseband spectral components in the mixed 

echoes. For this downmixing process, the in-phase/quadrature (I/Q) demodulation scheme is 

often implemented so that the analytic form of the pulse echoes can be obtained for subsequent 

analysis. After the pulse echoes are downmixed, they are sampled at time points that correspond 

to various depths in order to study the inter-pulse phase changes or time shifts at various axial 

and lateral positions (i.e. sample volumes) within the field of view. The resulting data samples 

can then be arranged into a three-dimensional (3D) data array: one dimension for pulse number, 

one for depth/axial position, and the other for line/lateral position. Since the pulse-number 

dimension of the 3D data array essentially contains information on the inter-pulse phase changes, 

this dimension is also referred to as the Doppler axis † . As well, the ensemble of pulse echo 

samples for a particular sample volume is referred to as the Doppler signal. 

Signal Processing 

In the signal processing stage of ultrasound color flow imaging, a Doppler power map 

and a flow velocity map are computed from the array of Doppler signals acquired through the 

pulse firings. Most commercial systems compute these two forms of flow information through a 

frequency-based estimation approach (i.e. by studying the inter-pulse phase changes) because of 

its relatively simpler computation complexity as compared to time-shift-based estimation 

approaches (Evans and McDicken 2000, Sec. 4.5.3). As shown in Fig. 1-3, the frequencydomain 

flow estimation process first involves the use of a digital highpass filter to suppress lowfrequency 

echoes that may be present in each Doppler signal. This filtering operation is carried 

out in attempt to distinguish the Doppler frequency shifts of blood scatterers from those that may 

arise due to acoustic reverberations of nearby tissues (whose strength can reach 60-80 dB greater 

than blood depending on the scanner’s dynamic range). After the filtered data samples are 

obtained, the average power of each post-filter Doppler signal is estimated by simply finding the 

† Some authors refer to the pulse-number dimension as the “slow-time” axis to avoid the Doppler misnomer. Also, 

the depth dimension is sometimes called the “fast-time” axis since it is the relative time axis of each pulse echo.


Power Estimator 

lateral 

(or time) 

depth 

Power Map 

Memory 

Doppler 

Signal 

Highpass Clutter 

Filter 

(cutoff freq. f D,thr ) 

Spurious Pixel 

Removal 

Velocity 

Estimator 

lateral 

(or time) 

depth 

Velocity Map 

Memory 

Display 

Fig. 1-3. The signal processing scheme used in the frequency-domain flow estimation 

approach. Note that the post-filter Doppler power estimates are used for both generating the 

power map as well as deciding whether a map pixel corresponds to real flow information. 

mean-squared signal value. The velocity estimate of each post-filter Doppler signal is also 

determined by first finding the mean Doppler frequency of the filtered signal and then converting 

this estimate into a velocity value via the Doppler equation given in (1-3) † . Once the power and 

velocity estimates are computed for all the Doppler data, they are turned into image pixels by 

respectively mapping the estimates onto a power map color scale and a velocity map color scale. 

Since blood flow may not be present in all places within the field of view, spurious pixels are 

removed from the power and velocity maps if their corresponding filtered Doppler power is 

below a threshold value. Finally, depending on which display mode is used, the resulting 

Doppler power map or flow velocity map is superimposed onto a B-mode image of the 

underlying structure to form a duplex display. 

1.4.2 Imaging Considerations 

Transducer Type and Transmit Pulse Characteristics 

In color flow imaging, the ability to provide real-time visualization of flow dynamics is 

often considered as an important diagnostic feature. To realize this real-time functionality, it is 

necessary to use an ultrasound transducer that is capable of transmitting all the pulse firings and 

† During the conversion, the parameters θ, f o , and c o in the Doppler equation are assumed to be constant.


collecting the returned echoes for each image frame within a short frame period (at most on the 

order of 100 ms). Hence, as similar to grayscale ultrasound imaging (i.e. B-mode scans), color 

flow imaging typically uses an array transducer such as linear array or curved array for the data 

acquisition. The advantage of using the array transducer is that it can acquire pulse echoes along 

various beam lines in real-time by sequentially exciting different groups of transducer elements. 

The array transducer is also able to acquire data at more acute beam-flow angles through 

electronic beam steering and in turn avoid Doppler frequency detection problems at neartransverse 

beam-flow angles. A further advantage of the array transducer is that it can acquire 

the pulse echoes from the different beam lines in an interleaved order to increase the coherence 

in the flow estimates between beam lines. In terms of their operating frequencies, commercially 

available array transducers generally operate in the frequency range between 1-17 MHz † . As 

will be discussed later in this section, such frequency range is generally sufficient for flow 

studies in large blood vessels, but higher frequencies are needed in order to study blood flow in 

the microcirculation. For data acquisition at higher ultrasound frequencies, current systems use a 

single-element transducer to obtain pulse echoes along one beam line and mechanically move the 

transducer to collect echoes from different beam lines. 

Unlike B-mode imaging that makes use of the backscattered echoes from tissues, color 

flow imaging relies on the blood scatterers inside the circulation as the primary signal targets. 

The strengths of backscattered blood echoes, however, are at least 10-20 dB lower than the 

backscattering strengths of tissues (Jensen 1996, Sec. 2.2). To compensate for this significant 

drop in the echo strength, it is not suitable to simply increase the intensity of the transmit pulses 

because of tissue heating concerns and safety regulations. Instead, the transmit pulses are made 

longer in duration than the ones used in B-mode scans so that the blood echo sensitivity and the 

blood-signal-to-noise ratio can be improved. Correspondingly, the transmit pulses for color flow 

imaging are often in the form of multi-cycle sinusoidal bursts, and they are more narrowband in 

nature than the ones used for B-mode imaging. Note that the pulse duration T P can generally be 

approximated as follows: 

N 

cyc 

T 

P 

= , (1-4) 

f 

o 

† This statement is made in reference to the array probes used in current top-of-the-line scanners including Philips 

iU22, Siemens Sequoia 512, and GE LOGIQ 9.


where N cyc denotes the number of sinusoidal cycles in the transmit pulse and f o is the pulse 

carrier frequency. 

Image Resolution and Penetration Depth 

The image resolution of color flow images, like B-mode images, is mainly defined by the 

type of transmit pulse used for the firings. In particular, the lateral resolution of the image 

depends on the lateral width of the beam profile generated from the transmit pulse, while the 

axial resolution depends on the duration of the transmit pulse. As described in ultrasound 

textbooks (e.g. see Cobbold 2006, Ch. 3), the lateral resolution l res corresponding to various 

focused beam patterns can be approximated as follows if the aperture width W is much greater 

than the pulse center wavelength λ o (i.e. W >> λ o ): 

F 

l = 

Fc 

o o 

res 

≈ λ , (1-5a) 

W Wf 

o 

where F is the beam focal length and it is well-known from wave physics that λ o equals to c o /f o . 

On the other hand, the axial resolution d res corresponding to a multi-cycle sinusoidal pulse can be 

approximated by: 

c T co 

N 

cyc 

≈ , (1-5b) 

2 2 f 

o P 

d 

res 

= 

where T P is as defined in (1-4) and the factor of one-half is included to account for the round-trip 

distance between the beam origin and a particular depth. As seen in (1-5a) and (1-5b), the lateral 

and axial resolutions both become finer at higher ultrasound frequencies. In general, the image 

resolution is between 0.1-1.0 mm for ultrasound frequencies in the operating range of array 

transducers (between 1-17 MHz). This resolution is sufficient to study the flow dynamics in 

large blood vessels and body organs, but a higher resolution is needed to study blood flow in the 

microcirculation where vessel diameters are typically less than 50 μm. 

Unlike the image resolution, the penetration depth of a pulse firing decreases at higher 

frequencies because the acoustic attenuation in tissues increases roughly as a linear function of 

the ultrasound frequency. In other words, any gain in the image resolution via an increase in the 

ultrasound frequency is accompanied by a concomitant decrease in the penetration depth. For a 

given attenuation factor α dB , the penetration depth d pen can be approximated as follows provided 

that the acoustic intensity decays exponentially with depth (Cobbold 2006, Sec. 1.8): 

o


d 

pen 

I 

pen(dB) 

≈ , (1-6) 

α f 

where I pen(dB) is the intensity attenuation (in dB) at the penetration depth. In soft tissues and 

blood, the attenuation coefficient usually ranges between 0.2-0.7 dB/(cm⋅MHz) (Evans and 

McDicken 2000, pp. 38), and thus the 20-dB penetration depth is between 1-20 cm for 

ultrasound frequencies less than 20 MHz. 

Velocity Resolution and Aliasing Limit 

The pulse-echo data acquisition process in color flow imaging can be considered as a 

sampling mechanism in which a single Doppler data sample is collected for each sample volume 

during each pulse firing. Owing to this sampling mechanism, the resulting flow estimates are 

inherently limited to a finite spectral resolution if they are derived from analyzing the inter-pulse 

phase changes. From sampling theory, it is well-known that the spectral resolution inversely 

depends on the ensemble period or, equivalently, the product of ensemble size and sampling 

period. Thus, the Doppler spectral resolution f D(res) can be approximated as follows: 

dB 

D 

o 

f = 1 

D(res) 

N T 

, (1-7a) 

where N D and T PRI are the Doppler ensemble size and the pulse repetition interval respectively 

(parameters to be described in the next subsection). Using the Doppler equation, the expression 

for Doppler resolution can be translated into the following expression for velocity resolution v res : 

PRI 

co 

v 

res 

= . (1-7b) 

f N T cosθ 

2 

o D PRI 

As seen in (1-7b), an increase in the ensemble period or ultrasound frequency can result in an 

improved velocity resolution, which is often important for studying slow flow dynamics. 

Amongst these factors, it is more suitable to increase the ultrasound frequency when imaging 

microcirculatory flow near the skin surface because a gain in both velocity resolution and image 

resolution can be achieved. On the other hand, it is necessary to use a longer ensemble period 

when imaging low-velocity flow at greater depths because high-frequency ultrasound cannot 

penetrate well into tissues. 

Another issue arising from the sampled nature of Doppler data is related to the maximum 

detectable frequency. From sampling theory, it is well-established that the maximum detectable


frequency without aliasing is equal to half the sampling rate. Hence, the velocity aliasing limit 

v alias has the following form (Cobbold 2006, Sec. 10.2): 

co 

v 

v 

alias 

= 

= 

4 f T cosθ 

o 

PRI 

res 

N 

2 

D 

, (1-8) 

where (1-7b) has been substituted into the above. As seen in (1-8), there is an inherent tradeoff 

in the maximum detectable velocity whenever a finer velocity resolution is achieved by 

increasing the ultrasound frequency or the pulse repetition interval. In order to detect flow 

beyond the velocity aliasing limit, it is necessary to derive the flow estimates from analyzing the 

inter-pulse time shifts instead of the inter-pulse phase changes. Nevertheless, the time-shiftbased 

estimation approach is often not necessary in color flow imaging because the presence of 

aliasing can actually enhance the visualization of blood flow dynamics (Tamura et al. 1991). 

Pulse Repetition Interval and Doppler Ensemble Size 

In color flow imaging, the range of pulse repetition intervals that can be used for data 

acquisition is bounded by two factors: real-time imaging requirements and imaging depth limits. 

In particular, the maximum pulse repetition interval mainly depends on real-time imaging 

constraints because all the pulse echoes from every beam line need to be acquired within a frame 

period that corresponds to a real-time frame rate (i.e. at least 5-10 Hz). On the contrary, the 

minimum pulse repetition interval depends primarily on the maximum imaging depth (which 

may not necessarily be the penetration depth) because sufficient time between firings is needed 

to collect pulse echoes from the field of view. Based on the two constraints, the pulse repetition 

interval T PRI must be defined within the following range for a given maximum depth d max , 

acoustic speed c o , frame period T F , number of beam lines N L , and Doppler ensemble size N D : 

2d 

c 

T 

max 

F 

≤ TPRI 

≤ . (1-9) 

o 

N 

L 

N 

D 

In the above expression, the left side is essentially the round-trip propagation time with respect to 

the maximum imaging depth, and the right side is the maximum allowable time between each 

firing in order for all the pulse echoes to be acquired. 

Amongst the three dependent factors seen in the upper limit of (1-9), the Doppler 

ensemble size is the only factor that can be modified for a given imaging dimension and frame 

rate. Typically, for a field of view with 30-60 beam lines, the ensemble size available for each


Doppler sample volume is limited to fewer than 20 samples so that the pulse repetition interval 

can be set to a value that satisfies both real-time imaging constraints and imaging depth limits. 

Note that, as mentioned earlier, the Doppler ensemble size also has an effect on the Doppler 

spectral resolution. Hence, one way to improve the Doppler spectral resolution is to use a larger 

Doppler ensemble size provided that the pulse repetition interval can still satisfy the upper limit 

of (1-9). 

Parameter Optimization 

Optimization of the system parameters for color flow imaging is generally a difficult task 

because of the various imaging considerations that need to be accounted for. Hence, solutions to 

this optimization problem are often non-trivial and involve many degrees of freedom. In order to 

simplify the design considerations, commercial systems often start with a few user-based 

acquisition parameters and then optimize the other ones internally. For instance, most systems 

allow the user to adjust parameters such as pulse repetition interval, ultrasound frequency, and 

field of view dimensions; from these parameters, the system then internally selects other 

parameters such as pulse duration, focal length, aperture width, frame rate, and Doppler 

ensemble size (i.e. the number of firings per beam line). The effects of different user-defined 

data acquisition parameters on the other imaging parameters are summarized in Table 1-1. In 

this thesis, we shall focus our effort on addressing the effects of Doppler ensemble size and pulse 

repetition interval on the performance of color flow signal processors. As will be noted in 

Section 1.5, these two parameters often have a significant impact on the efficacy of the Doppler 

signal processing strategies. 

1.4.3 Advanced Flow Imaging Approaches 

Three-Dimensional Mapping 

Over the past decade, various advanced data acquisition approaches have been proposed 

to expand upon the flow information provided in color flow images. One representative example 

of these advanced approaches is three-dimensional (3D) color flow imaging (Fenster et al. 2001). 

The main motivation for the development of 3D flow imaging is that conventional color flow 

images can only provide two-dimensional (2D) mono-planar flow maps along the transducer’s 

field of view. Such limitation may mask out important flow information under disturbed flow


Table 1-1. Effects of User-Defined Data Acquisition Parameters 

Parameter 

Increased ultrasound frequency 

(↑ f o ) 

Increased pulse repetition interval 

(↑ T PRI ) 

Increased lateral field of view 

Increased axial field of view 

Effects 

• Finer image resolution (↓ l res , d res ) 

• Lower penetration depth (↓ d pen ) 

• Finer velocity resolution (↓ v res ) 

• Lower velocity aliasing limit (↓ v alias ) 

• Finer velocity resolution (↓ v res ) 

• Lower velocity aliasing limit (↓ v alias ) 

• More beam lines (↑ N L ) 

• Longer frame period and/or smaller Doppler 

ensemble size (↑ T F , ↓ N D ) 

• Coarser velocity resolution (if Doppler 

ensemble size is decreased) (↑ v res ) 

• Increased maximum imaging depth (↑ d max ) 

• Longer pulse duration (to improve blood 

echo sensitvity) (↑ T P ) 

• Coarser axial resolution (↑ d res ) 

scenarios, as a mono-planar flow map inherently lacks the ability to reveal the 3D nature of 

turbulent flow dynamics. To overcome this difficulty, some research studies have attempted to 

use special transducer setups to acquire multiple slices of mono-planar color flow images and 

stack them together as a 3D volumetric display. For instance, Picot et al. (1993) have produced 

3D flow maps by physically moving the transducer to acquire 2D flow images along different 

scan planes. Similar flow images can also be generated through electronic scan plane steering 

with 2D array transducers (von Ramm et al. 1991). This latter approach appears to be more 

robust because it is more capable of acquiring 3D flow maps in real time. 

Vector Flow Imaging 

Vector-based flow imaging is another advanced scheme that is useful in vascular flow 

studies. As reviewed by Dunmire et al. (2000), the objective of vector flow imaging is to obtain 

consistent color flow estimates without prior knowledge of the beam-flow angle. This approach 

is beneficial to the study of complex flow patterns because the beam-flow angle may vary over 

the imaged vessel, and thus the velocity estimates may be less accurate if a fixed beam-flow 

angle is assumed. In terms of its principles, most vector flow imaging methods work by


combining the Doppler frequency shifts estimated from different beam directions based on 

geometric reconstruction techniques. This multi-beam configuration can be implemented 

physically using two different approaches. The first is to use multiple transducers for excitation 

and reception, such as that used in the study by Overbeck et al. (1992). The second is to use a 

single array transducer that has beam steering capabilities, such as the one used by Maniatis et al. 

(1994). Based on the second approach, Capineri et al. (2002) have developed an experimental 

vector flow imaging system that can provide angle-independent flow maps in real time. Aside 

from the use of multiple-beam setups, Jensen and Munk (1998) have proposed to obtain vector 

flow estimates using a single ultrasound beam that has a modified beamforming pattern. This 

single-beam vector flow imaging approach has recently been implemented on an experimental 

scanner (Udesen and Jensen 2006). 

High-Frequency Flow Imaging 

A third advanced imaging scheme of clinical interest is high-frequency color flow 

imaging that involves the use of ultrasound waves in the 30-100 MHz range (Foster et al. 2000). 

The primary advantage of using high-frequency ultrasound for color flow imaging is its superior 

spatial resolution as compared to conventional imaging that is typically done at frequencies 

below 15 MHz. With this improved spatial resolution, the resulting color flow images can 

provide visualization of the flow dynamics inside the microvasculature and hence obtain new 

insights on the physiology of microcirculation. In terms of its development, Ferrara et al. (1996) 

have developed a 38 MHz experimental flow estimation system, while Christopher et al. (1997) 

have reported on a similar apparatus that operates at 50 MHz. More recently, Goertz et al. 

(2000) have expanded upon the previous designs by building a 20-100 MHz color flow imaging 

system prototype. Kruse and Ferrara (2002) have also reported the development of a 25-38 MHz 

color flow imaging system that is based on sweep-scanning principles. 

Microbubble Contrast Agents 

The use of microbubble contrast agents for flow imaging is another advanced technique 

that has many potential clinical applications (Frinking et al. 2000). The aim of this technique is 

to facilitate better flow detection through the nonlinear scattering properties of microbubbles. 

Specifically, flow detection can be improved by injecting microbubbles into the vasculature and 

subsequently processing the fundamental and higher-harmonic echoes returned from these


contrast agents during their passage through the imaged vessel. Microbubbles and harmonic 

flow imaging are particularly useful in the visualization of flow dynamics inside small vessels 

where echoes from surrounding tissues tend to mask out the ones from red blood cell scatterers. 

Regarding the efficacy of microbubbles, several flow estimation studies (e.g. Schrope and 

Newhouse 1993, Chang et al. 1996) have experimentally evaluated the strength of harmonic 

echoes at different contrast agent concentrations. From these experimental findings, Hope 

Simpson et al. (1999) have subsequently proposed the use of a pulse inversion Doppler imaging 

scheme to further amplify the second harmonic echoes returned from microbubbles. This 

scheme works by transmitting a pulse as well as its negated form in sequence and summing the 

two complementary pulse echoes to cancel out the odd-harmonic signals while retaining the 

even-harmonic ones. With the pulse inversion method, Bruce et al. (2004) have reported 

significant improvements in the Doppler signal quality when microbubbles are used during data 

acquisition. 

Additional Remarks 

Even though their data acquisition principles are more advanced, the new flow imaging 

approaches described above (perhaps with the exception of pulse inversion Doppler) still make 

use of the same signal processing strategies as used in conventional color flow imaging. As 

such, their usefulness to vascular flow studies inherently depends on whether the clutter filters 

and flow estimators can perform well under various imaging conditions. Because of this 

dependence, it is important for the clutter filters to be able to adequately suppress low-frequency 

clutter and for the flow estimators to be able to obtain consistent velocity and power estimates. 

Otherwise, there may be noticeable errors in the additional flow information provided by the 

advanced flow imaging approaches. 

1.5 Signal Processing Challenges in Color Flow Imaging 

1.5.1 Background Considerations 

As pointed out earlier, the primary signal targets for ultrasound-based flow estimation 

schemes are the red blood cells moving inside the vessel. As such, in the ideal case, the acquired 

Doppler signal for a sample volume (or map pixel location) encompassing blood flow should 

only comprise of non-zero-frequency components that arise from the inter-pulse phase changes


Clutter 

Spectrum 

Filter Response 

Noise Floor 

0 

Blood 

Spectrum 

f D 

Fig. 1-4. Illustration of the typical spectral contents in a Doppler signal before digital 

sampling. Prior to flow estimation, it is often necessary to use a highpass filter to suppress 

the high-energy and low-frequency clutter in the Doppler signal. 

induced by moving blood scatterers. However, since the vessel is generally surrounded by other 

scattering sources like tissues and vessel walls, the Doppler signal may also contain undesired 

components that are due to reverberations from non-blood scatterers. In this thesis, these 

undesired Doppler signal components will be referred to as clutter, and they should be 

distinguished from the background white noise that naturally arises from random fluctuations in 

the data acquisition electronics. It is worth pointing out that, even though a sample volume may 

be nominally positioned inside the blood vessel, clutter may still be seen in the resulting Doppler 

signal because the diffractive nature of ultrasound beams inherently gives rise to beam sidelobes 

(sometimes referred to as leakage lobes) that encompass non-vessel regions outside the nominal 

sample volume range. 

In contrast to the moving blood scatterers, tissues and vessel walls tend to exhibit a 

significantly smaller degree of movement, and thus they often induce smaller degrees of interpulse 

phase changes. Based on this premise, the corresponding Doppler clutter is usually lowfrequency 

in nature and does not overlap with the blood spectral components on the Doppler 

frequency axis (as illustrated in Fig. 1-4). Nevertheless, since tissues have a stronger scattering 

strength than blood, the magnitude of Doppler clutter is usually much higher than that of the 

desired blood Doppler echoes. In fact, as suggested in various studies (Kadi and Loupas 1995, 

Torp 1997, Bjaerum et al. 2002a), the clutter strength can be as much as 60-80 dB greater in 

magnitude depending on the dynamic data range supported by the scanner. Consequently, their 

presence in the Doppler signal makes it more challenging to extract accurate blood flow 

information from the acquired data.


1.5.2 Challenges in Clutter Suppression 

Doppler Ensemble Size Limitations 

As already shown in Fig. 1-3, the first step in color flow signal processing is to apply a 

highpass filter to each Doppler ensemble so that any low-frequency clutter can be adequately 

suppressed prior to the computation of blood flow information. However, this clutter filtering 

operation is a rather challenging task in color flow imaging because less than 20 samples of 

Doppler data are usually available for processing owing to real-time imaging constraints (see 

Section 1.4.2). With such a small data size, the clutter filter’s transient characteristics tend to 

dominate the filter output and in turn reduce the efficacy of clutter suppression. 

To effectively remove clutter from Doppler signals that have small ensemble sizes, 

studies have considered various ways of minimizing the transient response of a clutter filter. For 

instance, Bjaerum et al. (2002a) have reported that the transient effects of finite impulse response 

(FIR) filters can be mitigated by implementing the minimum-phase equivalent of this filter. In 

contrast, for infinite impulse response (IIR) filters, studies have shown that filter initialization 

schemes like step initialization, projection initialization, and exponential initialization can be 

used to reduce the filter’s transient characteristics (Peterson et al. 1994, Kadi and Loupas 1995). 

Note that, as another way to circumvent filter transient problems, it is also possible to use 

polynomial regression filters to suppress clutter in Doppler ensembles (Hoeks et al. 1991, Kadi 

and Loupas 1995, Torp 1997). This type of filter, whose response is time-variant in nature, 

works by finding the least-squares fit of the Doppler signal onto a polynomial subspace and 

subtracting this fit from the Doppler signal. In examining the efficacy of these filtering 

approaches, Bjaerum et al. (2002a) have concluded that the clutter suppression performances of 

the projection-initialized IIR filter and the polynomial regression filter are generally better than 

the other ones because of their time-variant filter characteristics. Nevertheless, it should be 

noted that these filters all require an appropriate selection of the filter stopband in order to 

achieve adequate clutter suppression. To address this issue, Yoo et al. (2003) have described an 

adaptive approach to specify the filter stopband based on the mean clutter frequency. 

Wideband Clutter Problems 

Another challenge that affects the efficacy of clutter suppression is related to the use of a 

long ensemble period for pulse echo data acquisition. As discussed in Section 1.4.2, a longer


ensemble period sometimes needs to be used to detect low flow velocities in the deeper sample 

volumes. However, one major problem of lengthening the ensemble period for data acquisition 

is that tissues may undergo substantial motion over this extended period, and consequently the 

Doppler signal may contain clutter echoes that are more wideband in nature and that are not 

centered around zero frequency (Heimdal and Torp 1997). To suppress clutter in Doppler 

signals acquired over a lengthened period, it is necessary to widen the stopband of the clutter 

filter so that the possible wideband nature of clutter can be properly accounted for. Nevertheless, 

a widened filter stopband may concomitantly suppress a substantial portion of the blood echoes. 

To improve the suppression of wideband clutter, some studies have considered the use of 

more advanced filtering approaches. For instance, Thomas and Hall (1994) as well as Brands et 

al. (1995) have proposed an approach that first involves the downmixing of the Doppler signal 

with the mean clutter frequency estimate prior to the clutter filtering operation. As a result of 

this downmixing step, the clutter spectral components are shifted toward zero Doppler 

frequency, and hence it is more likely that the clutter echoes can be removed by a conventional 

clutter filter that has a narrower stopband. Another way of suppressing wideband clutter, as 

described by Ledoux et al. (1997), is to make use of a principal component analysis framework 

whereby the clutter echoes are adaptively filtered out by removing the most dominant 

components in the singular value decomposition of the Doppler data at various depths. Using the 

same principal subspace concept, Bjaerum et al. (2002b) have reported an eigen-based regression 

filter that is based on an eigendecomposition of the Doppler correlation statistics. Similarly, 

Kruse and Ferrara (2002) have developed an eigen-filtering strategy for use in swept-scan flow 

imaging applications, while Kargel et al. (2003) have applied the same approach to strain-flow 

imaging studies. An extension to the eigen-regression filtering approach has recently been 

considered by Gallippi et al. (2003), who made use of an independent component analysis 

framework to suppress Doppler clutter in acoustic radiation force imaging studies. As compared 

to the clutter-downmixing filter, the eigen-regression filter seems to be more capable of 

suppressing Doppler clutter while preserving blood flow echoes. However, the primary 

challenge of using this type of filter is that the Doppler correlation matrix often needs to be 

estimated from multiple Doppler signal snapshots that are statistically stationary. As such, this 

filter may be challenging to implement in some vascular imaging studies where it is difficult to 

segment out regions with similar Doppler data statistics. Another limitation of the eigen-


regression filter is that its efficacy is rather sensitive to the choice of the clutter eigen-space 

dimension (as analogous to the filter stopband of FIR/IIR filters) owing to the filter’s adaptive 

nature. Hence, it is necessary to develop an effective algorithm for this filter to select the clutter 

eigen-space dimension. 

1.5.3 Challenges in Velocity Estimation 

Carrier Frequency Variations 

Given the filtered Doppler signal, one way of estimating the mean flow velocity is to 

compute the mean Doppler frequency from the Doppler spectral moments and apply the Doppler 

equation to convert the frequency estimate into a velocity value. Based on this notion, Kasai et 

al. (1985) have reported a closed-form mean velocity estimator that works by analyzing the lagone 

autocorrelation of the filtered Doppler signal. This estimator has low computational 

demands, and it is generally unbiased in the presence of white noise. However, its mean Doppler 

frequency estimates may be biased when there are variations in the carrier frequency of the 

returned echoes simply because the pulse carrier frequency is one of the governing factors in the 

Doppler equation (Ferrara et al. 1992). As pointed out in a few studies (Round and Bates 1987, 

Embree and O’Brien 1990), attenuation and scattering are the two primary distortion 

mechanisms of the pulse carrier frequency. They tend to bring about a gradual shift in the pulse 

carrier frequency as depth increases, and the shift magnitude per unit depth is mainly dependent 

on the bandwidth of the transmit pulse. Fortunately, this problem is usually of less concern in 

color flow imaging because narrowband transmit pulses typically need to be used in order to 

improve the blood echo sensitivity and maintain adequate blood-signal-to-noise ratios (as noted 

in Section 1.4.2). Nonetheless, variations in the carrier frequency can become a significant 

velocity biasing source when wideband transmit pulses are used for data acquisition. 

To account for biases due to carrier frequency variations, one straightforward solution is 

to estimate the carrier frequency shifting rate as a function of depth and apply a correction factor 

to the mean Doppler frequency estimates based on the shifting rate. Alternatively, a number of 

some studies have considered the development of more advanced flow estimation strategies to 

account for carrier frequency variations. For instance, Loupas et al. (1995a) have reported a 2D 

autocorrelation estimator that can compute both the mean Doppler frequency and the mean echo 

carrier frequency based on the Doppler signals of adjacent sample volumes along the same beam


line. On the other hand, some studies have proposed the use of target tracking methods such as 

cross-correlation (Bonnefous and Pesque 1986) and maximum likelihood estimation (Ferrara and 

Algazi 1991a, Alam and Parker 1995). These approaches work by analyzing the inter-pulse 

displacement of the complex echo envelope, and thus they are inherently unaffected by 

variations in the carrier frequency. However, these correlation-based estimation approaches 

remain susceptible to biases originating from clutter filter distortions. 

Clutter Filter Distortions 

The primary side effect of using a filter to suppress low-frequency clutter is that the 

blood echoes may be concomitantly distorted due to non-idealities in the filter’s frequency 

response (Willemetz et al. 1989, Nowicki et al 1990, Tysoe and Evans 1995). The distortions 

can be particularly severe if the blood echoes’ Doppler spectral components are located near the 

filter’s cutoff frequency and its transition region. As a result of these potential distortions, there 

may be a bias in the mean frequency estimates computed from the filtered Doppler signal. Note 

that, at low blood-signal-to-noise ratios, the biasing problem becomes worse because clutter filter 

also distorts the background white noise and thereby adds further bias to the flow estimates. As 

pointed out by Rajaonah et al. (1994), post facto bias correction may be applied to the mean 

frequency estimates if a time-invariant filter (e.g. an FIR filter or non-initialized IIR filter) is 

used to suppress clutter. Nevertheless, for time-variant filters like the ones discussed earlier, this 

method itself may not be adequate to account for the filter biases. 

To avoid biases due to clutter filtering, some studies have considered the development of 

parametric spectral estimators that can compute the principal Doppler frequency of blood echoes 

in the presence of clutter. For instance, Ahn and Park (1991) have applied the autoregressive 

(AR) modeling method originally reviewed by Vaitkus and Cobbold (1988) to simultaneously 

extract both the principal clutter and blood Doppler frequencies from the raw (i.e. unfiltered) 

Doppler signal. In their approach, the raw Doppler signal is modeled as a two-pole signal where 

one of the signal poles corresponds to clutter and the other corresponds to blood signals. The 

signal poles are then solved to obtain the principal frequency estimate of clutter and blood 

signals. In contrast, Vaitkus and Cobbold (1998) have proposed the use of an eigen-analysis 

framework called “multiple signal classification” (MUSIC) for flow estimation without prior 

clutter filtering. This approach, which was originally examined by Allam and Greenleaf (1996),


models the Doppler signal as a summation of two principal harmonics, and it solves for the 

harmonics from the eigen-decomposition of the Doppler signal’s correlation matrix statistics. 

The primary limitation of the MUSIC approach and the AR method, however, is that the Doppler 

signal cannot always be modeled as a two-pole or two-harmonic signal (especially when the 

clutter is wideband in nature). As such, an adaptive model order selection algorithm is needed in 

order for these two estimators to be effective in general. 

1.6 Concluding Remarks 

In clinical diagnostic procedures, ultrasound color flow imaging is primarily used as a 

qualitative way of visualizing blood flow dynamics. Nevertheless, it is still important to provide 

accurate flow information in color flow images so that the risk of misdiagnosis can be reduced. 

In order to maintain the reliability of information provided in color flow images, however, there 

are a few signal processing challenges that first need to be overcome during clutter suppression 

and flow estimation. As discussed in the previous section, the major challenges encountered 

during clutter suppression are mainly associated with the lack of abundant Doppler samples 

available for data processing as well as the possible presence of wideband clutter when using 

longer data acquisition periods. On the other hand, the primary challenge for flow estimation is 

related to the flow signal distortions originating from clutter suppression. To address these 

challenges, the subsequent chapters of this thesis will present new eigen-based color flow signal 

processing methods that are based on the eigen-space properties of two matrix forms: the Hankel 

matrix and the matrix pencil. In particular, the remaining chapters have been organized as 

follows: 

• Chapter 2 provides a theoretical treatment of the principles behind the Hankel-SVD filter 

and describes this filter’s ability to suppress clutter in the Doppler signal; 

• Chapter 3 presents the theory and principles of the Matrix Pencil estimation framework and 

discusses this approach’s application in color flow data processing; 

• Chapter 4 describes how the Hankel-SVD filter and the Matrix Pencil estimator can be 

combined into a single-module color flow signal processor; 

• Chapter 5 summarizes the contributions of this thesis study and discusses some possible 

future directions in this research area.

CHAPTER 2 

Clutter Suppression in Color Flow Imaging: 

A Hankel-SVD Approach 

2.1 Chapter Overview 

As described in the introductory chapter, there are two problems that significantly 

increase the difficulty of clutter suppression in color flow imaging. The first is related to the 

small Doppler ensemble size (usually less than 20 samples) that is available for flow estimation 

because of the need to satisfy real-time imaging constraints. Due to such a small sample size, the 

filtered Doppler signal can be significantly affected by the transient characteristics of the clutter 

filter. The second problem is concerned with the longer data ensemble period (on the order of 

100 ms) that is sometimes required to detect slow flow dynamics at greater depths. Since tissues 

may undergo substantial motion over this extended period, the clutter component of the Doppler 

signal may be more wideband in nature and may not be centered around zero frequency. 

In attempt to address the problems in Doppler clutter suppression, this chapter presents 

the development of a new eigen-based, adaptive clutter filter design for color flow imaging 

applications. This new filter is based on a singular value decomposition (SVD) analysis of the 

Hankel matrix constructed from the Doppler signal of each sample volume, and hence it will be 

referred to as the Hankel-SVD filter from hereon. To formulate discussion on the Hankel-SVD 

filter, the rest of this chapter has been organized as follows: 

• Section 2.2 surveys the highlights of existing clutter filter designs and discusses their 

principles used to suppress Doppler clutter; 

• Section 2.3 presents the theoretical principles of the Hankel-SVD filter, describes its 

adaptability in suppressing wideband clutter, and discusses the computational considerations 

involved with the implementation of this filter; 

• Section 2.4 describes the details of Doppler signal synthesis method used to study the new 

filter’s flow detection performance; 

- 25 -

Chapter 2. Clutter Suppression in Color Flow Imaging: A Hankel-SVD Approach 26 

• Section 2.5 presents the analysis results for the Hankel-SVD filter’s performance and 

compares them with the ones obtained from other types of clutter filters; 

• Section 2.6 summarizes the highlights of the new filter and further discusses some practical 

considerations on the new filter design. 

2.2 Existing Clutter Filter Designs 

2.2.1 Non-Adaptive Filters 

Background Considerations 

To facilitate description of clutter filter designs, we first consider the Doppler signal 

model for a sample volume (or map pixel location) that encompasses blood flow. In this case, 

the Doppler signal generally comprises low-frequency tissue clutter, high-frequency blood 

echoes, and random white noise. For a given Doppler ensemble size N D , these signal 

components can be expressed in the following vector form: 

T 

[ ( 0), x(1), 

, x( 

N −1 

] = b + c w 

x = x K + (in vasculature), (2-1a) 

D 

) 

where x(n) is the n th Doppler data sample, while x, b, c, and w are vectors of length N D for the 

Doppler signal, blood echoes, clutter, and white noise respectively. On the other hand, for a 

sample volume located in the tissue region, its Doppler signal should only comprise of tissue 

clutter and random white noise (since blood flow is not present). Correspondingly, the signal 

equation for such a sample volume is given by: 

T 

[ ( 0), x(1), 

, x( 

N −1 

] = c w 

x = x K + (in tissue). (2-1b) 

D 

) 

To study whether blood signal is present in a Doppler ensemble, it is necessary to first 

use a highpass filter to suppress the low-frequency clutter whose strength is usually much higher 

(greater than 20 dB) than the other Doppler signal components. From a vector space perspective, 

this filtering task can be considered as the application of a linear matrix operator on the Doppler 

signal vector. Thus, the filtered Doppler signal vector y can be written as follows: 

T 

[ ( 0), y(1), 

, y( 

N −1 

] Fx 

y = y K = , (2-2) 

D 

) 

where y(n) is the n th filtered sample and F is an N D ×N D filter matrix. Note that, depending on 

how the filter matrix is formed, the clutter filter may have a time-variant frequency response. 

Thus, as pointed out by Torp (1997), the filter’s frequency response may not simply correspond


to the Fourier transform of the filter’s impulse response. Nevertheless, it is still possible to 

numerically compute the frequency response of these time-variant clutter filters by finding the 

filter’s output power for different complex sinusoid inputs. Based on this notion, the filter 

response H(f D ) at a particular Doppler frequency can be expressed as: 

2 H H 

( f 

D 

) = Fv 

CS 

= v 

CSF 

Fv 

CS 

H , (2-3a) 

where v CS is the following complex sinusoid vector of length N D : 

CS 

j 2πf 

2 ( 2) 2 ( 1) 

[ ] T 

D j πf 

D ND 

− 

, j πf 

D ND 

− 

1, e , K, 

e e 

v = 

. (2-3b) 

FIR and IIR Filters 

It is well-known that the finite impulse response (FIR) filter and the infinite impulse 

response (IIR) filter are the two common types of linear filters used in signal processing. As 

described in various signal processing textbooks (e.g. Oppenheim et al. 1999, Ch. 6 and 7), the 

FIR filter is a non-recursive filter whose output samples only depend on the input data samples, 

and hence its impulse response has a finite duration. On the other hand, the IIR filter is a 

recursive filter whose output samples depend on both input data samples and past output, and 

correspondingly its impulse response carries on for an infinite time. For both types of filters, the 

filter order is essentially equivalent to the number of input samples that each filter output 

depends on. A higher filter order can yield a narrower transition band and a more uniform 

passband in the filter’s steady-state frequency response, but it also brings about a lengthening of 

the filter’s transient response. Note that, for a K th -order IIR filter, its n th output sample can be 

expressed as follows: 

K 

∑ 

k = 1 

K 

∑ 

y( n) 

= − c y( 

n − k) 

+ b x( 

n − k) 

, (2-4a) 

k 

where c k and b k are the respective weights of past output samples and input samples used in the 

filter, and they can be found using approaches like Butterworth and Chebyshev design methods. 

The output for FIR filters essentially has the same form as shown in (2-4a) except that all the 

weights of past output samples are set to zero (i.e. c k = 0). The input weights for FIR filters can 

be found using approaches such as the windowing method and the Parks-McClellan algorithm. 

From a state-space perspective, the output of both FIR and IIR filters can be considered 

as a joint contribution from the filter’s initial state and the input samples. In particular, as 

k = 0 

k


described by Bjaerum et al. (2002a), the filter output can be expressed in the following vector 

form: 

y = Tv 

init 

+ Sx , (2-4b) 

where v init is the initial filter state vector of length K (i.e. the filter order), while T and S are 

respectively the transient filter matrix (of size N D ×K) and the steady-state filter matrix (of size 

N D ×N D ) whose entries depend on the filter weights c k and b k shown in (2-4a). For FIR filters, the 

initial filter state vector only affects the filter output for a finite duration; however, for IIR filters, 

this vector has an effect on the filter output all the time since the filter’s impulse response never 

vanishes. As such, the initial state vector should be defined in a way so that the filter’s transient 

effects can be mitigated. Based on this notion, a few studies have considered the use of timevariant 

initialization approaches such as step initialization and projection initialization (Peterson 

et al. 1994, Kadi and Loupas 1995). Recently, Bjaerum et al. (2002a) have shown that projection 

initialization is more effective because it gives a stopband frequency response that looks the most 

similar to the steady-state response. This initialization approach, as originally proposed by 

Chornoboy (1992), works by setting the initial state vector as the complement of the leastsquares 

fitting coefficients between the transient filter matrix and the steady-state filter output. 

In particular, the initial state vector for projection initialization can be expressed as follows: 

v 

H −1 

H 

+ 

= −( T T T Sx = −T 

Sx , (2-5a) 

init 

) 

where the ‘+’ superscript denotes a pseudo-inverse operation (i.e the singular matrix equivalent 

of a square matrix inverse). By combining (2-4b) and (2-5a), the overall filter matrix with 

projection initialization can then be written in the following form: 

+ 

F = [ I − TT ] S , (2-5b) 

where I is an N D ×N D identity matrix. Fig. 2-1 shows an example of the frequency response for 

projection-initialized IIR filters. This figure was computed by substituting (2-5b) into (2-3a), 

and it shows that the actual response of the projection-initialized IIR filter is suboptimal (but still 

close) to the steady-state filter response. Note that, besides the use of filter initialization 

schemes, minimum-phase filters can also be used to reduce transient filtering effects. Bjaerum et 

al. (2002a) have studied the use of minimum-phase FIR filters and have shown that they can 

improve the filtering performance.


(a) 5 th -order Filter 

(b) 8 th -order Filter 

Magnitude [dB] 

0 

-10 

-20 

-30 

-40 

-50 

-60 

Steady-State 

Actual 

0 50 100 150 200 250 

Doppler Frequency (f D 

) [Hz] 

0 

-10 

-20 

-30 

-40 

-50 

Steady-State 

-60 

Actual 

0 50 100 150 200 250 


) [Hz] 

Fig. 2-1. Actual and steady-state filter responses of the projection-initialized IIR filter. The 

responses are shown for: (a) 5 th -order filter and N D =10; (b) 8 th -order filter and N D =20. The 

Doppler sampling frequency (i.e. 1/T PRI ) used in both plots was 500 Hz, and the IIR filters 

were designed using the Chebychev method with a nominal cutoff frequency of 50 Hz. 

Magnitude [dB] 

Regression Filters 

As another way of suppressing Doppler clutter, a regression filter may be used in place of 

FIR and IIR filters. This type of filter works by first modeling Doppler clutter as a series of 

curve shapes and then computing the least-squares fitting residual between the Doppler signal 

and the given clutter model. From a subspace perspective, such regressive fitting is equivalent to 

a least-squares projection of the Doppler signal vector onto the orthogonal complement of the 

clutter model subspace. As described in various algebra textbooks (e.g. Moon and Stirling 2000, 

Sec. 3.4.1), the least-squares projection of a vector x onto the orthogonal complement of a 

subspace matrix C is given by: 

H −1 

H 

+ 

y = [ I − C( 

C C) 

C ] x = [ I − CC ] x . (2-6a) 

Consequently, the filter matrix for a regression filter can be expressed as follows: 

+ 

F = I − CC . (2-6b) 

In terms of the curve shapes used for the clutter model, low-order polynomials are often suitable 

because clutter echoes are low frequency components in nature. Thus, polynomial basis vectors 

are typically used for the columns of the clutter subspace matrix. For a K th -order polynomial 

clutter model ĉ, the subspace matrix C is an N D ×(K+1) matrix of the following form: 

⎡ | | | ⎤ 

K 

cˆ = ⇔ = 

⎢ 

⎥ 

∑ χ 

kp 

k 

C 

⎢ 

p 

0 

p1 

L p 

K 

⎥ 

, (2-7) 

k = 0 

⎢⎣ 

| | | ⎥⎦


where χ k and p k are the respective least-squares fitting coefficient and the length-N D basis vector 

for the k th -order polynomial (usually ranged between –1 and +1). Based on this notion, Hoeks et 

al. (1991) have considered using first- and second-order polynomial clutter models, while Kadi 

and Loupas (1995) have evaluated the use of polynomial models up to the fourth order. Torp 

(1997) has subsequently generalized this filtering approach by using orthogonal polynomials 

such as Legendre or Chebyshev polynomials † for the clutter model. In terms of its performance, 

Bjaerum et al. (2002a) have shown that, for the same filter order, the polynomial regression filter 

have a similar frequency response as the one for the projection-initialized IIR filter at low 

nominal cutoff frequencies. However, the polynomial regression filter has lesser flexibility in 

adjusting the width of the filter stopband since the filter response can only be changed by varying 

the filter order (whereas IIR filters can also vary the nominal cutoff frequency). Note that, 

besides the use of polynomials, other basis sets such as wavelets can also be used for the 

regression filter. For instance, Cloutier et al. (2003) have modeled clutter as a series of Gabor 

wavelets (i.e. orthogonal bases with different frequencies and Gaussian-shaped envelopes) that 

match the most dominant Doppler signal contents. 

2.2.2 Adaptive Filters 

Background Considerations 

When using non-adaptive filters for clutter suppression, it is inherently assumed that the 

clutter spectral contents of all the sample volumes would fall within a certain low-frequency 

band. Hence, the same clutter filter is often applied to the Doppler signal of all sample volumes. 

Nonetheless, clutter suppression would likely be more effective if the filter stopband can be 

selected adaptively according to the local clutter characteristics of individual sample volumes. 

For instance, in performing clutter filtering on Doppler data acquired under substantial tissue 

motion, the filter stopband may need to be dynamically adjusted because the Doppler clutter of 

some sample volumes may not be centered around zero frequency. Based on this principle, Yoo 

et al. (2003) have proposed the use of a filter bank approach whereby the width of the filter 

stopband can be changed depending on the magnitude of the mean clutter frequency. However, 

when a wider filter stopband is used, this clutter suppression approach may concomitantly 

† Legendre and Chebyshev polynomials can be obtained by orthogonalizing the polynomial set {1, n, n 2 , …}. The 

only difference between these two types of polynomials is their normalization weights.


(a) Clutter-Downmixing Filter 

Downmix 

clutter to f D 

=0 


(b) Non-Adaptive Filter 

Clutter 

Spectrum 


Shifted Clutter 

Spectrum 

Shifted Blood 

Spectrum 

Blood 

Spectrum 

0 

f D 

0 

f D 

Fig. 2-2. Illustration of the difference between (a) an adaptive filter with clutter downmixing 

and (b) a non-adaptive clutter filter. For the non-adaptive filter, a wider stopband is needed 

to suppress clutter, but its use causes parts of the blood spectrum to be attenuated. 

suppress a substantial portion of the blood echoes in the Doppler spectrum, and as a result the 

sensitivity of flow detection may be decreased. In order to effectively suppress clutter 

originating from tissue motion, it is beneficial to design filters that can adapt its stopband to the 

clutter spectral characteristics. This rationale is the fundamental basis of adaptive clutter filters. 

Clutter-Downmixing Filters 

In the presence of tissue motion, one intuitive way of adaptively suppressing Doppler 

clutter is to design and use a bandpass filter whose stopband is centered at the mean clutter 

frequency. To implement this approach, however, high computation power would be required 

because of the need to design a different bandpass filter for each sample volume. As an 

equivalent way of realizing the same bandpass filter operation, it is possible to first downmix the 

Doppler signal with the mean clutter frequency before carrying out a highpass filtering operation. 

An illustration of this clutter downmixing strategy, which was originally introduced by Thomas 

and Hall (1994) and Brands et al. (1995), is shown in Fig. 2-2. Mathematically, the n th sample of 

the downmixed Doppler signal x DM (n) can be expressed as: 

x 

− jφ 

( n) 

DM 

n) 

= x( 

n) 

e for φ( 

n) 

= 2 

( πf 

nT 

D(c) 

PRI 

, (2-8a) 

where φ(n) is the instantaneous phase of the downmixing carrier, while f D(c) denotes the mean 

clutter frequency that can be estimated using closed-form methods like the lag-one autocorrelator 

(Kasai et al. 1985). From a vector space perspective, the downmixing operation is equivalent to 

applying a diagonal matrix operator Θ (of size N D ×N D ) to the Doppler signal vector x. As such, 

the downmixed Doppler signal vector x DM can be written as:


DM 

− jφ 

(0) − jφ 

(1) − jφ 

( ND 

−1) 

[ e , e , K, 

e ] 

x = Θx for Θ = diag 

. (2-8b) 

After the downmixing, any type of non-adaptive filter can generally be used to suppress the zerofrequency 

centered Doppler clutter. Consequently, the overall filter matrix of a filter with clutter 

downmixing is simply equal to the multiplication between the original filter matrix F and the 

downmixing operator Θ. As seen in Fig. 2-2, this filtering strategy is particularly useful in 

situations where tissue and blood scatterers move in opposite directions (e.g. when arterial 

distension pushes tissue towards surface while blood scatterers flow away from surface), because 

in this case the blood spectrum can more likely be preserved after the clutter downmixing 

process. Note that, as described by Bjaerum et al. (2002b), the performance of the downmixing 

filter can be further improved if instantaneous clutter frequencies are used for the phase terms in 

the downmixing operator. Nevertheless, to accurately estimate the instantaneous clutter 

frequencies, one would require the availability of multiple Doppler signal vectors that are 

statistically stationary. 

Eigen-Based Filters 

Another way of adaptively suppressing Doppler clutter is to directly analyze the 

composition of the Doppler signal and remove a composition subset that corresponds to clutter. 

This filtering strategy is essentially the same as using a regression filter whose clutter model 

consists of a subset of components seen in the Doppler signal composition. In terms of its 

implementation, the signal analysis can be effectively carried out by decomposing the Doppler 

data into a series of adaptable, orthogonal basis functions (as opposed to fixed ones like the 

Fourier expansion). As discussed in algebra textbooks (e.g. see Moon and Stirling 2000, Sec. 

6.8), such decomposition is often referred to as the Karhunen-Loeve (KL) expansion or principal 

component analysis, and it can be computed through an eigen-decomposition of the Doppler 

signal’s correlation matrix. For a Doppler ensemble size N D , the KL expansion is given by: 

ND 

⎧λk 

, k = l 

x = ∑γ ke 

k 

for E{ γ 

kγ 

l} 

= ⎨ . (2-9a) 

k = 1 

⎩ 0 , k ≠ l 

where γ k is the k th expansion coefficient, while λ k and e k are the eigenvalue and eigenvector for 

the k th orthogonal basis function. Note that λ k and e k are related to the Doppler signal as follows: 

ND 

∑ 

k = 1 

λ R , (2-9b) 

H 

H 

ke ke 

k 

= E{ xx } = 

x


in which R x =E{xx H } refers to the statistical correlation matrix of the Doppler signal. In practice, 

this matrix can be estimated via ensemble averaging as follows: 

M 

1 

R ≈ x x , (2-9c) 

x 

∑ 

H 

m m 

M m= 

1 

where M is the number of Doppler signal vectors (or snapshots) that are statistically stationary 

and x m indicates the Doppler signal vector for the m th realization. 

Since clutter often has a much higher strength than blood echoes and white noise, the 

signal decomposition given in (2-9a) would have high-energy basis functions (or principal 

components) that correspond to clutter. Hence, as pointed out by Bjaerum et al. (2002b), 

Doppler clutter can be suppressed by using the basis vectors of these high-energy components as 

the clutter model for a regression filter. In particular, when K of the N D basis functions 

correspond to clutter (i.e. with a clutter eigen-space dimension equal to K), the resulting clutter 

model and the clutter subspace matrix for an eigen-based regression filter can be expressed as: 

⎡ | | | ⎤ 

K 

cˆ = ⇔ = 

⎢ 

⎥ 

∑ χ 

ke 

k 

C 

⎢ 

e1 

e 

2 

L e 

K 

⎥ 

, (2-10) 

k = 1 

⎢⎣ 

| | | ⎥⎦ 

and the filter matrix would carry the same form shown in (2-6b). Similar forms of this eigenbased 

filter have been reported in studies on swept-scan-based flow imaging (Kruse and Ferrara 

2002) as well as strain-flow imaging (Kargel et al. 2003). Note that, as described by Ledoux et 

al. (1997), the same filtering strategy can also be performed by applying singular value 

decomposition (SVD) to a multi-snapshot data matrix constructed from stacking together a 

number of stationary Doppler signal vectors. As well, as reported by Gallippi and Trahey 

(2002), the Doppler spectral analysis can be carried out using another approach called 

independent component analysis that decomposes a signal into a series of statistically 

independent basis functions (as opposed to orthogonal ones). This latter approach has been 

applied to suppress Doppler clutter in acoustic radiation force imaging (Gallippi et al. 2003). 

2.2.3 Comparison of Clutter Filters 

Table 2-1 summarizes and compares the main features of the four types of clutter filters 

described in this section. From this table, it can be seen that the two non-adaptive filters 

(projection-initialized IIR and polynomial regression) are simpler to implement than adaptive


Type 

IIR + Proj. 

Initialization 

Table 2-1. Comparison of Existing Clutter Filter Designs 

Stopband 

Adjustment 

Through changes 

in filter order and 

nominal cutoff 

Advantage 

Simple to carry out once 

filter coefficients are found 

Limitation 

Does not adapt to Doppler 

signal; efficacy depends on 

choice of filter parameters 

Polynomial 

Regression 

By choice of 

clutter polynomial 

order 

Same as above 

Same as above 

Clutter 

Downmixing 

From changes to 

stopband of nonadaptive 

filter 

Improves highpass filtering 

by first downmixing clutter 

to zero frequency 

Efficacy depends on 

choice of non-adaptive 

filter and accuracy of 

downmixing matrix 

Eigen- 

Regression 

Via selection of 

clutter eigen-space 

dimension 

Adapts filter matrix to 

Doppler signal contents via 

KL expansion 

Needs multiple snapshots 

to find correlation matrix; 

efficacy depends on choice 

of clutter dimension 

filters because their filter matrix remains unchanged for a given set of filter parameters. 

However, the primary shortcoming of these non-adaptive filters is that they are not adapted to the 

Doppler signal contents. Because of this limitation, they inherently need a wider stopband to 

suppress Doppler clutter that has wideband characteristics and ones that are shifted away from 

zero frequency. On another note, it is worth pointing out that amongst the two non-adaptive 

filters, the projection-initialized IIR filter seems to have more flexibility in defining the stopband 

since it can modify the filter response via changes in both the filter order and the nominal cutoff 

frequency. 

In contrast to the two non-adaptive filters, the clutter-downmixing filter can be perceived 

as a partially adaptive filter. Specifically, its filter matrix is a joint product between a fixed 

matrix operator that makes use of non-adaptive filtering principles as well as an adaptive matrix 

operator that is intended to downmix the Doppler clutter to zero frequency. The advantage of 

this filtering approach is that the highpass filtering operation can likely be improved since the 

downmixed Doppler clutter is supposedly centered at zero frequency. Nevertheless, because of 

the hybrid nature of its filter matrix, the clutter-downmixing filter’s efficacy is inherently 

dependent on two factors: 1) the choice of the non-adaptive filter used for the fixed matrix


operator, and 2) the accuracy of the downmixing matrix operator found by estimating the mean 

or instantaneous clutter frequencies of the Doppler data. 

Amongst the four classes of filters considered, the eigen-regression filter seems to have 

the best adaptability to the Doppler signal contents. In particular, the filter matrix of this filter is 

defined adaptively according to the most principal basis vectors in the Doppler signal’s KL 

expansion. In turn, the eigen-regression filter can more likely preserve non-clutter components 

of the Doppler signal unless they share the same principal basis vectors with clutter components. 

However, the formulation for this filter inherently requires the availability of Doppler data from 

multiple sample volumes with similar signal characteristics (i.e. statistically stationarity). As 

such, this filter may be challenging to implement in some vascular imaging studies where it is 

difficult to segment out regions with similar Doppler data statistics. Another limitation of the 

eigen-regression filter is that its efficacy is rather sensitive to the choice of the clutter eigenspace 

dimension owing to the filter’s adaptive nature. Hence, it is necessary to develop an 

effective algorithm for this filter to select the clutter eigen-space dimension. 

2.3 The Hankel-SVD Filter 


Design Motivations 

Despite its adaptability to the Doppler signal contents, the eigen-based filtering strategy 

embodied in (2-9) and (2-10) is often faced with two types of challenges in practice (as noted in 

the previous section). First, consistent estimation of Doppler correlation matrix R x seen in (2-9c) 

generally requires multiple Doppler signal vectors that are statistically stationary. In the reported 

eigen-based filter designs, this estimation is often performed by using the Doppler signal vectors 

along the same beam line (with the inherent assumption that the clutter statistics are stationary 

over the depth of view). Such an approach is generally effective in microvasculature flow 

studies where the imaging depth is only of a few millimeters, but it might not be valid for general 

vascular studies where the imaging depth can be several centimeters. Second, precise 

determination of the clutter eigen-space dimension K as seen in (2-10) is essential to the efficacy 

of the eigen-based filter. In most of the previously reported eigen-based filter designs, this 

parameter was manually selected by the user during analysis according to some prior knowledge


of the clutter spectral characteristics. Such a manual way of specifying the clutter dimension 

may reduce the filtering performance in settings where the clutter spectral characteristics 

generally vary over time. To avoid these practical challenges, a new eigen-based filtering 

strategy will be developed in this section by exploiting the eigen-space properties of the Hankel 

matrix (which has constant entries along its reverse diagonals). 

Fundamental Principles 

The new filtering approach, which we refer to as the Hankel-SVD filter, is derived from 

insights † drawn upon a principal Hankel component analysis framework that is often used in 

time series analysis (van der Veen et al. 1993). Like the KL expansion, this new method 

attempts to decompose the Doppler signal as a sum of adaptable, orthogonal basis functions. 

However, instead of relying on multiple snapshots of Doppler signal to estimate the basis 

functions, the Hankel-SVD approach computes the basis functions through SVD analysis of the 

Hankel data matrix created from the Doppler signal of individual sample volumes. It should be 

noted that, because of the Hankel matrix’s inherent structure, the resulting Doppler signal 

composition would only have P orthogonal bases (with P less than N D ). As such, the signal 

synthesis equation for the Hankel-SVD approach can be expressed as: 

P 

x = ∑γ φ for γ ⊥ γ , (2-11) 

k = 1 

k 

k 

where γ k and ϕ k are respectively the k th expansion coefficient and orthonormal basis vector in the 

Hankel component approximation. This formulation should be distinguished from the SVD 

analysis framework reported by Ledoux et al. (1997), who used multiple Doppler signal vectors 

to construct the data matrix. 

2.3.2 Theoretical Formulation 

Construction of Hankel Data Matrix 

An overview of the new eigen-based filtering strategy is depicted in Fig. 2-3. In terms of 

its details, the Hankel-SVD filter begins with the formation of a Hankel data matrix by dividing 

the Doppler signal into partially overlapping segments and rearranging them into an array. For a 

k 

l 

† This new framework is also motivated by our investigations on the Matrix Pencil estimation framework (to be 

presented in Chapter 3).


Doppler 


x 

Create Hankel 

Matrix 

A = 

0 1 

2 

3 

1 

2 

3 

4 

2 

3 

3 

4 

4 

5 

5 

6 

4 5 6 

7 

Perform SVD 

P 

A = Σσ k 

u k 

v 

H 

k 

1 

Find Principal Basis 

Functions 

γ k ϕ k (n) = 

ΣΣa k (i,j) δ n,i+j–1 

ΣΣ δ n,i+j–1 

Filter Output 

y 

Reconstruct 

Filtered Signal 

P 

y = Σ γ k ϕ k 

K c +1 

Estimate Clutter Eigen-Space 

Dimension 

Is |f D,1 | > f thr(clut) ? 

No 

Yes 

K c = 0 

K c =argmax{|f D,k – f D,1 |


Doppler signal vector x 

0 

1 

2 

3 

4 

5 

6 

7 

Hankel matrix A 

0 1 2 3 

1 2 3 4 

2 

3 

3 

4 

4 

5 

5 

6 

4 5 6 7 

Fig. 2-4. Illustration of how the Hankel matrix A is constructed for the case where the 

Doppler ensemble size is eight (N D =8) and the dimension parameter is four (P=4). 

where A k is the k th rank-one Hankel component in the decomposition, while σ k , u k , and v k are 

correspondingly the singular value, left singular vector (with dimension N D –P+1), and right 

singular vector (with dimension P) of A k . As discussed in algebra textbooks (e.g. Moon and 

Stirling 2000, Sec. 7.1), the P singular values in (2-13) are ordered from largest to smallest by 

definition, and hence A k can be considered as the k th -order principal Hankel component. Note 

that, by using these principal Hankel components, it is possible to reconstruct the orthogonal 

basis functions γ k ϕ k as seen in (2-11). One particular way to perform this reconstruction process, 

as originally described by Poon et al. (1993), is to sum and average the matrix elements along the 

reverse diagonals of A k . Based on this notion, the n th sample of γ k ϕ k (n) can be mathematically 

expressed as: 

N −P 

P−1 

D 

∑∑ 

n= 

i+ 

j 

i= 

0 j= 

0 

= 

N −P 

P−1 

D 

∑∑ 

i= 

0 

δ 

j= 

0 

a ( i, 

j) 

k 

γ 

kϕk 

( n) 

, (2-14) 

δ 

where a k (i, j) is the entry of A k at row i and column j and δ n=i+j is a Kronecker delta function that 

equals to one if n = i+j and equals to zero otherwise. 

Estimation of Clutter Eigen-Space Dimension 

Since the aim of the Hankel-SVD filter is to suppress clutter in the Doppler signal, it is 

necessary to determine whether a principal Hankel component A k is part of the clutter eigenspace. 

In general, there are two types of approaches to carry out this analysis. First, given that 

clutter often has higher energy than blood echoes and white noise, a principal Hankel component 

can be considered as being part of the clutter eigen-space if its singular value magnitude σ k is 

n= 

i+ 

j


1. Test clutter presence with f thr(clut) 

2. Identify clutter dimension with Δf thr 

Component Mean Freq. (f D,k 

) 

f D,1 

0 

+f thr(clut) 

k 

–f thr(clut) 

Component Mean Freq. (f D,k 

) 

f D,1 

0 

Clutter Band 

k 

Δf thr 

Fig. 2-5. Steps used by the Hankel-SVD filter to estimate the clutter eigen-space dimension. 

First (left figure), clutter presence is examined by testing whether the most dominant 

component has a mean frequency (marked as black ⊗) less than a threshold f thr(clut) . After that 

(right figure), clutter dimension is determined by finding largest component order with mean 

frequency estimate inside the clutter band Δf thr (in this case, the clutter dimension is three). 

larger than a given value. Examples of clutter eigen-space estimation algorithms using this 

analysis approach can be found in a few previously reported eigen-filter designs (Kruse and 

Ferrara 2002, Lovstakken et al. 2006). Alternatively, since clutter generally consists of lowfrequency 

contents, it is possible to identify a clutter eigen-space component based on the 

frequency contents of each principal basis vector ϕ k . This latter approach is used in the Hankel- 

SVD filter to determine the clutter eigen-space dimension, since it can be implemented in a way 

that gives filtering characteristics similar to the stopband of a conventional bandpass filter. 

The clutter eigen-space analysis algorithm used by the Hankel-SVD filter is illustrated in 

Fig. 2-5. This algorithm is fundamentally based on two assumptions: 1) Doppler clutter is 

contained in the more dominant Hankel components (i.e. ones with larger singular values); 2) the 

blood flow component of the Doppler signal is contained in the Hankel components with high 

Doppler frequencies. In terms of its details, the analysis begins with an assessment of whether 

clutter is present in the Doppler signal by finding whether the most dominant Hankel component 

has a mean frequency f D,1 that is less than a spectral threshold f thr(clut) . If clutter is considered to 

be present, then the mean frequency of the other principal basis functions are estimated. Note 

that, by using the well-known single-lag autocorrelator (Kasai et al. 1985), the mean Doppler 

frequency f D,k corresponding to each principal basis function ϕ k can be found to be equal to:


f 

D, k 

ND 

⎡ 

⎤ 

= ⎢ ∑ − 1 

1 

* 

arg ϕ 

k 

( n) 

ϕ 

k 

( n −1) 

⎥ . (2-15) 

2πTPRI 

⎣ n= 

1 

⎦ 

From these estimates, the clutter eigen-space dimension is found by searching for the largest 

component order whose mean frequency falls within a clutter band Δf thr that is centered at f D,1 (as 

shown in Fig. 2-5). Consequently, the estimated clutter dimension K c can be expressed as: 

K 

c 

⎪ 

⎧ 

= ⎨arg max 

⎪⎩ k 

{ f − f < Δf 

/ 2} 

D, k 

0 

D,1 

thr 

,for 

, for 

f 

f 

D,1 

D,1 

> 

< 

f 

f 

thr(clut) 

thr(clut) 

. (2-16) 

Once the clutter dimension has been estimated, the filtered Doppler signal can be found as 

follows by summing the principal basis functions beyond the estimated clutter dimension: 

P 

∑ 

y = γ φ . (2-17) 

k 

k = KC 

+ 1 

It is worth noting that the average power of the filtered Doppler signal can be estimated from the 

singular values of the principal Hankel components beyond the clutter dimension. In particular, 

since the squared sum of singular values is equal to the Frobenius matrix norm (Moon and 

Stirling 2000, Sec. 7.1), the post-filter Doppler power can be estimated to equal to: 

P 

2 

k 

∑ 

k = K + 1 P( 

N 

D 

− P + 

k 

σ 

ρ y 

= 

. (2-18) 

1 

C 

) 

where the normalization factor in the denominator stems from the fact that each Hankel 

component A k has P(N D –P+1) entries. 

Computational Considerations 

As can be expected from its formulation, the computation load of the Hankel-SVD filter 

is rather high comparing to non-adaptive filters like the IIR filter. In particular, this new filtering 

strategy requires the computation of an SVD on the Hankel matrix formed from each Doppler 

signal, and after the SVD is computed, a series of vector operations is needed to reconstruct the 

principal Hankel components. To estimate the overall computation burden of the new filter, an 

analysis can be performed to determine the number of floating point operations † (flops) required 

during various stages of the filter. Such an analysis is provided in Table 2-2. As can be seen, 

applying the Hankel-SVD filter to each Doppler signal would need at least on the order of P 3 

† In this study, both multiplication and addition are considered as floating point operations.


Table 2-2. Summary on the Flops Needed in the Hankel-SVD Filter 

Step Flops Rationale 

1. Form Hankel matrix 

of size (N D –P+1)×P 

2. Compute SVD of the 

Hankel matrix 

3. Reconstruct the 

principal basis 

functions 

0 

O[P 3 ] 

O[P 3 ] 

• No computations required to form the Hankel data 

matrix 

• According to Golub and Van Loan (1996, Sec. 8.3), 

SVD of an (N D –P+1)×P matrix needs 

O[(N D –P+1)P 2 ] flops 

• For P=N D /2, O[(N D –P+1)P 2 ] ≈ O[P 3 ] 

• O[(N D –P+1)P] flops needed to reconstruct ϕ k and 

A k from σ k u k v k 

H 

• P principal components to reconstruct, so 

O[(N D –P+1)P 2 ] flops needed in total 

• For P=N D /2, O[(N D –P+1)P 2 ] ≈ O[P 3 ] 

4. Estimate component 

mean frequencies 

5. Estimate clutter 

eigen-space dimension 

6. Reconstruct filter 

output 

O[P 2 ] 

0 

O[N D ] 

• O[N D ] flops needed to compute the lag-one 

autocorrelation estimate 

• With P components to estimate, O[N D P] flops 

needed in total 

• For P=N D /2, O[N D P] ≈ O[P 2 ] 

• No computations required to estimate clutter 

dimension 

• O[N D ] flops required to sum all principal 

components beyond the clutter dimension 

flops (when P is set to N D /2). This computational demand is one exponential order higher than 

that required for typical non-adaptive filters (by inspection of (2-2), such a filter only needs on 

the order of N 2 D flops). Nevertheless, since N D is often kept below 20 samples in color flow 

imaging, the additional burden needed by the Hankel-SVD filter is less substantial. As well, 

with the persymmetric structure of the Hankel data matrix, the SVD can be computed with lesser 

flops via the use of more efficient algorithms (Badeau et al. 2004). 

2.4 Simulation Method 

2.4.1 Signal Synthesis Model 

General Principles 

To facilitate analysis of the Hankel-SVD filter’s performance, a signal synthesis model 

was first developed to generate Doppler signals with various Doppler spectral characteristics. In


this synthesis model, the clutter and blood components of the Doppler signal were separately 

generated using two different approaches. For the clutter components, a phase modulation 

model was used to synthesize ensembles with non-stationary clutter spectral characteristics. As 

first introduced by Heimdal and Torp (1997), the physical notion behind this synthesis approach 

is that, due to the contractile nature of muscles, body tissues tend to move in a cyclic pattern and 

in turn give rise to Doppler clutter with phase modulation features. In contrast, for the blood 

flow components of the Doppler signal, they were generated by feeding complex Gaussian noise 

through a linear filter whose impulse response corresponds to the Doppler signal for a single 

moving scatterer. As originally described by Kristoffersen and Angelsen (1988), the physical 

rationale behind this filter-based synthesis approach is that, due to the random nature of blood 

scatterer distribution, the scattering strength at each Doppler sampling instant is essentially a 

complex Gaussian random variable. As such, the overall Doppler signal from various blood 

scatterers can be modeled as a superposition of various randomly time-lagged and amplitudescaled 

replica of the single-scatterer echo template (i.e. similar to a convolution or linear filtering 

operation). Note that, as reviewed by Mo and Cobbold (1992), this blood signal synthesis model 

has the advantage of allowing the user to directly control the Doppler spectral characteristics of 

blood scatterers, and thus it is useful for testing various Doppler signal processing strategies. 

Synthesis Procedure 

Fig. 2-6 shows a system-level schematic of the procedure used in the synthesis model. In 

general, the entire synthesis procedure can be divided into three main parts. In the first part, the 

clutter component of Doppler signal was generated. This process begins by constructing a 

clutter vibration pattern to represent the instantaneous phase of the Doppler clutter. For our 

study, the vibration pattern was modeled as a single-tone waveform whose frequency and peak 

amplitude were set based on the specified clutter parameters (to be described in the next 

subsection). The clutter signal template was then obtained by setting the vibration pattern as the 

phase argument of a complex exponential operator, and correspondingly the n th sample of this 

template can be expressed as follows: 

h ( n) 

c 

jφ 

sin(2πf 

nT 

) 

c,max vib PRI 

= e 

, (2-19) 

where φ c,max is the peak phase deviation in the phase-modulated signal and f vib is the vibration 

frequency of the clutter vibration pattern. To model the random magnitude and phase of the


Clutter 

Vibration 

Pattern 

Complex 

Exponential 

Operator 

Clutter 

Strength 

Σ 

Clutter 

Doppler 


Gaussian 

Noise 

Blood 

Scatterer 

Distribution 

Single-Scatterer 

Echo Template 

Noise 

Strength 

Σ 

Clutter + Blood 

Doppler 


Blood 

Strength 

Fig. 2-6. System-level schematic of the Doppler signal synthesis approach. In this study, the 

single-scatterer blood echo template carried the form of a Gaussian-shaped complex sinusoid, 

while the clutter vibration pattern was a single-tone waveform. 

tissue scatterer distribution, the clutter signal template was subsequently multiplied with a 

complex Gaussian random number. 

In the second part of the synthesis process, the blood component of the Doppler signal 

was generated. This process first involves the creation of an echo template to model the Doppler 

signal originating from a single moving scatterer. The echo template used in this study was 

defined as a Gaussian-enveloped complex sinusoid whose modulating frequency corresponds to 

the mean Doppler frequency calculated from the specified blood flow parameters (to be 

described in the next subsection). For a given modulating frequency f D(b) and a temporal width 

parameter T dur , the n th sample of the blood echo template can be expressed as follows: 

h 

b 

( n) 

2 

−( 

nT 

2 

PRI / Tdur 

) j πfD(b) 

nTPRI 

= e e . (2-20) 

To account for the finite transit time of blood scatterers (as recently reviewed by Yu et al. 

(2006)), the temporal width of the blood echo template was defined as a function of blood flow 

velocity. From the single-scatterer echo template, the blood signal for multiple scatterers was 

then synthesized by convolving complex Gaussian random samples with the template. Note that, 

during the simulations, the transient samples in the convolution were discarded to maintain


statistical uniformity in the synthesized data. As well, multiple realizations of the blood signal 

were generated at once by convolving a large number of random samples with the singlescatterer 

echo template. 

In the last part of the synthesis process, the blood and clutter components generated from 

the previous parts were combined to form the Doppler signal. To carry out this process, the 

amplitudes of the blood and clutter components were first scaled according to the specified 

clutter and blood scattering strengths. After that, the scaled components were then summed 

together along with white noise samples that were generated separately. For analysis purpose, 

the clutter-only Doppler signal (i.e. the signal for a sample volume located in the tissue region) 

was also obtained by only summing the scaled clutter components and white noise samples. 

Note that the n th sample of the synthesized Doppler signal and its clutter-only counterpart can be 

expressed as follows: 

[ g h ( n) 

] + κ [ g ( n) 

∗ h ( n) 

] κ w( 

n) 

x( n) 

= κ 

c c c 

b b b 

+ 

w 

, (2-21a) 

[ g h ( n) 

] κ w( 

n) 

xc 

( n) 

= κ 

c c c 

+ 

w 

, (2-21b) 

where g c and g b (n) are respectively the complex Gaussian random samples used to model the 

tissue and blood scatterer distributions; also, κ c , κ b , and κ w respectively denote the amplitude 

scaling coefficients for clutter, blood echoes, and white noise. Prior to using the signals 

synthesized from (2-21a) and (2-21b) for filter performance analysis, they were quantized to a 

certain number of bits as set forth by the specified dynamic range, and they were divided into 

non-overlapping segments to yield multiple signal realizations. 

Model Parameters 

In the simulation model, four main classes of parameters can be specified during the 

synthesis process so that Doppler signals with various spectral characteristics can be generated. 

The first class of parameters is associated with the ultrasound machine and the synthesis process. 

These parameters include acoustic speed (c o ), pulse carrier frequency (f o ), pulse repetition 

interval (T PRI ), F-number (i.e. ratio of the aperture width to the focal length, F/W), Doppler 

ensemble size (N D ), dynamic range of data sampling, and number of realizations to be 

synthesized. As already discussed in Section 1.4.2, most of these system parameters have a 

direct influence on the velocity aliasing limit and the velocity resolution of the synthesized 

Doppler signals.


The second class of parameters is related to the generation of the clutter signal. In 

particular, the two clutter parameters of interest are clutter vibration frequency (f vib ) and 

maximum tissue velocity (v c,max ). As its name implies, the clutter vibration frequency controls 

the rate of change of the clutter signal’s instantaneous phase. This parameter, along with the 

maximum tissue velocity, also influences the peak phase deviation in the phase-modulated 

clutter signal. Following a similar derivation by Heimdal and Torp (1997), it can be shown that 

the two clutter parameters influence the peak clutter phase deviation (φ c,max ) according to the 

following relationship: 

2 f 

ovc,max 

φ 

c,max 

= . (2-22) 

c f 

Note that, as discussed in various communications textbooks (e.g. see Haykin 2001, Sec. 2.7), 

the phase-modulated signal can have time-varying spectral components anywhere over spectral 

range ±2f vib (1+φ c,max ). 

The third class of parameters is concerned with the synthesis of the blood signal. 

Specifically, the two blood parameters of interests are mean flow velocity (v b ) and beam-flow 

angle (θ). According to the Doppler equation given in (1-1), these two parameters directly 

govern the Doppler modulation frequency f D(b) in the single-scatterer blood echo template. Also, 

for large beam-flow angles, it can be shown that the two parameters affect the bandwidth of the 

blood echo template as follows due to transit time broadening (Yu et al. 2006): 

2Wv 

o 

f 

vib 

sinθ 

b o 

β 

b 

= . (2-23a) 

Fco 

Assuming that this value represents the –20 dB spectral broadening bandwidth, the temporal 

width parameter given in (2-20) can then be shown to be equal to the following based on the 

Fourier transform of Gaussian functions and some trivial derivations: 

2 ln10 

Fco 

T 

width 

≈ = 0.483 

. (2-23b) 

πβ Wv f sinθ 

b 

The last class of parameters is related to the relative strengths of the clutter, blood echoes, 

and white noise. In the simulation model, these relative strengths are characterized according to 

the clutter-to-blood signal ratio (CBR) and the blood-signal-to-noise ratio (BSNR), which in turn 

are defined as follows: 

b 

o


Table 2-3. Simulation Parameters Used in the Clutter Filter Study 

Parameter Arterial flow Low-velocity flow 

Fixed Machine and Synthesis Parameters 

Acoustic speed, c o 

1540 m/s 

F-number, F/W 4 

Doppler ensemble size, N D 10, 20 

Dynamic range 

14 bits 

Number of ensembles per dataset 10000 (repeated for every increment in v b ) 

Signal Strength Parameters 

Clutter-to-blood signal ratio 

30 dB 

Blood-signal-to-noise ratio 

10 dB 

Average noise strength, κ n 

10 dB 

Blood Parameters 

Beam-flow angle, θ 60° 

Flow velocity, v b 

Varying from 0 to aliasing velocity 

Manipulated Machine Parameters 

Pulse carrier frequency, f o 5 MHz 2 MHz 

Pulse repetition interval, T PRI 0.4 ms 2.0 ms 

Clutter Parameters 

Maximum tissue velocity, v c,max 2 mm/s 20 mm/s 

Vibration frequency, f vib 5 Hz 5 Hz 

Other Responding Parameters 

Aliasing velocity, v alias 38.5 cm/s 19.2 cm/s 

Clutter phase modulation index, φ c,max 2.6 radians 10.4 radians 

Peak clutter frequency 36 Hz 114 Hz 

( κ κ ) 

CBR = 20log 10 

/ , (2-24a) 

c 

b 

( κ κ ) 

BSNR = 20log 10 

/ , (2-24b) 

where κ c , κ b and κ n are the amplitude coefficients as seen in (2-21a) (i.e. the signal equation for 

the synthesized Doppler signal). Note that, in this study, the white noise scaling coefficient κ n 

was set such that the synthesized signals were contained within the system’s dynamic range. 

2.4.2 Method of Analysis 

General Overview 

To assess the efficacy of the Hankel-SVD filter, various sets of Doppler signals and their 

clutter-only components (each with 10000 realizations) were generated using the above 

described synthesis model and the parameters listed in Table 2-3. The synthesized data were 

intended to model the received Doppler signal in a normal arterial flow scenario with minor 

tissue motion as well as a low-velocity flow scenario with significant tissue motion and longer 

b 

w


ensemble periods. As an initial assessment, particular signal realizations were selected from the 

synthesized data to analyze the characteristics of the principal Hankel components. To more 

quantitatively assess the filter’s performance, the Hankel-SVD filter was applied to all the 

synthesized Doppler signals and their resulting post-filter power was computed. As a benchmark 

for the post-filter power estimates, the same procedure was repeated with the clutter-only signals. 

Note that during the analysis, the dimension parameter P of the Hankel-SVD filter was set as 

N D /2, while the two thresholds f thr(clut) and Δf thr were respectively set to 100 Hz and 50 Hz. 

Performance Measure 

In this study, the post-filter clutter-to-blood signal ratio (CBR) was analyzed at various 

blood velocities to quantitatively examine the efficacy of the Hankel-SVD filter. This quantity 

was computed by finding the square-rooted ratio between the average post-filter power of 

clutter-only signals and that of Doppler signals corresponding to a particular blood velocity. If a 

filter can effectively suppress clutter, the post-filter CBR should be less than one in linear scale 

(or be negative in dB scale) because blood echoes should dominate the filtered Doppler signal. 

To facilitate comparative assessment, the above filter analysis was also carried out with 

an IIR-based (with projection-initialization) clutter-downmixing filter. For the arterial flow 

scenario, the IIR filter used was either a third-order (for N D =10) or fourth-order (for N D =20) 

Chebychev filter with 100 Hz cutoff; as for the low-velocity flow scenario, filter orders of five 

(for N D =10) and eight (for N D =20) were used. Note that these filter orders were chosen because 

their stopband width is similar to the cutoff bandwidth used by the Hankel-SVD filter. It is also 

worth pointing out that the eigen-regression filter was not examined in these simulations because 

our signal synthesis model does not generate statistically-stationary Doppler ensembles for 

multiple sample volumes (as needed by the eigen-filter to estimate the correlation matrix). 

Analysis of this filter will be considered in the experimental studies presented in Chapter 4. 

2.5 Simulation Results 

2.5.1 Characteristics of Principal Hankel Components 

Arterial Flow Scenario 

To study the characteristics of principal Hankel components, one particular realization of 

Doppler signal (with an ensemble size of 20) was selected from the dataset synthesized with


(b) Singular Value Spectrum 

Relative Power [dB] 

(a) Theoretical Spectrum 

80 

70 

60 

50 

40 

30 

20 

10 

0 

-100 0 100 200 300 400 500 


) [Hz] 

Dopp. Freq. (f D 

) [kHz] 

Singular Value (σ k 

) [dB] 

60 

50 

40 

30 

20 

10 

0 

1.25 

0.75 1 

0.5 

0.25 

-0.25 0 

-0.5 

-0.75 

-1.25 

-1 

1 2 3 4 5 6 7 8 9 10 

Component Order (k) 

(c) Mean Frequency Estimates 

1 2 3 4 5 6 7 8 9 10 


Fig. 2-7. Characteristics of principal Hankel components for a signal realization in the 

arterial flow scenario with 10 cm/s blood velocity: (a) the theoretical Doppler spectrum; (b) 

singular value distribution; (c) mean Doppler frequency estimates (marked as ⊗). 

arterial flow parameters. The blood velocity used to synthesize this signal realization is 10 cm/s 

(i.e. a mean Doppler frequency of 325 Hz). Also, according to the responding parameters listed 

in Table 2-3 (bottom of middle column), this signal realization has a peak clutter phase deviation 

of 2.6 radians, and it can have time-varying clutter components over the spectral range ±36 Hz. 

The theoretical Doppler spectrum for this simulation flow scenario is shown in Fig. 2-7a. As can 

be seen, the clutter phase modulation gives rise to low-frequency impulses that are closely 

spaced in the Doppler spectrum. 

Figs. 2-7b and 2-7c respectively show the singular value and mean frequency estimate of 

the principal Hankel components that correspond to the selected signal realization. From these 

plots, it can be seen that the first principal component corresponds to the clutter eigen-space 

based on its distinctly higher singular value magnitude and low mean frequency estimate. Also, 

the second principal component appears to belong to the flow eigen-space because its mean 

Doppler frequency estimate is close to the blood’s actual mean frequency. The rest of the


(b) Singular Value Spectrum 

Relative Power [dB] 

80 

70 

60 

50 

40 

30 

20 

10 

0 

(a) Theoretical Spectrum 

-200 -100 0 100 200 


) [Hz] 

Dopp. Freq. (f D 

) [Hz] 

Singular Value (σ k 

) [dB] 

70 

60 

50 

40 

30 

20 

10 

0 

250 

200 

150 

100 

50 

-50 0 

-100 

-150 

-200 

-250 

1 2 3 4 5 6 7 8 9 10 


(c) Mean Frequency Estimates 

1 2 3 4 5 6 7 8 9 10 


Fig. 2-8. Characteristics of principal Hankel components for a signal realization in the lowvelocity 

flow scenario with 10 cm/s blood velocity. Descriptions are the same as Fig. 2-7. 

principal Hankel components appear to correspond to the noise floor since their singular values 

are relatively small and their mean frequency estimates do not correspond well with the 

predefined spectral characteristics. 

Low-Velocity Flow Scenario 

As a comparison with the arterial flow example, a different realization of Doppler signal 

was selected from the low-velocity Doppler dataset. For this realization (also with an ensemble 

size of 20), the blood velocity was 10 cm/s, which corresponds to a mean Doppler frequency of 

130 Hz. As well, the signal’s peak clutter phase deviation is 10.4 radians, and it has timevarying 

clutter components over the spectral range ±114 Hz (see right column of Table 2-3). An 

illustration of the Doppler clutter’s wideband spectral characteristics is provided in Fig. 2-8a, 

where it can be seen that the low-frequency impulses due to clutter phase modulation are more 

significant than the ones in the previous scenario. 

Figs. 2-8b and 2-8c show the corresponding singular values and mean frequency 

estimates for the selected signal’s principal Hankel components. It can be seen that the first two


principal components have distinctly higher singular values and their frequency estimates are 

within the clutter spectral range; as such, they appear to correspond to the clutter eigen-space. 

On the other hand, the third to fifth principal components appear to belong to the flow subspace 

because its mean frequency estimate is near the predefined flow Doppler frequency. As for the 

sixth to tenth principal components, they appear to be describing the noise floor in view of their 

small singular value magnitudes and their spurious mean frequency estimates. Based on the 

results seen in Figs. 2-7 and 2-8, it can generally be concluded that the clutter and blood 

components are likely to be contained within the first few principal Hankel components. 

Nevertheless, their actual eigen-space dimensions may vary depending on their respective 

Doppler bandwidths and the random variations inherent in each signal realization. 

2.5.2 Post-Filter Clutter-to-Blood Signal Ratios 

Arterial Flow Scenario 

As a quantitative assessment of the new filter’s performance in the arterial flow scenario, 

the top part of Figs. 2-9 shows the post-filter CBR as a function of the actual blood velocity for 

Doppler ensemble sizes of 10 and 20 samples. The shown results are the 25 th , 50 th , and 75 th - 

percentile post-filter CBR estimates obtained from the 10000 signal realizations synthesized at 

each blood velocity. The primary observation to be noted from these plots is that the 50 th - 

percentile (i.e. median) post-filter CBR curves approached the reciprocal of the expected BSNR 

of –10 dB in the higher flow velocity range. This result is expected because, if clutter is 

adequately suppressed without distorting the blood signals, the filtered clutter-only signal and the 

filtered Doppler signal should respectively be dominated by white noise and blood echoes. As a 

comparison for these findings, the middle part of Fig. 2-9 shows a similar set of post-filter CBR 

curves obtained using a clutter-downmixing filter with similar stopband size; also, the bottom 

part of Fig. 2-9 gives a comparison of the median post-filter CBR curves for the two filters. It 

can be seen that for all the shown post-filter CBR curves (i.e. the 25 th , 50 th , and 75 th -percentile 

curves), the Hankel-SVD filter has a narrower transition region than the clutter-downmixing 

filter in the post-filter CBR curves for both Doppler ensemble sizes used in this study. Since a 

similar cutoff bandwidth was used for the two filters, this result suggests that the Hankel-SVD 

filter has a better clutter suppression performance in the case where the Doppler clutter is 

relatively narrowband with respect to the Doppler sampling rate.


(a) N D = 10 (b) N D = 20 

Hankel-SVD 

Percentile CBR [dB] 

0 

-5 

-10 

-15 

25th 

50th 

75th 


0 

-5 

-10 

-15 

25th 

50th 

75th 

0 4 8 12 16 20 24 28 32 36 

Actual Blood Velocity [cm/s] 

0 4 8 12 16 20 24 28 32 36 


Clutter-Downmixing 


0 

-5 

-10 

-15 

25th 

50th 

75th 

0 4 8 12 16 20 24 28 32 36 



0 

-5 

-10 

-15 

25th 

50th 

75th 

0 4 8 12 16 20 24 28 32 36 


Filter Comparison 

50 th Pecentile CBR [dB] 

0 

-5 

-10 

-15 

Hankel-SVD 

Downmix 

0 4 8 12 16 20 24 28 32 36 



0 

-5 

-10 

-15 

Hankel-SVD 

Downmix 

0 4 8 12 16 20 24 28 32 36 


Fig. 2-9. Post-filter CBR as a function of blood velocity in the arterial flow scenario for 

Doppler ensemble sizes of (a) 10 and (b) 20 samples. Shown in the top and middle plots 

respectively are the 25 th , 50 th , and 75 th -percentile post-filter CBR estimates for the Hankel- 

SVD filter and the clutter-downmixing filter. Also, a comparison of the 50 th -percentile postfilter 

CBR for the two filters is given in the bottom plots. The filter parameters used to obtain 

these results are discussed in Section 2.4.2. 

Low-Velocity Flow Scenario 

In contrast to the results for the arterial flow scenario, Fig. 2-10 shows the post-filter 

CBR curves in the low-velocity flow scenario as a function of the true blood velocity. As can be 

expected, the transition regions in all the post-filter CBR curves of both filters are wider in this 

case because the Doppler clutter is significantly more wideband in nature (as notable from Fig.


(a) N D = 10 (b) N D = 20 

Hankel-SVD 


0 

-5 

-10 

-15 

25th 

50th 

75th 


0 

-5 

-10 

-15 

25th 

50th 

75th 

0 2 4 6 8 10 12 14 16 18 


0 2 4 6 8 10 12 14 16 18 


Clutter-Downmixing 


0 

-5 

-10 

-15 

25th 

50th 

75th 

0 2 4 6 8 10 12 14 16 18 



0 

-5 

-10 

-15 

25th 

50th 

75th 

0 2 4 6 8 10 12 14 16 18 


Filter Comparison 


0 

-5 

-10 

-15 

Hankel-SVD 

Downmix 

0 2 4 6 8 10 12 14 16 18 



0 

-5 

-10 

-15 

Hankel-SVD 

Downmix 

0 2 4 6 8 10 12 14 16 18 


Fig. 2-10. Post-filter CBR as a function of actual blood velocity in the low-velocity flow 

scenario. Descriptions are the same as Fig. 2-9. 

2-8a). Nevertheless, within the transition region, the Hankel-SVD filter appears to be able to 

attain the steady-state CBR value at lower velocities (near 8-10 cm/s) as compared to the clutterdownmixing 

filter (which reaches steady-state CBR near 12-14 cm/s). This result is consistent 

with the findings in the arterial flow scenario. Hence, it seems that the Hankel-SVD filter is 

generally more capable of suppressing slow-time clutter without concomitantly attenuating the 

blood signal components. Another observation to be noted from the plots in Fig. 2-10 is that for 

the case with Doppler ensemble size of 20 samples, the post-filter CBR estimates obtained from 

the Hankel-SVD filter appear to have reached a lower steady-state value than the ones found


using the clutter-downmixing filter. This result suggests that, at larger ensemble sizes, the 

Hankel-SVD filter may be able to provide a more thorough suppression of the Doppler clutter 

than that achievable using a clutter-downmixing filter with the same stopband size. Perhaps a 

physical way of perceiving such notion is to recognize that when a larger ensemble size is used, 

the Hankel-SVD filter can generally span wideband Doppler clutter into more orthogonal 

components because of the increased data resolution, and thus it may be able to remove clutter 

more adequately. 


Existing eigen-based clutter filter designs generally require multiple snapshots of Doppler 

data that are statistically stationary in order to carry out the eigen-decomposition analysis. To 

avoid such requirement, this chapter has presented a new eigen-based filtering strategy – the 

Hankel-SVD filter – that is intended to work with the Doppler signal of individual sample 

volumes. This new filtering approach can generally be beneficial in vascular imaging studies 

where it is difficult to segment out image regions with Doppler sample volumes that have similar 

signal characteristics (such as the case where spatially-varying tissue motion is present). 

Nevertheless, it may not be as effective as existing eigen-based filters at very small Doppler 

ensemble sizes (e.g. when N D = 4) because the Hankel-SVD analysis only decomposes the 

Doppler signal into, at most, N D /2 orthogonal basis functions. 

A possible limitation of the Hankel-SVD filter is that, during the clutter eigen-space 

analysis, Doppler clutter is presumed to be contained in the more dominant Hankel components. 

This assumption is generally valid when clutter is significantly higher in strength as compared to 

the blood echoes (as is the case in general vascular imaging). However, the assumption would 

be invalid if the Doppler clutter and the blood echoes are similar in strength (e.g. in highfrequency 

imaging studies where the blood scattering strength is higher). In this case, a more 

advanced algorithm is needed to precisely determine the clutter eigen-space dimension. One 

possible way of developing such an algorithm is to make use of fuzzy logic and pattern 

recognition principles, like the one used by Hu and Gosine (1997) to perform time-series 

analysis. Alternatively, it may be possible to use a multi-modal spectral estimator to first extract 

a number of principal frequency estimates in parallel and then determine the clutter eigen-space 

dimension based on the spectral spread of the estimates.

CHAPTER 3 

Velocity Estimation in Color Flow Imaging: 

A Matrix Pencil Approach 


It has been mentioned in the introductory chapter that there are two signal processing 

challenges when computing velocity estimates from color flow imaging data. The first is related 

to the carrier frequency variations that may be present in the received echoes due to physical 

mechanisms such as frequency-dependent attenuation and scattering. Given that the carrier 

frequency is one of the factors in the Doppler equation, these variations may lead to biases in the 

Doppler frequency estimates. The second challenge is associated with the non-idealities in the 

transfer response of the clutter suppression filter. As flow estimates are often derived from the 

filtered Doppler signal, these filter non-idealities may also give rise to Doppler frequency biases 

during flow estimation. Between the two signal processing challenges, the one due to carrier 

distortions is less significant in practice because the transmit pulses used in color flow imaging 

are relatively narrowband in nature (see Section 1.4.2) and are inherently less prone to carrier 

frequency shifts. On the other hand, biases due to clutter filter non-idealities can be significant 

depending on the filter characteristics and the Doppler spectral contents of blood echoes. 

To address the biasing problems originating from the clutter filtering operation, this 

chapter presents a new parametric velocity estimation framework called the Matrix Pencil. This 

new estimator works by exploiting the properties of an algebraic form known as matrix pencil 

and in turn treating flow estimation as a generalized eigenvalue (GE) problem. To formulate 

discussion on the Matrix Pencil, the rest of this chapter is organized as follows: 

• Section 3.2 reviews the principles of existing flow estimation strategies and discusses their 

advantages and limitations; 

• Section 3.3 presents the theoretical formulation of the Matrix Pencil estimation framework 

and describes its applications in color flow data processing; 

- 54 -

Chapter 3. Velocity Estimation in Color Flow Imaging: A Matrix Pencil Approach 55 

• Sections 3.4 and 3.5 discuss the simulation approach used to analyze the new estimator’s 

performance and show the corresponding results obtained from this study; 

• Section 3.6 summarizes the highlights of the new flow estimation framework and its 

applicability to color flow imaging. 

3.2 Existing Flow Estimation Strategies 

3.2.1 Non-Parametric Estimators 

Fundamental Considerations 

In color flow imaging, a two-stage process is often used to compute velocity estimates. 

First, a clutter filter is applied to the Doppler signal of each sample volume to remove lowfrequency 

echoes originating from tissues and vessel walls. The filtered signal is then passed 

into a non-parametric flow estimator to compute the mean flow velocity. For this two-stage 

estimation approach, it is presumed that the clutter filter can adequately suppress clutter so that 

the filtered Doppler data is only consisted of blood echoes and filtered white noise. Based on 

such notion, the filtered Doppler data vector y (of length N D ) can be described by the following 

signal model: 

T 

[ ( 0), y(1), 

, y( 

D 

−1) 

] = b w 

f 

y = y K N + , (3-1) 

where y(n) is the n th filtered sample, while b and w f respectively denote the signal vectors for 

blood echoes and filtered white noise. This signal model often serves as the starting point in the 

derivation of various non-parametric mean velocity estimators. 

Frequency-Based Methods 

One approach for finding the mean flow velocity is to first estimate the mean frequency 

of the filtered Doppler signal and then use the Doppler equation to convert the frequency 

estimate into a velocity value. In this estimation approach, computation of the mean Doppler 

frequency can theoretically be performed by dividing between the first-order and zeroth-order 

Doppler spectral moments. Alternatively, by making use of the Wiener-Khinchin relation † , the 

same computations can be carried out in a more efficient manner through an analysis of the 

† The Wiener-Khinchin relation states that the autocorrelation function of a signal is equivalent to the inverse 

Fourier transform of the signal’s power spectral density.


Doppler autocorrelation function (i.e. without directly analyzing the Doppler spectrum). As first 

described by Kasai et al. (1985) in the context of color flow imaging, the autocorrelation-based 

approach would lead to the following expression for the mean Doppler frequency: 

f 

D(est) 

1 

= 

2πT 

PRI 

arg 

ND 

{ } ∑ − 

R (1) for R (1) = 

y 

y 

1 

n= 

1 

y( 

n) 

y 

∗ 

( n −1) 

, (3-2) 

where R y (1) is the Doppler autocorrelation function evaluated at a lag of one sample. Based on 

this equation, it can be seen that the lag-one autocorrelation phase is the primary factor in the 

mean Doppler frequency estimate. The phase quantity, however, can be affected by signal 

aliasing due to the sampled nature of the Doppler data (see Section 1.4.2), and such problem is 

often considered as a theoretical limitation of the autocorrelation estimator. Nonetheless, the 

aliasing limit is actually appreciated in practice because the unaliased velocity range is often 

used to define the mapping scale when designing the color map. Also, as pointed out by Tamura 

et al. (1991), aliasing is sometimes introduced intentionally in color flow images to improve the 

visualization of laminar flow patterns. 

The estimation variance of the lag-one autocorrelator shown in (3-2) can be reduced by 

expanding the Doppler autocorrelation into a two-dimensional (2D) function based on multiple 

Doppler signal snapshots within a given depth range (Torp et al. 1994). However, its use in 

color flow data processing is often not vital because the estimation variance can also be reduced 

via spatial averaging of the velocity estimates from adjacent sample volumes within the imaging 

view. As pointed out by Loupas et al. (1995a, 1995b), another possible advantage of the 2D 

autocorrelation approach is that it can be used to jointly estimate both the mean Doppler 

frequency and the instantaneous carrier frequency. Such feature is useful for correcting Doppler 

frequency biases originating from carrier frequency variations that arise when wideband pulses 

are used to acquire color flow data (even though wideband data acquisition is not common in 

color flow imaging as we noted in Section 1.4.2). 

Time-Shift-Based Methods 

Another approach for estimating the mean flow velocity is to find the average inter-pulse 

time shift between post-filter echo envelopes of the same beam line and then apply (1-2) to 

convert the time shift estimate into a velocity value. As reviewed by Alam and Parker (2003), 

this time-shift-based estimation approach is essentially derived from target tracking principles


that are used in radar and sonar. In terms of its implementation, time shift estimation can be 

performed in two different ways. The first way, as initially applied to color flow imaging by 

Bonnefous and Pesque (1986), involves finding the average time lag that yields the maximum 

cross-correlation between successive echo envelopes of the beam line. The second way, as 

proposed by Ferrara and Algazi (1991a, 1991b) as well as Alam and Parker (1995), involves 

computing a time-shift likelihood function from geometric projection principles and then 

searching for its location of maximum. For both methods of implementation, their performance 

is not affected by carrier frequency distortions because inter-pulse time shifts are not physically 

dependent on the carrier frequency (as seen in (1-2)). They also do not suffer from aliasing 

problems since they are not based on the analysis of inter-pulse phase changes. However, the 

theoretical advantages offered by these estimators appear to be non-essential to color flow data 

processing (e.g. aliasing is indeed preferred in color flow imaging as noted earlier). As such, 

time-shift-based estimators are often not used in color flow imaging, even though they have been 

a significant part of early research efforts on color flow signal processing. 

3.2.2 Parametric Estimators 

Fundamental Considerations 

In using non-parametric estimators to find flow velocities, estimation biases can be 

expected whenever the clutter filter distorts parts of the blood signal or fails to adequately 

suppress clutter. An illustration of this problem for a frequency-based non-parametric estimator 

is given in Figure 3-1a. Note that, at low blood-signal-to-noise ratios (BSNR), further estimation 

biases can be anticipated because the filtered white noise (i.e. colored noise) becomes a 

significant part of the filtered Doppler signal and in turn adds bias to the flow estimates. One of 

the ways to account for these clutter filter biases is to apply a post facto correction factor to the 

flow estimates. For instance, Rajaonah et al. (1994) have reported a frequency-based bias 

correction scheme that applies a correction factor based on the mean Doppler frequency of 

colored noise samples. Their approach, however, is intended to work with time-invariant clutter 

filters such as FIR filters, and thus it remains unclear as to whether this method can sufficiently 

account for biases due to time-variant filters like regression filters and initialized IIR filters. 

In order to effectively account for clutter filter biases, it is theoretically advantageous to 

use estimation strategies that can be directly applied to the raw (i.e. unfiltered) Doppler signal.


(a) Non-Parametric Spectral Estimator 

Clutter 

Spectrum 

Filter 

Stopband 

0 

Mean Flow 

Freq (Biased) 

Blood 

Spectrum 

f D 

(b) Parametric Spectral Estimator 

Clutter 

Spectrum 

Clutter 

Mode 

0 

Flow Signal 

Mode 

Blood 

Spectrum 

f D 

Fig. 3-1. Difference between the spectral estimates of (a) non-parametric and (b) rank-two 

parametric flow estimators. Note that, in this example, the mean estimate in (a) is biased 

away from the blood spectral peak because of filter distortions, while the two spectral modes 

in (b) are relatively unbiased. 

This rationale has motivated the development of parametric estimators that work by analyzing 

the principal Doppler spectral contents (i.e. they are frequency-based estimators). For these 

estimators, their principles are generally related to the following principal-component signal 

model: 

T 

[ x( 0), x(1), 

, x( 

N −1 

] = b + c + w ≈ χ v w 

x = K 

D 

) ∑ k CS( k ) 

+ . (3-3a) 

In the above expression, x(n) is the n th raw Doppler data sample, while x, b, c, and w respectively 

denote the length-N D vectors for raw Doppler signal, blood echoes, clutter, and white noise; also, 

K is the number of principal components in the signal model, whereas χ k and v CS(k) are the 

weight and complex sinusoid vector of the k th component. Note that v CS(k) essentially has the 

following vector form: 

CS( 

k ) 

K 

k = 1 

2 ( ND −1) 

2 D, PRI 

[ 1, , , , ] T 

j πf 

kT 

z z K z for z = e 

v = 

, (3-3b) 

k 

k 

k 

where f D,k is the k th principal Doppler frequency. It should be pointed out that, from a subspace 

perspective, this principal-component signal model is equivalent to the eigen-structure of a raw 

Doppler signal whose rank is equal to K. An illustration of the principal frequencies found using 

a rank-two parametric estimator is shown in Figure 3-1b. 

Autoregressive Modeling 

Autoregressive (AR) modeling (also known as linear prediction) is a type of parametric 

estimation approach that computes the principal Doppler frequencies through an all-pole signal 

k


analysis. As reviewed by Vaitkus and Cobbold (1988) in the context of Doppler ultrasound, this 

approach begins by expressing each Doppler data sample as a linear combination of white noise 

and past data samples. In particular, for a K th -order AR model, the n th sample of the Doppler 

signal x(n) is mathematically described by the following difference equation: 

K 

∑ 

x( n) 

= w( 

n) 

− c x( 

n − k) 

, (3-4a) 

k = 1 

where w(n) and c k are respectively the n th white noise sample and the k th model fitting 

coefficient. Note that this difference equation has the following all-pole form in the discrete 

frequency domain: 

X 

1 

K 

−1 

AR 

( f 

D 

) = 

= ( z z ) 

−1 

−K 

∏ − 

k 

1+ 

c1z 

+ K + cK 

z k = 1 

k 

j 2πf 

DT 

z= 

e PRI 

, (3-4b) 

where z k is the k th characteristic mode (i.e. the k th root of the denominator polynomial). From 

discrete-time signal theory, it is well-known that the difference equation’s natural response takes 

on the same form as the principal-component signal model given in (3-3a) when no repeated 

modes are present. As such, the characteristic modes of the difference equation can be used to 

estimate the principal frequencies of the Doppler signal. Specifically, since z is defined as 

exp{j2πf D T PRI }, the k th principal Doppler frequency can be found from the k th characteristic mode 

as follows: 

f 

1 

= arg . (3-5a) 

{ } 

D, k 

z k 

2πTPRI 

In turn, the modal flow frequency can be identified from the principal spectral estimate that has 

the largest magnitude: 

{ f ∀ k ∈[1, 

]} 

f 

D(est) 

= f 

D, k 

for k 

b 

= arg max 

b D, 

k 

K . (3-5b) 

In practice, the coefficients c k in the AR model are the main parameters that characteristic 

modes depend on, and they are typically found by fitting the Doppler samples onto the data 

model. As discussed in spectral analysis textbooks (e.g. see Stoica and Moses 1997, Ch. 3), an 

effective way of optimizing these coefficients is to minimize the mean-squared fitting error from 

both forward and backward regression perspectives. Such a least-squares fit can be computed 

using the Prony forward-backward fitting method (also known as modified covariance method), 

which is described by the following system of equations given a Doppler ensemble size of N D : 

k


⎡ x( 

K −1) 

⎢ 

x( 

K) 

⎢ 

⎢ M 

⎢ 

⎢ x( 

N 

D 

− 2) 

⎢ 

∗ 

x (1) 

⎢ 

∗ 

⎢ x (2) 

⎢ M 

⎢ 

∗ 

⎢⎣ 

x ( N 

D 

− K) 

x( 

K − 2) 

x( 

K −1) 

x( 

N 

x 

x 

∗ 

x ( N 

D 

∗ 

D 

∗ 

M 

− 3) 

(2) 

(3) 

M 

− K + 1) 

L 

L 

O 

L 

L 

L 

O 

L 

x(0) 

⎤ ⎡ 

x(1) 

⎥ ⎢ 

⎥ ⎢ 

M ⎥⎡ 

c ⎢ 

1 ⎤ 

⎥ ⎥ ⎢ 

x( 

N 

⎢ 

D 

− K −1) 

⎥⎢ 

c2 

⎥ = −⎢ 

∗ 

x ( K) 

⎥⎢ 

M ⎥ ⎢ 

∗ 

⎥⎢ 

⎥ ⎢ 

x ( K + 1) ⎥⎣cK 

⎦ ⎢ 

M ⎥ ⎢ 

⎥ ⎢ 

∗ 

x ( N 

D 

−1) 

⎥⎦ 

⎢⎣ 

x 

∗ 

x( 

K) 

⎤ 

x( 

K + 1) 

⎥ 

⎥ 

M ⎥ 

⎥ 

x( 

N 

D 

−1) 

⎥ . (3-6) 

∗ 

x (0) ⎥ 

∗ 

⎥ 

x (1) ⎥ 

M ⎥ 

⎥ 

( N 

D 

− K −1) 

⎥⎦ 

In this system of equations, the upper and lower halves respectively account for the forward and 

backward regressions of the Doppler data. Note that these equations may be recursively solved 

via an approach known as the Burg method. 

In the context of color flow data processing, Loupas and McDicken (1990) first 

considered the use of AR modeling in flow estimation. They showed that a first-order AR 

estimator, which only finds one Doppler spectral mode, is essentially equivalent to the lag-one 

autocorrelator shown in (3-2). Subsequently, Ahn and Park (1991) attempted to use AR 

modeling in flow estimation studies that work with raw Doppler data. In particular, they 

developed a second-order AR estimator to simultaneously find the principal Doppler frequency 

of clutter and blood echoes. However, in this estimator, clutter is inherently assumed to be a 

single low-frequency complex sinusoid. Such an assumption may not always be valid, especially 

in cases with wideband clutter. 

Multiple Signal Classification 

Another way of computing principal Doppler frequencies is to perform eigen-analysis on 

the correlation statistics of the Doppler signal. In particular, a form of eigen-analysis called 

multiple signal classification (MUSIC) has demonstrated potential in obtaining flow estimates 

from the raw Doppler data. As first applied to Doppler ultrasound by Allam and Greenleaf 

(1996), the MUSIC approach begins by decomposing the Doppler signal into a set of orthogonal 

basis functions though an eigen-decomposition of the correlation matrix (just like the eigenregression 

filter as described in Section 2.2.2). For a Doppler ensemble size N D , such 

decomposition can be described by the following set of expressions † : 

† This decomposition, sometimes referred to as the Karhunen-Loeve (KL) expansion, is essentially the same set of 

equations given in (2-9) of Section 2.2.2. It is restated here for convenience of discussion.


D 

∑ 

H 

H 

R = E{ xx = λ e e , (3-7a) 

x 

} 

ND 

⎧λk 

, k = l 

x = ∑γ ke 

k 

for E{ γ 

kγ 

l} 

= ⎨ , (3-7b) 

k = 1 

⎩ 0 , k ≠ l 

where R x is the correlation matrix of the Doppler signal, while γ k , λ k , and e k are the respective 

expansion coefficient, eigenvalue, and eigenvector of the k th orthogonal basis function. In 

practice, the Doppler correlation matrix in (3-7a) can be computed through ensemble averaging 

of multiple Doppler snapshots over a range of depth with similar signal characteristics. 

Within the eigen-decomposition given in (3-7), the Doppler signal contents are mainly 

contained in the high-energy basis functions since their strength is generally greater than 

background white noise. As such, the basis functions of the decomposition can be separated into 

signal and noise components, thereby forming two mutually orthogonal subspaces. From this 

signal separation, a frequency pseudo-spectrum can then be computed by finding the reciprocal 

of the cross-correlation between noise components and various complex sinusoids (with spectral 

peaks appearing at frequencies where the cross-correlation approaches zero). Specifically, for a 

Doppler signal with K principal bases, the pseudo-spectrum X MUSIC (f D ) can be expressed as: 

X 

f 

N 

k = 1 

⎡ | | 

1 

1 

) = = 

for E = 

⎢ 

n 

⎢ 

e 

2 H H 

K + 1 

e 

K + 

H 

En 

v v 

CS 

CSEnEn 

v 

CS 

⎢⎣ 

| | 

MUSIC 

( 

D 

2 

D 

k 

k 

k 

| ⎤ 

L e 

⎥ 

N 

⎥ 

, (3-8a) 

| ⎥⎦ 

where E n is the noise subspace matrix (which consists of the N D –K+1 least dominant 

eigenvectors) and v CS is a length-N D complex sinusoid vector of frequency f D . As shown in 

spectral analysis textbooks (e.g. Stoica and Moses 1997, Sec. 4.5), the denominator in (3-8a) is 

essentially equal to the following (2N D –1) th -order polynomial whose coefficients are given by the 

sum of elements along each diagonal of the matrix E n E H n : 

E 

H 

n 

v 

CS 

2 

= c 

ND 

−1 

z 

ND 

−1 

+ c 

ND 

−2 

z 

ND 

−2 

+ K + c 

−( 

ND 

−1) 

z 

−( 

ND 

−1) 

for c 

k 

j 2πf 

z= 

e DTPRI 

= 

N − 1N 

−1 

D D 

∑∑ 

i= 

0 

j= 

0 

δ 

k = j−i 

. (3-8b) 

e ( i, 

j) 

n 

In the above expression, c k is the k th polynomial coefficient, e n (i, j) is the entry of E n E n H at row i 

and column j, and δ k=j–i is a Kronecker delta function that equals to one if k = j–i and equals to 

zero otherwise. Note that the polynomial only has K non-spurious modes.


Table 3-1. Comparison of Existing Frequency-Based Velocity Estimators 

Type Advantage Limitation 

Lag-one 

Autocorrelator 

AR Estimator 

MUSIC 

Simple to implement with low 

computation demand 

Finds flow velocity directly from 

raw Doppler data without clutter 

filtering 

Same as AR estimator, but more 

resilient to white noise 

Only works on filtered Doppler data; 

prone to biases from filter distortions 

Least-squares fit of c k fails at high 

white noise levels; efficacy depends 

on choice of model order 

Needs multiple data snapshots to 

find correlation matrix; efficacy 

depends on eigen-structure rank 

Like AR modeling, the eigen-modes in the MUSIC pseudo-spectrum can be found 

numerically by solving for the denominator roots, and correspondingly the modal flow frequency 

can be found from the principal frequency estimates using (3-5). Based on this root-finding 

principle, Vaitkus and Cobbold (1998) developed a closed-form parametric estimator called 

Root-MUSIC to estimate principal flow velocities from raw Doppler data. Their approach, 

which uses a rank-two eigenstructure to model the raw Doppler signal, was analyzed using in 

vivo Doppler data whose clutter can be sufficiently modeled as a single complex sinusoid 

(Vaitkus et al. 1998). Note that another way to solve for the MUSIC eigen-modes is to use a 

peak searching algorithm to find the principal peaks in the pseudo-spectrum. Such approach was 

used by Allam et al. (1996) to process Doppler data acquired from a string phantom and a 

reflective surface that respectively simulate blood flow and stationary clutter. 

3.3.2 Comparison of Frequency-Based Velocity Estimators 

Table 3-1 summarizes the advantages and limitations of the three frequency-based 

velocity estimators covered in this section. As indicated, the lag-one autocorrelator has the 

advantage of being computationally efficient. Indeed, the low computing burden of this 

estimator is well appreciated in early research and developments because of processing power 

limitations. However, as already pointed out, the lag-one autocorrelator can only be applied to 

filtered Doppler data, and thus, its velocity estimates are inherently prone to biases from clutter 

filter distortions. 

Contrary to the lag-one autocorrelator, the AR estimator has the theoretical advantage of 

being able to obtain velocity estimates by directly processing the raw Doppler data. This


estimator first uses a least-squares fitting procedure to compute the characteristic modes of the 

raw Doppler signal and then identifies the modal flow component based on the largest-frequency 

mode, thereby avoiding the need for clutter filtering. Nevertheless, the primary shortcoming of 

the AR estimator is that the least-squares fitting solution to the model coefficients assumes that 

the Doppler data samples are free of noise perturbations. Thus, its accuracy tends to degrade 

significantly as the noise level increases. Another limitation with the AR estimator is that its 

flow estimation performance is dependent on the choice of the model order (or eigen-structure 

rank). Specifically, a wrong choice of model order would give rise to spurious spectral modes 

and in turn give incorrect modal velocity estimates. 

Unlike the AR estimator, the MUSIC estimator is more resilient to white noise because it 

first makes use of an eigen-decomposition procedure to separate the signal bases from noise floor 

components. Therefore, when applied to raw Doppler data with high noise levels, this estimator 

can obtain modal velocity estimates that are less biased than the ones found from AR modeling. 

On a different note, it is worth pointing out that the formulation of MUSIC begins with the same 

eigen-decomposition step as seen for the eigen-regression filter. Based on this notion, it follows 

that MUSIC has limitations similar to the eigen-regression filter. In particular, MUSIC would 

need multiple signal snapshots that are statistically stationary in order to form an accurate 

estimate of the Doppler correlation matrix. One further limitation of the MUSIC approach is 

that, as similar to AR modeling, the performance of this estimator is contingent upon suitable 

choice of the eigen-structure rank. As such, an adaptive rank selection algorithm is needed in 

order for this estimator to be effective in general. 

3.3 The Matrix Pencil Estimator 


Design Motivations 

From the previous work on AR modeling and MUSIC, it can be seen that parametric 

estimation strategies have potential in obtaining modal flow estimates in the presence of clutter. 

However, as noted earlier, these parametric estimators have certain individual limitations that 

reduce their efficacy in color flow data processing. For instance, the least-squares fitting 

procedure used by an AR estimator to compute the model coefficients is only effective at high


blood-signal-to-noise ratios. As for MUSIC, its multi-snapshot way of estimating the Doppler 

correlation matrix inherently requires statistical stationarity amongst the Doppler signal vectors 

over the specified range of depth, and such requirement cannot always be satisfied due to the 

spatially varying nature of clutter and flow dynamics. In view of these theoretical limitations, 

we considered the development of a new parametric flow estimator that is resilient to white noise 

and that does not require multiple Doppler signal ensembles. 


The new parametric estimation approach to be presented in this section is based on a 

framework originally developed for pole estimation in control systems as well as direction-ofarrival 

estimation in array processing (Hua and Sarkar 1990). It works by exploiting the 

properties of an algebraic form called matrix pencil, and hence it will be referred to as the Matrix 

Pencil estimator. In terms of its principles, this new estimator is primarily concerned with the 

solution to the following generalized eigenvalue (GE) problem: 

A q = λ A q ⇔ A − A ) q = 0 . (3-9) 

1 0 

( 

1 

λ 

0 

In this equation, A 1 and A 0 are singular matrices of the same dimensions, and q is the 

generalized eigenvector for a particular eigenvalue λ. As first extensively discussed by 

Gantmacher (1960, Ch. XII), the set of matrices A 1 –λA 0 created from all values of λ is referred 

to as a matrix pencil † . The GEs of this matrix pencil are the particular values of λ that yield nonzero 

solutions to the eigenvector q in (3-9). Such values can also be regarded as those that 

reduce the rank of the matrix pencil so that a non-empty nullspace exists for A 1 –λA 0 . 

3.3.2 Theoretical Formulation 

Construction of Data Matrix Pair 

The overall formulation of the Matrix Pencil estimator can be summarized in the 

flowchart given in Fig. 3-2. In terms of its details, this estimator begins by constructing the 

matrix pair in (3-9) in a way so that its GEs contain information about the principal Doppler 

frequencies. One particular way to form the matrix pair is to define A 1 and A 0 such that their 

structure only differs from each other by a diagonal matrix operator (Hua and Sarkar 1990). 

† In mathematics, the term pencil generally refers to a set of entities that share a common property, such as passage 

through the same given point.


Doppler 


Create Hankel 

Matrix 

A = 

0 1 

2 

1 

2 

3 

2 

3 

3 

4 

4 

5 

4 5 6 

5 6 7 

Perform SVD 

P 

A = Σσ k 

u k 

v k 

H 

1 

Obtain Filtered Data 

Matrix 

Ã= 

~ ~ ~ 0 1 2 ~ ~ ~ 1 2 3 ~ ~ ~ 2 3 4 ~ ~ ~ 3 4 5 ~ ~ ~ 4 5 6 

~ ~ ~ 5 6 7 

Flow 

Frequency 

Estimate 

f D(est) 

Find Largest 

Frequency 

f D(est) = max{|f D,k |} 

Find Generalized 

Eigenvalues 

P 

Ã Ã = Σλ e e H 

0+ 1 k k k 

1 

Create Matrix Pencil 

~ ~ ~ 1 2 3 ~ ~ ~ 2 3 4 ~ ~ ~ 3 4 5 ~ ~ ~ 4 5 6 

Ã 1 = 

Ã 0 = 

~ ~ ~ 5 6 7 

~ 5 

~ 6 

~ 7 

~ ~ ~ 0 1 2 ~ ~ ~ 1 2 3 ~ ~ ~ 2 3 4 ~ ~ ~ 3 4 5 ~ ~ ~ 4 5 6 

Fig. 3-2. Block diagram overview of the Matrix Pencil estimator with low-rank filtering. 

During operation, this estimator is applied to the Doppler signal of each sample volume. 

Specifically, for a Doppler ensemble size of N D (indexed from 0 to N D –1), A 1 and A 0 can be 

defined as the following Hankel data matrices: 

⎡ x(1) 

x(2) 

L x( 

P) 

⎤ 

⎢ 

x 

x 

x P 

⎥ 

⎢ 

(2) (3) L ( + 1) 

A ⎥ 

1 

= 

, (3-10a) 

⎢ M 

M O M ⎥ 

⎢ 

⎥ 

⎣x( 

N 

D 

− P) 

x( 

N 

D 

− P + 1) L x( 

N 

D 

−1) 

⎦ 

( N −P) 

× P 

⎡ x(0) 

x(1) 

L x( 

P −1) 

⎤ 

⎢ 

x 

x 

x P 

⎥ 

⎢ 

(1) (2) L ( ) 

A ⎥ 

0 

= 

. (3-10b) 

⎢ M 

M O M ⎥ 

⎢ 

⎥ 

⎣x( 

N 

D 

− P −1) 

x( 

N 

D 

− P) 

L x( 

N 

D 

− 2) ⎦ 

D 

( N −P) 

× P 

An example on the construction of this matrix pair is given in Fig. 3-3. Note that, in A 1 and A 0 , 

P is a pencil dimension parameter that is similar to the one defined for the full data matrix given 

in (2-12) of Chapter 2; for a K th -order signal eigen-structure, this parameter must be greater than 

K but less than ⎡N D /2⎤ (where the ⎡.⎤ operator denotes the smallest integer greater than or equal 

to N D /2). It should be also pointed out that the two matrices defined in (3-10) are essentially 

overlapping subsets of the full data matrix A defined in (2-12). Hence, like the full data matrix, 

they assume that the samples within the Doppler signal vector are statistically stationary. 

To appreciate how the matrix pencil A 1 –λA 0 gives GEs that correspond to the principal 

frequencies, it is necessary to consider the decomposition properties of the two data matrices. In 

D


1 2 3 

2 3 4 

3 

4 

4 

5 

5 

6 

5 6 7 

Data matrix A 1 

Doppler signal vector x 

0 

1 

2 

3 

4 

5 

6 

7 

Data matrix A 0 

0 1 2 

1 2 3 

2 

3 

3 

4 

4 

5 

4 5 6 

Fig. 3-3. Illustration of how to form the data matrices A 1 and A 0 in the Matrix Pencil 

estimator. In this example, the case where N D =8 and P=3 is assumed. 

particular, with reference to the signal model given in (3-3), the data matrices A 1 and A 0 in the 

absence of noise can be decomposed into the following forms: 

A = , (3-11a) 

1 

Z 

LDΦ 

Z 

R 

A = , (3-11b) 

0 

Z 

LDZ 

R 

where the matrices Z L , Z R , D, and Φ are defined as follows: 

⎡ 

⎢ 

⎢ 

⎢ 

⎢ 

⎣z 

1 

2 

K 

Z 

L 

= 

, (3-12a) 

( N 

1 

1 1 L 

z 

D 

M 

−P−1) 

z 

( N 

2 

z 

D 

M 

−P−1) 

L 

O 

L 

z 

( N 

K 

1 ⎤ 

z 

D 

M 

−P−1) 

⎥ ⎥⎥⎥ ⎦ 

( N 

−P) 

× K 

z k = e 

D 

j 2πf 

D,kTPRI 

⎡1 

( P−1) 

1 

1 

⎢ 

( P−1) 

⎥ 

⎢1 

z2 

L z2 

Z ⎥ 

R 

= 

, (3-12b) 

⎢M 

⎢ 

⎢⎣ 

1 

z 

z 

M 

K 

L 

O 

L 

z 

z 

M 

( P−1) 

K 

⎤ 

⎥ 

⎥ 

⎥⎦ 

K× 

P z k = e 

j 2πf 

D,kTPRI 

D = diag χ , χ , K, 

χ } , (3-12c) 

{ z 

2 

{ 

1 2 K 

Φ = diag K 

. (3-12d) 

1, 

z , , z K 

} j 2πfD,kTPRI 

z k e 

Given these decomposed forms, the matrix pencil A 1 –λA 0 can be expressed as: 

A 

1 

λA 

0 

= Z 

LD Φ − λI] 

= 

− [ Z . (3-13) 

From the Φ–λI term in the above expression, it can be seen that if λ equals to any of the main 

diagonal entries in Φ (i.e. any of z k ), then the rank of A 1 –λA 0 is reduced by one. Hence, in the 

absence of noise, values of the set {z k } are indeed the GEs of A 1 –λA 0 , and correspondingly the 

k th principal Doppler frequency f D,k can be found from the k th GE as follows: 

R


f 

1 

{ λ } 

D , k 

= arg k 

. (3-14) 

2πTPRI 

As reviewed by van der Veen et al. (1993), the solution for Φ in (3-13) is equivalent to finding a 

rotational matrix operator such that the rotation is invariant from a subspace perspective (i.e. the 

two data matrices A 1 and A 0 span the same subspace). Hence, the Matrix Pencil estimator can be 

considered as an estimation method that is based on rotational invariance principles. 

Low-Rank Matrix Reduction 

In the general case where white noise is present, the decomposed matrix forms shown in 

(3-11) would still hold if low-rank matrix reduction is applied to A 1 and A 0 before finding the 

GEs. The low-rank reduction can generally be performed by using singular value decomposition 

(SVD) to create A 1 and A 0 from only their K largest singular components (Hua and Sarkar 1990). 

As pointed out by van der Veen et al. (1993), such approach is equivalent to a total-least-squares 

formulation of the estimation problem from a data fitting perspective. In terms of its 

implementation, an efficient way of realizing low-rank reduction is to first compute the rankreduced 

equivalent of a full Hankel data matrix (as defined in (2-12)) and then create the matrix 

pair from subsets of the rank-reduced matrix (Hua and Sarkar 1991). As originally derived using 

state-space estimation principles (Kung et al. 1983), this method is based on the following 

relation between the full data matrix A and its rank-reduced equivalent Ã: 

~ 

A = A + Δ = 

K 

∑ 

k = 1 

P 

∑ 

H 

H 

σ u v + σ u v , (3-15) 

k 

k 

k 

k 

k = K + 1 

where Δ represents the noise perturbation of the full data matrix, while σ k , u k , and v k are 

respectively the singular value, left singular vector (of size N D –P+1), and right singular vector 

(of size P) for the k th SVD component. Since the matrix elements of A 1 and A 0 only differ by 

one row (see (3-10)), it follows that the rank-reduced matrices Ã 1 and Ã 0 can be expressed as: 

~ 

A 

1 

K 

~ 

= last ( N 

D 

− P) 

rows of A = ∑σ 

ku 

for u 

1, k 

= 

k = 1 

1, k 

k 

T 

[ u (1), u (2), K, 

u ( N − P) 

] , 

k 

k 

v 

H 

k 

k 

k 

D 

(3-16a) 

~ 

A 

0 

K 

~ 

= first ( N 

D 

− P) 

rows of A = ∑σ 

ku 

for u 

0, k 

= 

k = 1 

0, k 

T 

[ u (0), u (1), K, 

u ( N − P −1) 

] , 

k 

k 

v 

H 

k 

k 

D 

(3-16b)


where u 0,k and u 1,k are the first and last (N D –P) samples of the k th left singular vector. The 

efficacy of low-rank reduction improves in general when there is a large difference between the 

signal eigen-rank K and the matrix dimension parameter P. In fact, as shown by Hua and Sarkar 

(1990) for rank-one signals, better reduction results can be achieved when P lies between the 

range from N D /3 to N D /2. 

To further improve the estimation accuracy, an additional round of low-rank matrix 

reduction may be carried out through a joint SVD truncation of the reduced matrix pair [Ã 1 , Ã 0 ]. 

Nevertheless, this extra reduction does not lead to substantial performance improvement because 

its noise perturbation characteristics are asymptotically equivalent to the initial round of 

reduction (Hua and Sarkar 1991). It is also worth noting that, if the signal is purely consisted of 

undamped sinusoids, low-rank reduction can be more accurately carried out by using a forwardbackward 

approach that applies SVD truncation to an augmented data matrix formed from the 

signal samples and their complex conjugates (del Rio and Sarkar 1996). However, we found that 

the forward-backward approach is not significantly useful in color flow data processing since the 

intrinsic presence of transit-time broadening effects gives rise to damping characteristics in the 

Doppler signal. 

Numerical Computation 

To solve for the GEs of the matrix pair defined by (3-10) or (3-16), we can first rewrite 

the GE equation in (3-9) as a standard eigenvalue equation. This algebraic manipulation can be 

done by multiplying both sides of the GE equation with A H 0 and moving all matrix terms to one 

side of the equation. The resulting eigenvalue equation is then given by: 

H −1 

H 

+ 

A A ) A A q = λ q ⇔ [ A A − I] 

q = 0 , (3-17) 

( 

0 0 0 1 

0 1 

λ 

where the ‘+’ superscript refers to a pseudo-inverse that can be computed from a matrix’s 

singular value components (Moon and Stirling 2000, Sec. 7.3). Based on this expression, it can 

be seen that the GEs of A 1 –λA 0 (or Ã 1 –λÃ 0 ) are essentially the eigenvalues of A + 0 A 1 (or Ã + 0 Ã 1 ). 

Hence, it is possible to use a standard eigenvalue solving algorithm such as QR factorization or 

power iterations to compute the GEs (Golub and van Loan 1996, Ch. 7 and 8). Once the GEs are 

found, the K principal Doppler frequencies can be calculated using (3-14). Subsequently, the 

frequency estimates can be classified as to whether they correspond to clutter or blood echoes. 

As already described in Section 3.2.2, one intuitive way to perform this classification is to make


Table 3-2. Summary on the Flops Needed in the Matrix Pencil Estimator 

Step Flops Rationale 

1. Form Hankel matrix 

of size (N D –P+1)×P 

0 • No computations needed for data structuring 

2. Compute SVD of the 

Hankel matrix 

O[P 3 ] • Same derivation as Step 2 in Table 2-2 

3. Obtain the filtered 

data matrix Ã 

O[P 2 ] 

• O[(N D –P+1)P] flops needed to obtain Ã from 

Σ Κ σ k u k v k 

H 

• For P=N D /2, O[(N D –P+1)P] ≈ O[P 2 ] 

4. Form the matrix pair 

[Ã 1 , Ã 0 ] 

0 • No computations needed for data structuring 

5. Find the GEs of the 

matrix pencil Ã 1 –λÃ 0 

O[P 3 ] 

• An eigenvalue solver such as QR algorithm needs 

O[P 3 ] flops for the P×P matrix Ã 0 + Ã 1 (see Golub 

and Van Loan 1996, Ch. 7 and 8) 

use of the fact that blood echoes generally give rise to higher Doppler frequencies. As such, it is 

possible to identify the modal frequency of blood echoes from the principal spectral estimate that 

has the largest magnitude. This notion can be mathematically expressed as follows: 

{ f ∀ k ∈[1, 

]} 

f 

D(est) 

= f 

D, k 

for kb 

= arg max 

b D, 

k 

K . (3-18) 

In terms of its computational load, it can be expected that the Matrix Pencil estimator has 

a higher burden than the lag-one autocorrelator. In particular, the Matrix Pencil involves the use 

of SVD in order to carry out low-rank reduction on the matrix pair [A 1 , A 0 ], and it also requires 

the use of an eigenvalue solver to compute the principal frequency estimates. A breakdown of 

this estimator’s overall computational load is provided in Table 3-2, which shows the number of 

floating point operations (flops) needed for each step in the estimator. As can be noted, the 

Matrix Pencil in theory needs on the order of P 3 flops for each estimation run. Therefore, this 

estimator is slightly more efficient than other advanced flow estimators that generally require on 

the order of N 3 D flops (Vaitkus et al. 1998), but it is two orders of magnitude more complex than 

the lag-one autocorrelator that only requires on the order of N D flops (by inspection of (3-2)). To 

improve the computational efficiency of the Matrix Pencil, it is possible to use more efficient 

SVD algorithms and eigenvalue solvers that exploit the Hankel structure inherent in the data 

matrices of this estimator (e.g. see Badeau et al. 2004 for a recent reference). 

k


Relationship to Eigen-Analysis 

The theoretical formulation of the Matrix Pencil estimator can actually be linked to 

eigen-analysis. In particular, Matrix Pencil can be considered as a signal-domain formulation of 

a modified eigen-based estimator that involves data smoothing. This relationship can be seen by 

first noting that the Hankel data matrix A in the Matrix Pencil can be converted into the 

following correlation matrix: 

R 

x 

≈ A 

T 

A 

∗ 

⎡ R0 

(0) 

⎢ 

R0 

(1) 

= ⎢ 

⎢ M 

⎢ 

⎣R0 

( P −1) 

R ( −1) 

R (0) 

R ( P − 2) 

1 

1 

1 

M 

L 

L 

O 

L 

RP− 

1( 

−P 

+ 1) ⎤ 

RP 

1( 

P 2) 

⎥ 

− 

− + 

⎥ 

M ⎥ 

⎥ 

RP− 

1(0) 

⎦ 

for R 

k 

( l) 

= 

P× 

P 

ND 

−P+ 

max[0, l] 

+ k 

∑ 

n= 

max[0, l] 

+ k 

x( 

n) 

x 

∗ 

( n − l) 

, 

(3-19) 

where R k (l) is the smoothed autocorrelation estimate for the l th lag and is computed by summing 

the single-sample correlation values over a window of N D –P+1 samples (with k being the first 

sample index in the window when the lag is zero). From the properties of SVD (Moon and 

Stirling 2000, Sec. 7.1), it is well-known that the eigen-decomposition of this smoothed 

correlation matrix is related to the SVD of the Hankel data matrix as follows: 

A = 

P 

P 

H T ∗ 

2 H 

∑σ ku 

k 

v 

k 

⇔ A A = ∑σ 

k 

v 

k 

v 

k 

, (3-20) 

k = 1 

k = 1 

where the eigenvalues and eigenvectors of A T A * are respectively the squared singular values and 

the right singular vectors of A. Based on such property and the principles of low-rank reduction, 

the Matrix Pencil can therefore be seen as being similar to an eigen-based estimator that involves 

the computation of GEs for a rank-reduced matrix pair formed from the K largest eigencomponents 

of the smoothed correlation matrix. Note that the eigen-decomposition equivalent 

of the Matrix Pencil was studied extensively by Stoica and Soderstrom (1991), and it is simply a 

data-smoothed version of the “estimation of signal parameters via rotational invariance 

techniques” (ESPRIT) framework originally proposed by Roy and Kailath (1989). Besides datasmoothed 

ESPRIT, it should be pointed out that a data-smoothed version of MUSIC also exists 

in practice (Stoica and Soderstrom 1991). In color flow data processing, this smoothed-MUSIC 

estimator is more useful than the original one described in Section 3.2.2 because multiple signal 

snapshots are generally not available to estimate the correlation matrix of each sample volume.


Table 3-3. Three Types of Matrix Pencil Estimators for Color Flow Data Processing 

Type Principle Main Usage Limitation 

Rank-One 

(K = 1) 

Obtain one principal 

frequency estimate 

from a data ensemble 

Find mode frequency 

of filtered Doppler data 

Cannot be applied 

directly to raw Doppler 

data to estimate flow 

Rank-Two 

(K = 2) 

Extract two principal 

frequencies from a data 

ensemble 

Find modal flow 

estimates in presence of 

narrowband clutter 

Estimates may be biased 

when applied to data 

with wideband clutter 

Rank-Adaptive 

(K = K nom ) 

Compute principal 

frequencies using an 

eigen-rank found from 

spectral spread analysis 

Find modal flow 

estimates in presence of 

any type of clutter 

Assume clutter spans a 

prescribed bandwidth 

3.3.3 Applications in Color Flow Data Processing 

Rank-One Matrix Pencil 

As outlined in Table 3-3, the Matrix Pencil estimator has several potential applications in 

color flow data processing. For instance, by assuming a rank-one signal eigen-structure (i.e. 

setting K = 1), this parametric estimator can be used to compute the mode frequency of any postfilter 

Doppler signal. Note that the rank-one Matrix Pencil estimator can be considered as a 

generalized form of the lag-one autocorrelator given in (3-2). Specifically, the rank-one 

estimator is a more advanced autocorrelator that has low-rank reduction capabilities depending 

on the choice of the pencil dimension P. As such, when the pencil dimension is greater than one, 

rank-one Matrix Pencil is less prone to noise perturbations than the original lag-one 

autocorrelator. It should be pointed out that the two estimators become equivalent in their form 

when the pencil dimension is set equal to one (i.e. when P = 1). This equivalence can be shown 

by noting that, for a given post-filter Doppler signal y(n), the matrices A 1 and A 0 in the Matrix 

Pencil estimator degenerate to the following vectors (of size N D –1) in the limiting case: 

[ y( 1), y(2), 

K, 

y( 

−1 

] T 

a , (3-21a) 

1 

= N 

D 

) 

[ y( 0), y(1), 

K, 

y( 

− 2 ] T 

a . (3-21b) 

0 

= N 

D 

) 

From the solution of (3-17) and the expression in (3-14), a 1 and a 0 would form a matrix pencil 

that gives the following modal frequency estimate:


H 

1 

a 

0 

a1 

f 

D 

= arg{} 

λ for λ = = 

H 

2πT 

a a 

PRI 

0 

0 

Ry 

(1) 

, (3-21c) 

R (0) 

y 

where R y (0) and R y (1) are respectively the lag-zero and lag-one autocorrelation functions of the 

filtered Doppler signal. Since R y (0) is always a real quantity (because it is simply the squarednorm 

of a 0 ), (3-21c) is essentially equivalent to (3-2), thus indicating that rank-one Matrix Pencil 

is the same as the lag-one autocorrelator when the pencil dimension is equal to one. 

Rank-Two Matrix Pencil 

Besides computing modal frequencies from filtered Doppler data, the Matrix Pencil can 

also be used to obtain flow estimates in the presence of clutter. In particular, by assuming a 

higher-rank signal eigen-structure, the Matrix Pencil can be applied directly to the raw Doppler 

data to extract the modal Doppler frequency of blood echoes. One type of higher-rank Matrix 

Pencil estimator that is useful in color flow data processing is the rank-two approach that models 

the raw Doppler signal as a sum of two principal complex sinusoids (i.e. it assumes K = 2). The 

rank-two Matrix Pencil, which is similar to second-order AR modeling and rank-two MUSIC, 

assumes that one of the principal complex sinusoids corresponds to clutter and the other 

corresponds to blood echoes. In turn, it obtains the modal frequency of blood echoes from the 

larger (magnitude-wise) of the two principal frequencies. In terms of its use in flow studies, this 

rank-two estimator is suitable for finding the modal frequency of blood echoes in studies where 

the clutter is narrowband in nature with respect to the Doppler spectral resolution. For instance, 

as recently demonstrated (Yu et al. 2005), it can be used to compute flow velocities from color 

flow data acquired from a flow phantom that produces stationary clutter. 

It should be pointed out that rank-two Matrix Pencil essentially degenerates to a secondorder 

AR estimator when the pencil dimension is set equal to two (i.e. when P = 2). Specifically, 

in this limiting case, the matrix pair [A 1 , A 0 ] for Matrix Pencil and the matrix A + 0 A 1 as seen in 

(3-17) would take on the following forms (Hua and Sarkar 1990): 

⎡ x(1) 

x(2) 

⎤ 

A 

⎢ 

⎥ 

1 

= 

⎢ 

M M 

⎥ 

, 

⎢⎣ 

x( 

N − 2) ( −1) 

⎥ 

D 

x N 

D ⎦ 

⎡ x(0) 

x(1) 

⎤ 

A 

⎢ 

⎥ 

0 

= 

⎢ 

M M 

⎥ 

, (3-22a)(3-22b) 

⎢⎣ 

x( 

N − 3) ( − 2) ⎥ 

D 

x N 

D ⎦ 

⎡0 

c 

+ 

1 ⎤ 

A 

0 

A1 

= ⎢ ⎥ , (3-22c) 

⎣1 

c2 

⎦


where c 1 and c 2 are respectively the product coefficients between A + 0 and the last column of A 1 . 

As described in algebra textbooks (e.g. see Moon and Stirling 2000, Sec. 8.5.1), the matrix 

shown in (3-22c) is the same as the companion matrix of a second-order AR polynomial, and its 

two eigenvalues are indeed the roots of the AR polynomial. Therefore, rank-two Matrix Pencil 

and second-order AR estimator are equivalent in principles when a pencil dimension of two is 

used. The theoretical advantage of using rank-two Matrix Pencil as opposed to second-order AR 

modeling is that the Matrix Pencil estimates can be made less sensitive to noise perturbations by 

performing low-rank reduction via the use of a larger pencil dimension (just like the rank-one 

formulation). Note that, to achieve the same advantage with AR modeling, the original Prony’s 

method given in (3-6) may be modified to include a SVD truncation step before calculating the 

model coefficients (Tufts and Kumaresan 1982). Nevertheless, Hua and Sarkar (1990) have 

shown that the SVD-based Prony’s method is not as effective as the Matrix Pencil. 

Rank-Adaptive Matrix Pencil 

As already pointed out, the rank-two Matrix Pencil estimator inherently assumes that 

clutter in the Doppler signal can be adequately modeled as a single complex sinusoid. However, 

such assumption may not always be valid because tissue motion can give rise to clutter that is 

more wideband in nature. Consequently, rank-two Matrix Pencil may sometimes give modal 

flow estimates that are inconsistent with the actual flow dynamics. The inconsistency problem is 

generally more significant for Doppler data acquired using higher frequencies or longer 

ensemble periods because the Doppler spectral resolution is finer in these cases. 

To address the theoretical limitation of rank-two Matrix Pencil, it is worthwhile to 

develop an algorithm for Matrix Pencil to adaptively select its eigen-structure rank based on the 

Doppler spectral characteristics. An intuitive way of designing such algorithm is to analyze the 

spectral spread of the Matrix Pencil frequency estimates for different eigen-structure ranks. In 

particular, it can be expected that, for cases with dominating wideband clutter, the spectral 

spread of lower-rank Matrix Pencil estimates should be smaller because the most dominant 

frequencies would likely correspond to clutter. Hence, as illustrated in the flowchart in Fig. 3-4, 

the rank selection algorithm can involve a search for the minimum eigen-structure rank that 

yields Matrix Pencil estimates with spectral spread greater than a certain threshold Δf thr . The 

nominal rank K nom obtained from this algorithm can mathematically be expressed as follows:


Initialize Rank 

K = 2 

Get Rank-Reduced 

Data Matrix Ã 

Form Matrix Pencil 

Ã 1 

–λ Ã 0 

Find Generalized 

Eigenvalues of Ã 0+ 

Ã 1 

Increment Rank 

K = K+1 

No 

Freq Spread 

> Δf thr 

? 

Yes 

Current Rank is 

the Nominal Rank 

Fig. 3-4. Flowchart of the rank selection algorithm used for rank-adaptive Matrix Pencil. 

The algorithm is based on analyzing the spectral spread of Matrix Pencil frequency estimates 

obtained for each eigen-structure rank. 

K 

( max{ f , f ,..., f } − min{ f , f f } > Δf 

) 

nom 

arg min 

D,1 D,2 D, K 

D,1 D,2 

,..., 

K 

= . (3-23) 

Note that the efficacy of this rank selection algorithm primarily depends on the choice of the 

spectral spread threshold Δf thr , which is a quantity analogous to the stopband of a clutter filter. 

Once the rank has been adaptively estimated, the modal frequency of blood echoes can then be 

set equal to the frequency with largest magnitude in the set of K nom Matrix Pencil estimates. 

It is worth pointing out that, in the signal processing field, there exist some other rank 

selection criteria such as the Akiake Information Criterion and the Minimal Description Length. 

These criteria generally work by finding the eigen-structure rank that gives the minimum fitting 

error with respect to the Doppler signal (Hayes 1996, Sec 8.5). However, they are not suitable 

for color flow data processing because parametric modeling is merely used in this application to 

obtain the modal flow estimate instead of finding the best model fit for the given Doppler data. 

Indeed, we recently showed that the Minimal Description Length criterion tends to overestimate 

the eigen-structure rank owing to the presence of transit-time broadening in the blood component 

of Doppler signals (Yu and Cobbold 2006). 

D, K 

thr


3.4 Simulation Methods 

3.4.1 Method of Study 

General Overview 

To examine the theoretical performance of the Matrix Pencil estimation framework, we 

made use of the simulation model that was previously developed to analyze the Hankel-SVD 

filter (see Section 2.4.1). In particular, two series of simulations were conducted to 

quantitatively analyze the efficacy of Matrix Pencil under various flow scenarios and noise 

levels. The first series of simulations, which involves the modeling of a flow scenario with no 

tissue motion, aims to evaluate the performance of rank-one and rank-two Matrix Pencil 

estimators at different BSNRs and flow velocities as well as to demonstrate these estimators’ 

low-rank reduction capabilities for various pencil dimensions. As a comparative assessment, we 

also studied the estimation performance of other frequency-based velocity estimators (as 

discussed in Section 3.2) in this flow scenario. With the insights gained from the initial 

simulations, a second series of simulations based on a flow scenario with moving tissue was then 

performed to study the efficacy of rank-adaptive Matrix Pencil and the spectral spread criterion 

proposed for rank selection. The performance analysis was also carried out with rank-one and 

rank-two Matrix Pencil estimators to gain comparative insights on the advantages of adaptive 

rank selection. 

Performance Measures 

In this simulation study, both first-order and second-order statistical measures were used 

to quantitatively assess Matrix Pencil’s estimation performance. In particular, the two 

performance measures that were considered are: 1) mean estimation bias, and 2) root-meansquared 

(RMS) estimation error. The mean estimation bias was computed by finding the average 

difference between the estimated velocity and the actual blood velocity used to synthesize the 

Doppler signal. It is well-established from estimation theory that a low bias is an indication of 

high estimation accuracy. On the other hand, the RMS estimation error was calculated by taking 

the square root of the average squared difference between the estimated velocity and the true 

blood velocity. For an estimator to have high accuracy and precision, its RMS estimation error 

should be low in general.


Table 3-4. Simulation Parameters Used in the Estimator Analysis 

Parameter 

Value 

Fixed Parameters 

Ultrasound propagation speed, c o 

1540 m/s 

F-number, F/W 4 

Pulse carrier frequency, f o 

5 MHz 

Pulse repetition interval, T PRI 

1.0 ms 


Doppler ensemble size, N D 10 

Dynamic range 

14 bits 

Number of realizations per dataset 10000 

Clutter-to-blood signal ratio 

30 dB 

Average noise strength, κ n 

10 dB 

Clutter vibration frequency, f vib 

5 Hz 

Variable Parameters 

Blood velocity, v b 

0 to v alias 

Blood-signal-to-noise ratio, BSNR -20 to +30 dB 

Maximum tissue velocity, v c,max 

0 or 1 cm/s 

Responding Parameters 

Aliasing velocity, v alias 

15.4 cm/s 

Clutter phase modulation index, φ c,max 0 or 13.0 radians 

Maximum clutter frequency 

0 or 70 Hz 

3.4.2 Simulation Parameters 

Data Synthesis 

The data synthesis parameters used in the simulations are listed in Table 3-4. For this 

study, blood velocity (v b ) and blood-signal-to-noise ratio (BSNR) are the two major variable 

parameters of interest. Specifically, color flow datasets (each with 10000 realizations) were 

synthesized for blood velocities ranging from zero to the aliasing limit (15.4 cm/s) as well as 

BSNRs ranging from -20 to 30 dB in order to assess the performance of the Matrix Pencil 

estimator. A third parameter that was varied in this study is the maximum clutter velocity 

(v c,max ). This parameter was either set to 0 or 1 cm/s to simulate narrowband Doppler clutter as 

well as wideband ones. In terms of the major fixed parameters, the ultrasound frequency (f o ) and 

the pulse repetition interval (T PRI ) were respectively set to 5 MHz and 1.0 ms in order to model 

data acquisition in a typical low-velocity vascular imaging scenario. As well, the average 

strength of Doppler clutter was defined to be 30 dB greater than that of blood echoes, and a 

Doppler ensemble size of 10 samples was used for the synthesized datasets.


Data Analysis 

For this study, flow estimation was performed using four different frequency-based 

methods: lag-one autocorrelation, AR modeling, Matrix Pencil, and data-smoothed Root- 

MUSIC. The estimation process was carried out on all the synthesized Doppler datasets to 

assess the mean bias and the RMS error of the four estimators. Note that for the autocorrelator 

and the rank-one Matrix Pencil estimator, a fifth-order, projection-initialized IIR filter with a 100 

Hz nominal cutoff was used to suppress Doppler clutter prior to flow estimation † . As for the 

rank-adaptive Matrix Pencil estimator, a two-way spectral spread threshold of 150 Hz was used 

for the rank selection algorithm (i.e. Δf thr = 150 Hz in (3-23)). It should also be pointed out that, 

to avoid aliasing problems during the analysis, all flow estimates were unwrapped in velocity so 

that they are bounded within the range v b ±v alias , where v alias is the aliasing velocity. 

3.5 Simulation Results 

3.5.1 Flow Scenario with Stationary Clutter 

Performance of Low-Rank Reduction 

As an assessment of the Matrix Pencil’s low-rank reduction capabilities, Fig. 3-5 shows 

the bias and the RMS error of this estimator for different pencil dimensions as a function of 

BSNR. Results are provided for both rank-one and rank-two Matrix Pencil when the Doppler 

ensemble size is 10 samples, and they were obtained by processing Doppler data synthesized 

with stationary clutter and a 5 cm/s blood velocity. As can be seen, the estimation bias of both 

rank-one and rank-two Matrix Pencil drops significantly at medium and low BSNRs as the pencil 

dimension increases. In addition, the RMS error appears to decrease gradually when the pencil 

dimension becomes larger. Both of these observations show that low-rank reduction allows 

Matrix Pencil to have better resilience against background white noise. They also suggest that 

the flow estimation performance is generally better at larger pencil dimensions: a result that can 

be expected because larger pencil dimensions give rise to more entries in the Hankel data matrix 

and in turn the rank reduction performance should be improved. Based on these grounds, it 

appears that the dimension parameter for the Matrix Pencil estimator should be set to the 

maximum possible value (i.e. N D /2). 

† Other types of filters may be used instead to obtain similar performance insights for these two estimators.


Estimation Bias [cm/s] 

(a) Bias (Rank-One Matrix Pencil) 

14 

12 

10 

8 

6 

4 

2 

P=1 

P=2 

P=3 

P=4 

P=5 

0 

-20 -15 -10 -5 0 5 10 15 20 25 30 

Blood-Signal-to-Noise Ratio [dB] 

RMS Error [cm/s] 

(b) RMS Error (Rank-One Estimator) 

14 

12 

10 

8 

6 

4 

2 

P=1 

P=2 

P=3 

P=4 

P=5 

0 

-20 -15 -10 -5 0 5 10 15 20 25 30 



14 

12 

10 

8 

6 

4 

2 

(c) Bias (Rank-Two Estimator) 

P=2 

P=3 

P=4 

P=5 

0 

-20 -15 -10 -5 0 5 10 15 20 25 30 



(d) RMS Error (Rank-Two Estimator) 

14 

12 

10 

8 

6 

4 

2 

P=2 

P=3 

P=4 

P=5 

0 

-20 -15 -10 -5 0 5 10 15 20 25 30 


Fig. 3-5. Estimation performance of rank-one and rank-two Matrix Pencil estimators in the 

flow scenario with stationary clutter. In (a) and (b), the estimation biases and the RMS errors 

are shown for the rank-one estimator at various BSNRs and pencil dimensions. The 

corresponding results for the rank-two estimator are shown in (c) and (d). Note that the true 

blood velocity used to synthesize the Doppler data is 5 cm/s. 

Comparative Assessment at Different BSNRs 

In Fig. 3-6, the estimation performance of Matrix Pencil at various BSNRs is compared 

against the autocorrelator, the second-order AR estimator, and the rank-two Root-MUSIC 

estimator (with data smoothing). The results shown are based on Doppler data synthesized with 

actual blood velocities of 5 and 10 cm/s (i.e. mean blood Doppler frequencies of 162 and 325 

Hz). In other words, these results respectively correspond to cases where the blood frequency



14 

12 

10 

8 

6 

4 

2 

(a) Bias (v b = 5 cm/s) 

ACR 

AR 

MUSIC 

RK-1 MP 

RK-2 MP 

0 

-20 -15 -10 -5 0 5 10 15 20 25 30 



14 

12 

10 

8 

6 

4 

2 

(b) RMS Error (v b = 5 cm/s) 

ACR 

AR 

MUSIC 

RK-1 MP 

RK-2 MP 

0 

-20 -15 -10 -5 0 5 10 15 20 25 30 



14 

12 

10 

8 

6 

4 

2 

(c) Bias (v b = 10 cm/s) 

ACR 

AR 

MUSIC 

RK-1 MP 

RK-2 MP 

0 

-20 -15 -10 -5 0 5 10 15 20 25 30 



14 

12 

10 

8 

6 

4 

2 

(d) RMS Error (v b = 10 cm/s) 

ACR 

AR 

MUSIC 

RK-1 MP 

RK-2 MP 

0 

-20 -15 -10 -5 0 5 10 15 20 25 30 


Fig. 3-6. Estimation performance of various frequency-based flow estimators in the flow 

scenario with stationary clutter. In (a) and (b), the estimation bias and the RMS error at 

various BSNRs are shown for the case where the actual blood velocity is 5 cm/s. The 

corresponding results for the 10 cm/s blood velocity case are shown in (c) and (d). 

contents are near and away from the stopband of the clutter filter. Note that, when using the 

Matrix Pencil estimator to process the synthesized Doppler data, the pencil dimension was 

defined as N D /2 to maximize the low-rank reduction performance. Likewise, when using the 

data-smoothed MUSIC estimator, a window size of N D /2 was chosen to form the Doppler 

correlation matrix. 

From the results shown, it can be seen that rank-one Matrix Pencil (black dashed line) is 

less biased than the autocorrelator (gray dashed line) at any BSNR. The rank-one Matrix



8 

6 

4 

(a) Bias (BSNR = 10 dB) 

ACR 

AR 

MUSIC 

RK-1 MP 

RK-2 MP 

2 

0 

-2 

-4 

0 2 4 6 8 10 12 14 



14 

12 

10 

(b) RMS Error (BSNR = 10 dB) 

8 

6 

4 

2 

ACR 

AR 

MUSIC 

RK-1 MP 

RK-2 MP 

0 

0 2 4 6 8 10 12 14 



8 

6 

4 

2 

0 

-2 

-4 

(c) Bias (BSNR = 0 dB) 

ACR 

AR 

MUSIC 

RK-1 MP 

RK-2 MP 

0 2 4 6 8 10 12 14 



14 

12 

10 

8 

6 

4 

2 

(d) RMS Error (BSNR = 0 dB) 

ACR 

AR 

MUSIC 

RK-1 MP 

RK-2 MP 

0 

0 2 4 6 8 10 12 14 


Fig. 3-7. Estimator performance as a function of blood velocity in the flow scenario with 

stationary clutter. In (a) and (b), the estimation bias and the RMS error are shown for the 

case where the BSNR is 10 dB. In (c) and (d), similar results are shown for a BSNR of 0 dB. 

Pencil also appears to be more precise than the autocorrelator when the blood frequency contents 

are close to filter stopband (as seen for the case where the blood velocity is 5 cm/s). Both of 

these findings are consistent with our previous observations on the advantages of low-rank 

reduction in the Matrix Pencil estimator. Another observation to be noted is that, for the 5 cm/s 

blood velocity case, rank-one Matrix Pencil and the autocorrelator both remain biased at high 

BSNRs because of distortions from the clutter filter. On the other hand, all three rank-two flow 

estimators are not prone to these distortions since they do not involve clutter filtering. In 

comparison between the rank-two estimators, rank-two Matrix Pencil (black solid line) is


significantly more accurate than the second-order AR estimator (gray dotted line) at medium and 

low BSNRs, and it appears to share similar performance trends with the rank-two Root-MUSIC 

estimator (gray solid line). This result can be expected because Matrix Pencil and Root-MUSIC 

both involve an orthogonal decomposition (either through SVD or eigen-decomposition) in their 

theoretical formulation to help separate signal components from noise (as noted in Section 3.3.2). 

Comparative Assessment at Different Blood Velocities 

As a generalization of the above observations, Fig. 3-7 shows the bias and the RMS error 

of various frequency-based flow estimators as a function of the true blood velocity. These 

results were found by processing Doppler data generated with BSNRs of 0 and 10 dB. As can be 

seen, the rank-one Matrix Pencil and the lag-one autocorrelator both exhibit substantial 

estimation bias at low blood velocities because of distortions from the clutter filter stopband. In 

fact, their bias is at a maximum for blood velocities near 4-5 cm/s (i.e. for blood Doppler 

frequencies around 130-160 Hz) since the blood spectral components in this case are mainly 

located inside the clutter filter’s transition region where signal distortions are more substantial. 

On the other hand, rank-two Matrix Pencil and rank-two Root-MUSIC both have significantly 

lower biases than the other estimators at any blood velocity. This observation is in accordance 

with our previous results that demonstrate the advantage of applying low-rank reduction to flow 

estimation. It is also worth pointing out that, at higher blood velocities, the precision of rank-one 

Matrix Pencil and the autocorrelator appears to be slightly better. This phenomenon can be 

explained by recognizing that, for these two estimators, the clutter filtering step prior to flow 

estimation has more or less suppressed some of the background white noise and hence the 

estimation variance is inherently lower. 

3.5.2 Flow Scenario with Non-Stationary Clutter 

Efficacy of Adaptive Rank Selection 

To demonstrate the need for adaptive rank selection when using Matrix Pencil, we first 

studied the estimation performance of rank-two Matrix Pencil isn the flow scenario with clutter 

generated by moving tissue. In this study, the flow estimates of rank-two Matrix Pencil were 

evaluated for cases with blood velocities of 5 and 10 cm/s as well as a BSNR of 10 dB. The 

upper half of Fig. 3-8 shows the corresponding distribution for 10000 estimates of the blood


(a) Rank-Two Matrix Pencil 

(v b = 5 cm/s) 

(b) Rank-Two Matrix Pencil 

(v b = 10 cm/s) 

Num. Estimates [1000's] 

1.4 

1.2 

1 

Clutter 

Range 

0.8 

0.6 

0.4 

0.2 

0 

-338 0 162 662 

Estimated Doppler Frequency [Hz] 


1.4 

1.2 

1 

Clutter 

Range 

0.8 

0.6 

0.4 

0.2 

0 

-175 0 325 825 


(c) Rank-Adaptive Matrix Pencil 

(v b = 5 cm/s) 

(d) Rank-Adaptive Matrix Pencil 

(v b = 10 cm/s) 


1.4 

1.2 

1 

Clutter 

Range 

0.8 

0.6 

0.4 

0.2 

0 

-338 0 162 662 



1.4 

1.2 

1 

Clutter 

Range 

0.8 

0.6 

0.4 

0.2 

0 

-175 0 325 825 


Fig. 3-8. Histograms of flow frequency estimates obtained from Matrix Pencil. The results 

for the rank-two estimator are shown in (a) and (b), while the ones for the rank-adaptive 

estimator are shown in (c) and (d). These histograms correspond to cases where the actual 

blood velocity is 5 or 10 cm/s. Note that the flow estimates are displayed over the unaliased 

Doppler spectral range, and 10000 estimates were used to form each histogram. 

Doppler frequency (prior to velocity conversion) in the two cases. As can be seen, a number of 

rank-two Matrix Pencil estimates are biased towards zero: a result that is inconsistent with the 

flow dynamics since blood velocities of 5 and 10 cm/s respectively give rise to mean blood 

Doppler frequencies of 162 and 325 Hz. In fact, when the blood velocity is 5 cm/s, it was found 

that 19.2% of the flow estimates in this dataset have a magnitude lower than the approximate 

clutter range of ±70 Hz; likewise, when the blood velocity is 10 cm/s, 13.9% of the estimates 

were less than the peak clutter frequency. In contrast to the rank-two estimation results, the 

bottom half of Fig. 3-8 shows the distribution of blood Doppler frequencies found from a rankadaptive 

Matrix Pencil estimator that uses a spectral spread threshold of 150 Hz during rank




8 

6 

4 

2 

0 

-2 

-4 

(a) Bias (BSNR = 10 dB) 

ACR 

RK-1 MP 

RK-2 MP 

RK-A MP 

0 2 4 6 8 10 12 14 


8 

6 

4 

2 

0 

-2 

-4 

(c) Bias (BSNR = 0 dB) 

ACR 

RK-1 MP 

RK-2 MP 

RK-A MP 

0 2 4 6 8 10 12 14 




14 

12 

10 

(b) RMS Error (BSNR = 10 dB) 

8 

6 

4 

2 

ACR 

RK-1 MP 

RK-2 MP 

RK-A MP 

0 

0 2 4 6 8 10 12 14 


14 

12 

10 

8 

6 

4 

2 

(d) RMS Error (BSNR = 0 dB) 

ACR 

RK-1 MP 

RK-2 MP 

RK-A MP 

0 

0 2 4 6 8 10 12 14 


Fig. 3-9. Estimation performance of flow estimators in the flow scenario with non-stationary 

clutter. In (a) and (b), the estimation bias and the RMS error as a function of blood velocity 

are shown for the case where the BSNR is 10 dB. Similar results for a BSNR of 0 dB are 

shown in (c) and (d). 

selection. Although favorable results were not observed (some overestimation were seen) for the 

case with 5 cm/s velocity, the histogram for the higher velocity case (10 cm/s) revealed that 

rank-adaptive Matrix Pencil is able to obtain flow estimates that are relatively centered at the 

mean blood frequency of the synthesized Doppler data. As such, rank-adaptive Matrix Pencil 

appears to have better estimation accuracy than the rank-two counterpart at high blood velocities. 

Comparative Assessment at Different Blood Velocities 

As an assessment of the general performance of rank-adaptive Matrix Pencil, Fig. 3-9 

shows the bias and the RMS error of this estimator (solid line) for various blood velocities and


two BSNR values. For comparison, the performance curves are also plotted for the 

autocorrelator (dash-dotted line) as well as the rank-one (dotted line) and rank-two Matrix Pencil 

estimators (dashed line). From these figures, it can be seen that in the higher blood velocity 

range (> ½v alias ), rank-adaptive Matrix Pencil is the most accurate (i.e. lowest bias) amongst the 

four estimators. However, in the lower blood velocity range (< ½v alias ), rank-adaptive Matrix 

Pencil appears to suffer a significant drop in the estimation precision. Such result can be 

expected though because the rank selection algorithm used in this study assumes that the Doppler 

clutter spans a prescribed bandwidth and therefore tends to exhibit performance characteristics 

similar to the stopband of a clutter filter. In fact, as can be seen from Fig. 3-9, the estimation 

precision of rank-adaptive Matrix Pencil is quite similar to that for the autocorrelator and rankone 

Matrix Pencil (both of which involves the use of a clutter filter). It is also worth pointing out 

that the better accuracy and precision seen for rank-two Matrix Pencil at lower velocities is a 

somewhat misleading result since this estimator gives flow estimates that are always biased 

towards zero and hence it tends to yield lower estimation errors when the actual blood velocity is 

low. Note that the use of rank-two Matrix Pencil in flow scenarios with non-stationary clutter 

can be considered as being analogous to applying a correlation-based estimator to color flow data 

whose clutter has not been adequately suppressed. 


It has been the intent of this chapter to investigate the use of the Matrix Pencil estimation 

framework in color flow data processing. In our theoretical formulation, we have first 

considered a type of fixed-rank Matrix Pencil estimator called rank-one Matrix Pencil and have 

shown that it is a generalized form of the lag-one autocorrelator that is often used in color flow 

data processing. To realize the potential of Matrix Pencil in obtaining flow estimates directly 

from raw color flow data (i.e. unfiltered Doppler data), we have considered the use of a rank-two 

Matrix Pencil estimator and have shown that it is suitable for use in flow scenarios where the 

Doppler clutter can be sufficiently modeled as a single complex sinusoid. Besides fixed-rank 

Matrix Pencil estimators, we have also developed an adaptive-rank Matrix Pencil estimator that 

can be used to obtain flow estimates in the presence of wideband Doppler clutter. The rank 

selection algorithm used in rank-adaptive Matrix Pencil is based on a search of the minimum 

eigen-structure rank that yields a Matrix Pencil spectral spread greater than a certain bandwidth.


Our simulation study has demonstrated that rank-one Matrix Pencil is more accurate and 

precise than the lag-one autocorrelator at various blood-signal-to-noise ratios. Results have also 

shown that, in flow scenarios with stationary Doppler clutter, rank-two Matrix Pencil is even 

more accurate than rank-one Matrix Pencil since the former does not suffer from potential biases 

due to clutter filtering. On the other hand, in flow scenarios with non-stationary Doppler clutter, 

rank-adaptive Matrix Pencil has shown to be more useful as it can give velocity estimates that 

are more consistent with the actual flow dynamics (provided that the blood spectral components 

are outside the specified clutter bandwidth). From these findings, it can be seen that the Matrix 

Pencil framework is potentially useful in color flow data processing.

CHAPTER 4 

Single-Module Approach to Color Flow Data 

Processing: An Experimental Study 


In the previous two chapters, we considered how clutter suppression and flow estimation 

in color flow imaging can be performed by making use of two algebraic quantities known as the 

Hankel matrix and the matrix pencil. Specifically, Chapter 2 has described an adaptive clutter 

filtering design that is based on principal Hankel component analysis of each Doppler signal 

vector. As well, Chapter 3 has presented a parametric flow estimation framework that treats 

Doppler spectral estimation as a generalized eigenvalue (GE) problem. In this chapter, the 

concepts presented in Chapters 2 and 3 are combined to form a color flow data processor that 

performs flow detection and flow estimation within the same processing module. The proposed 

processor is then applied to a few different in vitro and in vivo flow imaging studies to gain 

insights on its flow estimation accuracy and flow detection performance. 

The remainder of this chapter is aimed at providing a system-level description of the 

proposed color flow data processor and presenting this processor’s efficacy on experimentally 

acquired color flow imaging data. To formulate discussion, the chapter contents have been 

organized as follows: 

• Section 4.2 highlights the theoretical principles used by the new eigen-based color flow data 

processor and provides a conceptual overview of this design; 

• Section 4.3 presents results on the performance of the new color flow signal processor based 

on color flow imaging data acquired from an in vitro scenario; 

• Section 4.4 provides further results on the efficacy of the new color flow signal processor 

based on in vivo color flow imaging data acquired from two different cases; 

• Section 4.5 gives some additional comments on the proposed signal processor and 

summarizes the conclusions drawn from the experimental studies. 

- 86 -

Chapter 4. Single-Module Approach to Color Flow Data Processing 87 

(a) Conventional Processors 

Raw 

Doppler Data 

(b) Proposed Processor 

Raw 

Doppler Data 

Filter 

Parameters 

CLUTTER FILTER / 

FLOW DETECTION 

Filtered 

Doppler Data 

Clutter 

BW Thr. 

HANKEL-SVD + 

MATRIX PENCIL 

FLOW 

ESTIMATION 

Doppler 

Power 


Velocity 

Doppler 

Power 


Velocity 

Fig. 4-1. Block diagrams illustrating the difference between (a) conventional color flow 

data processors and (b) the proposed processor. Note that the proposed processor does not 

involve direct computation of the filtered Doppler data since flow detection and flow 

estimation are jointly performed within the same module. 

4.2 The Single-Module Color Flow Data Processor 


Design Motivation 

As shown in Fig. 4-1a, color flow signal processing is conventionally performed using a 

two-module approach that treats clutter filtering and flow estimation as independent processing 

modules. During operation, the clutter filtering module first attempts to distinguish the blood 

component of Doppler signal from low-frequency tissue clutter, and thus it can be considered as 

a flow detection module. From the filtered Doppler signal, the flow estimation module then 

finds the flow velocity and the flow signal power as quantitative indicators for the presence or 

absence of blood flow. The estimation performance of this module, however, much depends on 

the ability of the clutter filtering module to adequately suppress Doppler clutter without 

distorting blood echoes. Because of such dependence, the clutter filtering module is often 

regarded from a signal processing perspective as a principal source of error in the estimation of 

flow velocity and flow power. In attempt to avoid this problem associated with the two-module


color flow signal processing approach, we pursued the development of a new single-module 

signal processor that is based on a joint detection-estimation formulation. 


Before the details of the new color flow data processor are considered, it is worth 

reviewing the basic principles of the Hankel-SVD filter and the Matrix Pencil estimator, which 

are the two signal processing frameworks presented in the previous chapters. First, as described 

in Chapter 2, the Hankel-SVD filter is an adaptive clutter suppression strategy based on the 

singular value decomposition (SVD) of a Hankel matrix created from each Doppler signal 

vector. It works by identifying the principal Hankel components that correspond to clutter and 

subsequently removing these components to obtain the filtered Doppler signal. On the other 

hand, as described in Chapter 3, the Matrix Pencil approach is a parametric spectral estimation 

method whose formulation is based on the generalized eigenvalue (GE) problem and low-rank 

matrix reduction principles. During its operation, this estimator first creates a rank-reduced 

matrix pair from the dominant SVD components of the Hankel data matrix corresponding to a 

Doppler signal vector, and then it finds the principal frequencies of the Doppler signal by solving 

for the GEs of a matrix pencil formed from the rank-reduced matrix pair. As can be seen, the 

Hankel-SVD filter and the Matrix Pencil approach both begin with the same SVD step in their 

theoretical formulation, even though the aim of the two frameworks are somewhat different. 

Such similarity in principles makes it possible for these two frameworks to be combined together 

as a single processing module in the proposed color flow data processor. 

4.2.2 Processor Details 

Conceptual Overview 

To appreciate how the new color flow data processor makes use of the two frameworks 

presented earlier in this thesis, we first recall that the Hankel-SVD filter is essentially an adaptive 

flow detection approach that involves spectral estimation principles. As described in Section 

2.3.2, this approach uses the lag-one autocorrelator to compute the mean Doppler frequency of 

each principal Hankel component found from the SVD of the Hankel data matrix. The resulting 

spectral estimates are then used to determine the clutter eigen-space dimension: a quantity that is 

needed to find the flow signal power as seen in (2-18). Instead of using the lag-one auto-


correlator to find the mean frequency of each Hankel component individually, it is possible to 

exploit the Hankel matrix structure and apply the Matrix Pencil approach to simultaneously 

compute the modal frequencies for multiple Hankel components. The advantage of using Matrix 

Pencil, as we showed in Chapter 3, is that low-rank matrix reduction can be performed to give 

more consistent spectral estimates for use during clutter eigen-space analysis. In turn, this 

estimator can help the Hankel-SVD filter to more reliably determine the clutter eigen-space 

dimension and consequently obtain more accurate flow power estimates. Note that, since the 

Matrix Pencil spectral estimates can be used in parallel to find the modal flow velocity (as seen 

in (3-18)), it is not necessary to perform separate computations for velocity estimation when 

using Matrix Pencil as the spectral estimator in the Hankel-SVD filter. In other words, flow 

detection and flow estimation can be jointly performed in this combined formulation. 

Based on the above principles, we developed a new single-module signal processor for 

use in color flow data processing. This new processor, which performs flow detection and flow 

estimation on Doppler ensembles via a Hankel matrix formulation, can be considered as a fusion 

between the Hankel-SVD filter’s flow detection concepts and the Matrix Pencil approach’s 

multi-modal spectral estimation capabilities. Like the Hankel-SVD filter, the proposed processor 

computes the flow signal power via the squared sum of singular values for Hankel component 

orders higher than the clutter eigen-space dimension. As well, it estimates the modal flow 

velocity from the largest frequency in the set of Matrix Pencil spectral estimates obtained during 

clutter eigen-space analysis. Note that, in the proposed processor, the clutter eigen-space 

dimension is determined by analyzing the spectral spread of the Matrix Pencil frequency 

estimates. This analysis approach is similar to the one used to determine the nominal eigenstructure 

rank shown in Section 3.3.3. Therefore, the proposed processor is essentially a 

modified rank-adaptive Matrix Pencil estimator that can compute the flow signal power in 

addition to the modal flow velocity. 

Description of Processing Module 

A basic input-output diagram of the proposed color flow data processor is illustrated in 

Fig. 4-1b. The two major inputs to the processor are the raw Doppler ensemble and the clutter 

bandwidth threshold; in return, the processor computes estimates of the flow signal power and 

the modal flow velocity. Fig. 4-2 shows the actual sequence of operations performed by the


Doppler 


1. Create Hankel 

Matrix A 

2. Compute SVD 

A = Σσ k 

u k 

v k 

H 

3. Initialize Clut 

Dim: K c 

= 1 

4. Form Matrix Pair 

With K c 

Largest 

Hankel Components 

[Ã 0 

, Ã 1 

] 

Power Map 

Memory 

Velocity Map 

Memory 

8a. Find Flow Power 

P 

Σσ 2 

k 

K c 

ρ = 

P(N D –P+1) 

8b. Find Flow Velocity 

v = 

2f o cosθ 

c o 

fD,argmax{|fD,k f D,argmax{|fD,k |} 

No 

7a. Increment Clut 

Dim: K c 

=K c 

+1 

Yes 

7. Is Spread 

< Thr? 

5. Find K c 

Modal Freq. 

From GEs of Ã 1 

–λÃ 0 

6. Find Spectral Spread 

of K c 

Modal Freq. 

max{f D,k 

}–min{f D,k 

} 

Fig. 4-2. Flowchart showing the steps involved in the proposed color flow data processor. 

Note that the clutter eigen-space analysis performed in this processor is based on the 

adaptive rank selection algorithm used by rank-adaptive Matrix Pencil. 

proposed processor to obtain the flow estimates. As can be seen, the proposed processor begins 

by computing the principal Hankel components of the Doppler signal via an SVD of the Hankel 

data matrix. Note that these initial steps are essentially the same ones as seen in the formulation 

of the Hankel-SVD filter and the Matrix Pencil approach. With the SVD components, the 

processor then iteratively finds the clutter eigen-space dimension by searching for the largest 

eigen-structure rank that gives a set of Matrix Pencil frequency estimates with spectral spread 

smaller than the specified clutter bandwidth threshold. It is worth pointing out that the clutter 

eigen-space dimension found from this analysis is essentially similar to the nominal eigenstructure 

rank defined in (3-23). From the clutter eigen-space dimension estimate, the flow 

signal power is subsequently computed from the singular values via (2-18). In addition, the 

modal flow velocity is found from the set of Matrix Pencil frequency estimates via (3-18). As 

seen in this series of steps, a feedback approach is used by the proposed processor to jointly 

perform flow detection and flow estimation. With such feedback, more consistent estimates of 

the flow signal power and the flow velocity should be obtained as a result.


(a) Physical Setting 

(b) Theoretical Profile 

Fig. 4-3. Imaging scenario for the in vitro flow phantom study: (a) an illustration of the 

physical setting during data acquisition; (b) a color flow map showing the phantom’s 

theoretical flow profile (parabolic flow with a center-line velocity of 70 cm/s). 

4.3 In Vitro Color Flow Imaging Study 

4.3.1 Experimental Protocol 


As an initial performance assessment, an in vitro flow imaging study was performed to 

evaluate the proposed processor’s ability to obtain flow estimates in the presence of motionless 

clutter † . As shown in Fig. 4-3a, the imaging view of this experiment is an in-plane slice of a 

commercial flow phantom (Gammex-RMI, Middleton, WI, USA; Model 1425A) that has a 

5mm-diameter flow tube surrounded by rigid tissue-mimicking material and angled at 50° with 

respect to the surface. Note that the bottom-center of the imaging view was aligned with the 

flow phantom’s distal tube wall during the data acquisition. Based on this physical setting, five 

frames of raw color flow data were acquired for offline processing using an experimental scanner 

(ZONARE Medical Systems, Mountain View, CA, USA) that was equipped with an L10-5 

transducer probe. In terms of the acquisition parameters, the color flow dataset was collected by 

† The flow phantom setup and the data acquisition portion of this study were performed by Dr. Larry Mo at 

ZONARE Medical Systems.


Table 4-1. Experimental Parameters in the Flow Phantom Study 

Parameter 

Value 

Flow Phantom Parameters 


1540 m/s 


Tube diameter 

5 mm 

Flow Profile 

Steady, parabolic flow 

Center-line flow velocity 

70 cm/s 

Clutter-to-flow-signal ratio 

In the range 20–30 dB 

Data Acquisition Parameters 

Transmit pulse characteristics 

6-cycles, 5MHz sinusoid 

Depth sampling rate 

5 MHz 


175 μs 


Image Size Parameters 

Lateral field of view –19.2 to +19.2 mm (257 beam lines) 

Axial field of view +24.5 to +50.0 mm (166 depth samples) 

using a 5-MHz six-cycle sinusoidal pulse along with a 0.175 ms pulse repetition interval and 14 

firings per beam line. Also, the online settings of the flow phantom were adjusted to generate a 

parabolic flow profile with a center-line velocity of 70 cm/s. A summary of the parameters used 

in this in vitro study is provided in Table 4-1. 

Signal Processing Procedure 

The first step in the processing of each color flow data frame is to obtain the analytic 

Doppler signal of individual sample volumes from the raw ultrasound echoes. For the in vitro 

datasets used in this study, such pre-processing was internally carried out in the experimental 

scanner’s channel domain memory (Mo et al. 2003). Once the Doppler data ensembles were 

derived from the raw echoes, they were passed into the proposed signal processing module to 

estimate the flow signal power and flow velocity of each sample volume along all beam lines. In 

order to lower the estimation variance, a five-tap median filter and a five-tap mean filter were 

applied in sequence to both the flow power and flow velocity estimates. Subsequently, the 

smoothed power estimates were color-coded based on their magnitude in a decibel scale, while 

the smoothed velocity estimates were color-coded with respect to the unaliased velocity range. 

Note that spurious pixels were removed from the power and velocity maps if their corresponding 

post-filter signal power was below a given threshold value. In the last step, the non-spurious 

pixels were overlaid on top of a B-mode image for duplex display.


Data Analysis Method 

As an analysis benchmark, a theoretical color flow map of the imaging view was first 

generated based on the fact that the flow phantom was calibrated to form a parabolic flow profile 

with 70 cm/s center-line velocity. This theoretical profile, which is shown in Fig. 4-3b, was then 

used to evaluate the proposed processor’s ability to reconstruct the flow dynamics of the 

phantom. To facilitate quantitative evaluation, we computed the correlation coefficient between 

the theoretical flow profile and each flow velocity map obtained from the proposed processor. 

Such performance measure can be expected to be close to unity if the reconstructed flow maps 

were similar to the theoretical flow profile. Besides computation of the correlation coefficient, 

we also assessed whether consistent velocity profiles were obtained at all cross-sections of the 

flow tube. This analysis was carried out by first locating the flow tube position within the 

imaging view (based on the fact that the distal tube wall was at the bottom center and the beamflow 

angle was 50°) and subsequently computing the mean and variance of velocity estimates for 

each point across the 5 mm flow tube diameter. 

As a comparison for the proposed processor, the analysis procedure described above was 

repeated for two types of color flow data processors that make use of the conventional twomodule 

processing approach. The first comparison processor was based on the use of a clutterdownmixing 

filter for clutter suppression and a lag-one autocorrelator for velocity estimation, 

and hence it will be hereafter referred to as the “clutter-downmixing processor”. On the other 

hand, the second comparison processor was based on the use of a fixed-rank eigen-regression 

filter for clutter suppression and a lag-one autocorrelator for velocity estimation, and it will be 

denoted as the “fixed-rank eigen-processor” from hereon. Note that, for the eigen-regression 

filter of this processor, the Doppler correlation matrix was computed via ensemble averaging of 

all the Doppler signal vectors along the same beam line. Such computation approach inherently 

presumes that the clutter statistics were stationary over the depth range. 

4.3.2 Experimental Results 

Color Flow Images 

Fig. 4-4 shows the power maps and velocity maps obtained from the proposed processor 

for a representative frame in the in vitro dataset. These images were found by using a clutter 

bandwidth threshold of 50 Hz for the proposed processor, a second-order IIR filter (projection-


(a) Proposed Processor 

(b) Clutter-Downmixing 

Processor 

(c) Fixed-Rank 

Eigen-Processor 

Velocity Map 

Flow Power Map 

Corr. Coeff. = 0.978±0.004 Corr. Coeff = 0.972±0.004 Corr. Coeff = 0.979±0.004 

Fig. 4-4. Color flow images of the flow phantom for one of the five frames in the dataset. These 

mappings were obtained from: (a) the proposed processor, (b) the clutter-downmixing processor, 

and (c) the fixed-rank eigen-processor. The dynamic range of the flow power maps was set to 10 

dB, while the unaliased velocity range of the velocity maps was defined to be between –61.6 cm/s 

and 75.3 cm/s. Also, the same spurious flow power threshold was used for all flow maps. 

initialized with 100 Hz nominal cutoff) for the clutter-downmixing processor, and a first-order 

clutter eigen-space dimension for the fixed-rank eigen-processor. From these figures, two 

general observations can be made. First, the flow power maps of all three processors have 

indicated a presence of flow only within the same spatial region as depicted in the theoretical 

profile of Fig. 4-3b. Second, as suggested by their high correlation coefficients (greater than 

0.97), the three processors seem to be capable of reconstructing velocity maps that have good 

agreement with the theoretical flow profile. These two observations indicate that, for imaging 

scenarios with no tissue motion, the proposed processor is able to obtain flow estimates that are 

comparable to the ones found from conventional two-module processors. Indeed, as will be 

subsequently seen in the in vivo studies, the proposed processor can outperform the other two 

processors in imaging scenarios when tissue motion is present. 

Velocity Profiles 

As a more quantitative assessment, Fig. 4-5 shows the cross-sectional velocity profiles 

computed from averaging the velocity estimates at corresponding spatial points in the flow tube.



Proximal Wall 

Theory 

Estimated 

Tube Cross-Section 

(5mm) 

Distal Wall 

70 

60 

50 40 30 20 

Flow Velocity [cm/s] 

10 

0 

(b) Clutter-Downmixing Processor 

(c) Fixed-Rank Eigen-Processor 

Proximal Wall 

Theory 

Estimated 

Proximal Wall 

Theory 

Estimated 


(5mm) 


(5mm) 

Distal Wall 

Distal Wall 

70 

60 

50 40 30 20 


10 

0 

70 

60 

50 40 30 20 


10 

0 

Fig. 4-5. The flow phantom’s cross-sectional velocity profile as obtained from: (a) the 

proposed processor, (b) the clutter-downmixing processor, and (c) the fixed-rank eigenprocessor. 

Also shown in these figures are the theoretical profile and the error bars within 

one deviation. 

As can be seen, the velocity profiles for all three processors more or less resemble a parabolic 

shape in the central part of the flow tube. Such result is in agreement with the theoretical profile 

and helps explain why the flow velocity maps in Fig. 4-4 have a high correlation coefficient. 

Nevertheless, it should be noted that the velocity estimates are actually under-biased for spatial 

points near the tube center and are over-biased for spatial points close to the tube wall. This 

phenomenon can be explained by recognizing that the spatial points in fact covers a range of 

velocities along the parabolic flow gradient (i.e. they are sample volumes); as a result, their


modal flow velocity estimates tend to be below maximum near the tube center and above 

minimum near the tube wall. 

4.4 In Vivo Color Flow Imaging Studies 

4.4.1 Experimental Protocol 


To facilitate further evaluation of the single-stage processor’s performance, in vivo color 

flow imaging data were acquired from the carotid arteries of a healthy 25-year-old male 

volunteer. Each dataset captured the flow dynamics over a full cardiac cycle, and it was 

consisted of 15 frames (at 15 Hz frame rate, 1 sec total) of raw ultrasound echoes returned from 

different beam lines along the imaging view. During the acquisition of each dataset, the 

volunteer was asked to either hold his breath or deeply inhale so that the resulting data frames 

would contain non-stationary clutter dominated by either arterial distension or patient movement. 

In this in vivo study, all the color flow datasets were obtained using a Philips HDI-5000 scanner 

(Bothell, WA, USA) equipped with an L7-4 linear array probe and a special proprietary interface 

for raw data downloading. The main acquisition parameters of the two in vivo color flow 

datasets presented in this section are listed in Table 4-2. 

Signal Processing Procedure 

Similar to the in vitro study in Section 4.3, processing of the acquired in vivo data begins 

with the conversion of raw echoes into analytic Doppler data samples. Since the HDI-5000 

scanner does not perform this conversion internally, an in-phase/quadrature (I/Q) demodulation 

routine was used to perform such conversion offline. Data decimation was then carried out after 

I/Q demodulation so that the depth sampling rate of each pulse echo was down-converted from 

the system’s digital sampling rate to the pulse carrier frequency. Once the Doppler signals were 

obtained, they were processed using the proposed processor to compute the flow signal power 

and the flow velocity of each sample volume. Prior to display, the flow estimates were passed 

through a five-tap median filter and a five-tap mean filter to reduce the estimation variance. 

Finally, the smoothed estimates were converted into color pixels based on their respective power 

or velocity scale and superimposed onto a B-mode image for display. Note that pixels with flow 

signal power below a threshold were considered as spurious and were not displayed.


Data Analysis Method 

In each in vivo color flow imaging study, regions of blood vessel, static tissue, and 

moving tissue within each data frame were identified to study the proposed processor’s 

sensitivity to blood flow echoes as well as its specificity to stationary and non-stationary Doppler 

clutter. For each identified region, the average post-filter signal power was calculated and 

compared against each other. Note that, if the proposed processor has good flow detection 

capabilities, then its regional power estimate for blood vessel sections (where blood flow is 

present) should be higher than that for tissue sections (where only clutter is present). To study 

the proposed processor’s flow detection performance at various stages in the cardiac cycle, the 

regional power analysis was conducted on all frames in the dataset. Also, as a comparative 

assessment, the analysis was repeated with the clutter-downmixing processor and the fixed-rank 

eigen-processor that were used for the in vitro performance analysis (see Section 4.3.1). It 

should be noted that the accuracy of velocity estimates was not analyzed quantitatively in these 

in vivo imaging studies since the actual flow dynamics were unknown. 

4.4.2 Case Study with Vessel Wall Motion 

Overview of Scenario 

Table 4-2. Experimental Parameters for the In Vivo Imaging Studies 

Parameter 

Dataset # 1 (while subject 

was holding breath) 

Data Acquisition Parameters 

Dataset # 2 (while subject 

was deeply inhaling) 

Transmit pulse characteristics 

3-cycles, 4MHz sinusoid 

Depth sampling rate 

4 MHz 


333 μs 


Number of data frames 

15 (at 15 Hz frame rate, 1 sec total) 


1540 m/s (approximate) 

Beam-flow angle, θ 

70° (approximate) 

Image Size Parameters 

Lateral field of view –11.5 to +11.5mm (77 lines) –12.5 to +12.5mm (85 lines) 

Axial field of view +5 to +25 mm (115 samples) +5 to +25 mm (115 samples) 

We first considered a case where the field of view corresponds to a cross-section of the 

carotid arteries beyond bifurcation while the subject was holding his breath (see Fig. 4-6a). In 

this case, the tissue motion is primarily caused by arterial wall distension over the cardiac cycle.



(b) B-mode Image 

Fig. 4-6. Graphical overview of an in vivo imaging study on the human carotid arteries (past 

bifurcation) with tissue motion due to arterial wall distension. In (a), an illustration of the 

physical setting is shown. In (b), a B-mode image of the imaging view is shown with 

labeled sections of blood vessel (BV), moving tissue (MT), and static tissue (ST). The view 

is a cross-sectional slice of the arteries (range of interest marked by green box). 

Hence, for the regional power analysis, moving tissue sections were selected to be regions 

adjacent to the blood vessels (as reflected in the B-mode image in Fig. 4-6b), while a static tissue 

section was selected from a region located far away from the blood vessels. For the analysis, a 

100 Hz clutter bandwidth threshold was used by the proposed processor since it yielded highpower 

flow estimates in the larger blood vessel (the BV1 section in Fig. 4-6b) while giving lowpower 

ones in the static tissue section. As well, a second-order filter with 100 Hz nominal cutoff 

was used by the IIR-based clutter-downmixing processor to obtain similar power estimates in the 

larger blood vessel, while a second-order clutter eigen-space dimension was used by the fixedrank 

eigen-processor. 

Results: Color Flow Images 

Fig. 4-7a shows the flow power map and the velocity map obtained from the proposed 

signal processor design during peak systole where wall motion was the most significant. These 

two images should be compared against the ones obtained from the two comparison processors




Processor 



Velocity Map 


Fig. 4-7. Color flow images for the in vivo scenario depicted in Fig. 4-6 for a frame acquired 

during peak systole. The flow power map and velocity map obtained from the proposed 

processor are shown in (a), while the ones for the two comparison processors are shown in (b) 

and (c). The dynamic range of the flow power maps was 15 dB, while the velocity maps had a 

range of ±84.4 cm/s. Note that a high color display gain was used for these images. 

given in Figs. 4-7b and 4-7c. Note that, for this series of figures, a high color display gain (i.e. a 

low spurious power threshold) was used intentionally to help visualize the processors’ flow 

detection performance. As shown in these images, all three processors have correctly indicated 

the presence of flow inside the two blood vessels within the imaging view. Nevertheless, the 

flow maps produced by the proposed processor appear to have fewer spurious pixels (often 

referred to as flash artifacts) than the ones obtained from the two comparison processors. This 

observation suggests that the proposed processor is more capable of distinguishing blood 

components from non-stationary Doppler clutter when the processors are optimized to provide 

similar flow power estimates in the larger blood vessel. Another point worth noting is that,


amongst the three processors considered in this study, the fixed-rank eigen-processor actually 

produced flow maps with the most spurious pixels. The mediocre performance of the fixed-rank 

eigen-processor is likely due to two factors: 1) the use of a fixed clutter eigen-space dimension 

may not be effective since clutter statistics generally vary over time; 2) estimation of the Doppler 

correlation matrix via multi-depth ensemble averaging may be impractical given the spatial 

varying nature of clutter characteristics over a depth range of few centimeters. Based on these 

problems, it seems that modifications to the formulation of the fixed-rank eigen-processor are 

needed in order for the processor to be effective in in vivo imaging scenarios where tissue motion 

is generally present † . 

Results: Regional Power Comparison 

To provide further insights on the efficacy of the color flow data processors, Figs. 4-8a 

through 4-8c show the average post-filter Doppler power for different regions in the imaging 

view as a function of the frame number (with peak systole happening at frame # 6). In addition, 

Fig. 4-8d shows a cross-processor comparison on the power ratios between the larger vessel 

(BV1) and its adjacent moving tissue region (MT1). From these plots, it can be seen that during 

peak systole (around frame # 6), all three data processors considered in this study can more or less 

obtain flow power estimates that are distinguishable from the post-filter tissue power estimates; 

during cardiac diastole (i.e. other data frames besides frame # 6), a similar result can be seen as 

well. Nonetheless, as reflected in the figures, the proposed processor appears to have a better 

separation between the power estimates of blood and tissue regions during cardiac systole where 

substantial wall motion was present. This observation helps explain why the color flow images 

obtained from the proposed processor have fewer spurious pixels than the ones found from the 

other two processors. 

4.4.3 Case Study with Translational Tissue Motion 

Overview of Scenario 

In the second case study, we considered a scenario where both tissue and blood vessel 

were progressively pushed towards the transducer throughout the frame acquisition period. This 

translational tissue motion was caused by the deep inhalation that the subject undertook while 

† To our knowledge, an investigation of the problems associated with the eigen-regression filter is currently being 

pursued as a doctoral study in the Norwegian research group (e.g. see Lovstakken et al. 2006).


Signal Power [dB] 

(a) Regional Power (Proposed) 

BV1 BV2 MT1 MT2 ST1 

Relative Time 

60 

0 0.2 0.4 0.6 

55 

0.8 1 

50 

45 

40 

35 

30 

25 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

Frame Num 

(c) Regional Power (Fixed-Rank Eigen) 


(b) Regional Power (Clutter-Downmixing) 



Relative Time 

60 

0 0.2 0.4 0.6 

55 

0.8 1 

50 

45 

40 

35 

30 

25 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

Frame Num 

(d) Power Ratio: BV1/MT1 

Proposed Downmix Fixed-Eigen 


Relative Time 

60 

0 0.2 0.4 0.6 

55 

0.8 1 

50 

45 

40 

35 

30 

25 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

Frame Num 

BV1/MT1 Ratio [dB] 

0 0.2 

Relative Time 

0.4 0.6 0.8 1 

20 

15 

10 

5 

0 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

Frame Num 

Fig. 4-8. Regional post-filter power estimates for different flow image frames acquired for 

the in vivo scenario considered in Fig. 4-6. Results are shown for: (a) the proposed 

processor, (b) the clutter-downmixing processor, and (c) the fixed-rank eigen-processor. In 

(d), the power ratio between the BV1 and MT1 sections are shown for the three processors. 

Note that peak systole occurs at Frame # 6 in this sequence (marked by gray vertical line). 

color flow data was acquired. It essentially masked out the arterial wall distensions and gave rise 

to Doppler clutter that was mostly non-stationary throughout the cardiac cycle. 

As illustrated in Fig. 4-9a, the imaging view of this second case study is an in-plane slice 

of the subject’s common carotid artery. It is important to note that, during the data acquisition, 

most of the imaging view was affected by the translational tissue motion. Hence, when 

analyzing the post-filter power estimates, the two adjacent regions above and below the blood 

vessel were labeled as moving tissue regions (see Fig. 4-9b), while only a small region near the 

top edge of the color box was labeled as static tissue.



(b) B-mode Image 

Fig. 4-9. Illustration of a flow study on a human common carotid artery in the presence of 

translational tissue motion. The physical setting of this imaging scenario is shown in (a), 

and a labeled B-mode image of the imaging view is shown in (b). Note that the view is an 

in-plane slice of a subject’s common carotid artery. 

In processing this dataset, we found that a clutter bandwidth threshold of 150 Hz was 

needed by the proposed processor to discriminate between blood and clutter components while 

maintaining reasonable flow power estimates within the blood vessel. Also, for the comparison 

processors, it was found that the IIR-based clutter-downmixing processor needed to use a thirdorder 

filter with 150 Hz nominal cutoff, whereas the fixed-rank eigen-processor required a thirdorder 

clutter eigen-space dimension. Note that these threshold values and filter stopband sizes 

are larger than the one used in the previous case study, but such increase can be expected since 

non-stationary Doppler clutter is often more wideband in nature. 

Results: Color Flow Images 

To examine the worst-case scenario for this case study, we focused on a data frame 

acquired during cardiac diastole where blood and clutter components were less separated in 

Doppler frequency due to the slower blood flow. In Fig. 4-10, the color flow images 

corresponding to such data frame are shown for the proposed processor and the two comparison 

processors. Like the previous case study, a high color display gain was purposely used for these




Processor 



Velocity Map 


Fig. 4-10. Color flow images for the in vivo scenario shown in Fig. 4-8 for a data frame 

acquired during cardiac diastole. In (a), the flow power map and velocity map for the 

proposed processor are shown. In (b) and (c), the corresponding flow maps for the two 

comparison processors are shown respectively. The dynamic range and unaliased velocity 

range are the same as the ones used in Fig. 4-7. Also, a high color display gain was used for 

these images. 

images to help visualize the flow detection performance of the three processors. As can be seen, 

the quality of these color flow maps is generally degraded by the presence of many spurious 

pixels. This result indicates that all the color flow data processors had suffered a drop in the 

flow detection performance to some extent. Nevertheless, from the images, it can be seen that 

the proposed processor have produced flow maps with fewer spurious pixels than the ones 

obtained using the two comparison processors. Indeed, amongst the three processors, the 

proposed processor seems to be the only one that can give flow maps with color pixels located 

mainly inside the blood vessel. This observation suggests that the proposed processor has better



(a) Regional Power (Proposed) 

BV1 MT1 MT2 ST1 

Relative Time 

60 

0 0.2 0.4 0.6 

55 

0.8 1 

50 

45 

40 

35 

30 

25 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

Frame Num 

(c) Regional Power (Fixed-Rank Eigen) 


(b) Regional Power (Clutter-Downmixing) 



Relative Time 

60 

0 0.2 0.4 0.6 

55 

0.8 1 

50 

45 

40 

35 

30 

25 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

Frame Num 

(d) Power Ratio: BV1/MT2 

Proposed Downmix Fixed-Eigen 


Relative Time 

60 

0 0.2 0.4 0.6 

55 

0.8 1 

50 

45 

40 

35 

30 

25 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

Frame Num 

flow visualization performance than the two comparison processors in the worst-case scenario of 

the current in vivo study. 

Results: Regional Power Comparison 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

Frame Num 

As an analysis of the flow detection performance at various cardiac cycle phases in this 

case study, Figs. 4-11a to 4-11c show the regional post-filter power estimates of the proposed 

processor and the two comparison processors for different data frames (with systole occurring at 

the start and end of sequence). Also, a cross-processor comparison is provided in Fig. 4-11d for 

BV1/MT2 Ratio [dB] 

20 

15 

10 

5 

0 

Relative Time 

0 0.2 0.4 0.6 0.8 1 

Fig. 4-11. Regional post-filter power estimates for all the image frames in the in vivo 

scenario considered in Fig. 4-8. Results for the proposed processor, the clutter-downmixing 

processor, and the fixed-rank eigen-processor are respectively shown in (a), (b), and (c). 

For reference, the power ratios between the BV1 and MT2 sections are shown in (d) for the 

three processors. Note that peak systole occurs at Frames # 2 and # 14 of this sequence 

(marked by the gray vertical lines).


the power ratio between the blood vessel section (BV1) and the more significant moving tissue 

section (MT2). These figures indicate that during the cardiac systolic phase (near frames # 2 and 

# 14), all three processors are generally able to distinguish blood signals from Doppler clutter. In 

contrast, during cardiac diastole (between frames # 2 and # 14), the proposed processor is more 

capable of discriminating between blood and tissue regions, as both the clutter-downmixing 

processor and the fixed-rank eigen-processor seem to have difficulties in preserving blood 

signals while suppressing Doppler clutter. This observation is consistent with the insights drawn 

from the color flow images seen in Fig. 4-10. 


Color flow signal processing is conventionally performed using a two-module approach 

where flow detection is first carried out prior to flow estimation. The primary shortcoming of 

this two-module processing approach, however, is that the flow estimation accuracy inevitably 

depends on the efficacy of the flow detection module. To avoid problems with the two-module 

processing approach, this chapter has described a new single-module processor design that can 

perform flow detection and flow estimation in parallel. The proposed processor is based on the 

combination of detection and estimation principles drawn from the Hankel-SVD filter and the 

Matrix Pencil estimator, which are the two frameworks described in the previous chapters. 

In our steady-flow phantom imaging study, the proposed processor has demonstrated its 

ability in reconstructing velocity maps that more or less follow the theoretical profile. Also, for 

this in vitro scenario where the clutter is stationary, the proposed processor has shown that its 

flow estimation accuracy and flow detection performance are similar to two kinds of 

conventional two-module processors. On the other hand, when this new processor is applied to 

in vivo imaging scenarios with non-stationary clutter, it appears to have better flow detection 

capabilities than the conventional processors. In particular, fewer spurious pixels are seen in the 

flow maps obtained from the new processor, and more favorable regional power estimates are 

observed in the corresponding blood and tissue sections of the color flow images. From these 

findings, it seems that the single-module processor presented in this chapter has higher potential 

than conventional ones in distinguishing blood flow signals from Doppler clutter.

CHAPTER 5 

Thesis Summary and Future Directions 


The purpose of this short concluding chapter is to summarize the key features of the new 

color flow signal processing methods presented in the previous three chapters. In particular, it is 

our intent to review the theoretical advantages of the proposed methods over existing ones. In 

addition to the summary, we will also provide some suggestions on the potential future research 

directions that can be pursued along this line of work. As such, the rest of this chapter has been 

organized as follows: 

• Section 5.2 reviews the significance of this research work on color flow signal processing 

and lists the major research contributions; 

• Section 5.3 outlines some prospective research topics on three different aspects of color flow 

signal processing and suggests potential ways to approach these topics; 

• Section 5.4 gives some concluding remarks on the potential impact of this research. 

5.2 Summary of Thesis Study 

5.2.1 Overall Scope 

This thesis study has been devoted to the design of new eigen-based signal processing 

methods for use in color flow imaging. In particular, it has been our intent to develop a new 

eigen-based clutter filter and a new parametric flow estimator to effectively account for three 

main problems in color flow signal processing: 1) the lack of abundant Doppler samples 

available for processing, 2) the possible presence of wideband Doppler clutter due to tissue 

motion, and 3) the potential flow signal distortions that may arise during clutter suppression. In 

approaching this task, we made use of the eigen-space principles related to two matrix forms 

known as the Hankel matrix and the matrix pencil. To our knowledge, such formulation has not 

been introduced before in the research field of color flow signal processing. Indeed, it is our 

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Chapter 5. Thesis Summary and Future Directions 107 

understanding that none of the existing clutter filters and flow estimators involves the Hankel 

matrix or the matrix pencil as part of their formulation, even though a few types of eigen-based 

signal processing methods have been proposed previously. 

Common to the signal processing methods proposed in this thesis is their adaptability to 

the Doppler signal contents. Such adaptability is achieved via a singular value decomposition 

(SVD) analysis of the Hankel matrix created from each Doppler signal vector. It is worth noting 

that, because of their formulation through the Hankel data matrix, all of the proposed methods 

are intended to work with each Doppler ensemble separately. This formulation should be 

distinguished from existing eigen-based color flow data processing methods that require Doppler 

ensembles from multiple sample volumes to estimate the correlation matrix statistics. In fact, the 

proposed eigen-based methods should be more useful than the existing ones because multiple 

sample volumes with similar Doppler signal characteristics may not always be available 

(especially when spatially-varying tissue motion is present). 

In our study, the potential of the proposed color flow signal processing methods were 

examined using both simulation and experimental means. For the simulations, we developed an 

in-house Doppler signal synthesis model to generate Doppler signals with non-stationary clutter 

characteristics and transit-time-broadened blood flow echoes. For the experiments, we acquired 

color flow imaging datasets from a steady-flow phantom and the human carotid arteries. From 

these analyses, it was generally observed that the proposed color flow signal processing methods 

have better capabilities than existing ones in discriminating blood flow signals from Doppler 

clutter and in obtaining flow estimates that are less biased. Hence, the proposed methods can 

potentially improve the flow visualization performance of ultrasound color flow images. 

5.2.2 Specific Developments 

Hankel-SVD Filter 

The first signal processing method that we developed in this thesis is a new eigen-based 

clutter suppression method called the Hankel-SVD filter. This new clutter filter design, as 

described in Chapter 2, is adaptive to the Doppler signal contents, and it is derived using the 

notion of principal Hankel component analysis. Similar to existing eigen-regression filters 

(Bjaerum et al. 2002b), the Hankel-SVD filter attempts to decompose the Doppler signal as a 

series of adaptable and orthogonal components. However, instead of estimating the components


through an eigen-decomposition of the Doppler correlation matrix formed from multiple signal 

snapshots, the new filter computes these components through the SVD of a Hankel data matrix 

created from each Doppler signal. To achieve clutter suppression, the filter then removes the 

principal Hankel components that correspond to tissue clutter by analyzing their mean Doppler 

frequency. Note that, for this filter design, we have shown that the filtered Doppler power (i.e. 

the flow signal power when blood flow is present) can actually be estimated directly from the 

squared sum of singular values for the non-clutter Hankel components. Therefore, it is not 

necessary to directly reconstruct the filtered Doppler signal vector. 

To study the efficacy of the Hankel-SVD filter, simulations were carried out via the use 

of an in-house Doppler signal synthesis model (to be described in the last part of this section). In 

these simulations, multiple sets of Doppler data were synthesized for two scenarios: 1) a case 

with arterial flow (>5cm/s), slowly moving tissue (


Doppler signal. Hence, by including the clutter harmonics as part of the signal eigen-structure, 

this framework can be used to extract the mode flow velocity in the presence of clutter. Based 

on this notion, we have developed several forms of Matrix-Pencil-based flow estimators that can 

potentially be useful to color flow data processing. These include both fixed-rank flow 

estimators (rank-one/rank-two Matrix Pencil) as well as adaptive-rank flow estimators (rankadaptive 

Matrix Pencil). Note that, for the adaptive-rank flow estimator, the nominal eigenstructure 

rank was found using a new rank-selection algorithm that is based on the spectral 

spread of principal frequency estimates. 

To assess the performance of the Matrix Pencil flow estimators, we used our in-house 

Doppler signal synthesis model to generate datasets for two flow scenarios: a case with 

narrowband Doppler clutter (no tissue motion) and another with wideband Doppler clutter (up to 

1 cm/s tissue velocity). Note that the flow velocity in the simulations ranged between 0-15 cm/s, 

and the ensemble period was set at 10 ms. Based on these synthesized datasets, the estimation 

bias and the root-mean-squared error of the Matrix Pencil flow estimators were computed as a 

function of blood-signal-to-noise ratio and blood velocity. Results have indicated that the Matrix 

Pencil flow estimators are generally less biased than most of the existing frequency-based flow 

estimators such as the lag-one autocorrelator. As such, the Matrix Pencil method seems capable 

of improving the accuracy of flow estimates. 

Single-Module Color Flow Data Processor 

Based on the Hankel-SVD filter and the Matrix Pencil method, we have developed a new 

color flow signal processor to perform flow detection and flow estimation in parallel within the 

same processing module. As described in Chapter 4, this new processor works by combining the 

frequency-based flow detection concepts from the Hankel-SVD filter and the multi-modal 

spectral estimation capabilities in the Matrix Pencil method. Specifically, it computes the flow 

signal power in the same way as done in the Hankel-SVD filter (i.e. by calculating the squared 

sum of the non-clutter singular values), and it estimates the modal flow velocity in the same way 

as carried out in the Matrix Pencil method (i.e. by finding the largest frequency in the set of 

principal spectral estimates). Note that, for this single-module estimator, the clutter eigen-space 

dimension was determined using a frequency-based feedback approach that iteratively searches 

for the largest eigen-structure rank with a spectral spread smaller than the specified clutter


bandwidth threshold (just like the rank-adaptive Matrix Pencil). Such a feedback approach is 

intended to provide more consistent estimates of the flow signal power and the flow velocity. 

To examine its performance, the single-module processor was applied to color flow data 

acquired from in vitro and in vivo imaging scenarios. First, this processor was used to compute 

flow estimates from an in vitro dataset corresponding to a steady-flow phantom (parabolic flow, 

70 cm/s center-line velocity) with tissue-mimicking material surrounding the flow tube. For this 

study, it was found that the single-module processor can obtain flow power maps that resemble 

the flow tube’s shape and can reconstruct velocity maps that have high correlation with the 

theoretical flow profile. Aside from the in vitro study, the single-module processor was also 

applied to two in vivo imaging scenarios where the field of view corresponds to the human 

carotid arteries when different degrees of tissue motion were present. It was found that this 

processor can produce color flow images with fewer spurious pixels than the ones obtained from 

conventional two-module processors. These results suggest that the single-module processor is 

generally more effective than conventional ones in identifying blood flow in the presence of 

clutter (especially non-stationary ones). 

Doppler Signal Synthesis Model 

For the purpose of analyzing the theoretical performance of the Hankel-SVD filter and 

the Matrix Pencil estimator, a Doppler signal synthesis model was developed as part of the 

background work in this thesis. As described in Section 2.4, our synthesis model separately 

generates clutter and blood components of the Doppler signal based on two different signal 

models. In particular, the Doppler clutter is synthesized via a phase-modulated signal model, 

while blood echoes are generated using an amplitude-modulated signal model. The use of two 

different models to simulate Doppler signal is physically justified by recognizing that tissue and 

blood scatterers follow different movement mechanisms: tissue tends to move in a quasi-cyclic 

pattern while blood scatterers simply traverse through the sample volume. Note that, in the 

literature, such a hybrid approach for synthesizing Doppler data has not been considered before. 

Indeed, existing Doppler signal synthesis approaches often treat both clutter and blood echoes as 

amplitude-modulated signals (e.g. see Bjaerum et al. 2002a), despite the fact that amplitude 

modulation does not seem to be an appropriate way of modeling tissue clutter’s movement 

mechanism.


5.2.3 List of Primary Research Contributions 

The original contributions of this thesis are mainly in the form of theoretical development 

along with experimental assessment. In particular, the following is a list of the primary research 

contributions made in this body of work: 

(a) Development of a new class of eigen-based color flow signal processing methods that are 

intended to work with the Doppler signal of each sample volume separately; 

(b) Design of a new clutter filter (Hankel-SVD filter) that is adaptive to the Doppler signal 

contents via the use of an existing framework called principal Hankel component analysis; 

(c) Development of a new eigen-based velocity estimator (Matrix Pencil) based on the eigenspace 

principles of an existing array processing framework called the Matrix Pencil; 

(d) Creation and implementation of a new Doppler signal synthesis model that can simulate nonstationary 

Doppler clutter and transit-time-broadened blood flow echoes; 

(e) Analysis of the Hankel-SVD filter’s flow detection performance and the Matrix Pencil 

method’s flow estimation accuracy via the use of our simulation model mentioned in (d); 

(f) Proposal of a novel single-module color flow data processor that performs flow detection and 

flow estimation in parallel via the combination of principles from the Hankel-SVD filter and 

the Matrix Pencil estimator; 

(g) Offline demonstration of the single-module processor’s improved capability over existing 

ones to detect blood flow in in vivo imaging scenarios with significant tissue motion (i.e. 

non-stationary Doppler clutter). 

Note that point (a) can be considered as the overall contribution of this thesis, while the other six 

points can be considered as specific contributions. 

5.3 Future Research Directions 

5.3.1 Developmental Aspects 

Real-Time Computation Issues 

To extend beyond the findings presented in this thesis, there are quite a few research 

topics that are worth exploring in the future. One of these prospective topics is on addressing the 

computational aspect of eigen-based signal processing strategies. In particular, it is wellrecognized 

that the primary challenge for implementing eigen-based signal processors in real-


time is the high computational complexity involved with the SVD or eigen-decomposition step 

(which require on the order of P 3 flops). Hence, a more efficient computation algorithm should 

be developed to help reduce the overall complexity of eigen-based methods, thereby making 

them more feasible for real-time implementation. For the case of the Hankel-SVD filter and the 

Matrix Pencil estimator, a potential approach for designing such algorithm is to exploit the 

persymmetric structure of the Hankel matrix during SVD computation. This design approach 

has been previously explored in a few theoretical studies (e.g. see Bedeau et al. 2004), and thus it 

will be worthwhile to examine its applicability to color flow signal processing (possibly in 

collaboration with experts in the matrix computation field). 

Adaptive Thresholding 

Another topic that can be pursued in the future is on developing an algorithm to 

adaptively select the clutter bandwidth threshold used by the single-module processor proposed 

in this thesis. The motivations for designing such an algorithm can be perceived by first 

recalling that the clutter bandwidth threshold is the main parameter influencing the flow 

detection/estimation performance of the single-module processor. In our study, this threshold 

was designated as a user-defined parameter in which the user can adjust between zero and the 

Doppler aliasing frequency, and the same threshold value was used to process the Doppler data 

in all the sample volumes within the imaging view. As such, the availability of an adaptivethresholding 

algorithm should further improve the proposed processor’s performance in 

scenarios where the Doppler clutter bandwidth varies significantly over both space and time. 

Note that, in developing this algorithm, one of the features in which the adaptation can be based 

upon is the Doppler clutter’s center frequency. Specifically, it can be anticipated that the clutter 

bandwidth is larger when the clutter spectrum deviates substantially from zero frequency since 

tissue motion is more likely to be significant in this case. Such property was exploited in a 

recent investigation by Yoo et al. (2003) on adaptive clutter filter selection. 

5.3.2 Experimental Aspects 

Flow Imaging Phantom Design 

An experimental research direction that can be pursued in the future is on designing a 

flow imaging phantom that has an elastic flow tube padded with a layer of tissue mimicking 

material. The aim of developing such a phantom is to facilitate rational modeling of the Doppler


echoes originating from a pulsating artery surrounded by body tissues, thereby allowing us to 

acquire experimental color flow data with more realistic clutter characteristics and in turn study 

the performance of color flow signal processors as a function of tissue elasticity and flow 

parameters. It is worth noting that, in the ultrasound research field, few have considered the 

design of flow imaging phantoms with elastic tissues and flow tubes. Indeed, like the Gammex- 

RMI phantom used in our study, most existing flow imaging phantoms mainly comprise a rigid 

flow tube that is positioned below a layer of tissue-like gel. Therefore, development of an elastic 

flow imaging phantom can be a substantial contribution to flow imaging research and system 

testing. Note that Kargel et al. (2003) have recently reported a dual-tube flow imaging phantom 

design in which one of the tubes is made of elastic material. This design appears to be an 

encouraging first step towards the development of an elastic flow imaging phantom. 

Doppler Clutter Analysis 

In parallel to the design of a flow imaging phantom, it may be worth conducting a 

complementary study on the Doppler clutter characteristics for various in vivo scenarios. In 

particular, it should be useful to analyze how tissue motion can affect the Doppler clutter for 

different sample volumes within the imaging view and at different parts of a cardiac cycle. One 

way of performing such an analysis is to study the statistical distribution of the main Doppler 

clutter features like energy, bandwidth, and center frequency. Correspondingly, new statistical 

models and mathematical approximations may be developed to predict the space-time variations 

of Doppler clutter in color flow data, thereby allowing us to more appropriately define the clutter 

stopband of color flow signal processors and design more suitable adaptive-thresholding 

algorithms. It should be noted that such a space-time statistical relationship for Doppler clutter 

has not been reported in the literature, and perhaps the closest of such kind is a phase-modulated 

clutter model that was developed to describe the clutter behavior in spectral Doppler studies 

(Heimdal and Torp 1997). Hence, this research direction can potentially lead to significant 

contributions to the current understanding of Doppler clutter. 

5.3.3 Theoretical Aspects 

Processing of Non-Uniformly Sampled Color Flow Data 

Aside from experimental research topics, some theoretical topics in color flow signal 

processing are also worth exploring upon in future studies. One of these topics is related to the


processing of non-uniformly sampled color flow data. As recently demonstrated in a preliminary 

study (Wilkening et al. 2002), non-uniform color flow data sampling has the advantage of 

virtually extending the aliasing limit without losing velocity resolution, but its resulting flow 

estimates tend to be less accurate than the ones derived from uniformly sampled data. It is thus 

worthwhile to develop a more robust flow estimator in order for this non-uniform sampling 

technique to obtain accurate flow estimates. One potential way of designing such an estimator is 

to modify the Matrix Pencil approach considered in this thesis so that the framework can be used 

to process non-uniformly sampled data. Note that the estimator design may also be based on 

solutions adopted from the array processing literature, since it is well-known that spectral 

estimation with non-uniformly sampled data is essentially the same as direction-of-arrival 

estimation using a non-uniformly spaced antenna array. 

Contrast-Enhanced Color Flow Data Processing 

The use of parametric methods to account for nonlinear harmonics in contrast-enhanced 

color flow data is another theoretical topic that can be studied in the future. As described by 

Bruce et al. (2004), the presence of nonlinear harmonics in the pulse echoes acquired using 

microbubble contrast agents (in conjunction with a pulse-inversion firing scheme) is the main 

factor that complicates the processing of contrast-enhanced color flow data. From a twodimensional 

(2D) frequency perspective, these nonlinear harmonics give rise to multiple 

components on the fast-time/slow-time frequency space even for single-velocity flow profiles, 

and thus multi-stage filtering strategies are typically needed to process contrast-enhanced color 

flow data. As a more direct approach to account for the nonlinear harmonics, it may be possible 

to use 2D parametric estimation methods to simultaneously compute all the principal modes in 

the fast-time/slow-time frequency space. For instance, a 2D parametric estimator that can 

potentially be useful is the 2D formulation of Matrix Pencil (Hua 1992). Hopefully, the use of 

2D parametric estimators can simplify the conventional multi-stage approach to the processing of 

contrast-enhanced color flow data. 

Other Single-Snapshot-Based Flow Estimators 

As pointed out earlier, this thesis is the first attempt in the ultrasound research 

community to perform eigen-based, parametric flow estimation without involving the use of 

Doppler ensembles from multiple sample volumes. In approaching this task, we have made use


of the Matrix Pencil framework that is well-recognized in the array processing field and have 

developed three forms of flow estimators using such method. It is worth noting that the Matrix 

Pencil is not the only kind of single-snapshot-based parametric estimation method available, 

though this approach was chosen for use in our study because of its computational efficiency as 

compared to other parametric estimators. In the future, it will be worthwhile to study the 

applicability of other single-snapshot parametric estimation methods to color flow signal 

processing. One particular kind of these methods worth devoting attention to is the deterministic 

maximum likelihood method, which theoretically has a more precise estimation performance 

than the Matrix Pencil approach as demonstrated in array processing field (van der Veen et al. 

1993). It would be interesting to compare the performance of various parametric estimators in 

the context of color flow signal processing. 

Flow Analysis Using Image Processing Tools 

Instead of approaching color flow data processing from a one-dimensional signal analysis 

perspective, it may be possible to carry out the same task by making use of image processing 

tools that are well-established in the computer vision field. Indeed, as reviewed by Hein and 

O’Brien (1993), such an image-processing-based flow analysis approach has been attempted in 

some heuristic studies where simple motion-tracking methods like block matching and optical 

flow analysis have been used to examine the inter-pulse movement of blood scatterers. Although 

these early studies did not achieve favorable performance results under in vivo flow scenarios 

(because of problems caused by tissue motion), they have still demonstrated that image 

processing tools may offer an alternative way of computing flow estimates from color flow data. 

Hence, another research topic worth exploring in the future is on developing new imageprocessing-based 

flow analysis strategies for color flow imaging applications. One possible 

approach to this topic is to study whether more robust motion analysis methods like statistical 

modeling can be used to analyze color flow data. Such a study may potentially lead to new 

insights on how flow detection and flow estimation can be performed in color flow imaging. 

5.4 Final Remarks 

Color flow signal processing has been an active field of research ever since the initial 

development of ultrasound flow imaging methods in the 1980s. As an attempt to expand upon 

current state of the art in the research field, this thesis has pursued the development of an eigen-


based signal processing framework that is based on the properties of Hankel matrix and the 

Matrix Pencil formulation. Since color flow imaging is already widely available in commercial 

ultrasound systems, it is hoped that the theoretical framework presented in this thesis can 

eventually be implemented as a patch modification to the processor architecture of existing 

ultrasound systems. As a result of such effort, the overall reliability of flow information 

provided in color flow images can potentially be improved.

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