.

RICE UNIVERSITY

Noise Suppression and Motion Estimation in

Medical Ultrasound Imaging

by

Yong Yue

Doctor of Philosophy

Houston, Texas

May 2007

.

RICE UNIVERSITY

Noise Suppression and Motion Estimation in

Medical Ultrasound Imaging

by

Yong Yue

A THESIS SUBMITTEDIN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE

Doctor of Philosophy

Approved, Thesis Committee:

John W. Clark, Jr., Professor, ChairElectrical and Computer Engineering

Richard G. Baraniuk, ProfessorElectrical and Computer Engineering

Fathi Ghorbel, ProfessorMechanical Engineering and Materials Science

Dirar S. Khoury, Associate ProfessorMethodist Hospital Research Institute

Houston, Texas

May 2007

Abstract

Noise Suppression and Motion Estimation inMedical Ultrasound Imaging

by

Yong Yue

Echocardiographic imaging is a primary modality in the diagnosis of heart disease.

Compared to other imaging techniques, such as X-Ray, MRI, and PET, ultrasound

imaging owes its great popularity to the fact that it is a safe and non-invasive pro-

cedure for visualizing the heart and vasculature. The ultrasound image however is

corrupted by speckle noise, which is distinguished from Gaussian noise by its signal-

dependent nature. This dissertation focuses on two important issues for the clinical

applications of medical ultrasound images: speckle suppression and motion estima-

tion.

The dissertation first describes the statistics of speckle and ultrasound image mod-

els, which are important for performance evaluation and further algorithm develop-

ment. Secondly, a novel speckle suppression approach is developed for the purpose of

visualization enhancement and auto-segmentation improvement. This method is de-

signed to utilize the favorable denoising properties of two frequently used techniques:

wavelet and nonlinear diffusion. Speckle is iteratively reduced by the multiscale non-

linear diffusion via the framework of dyadic wavelet transform. With a noise adaptive

feature, our algorithm is versatile for both envelop-detected and log-compressed ul-

trasound image. We validate our method using synthetic speckle images and real

ultrasonic images. Performance improvement over other despeckling filters is quanti-

fied in terms of noise suppression and edge preservation indices.

We further extend the ultrasound statistical knowledge into the motion estima-

tion, and develop a speckle tracking algorithm for myocardial wall motion estima-

tion in intracardiac echocardiographic images. To achieve robust noise resistance,

we employ maximum likelihood estimation while fully exploiting ultrasound speckle

statistics, and treat the maximization of motion probability as the minimization of an

energy function. Non-rigid myocardial deformation is estimated by optimizing this

energy function within a framework of elastic registration. Accuracy of the method

is evaluated by using a computer model and an animal model, which provides con-

tinuous intracardiac echocardiographic images as well as reference measurements for

myocardial deformation. As a result, our approach achieves an accurate estimation of

regional myocardial deformation from intracardiac echocardiography. This approach

has important clinical implications for multimodal imaging during catheterization.

To my parents.

Acknowledgements

I would like to express my deeply gratitude to my advisor Dr. John W. Clark

Jr. First, for his constructive suggestion in writing papers. Second, for being open-

minded to give me a freedom that is so necessary for research. Finally, what I grateful

for the most is his outstanding guidance, patience, and dedication to my growth as a

researcher.

I would like to thank Dr. Dirar S. Khoury for his support, discussion, and guidance

to complete my second research project.

I would like to thank my thesis committee members Dr. Richard G. Baraniuk and

Dr. Fathi Ghorbel for their help and guidance.

I also would like to thank all my friends and the ECE professors for their help

and supports.

Finally, I would like to thank my parents, my wife, my baby, and my whole family

for their constant supports and encourage.

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Ultrasound Principles 6

2.1 Ultrasound Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Ultrasound Pulse . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Interactions with Matter . . . . . . . . . . . . . . . . . . . . . 8

2.2 Ultrasound Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Display Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Image Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Heart and Echocardiography 13

3.1 Heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Cardiac Conduction System and Electrocardiogram . . . . . . 15

3.1.2 Cardiac Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Echocardiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Transthoracic Echocardiography . . . . . . . . . . . . . . . . . 19

3.2.2 Intracardiac Echocardiography . . . . . . . . . . . . . . . . . . 21

3.2.3 Myocardial Segmentation and Nomenclature . . . . . . . . . . 22

4 Ultrasound Image Model 24

4.1 Statistical Model of Speckle . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.1 Analytic Ultrasound Signal . . . . . . . . . . . . . . . . . . . . 25

4.1.2 Speckle Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Ultrasound Image Model . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.1 Constructing the Ultrasound Image from the RF Signal . . . . 30

4.2.2 Constructing from Complex Tissue Scattering Function . . . . 31

4.2.3 Multiplicative Model . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.4 Dynamic Compressed Ultrasonic Image . . . . . . . . . . . . . 34

i

4.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.1 Simulation with RF Signals . . . . . . . . . . . . . . . . . . . 36

4.3.2 Simulation with the Multiplicative Model . . . . . . . . . . . . 37

4.3.3 Logarithmic Compressed Ultrasound Image . . . . . . . . . . 39

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Speckle Suppression in Ultrasound Images 41

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Wavelet Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2.1 Nonlinear Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2.2 Dyadic Wavelet Transform . . . . . . . . . . . . . . . . . . . . 46

5.2.3 Wavelet Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3 Speckle Suppression with Wavelet Diffusion . . . . . . . . . . . . . . 53

5.3.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3.2 Edge Detection with Normalized Modulus . . . . . . . . . . . 56

5.3.3 Diffusion Threshold . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4 Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.4.1 Denoising Results for the Simulated Image . . . . . . . . . . 69

5.4.2 Real Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6 Speckle Suppression for 3-D Ultrasound Images 83

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2 3-D nonlinear multiscale wavelet diffusion . . . . . . . . . . . . . . . . 84

6.3 Despeckling using 3-D NMWD . . . . . . . . . . . . . . . . . . . . . . 86

6.3.1 Normalized modulus . . . . . . . . . . . . . . . . . . . . . . . 87

6.3.2 Diffusion threshold . . . . . . . . . . . . . . . . . . . . . . . . 90

6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7 Speckle Tracking in Intracardiac Echocardiographic Images 98

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.2 Ultrasound Image Model . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.2.1 Tissue Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.2.2 Motion Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3 Maximum Likelihood Motion Estimation . . . . . . . . . . . . . . . . 107

ii

7.3.1 Image Sequences with Gaussian Noise . . . . . . . . . . . . . . 108

7.3.2 Ultrasound Image Sequences . . . . . . . . . . . . . . . . . . . 108

7.4 Ultrasound Elastic Speckle Tracking . . . . . . . . . . . . . . . . . . . 110

7.4.1 Robust Noise Resistance . . . . . . . . . . . . . . . . . . . . . 111

7.4.2 Deformable Registration . . . . . . . . . . . . . . . . . . . . . 113

7.4.3 Implementation Details . . . . . . . . . . . . . . . . . . . . . . 114

7.4.4 Motion field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.4.5 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.5 Experimental Validation by Computer Model . . . . . . . . . . . . . 118

7.5.1 Ultrasonic Image Phantom . . . . . . . . . . . . . . . . . . . . 118

7.5.2 Experiments on a Pair of Images . . . . . . . . . . . . . . . . 118

7.5.3 Experiments on Image Sequences . . . . . . . . . . . . . . . . 121

7.6 Experimental Validation by Animal Model . . . . . . . . . . . . . . . 125

7.7 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 129

8 Conclusion 134

iii

List of Figures

2.1 The resolution components in 3-D space [1]. . . . . . . . . . . . . . . 7

2.2 B-mode ultrasound imaging system [2]. . . . . . . . . . . . . . . . . 11

3.1 Anatomy of the human heart [3]. (a) External View. (b) Internal View. 14

3.2 (a) Conduction system of the heart, and (b) typical ECG during heart

cycle. [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Cardiac cycle. Phase 1. the isovolumetric contraction; 2. the ejection;

3. the isovolumetric relaxation; 4. the ventricular filling. [4] . . . . . . 17

3.4 Parasternal long axis view of a human heart. . . . . . . . . . . . . . 19

3.5 Parasternal short axial view of a human heart recording at the papillary

muscle level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6 Apical four-chamber view of a human heart. . . . . . . . . . . . . . . 21

3.7 An ICE image of a dog heart. . . . . . . . . . . . . . . . . . . . . . . 21

3.8 Standard definition of the left ventricular 17 segments by American

Heart Association. Left column: basal, mid, and apical-cavity of heart;

right column top: long axis view; bottom: a circumferential polar plot

of the 17-myocardial segments and the recommended nomenclature for

tomographic imaging of the heart (adapted from [5]). . . . . . . . . . 23

4.1 Simulation an ultrasound image of the short-axial view of a left ventricle. 36

4.2 The histogram and Rayleigh fitting of the background region in Fig. 4.1. . 36

4.3 Simulated ultrasound images with the multiplicative model. Speckle

was synthesized by (a) lowpass filtering method, and speckle model

with the number of scatterers (b) N=100, (c) N=3 per resolution cell 37

4.4 The histograms (solid, −), Rayleigh (dash, −−) and Nakagami (dash-

dot, −.) of speckle homogenous regions, generated with the scatterer

density (a) 100 and (b) 3. . . . . . . . . . . . . . . . . . . . . . . . . 38

4.5 The Nakagami parameter m for the different distributed speckle, gen-

erated by varying the scatterer density. . . . . . . . . . . . . . . . . 38

4.6 Dynamic compressed ultrasound images. (a) the logarithmic compressed

image of Fig. 4.1, (b) the result of (4.29) . . . . . . . . . . . . . . . . . 39

iv

4.7 The histogram and Fisher-Tippet fitting curves for the log-compressed

image Fig. 4.6(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1 Scheme for 3-level wavelet diffusion. Sjf and W dj f denote the filtered

wavelet coefficients at scale 2j. . . . . . . . . . . . . . . . . . . . . . 52

5.2 (a) Simulated envelope-detected speckle image. (b) Real echocardio-

graphic image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Histograms and the Rayleigh mixture model fitting of the normalized

modulus at scale 22 for the simulated envelope-detected speckle im-

age (top) and real echocardiographic image (bottom), shown in Figs.

5.2(a,b),respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.4 Diffusion thresholds λj(j = 1, 2, 3) estimated from the homogenous

region in Fig. 5.2(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.5 Classified homogenous speckle regions (white) at scale 21, 22, 23 (from

top to bottom) for the simulated envelop-detected speckle image (left

column) and real echocardiographic image (right column). . . . . . . 66

5.6 Denoising results for the simulated envelope-detected ultrasonic image

(Fig. 5.2(a)). (a) Echogeneity map. Results filtered by (b) GenLik,

(c) SRAD and (d) NMWD, respectively. . . . . . . . . . . . . . . . . 69

5.7 Denoising results for the simulated log-compressed ultrasonic image.

(a) Original image. Results filtered by (b) GenLik, (c) SRAD and (d)

NMWD, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.8 Image quality indices ρ (top) and FOM (bottom), after the simulated

envelope-detected image is filtered by NMWD with different values of

K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.9 Image quality indices ρ (top) and FOM (bottom), after the simulated

log-compressed image is filtered by NMWD with different values of K. 75

5.10 Denoising results for the echocardiographic image. (a) Original image.

Results filtered by (b) the GenLik method, (c) NMWD (K = 0.5), (d)

SRAD and (e) NMWD (K = 1.5), respectively. The profiles along the

highlight line of the original image (a) are shown in their following row. 78

5.11 Denoising results for a liver image. (a) Original image. Results filtered

by (b) the GenLik method, (c) SRAD and (d) NMWD, respectively.

The profiles along the highlight line of the original image (a) are shown

in their following row. . . . . . . . . . . . . . . . . . . . . . . . . . . 80

v

6.1 Histograms and fittings of normalized modulus at the first (top) and

second (bottom) scales of a 3-D liver image (shown in the first row of

Figure 6.2). The fittings are modeled by the Rayleigh-mixture (left)

and Maxwell-mixture (right), respectively. . . . . . . . . . . . . . . . 88

6.2 Top row: the arbitrary slices of a 3-D human liver ultrasound image

along YZ, XZ and XY planes (left to right). Bottom row: the corre-

sponding slices taken from the classified normalized modulus, where

homogenous speckle regions are shown in white and edges in black. . 92

6.3 (a) 3-D phantom, (b) synthetic ultrasound image, and the filtered re-

sults generated by (c) 3-D SRAD and (d) 3-D NMWD. . . . . . . . . 93

6.4 The slices of 3-D human liver ultrasound image along YZ, XZ and XY

planes (left to right). Top row: original image, middle: 3-D SRAD,

bottom: 3-D NMWD. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.5 Volume visualization of vessels in a 3-D liver image, which is pre-

processed with (a) lowpass, (b) BLTP, (c) SRAD and (d) NMWD. . 96

7.1 Two consecutive frames of ICE images acquired in mid left ventricle

of a dog. (a) Frame 1, and (b) Frame 2, where the recording stages

are indicated by N on ECG signals. (c) A region of myocardium in

(a). (d) A region of myocardium in (b) and corresponding to that

of (c). Going from (c) to (d), there is a [5,5] pixel shift toward the

right-bottom direction. . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.2 Correlation coefficients (a) and image quality as described by signal-

to-noise ratio (SNR) (b) in relation to speckle decorrelation index (λ). 105

7.3 Comparison of robustness between SSD and USST estimators. Top,

object functions of SSD and USST estimators in relation to residual

r. Bottom left, SSD influence function, Bottom right, USST influence

function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.4 Warped results of a pair of synthesized ultrasound images. (a) refer-

ence, and (b) test image. (c) superposition of (a) and (b). (d) true

deformation field. (e) and (f): warped results by the SSD method. (g)

and (h): warped results by the USST method. . . . . . . . . . . . . 119

7.5 Optimization processes of two registration methods: (a) SSD method,

(b) USST method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

vi

7.6 Histograms of average angular errors for sequences with different el-

evational speckle decorrelation index (λ). From top to bottom, λ =

0.0, 0.05, 0.1, 0.3, 0.6, 1.0. Left column: results of the SSD method, and

right column: results of the USST method. . . . . . . . . . . . . . . 123

7.7 Average angular error (a) and average magnitude errors (b) associ-

ated with SSD and USST methods, displayed as functions of speckle

decorrelation index λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.8 Schematic of locations of sonomicrometry crystals (marked by rect-

angles) in the left ventricle. Measures of circumferential and radial

distances are also illustrated. . . . . . . . . . . . . . . . . . . . . . . 126

7.9 Displacement determined by computed deformation field and sonomi-

crometry, where ‘R’ represents radial displacement, ‘C’ represents cir-

cumferential displacement, and ‘SM’ represents sonomicrometry. Dis-

placements are shown for both USST and SSD methods . . . . . . . . 127

7.10 ICE images in mid LV at end diastole (left) and end systole (right).

Arrows indicate displacement field of the LV myocardium during sys-

tole (left) and diastole (right). Intervals for displacement fields are

indicated by vertical bars on corresponding ECG. . . . . . . . . . . . 128

7.11 Radial and circumferential strains computed by the USST method at

three recording stages: baseline (BL), DOB1 and DOB2. Dobutamine

concentration in DOB1 was higher than DOB2. ‘R’ represents radial

strain, and ‘C’ represents circumferential strain. . . . . . . . . . . . . 129

7.12 Bland-Altman plots comparing circumferential strains as determined

by two methods: (a) SSD and sonomicrometry, and (b) USST and

sonomicrometry. Baseline (*), DOB1(4) and DOB2 (¤). . . . . . . . 130

vii

List of Tables

5.1 Correspondence between concepts used in different denoising techniques 54

5.2 Performance comparison for different denoising techniques . . . . . . 72

6.1 Despeckling Performance Comparison . . . . . . . . . . . . . . . . . . 94

viii

Chapter 1

Introduction

1.1 Motivation

Medical ultrasound is an important imaging modality in the clinical applications.

Compared to other medical imaging modalities, such as X-ray, MRI, and PET, diag-

nostic ultrasound imaging owes its great popularity to the fact that it is a safe and

non-invasive procedure for visualizing the interior of the body. Ultrasound provides

detailed imaging of soft tissues, which are difficult to depict using conventional X-ray

techniques. Unlike other tomographic techniques, ultrasound offers interactive visu-

alization of underlying anatomy in real time and can also image dynamically varying

structures within the body. Echocardiography, in particular, has evolved as a well-

established diagnostic imaging modality for cardiac function analysis, and it offers

significant advantages over the other imaging modalities in its safe, realtime, and

non-invasive evaluation of cardiac structure and ventricular wall motion.

Although ultrasound imaging has reached a high level of technical sophistica-

tion, speckle noise is a fundamental issue that has to be addressed in interpreting

ultrasound images. Solution of this problem is critical to further advancement of

the diagnostic capability of this imaging modality. A key problem is that speckle

originates from the same source as the signal, and hence is an inherent property of

1

the ultrasound image itself. Historically, speckle has been considered an undesirable

noise source and techniques have been developed to minimize its effects. It also has

been considered as a signal that carries information regarding the underlying scatter-

ing characteristics of the target. In speckle analysis, the particular approach taken

depends significantly on the application.

In applications to the areas of image visualization and auto-segmentation, speckle

is considered a contaminating factor that severely degrades image quality. Speckle

reduction is important in these applications. Most speckle filters are developed for the

purpose of enhancing visualization, and texture recovery is a highly desired feature of

the filtering process. In contrast, the goal of auto-segmentation favors image simpli-

fication. In this case, texture removal significantly improves the speed and accuracy

of automated object detection. Speckle filters designed for texture recovery have a

rather limited application in auto-segmentation improvement. Consequently, a versa-

tile speckle suppression algorithm is in order, one that mainly focuses on producing

a denoising result for auto-segmentation improvement, but is also able to provide

visualization enhancement.

Another important application of ultrasound imaging is the quantitative assess-

ment of tissue motion. Specifically, myocardial motion analysis utilizing an echocar-

diographic image sequence has become a major diagnostic tool for the identification

of pathological abnormalities, such as myocardial ischemia and infarction. In clinical

practice, the analysis mainly rests on visual inspection or manual measurements by ex-

perienced cardiologists. Likewise, the objective of an automated myocardial function

analysis system is the accurate detection of regions of mechanical malfunction attend-

ing myocardial infarction. Conventional motion estimation algorithms are developed

2

for general image sequences, which are assumed to be corrupted by Gaussian noise.

When these methods are applied to echocardiographic motion estimation, speckle

is considered as a spatial marker for the underlying tissue structure, and speckle

patterns of moving tissue are assumed stable under conditions of small amplitude

motion. Such temporal constancy is usually not valid in actual echocardiographic

image sequences due to non-uniform myocardial motion and speckle decorrelation.

Even for a myocardium with uniform structural/perfusion properties, its texture in

ultrasound images may appear different from frame to frame. Random variation of

the speckle pattern obviously complicates any tracking algorithm that relies on tex-

ture constancy. A prospective solution to this problem is to appropriately integrate

the speckle statistics into the motion estimation algorithm.

In this thesis work, we concentrate on the improvements of interpretation of ul-

trasound images by appropriately controlling the role of speckle in the analysis. We

separate our work into two major tasks: speckle suppression and myocardial motion

analysis. In the first task, we consider speckle as a heavy contaminating factor, and

develop a novel method to suppress speckle while enhancing edge of image struc-

tures. In the second task, we incorporate the statistical knowledge of speckle into a

motion estimation scheme, and develop a new speckle tracking algorithm to estimate

myocardial wall motion in echocardiographic images.

1.2 Thesis Organization

The thesis is organized as follows. Following this general introduction, Chapter 2

proceeds with a review of the basic principles of ultrasound and the instrumentation

3

techniques of ultrasound imaging.

Chapter 3 reviews the heart anatomy and cardiac physiology, and introduces tech-

nical terms that are frequently used throughout this work. An overview of echocar-

diography is given, as well as an introduction to the standard two-dimensional views

(acquired during clinical examinations). The echocardiographic nomenclature of the

myocardial segments is introduced in the last section.

In Chapter 4, we first review the statistics of the backscattering ultrasound signals,

and introduce the statistical models for both the envelope-detected and dynamic range

compressed speckle. Based on this statistical analysis, we provide several ultrasound

image simulation approaches, which are used for performance evaluation in our later

algorithm development.

Subsequent chapters present the scientific contributions of this work. Beginning

in Chapter 5, we present a novel speckle suppression method for the purposes of vi-

sualization enhancement and auto-segmentation improvement of an image. In this

approach, speckle is iteratively reduced by the nonlinear diffusivity function via the

framework of the dyadic wavelet transform. In our approach, we use the normalized

wavelet modulus to expose the intrinsic speckle/edge relation. Relying on the sta-

tistical analysis of this edge map, our method is able to classify homogenous speckle

regions in the image, and provide strong speckle suppression and boundary preser-

vation. With a noise-adaptive feature, the resultant algorithm is versatile for both

envelope-detected and log-compressed speckle images.

In Chapter 6, we extend our speckle suppression method to 3-D ultrasound image.

The method aims to enhance volume visualization of 3-D ultrasound images, and to

improve the accuracy of volume measurement. As an extension of 2-D nonlinear

4

multiscale wavelet diffusion, the proposed method is also developed on the basis of

integration of 3-D nonlinear diffusion and 3-D dyadic wavelet transform. We validate

our method using synthetic and real 3-D ultrasonic images. Performance improvement

over other filters is quantified by quality indices and the volume rendering technique.

In Chapter 7, we present a speckle tracking approach for myocardial motion es-

timation using intracardiac echocardiographic images. Our method incorporates the

statistical features of ultrasonic images into a maximum likelihood motion analysis,

and treats maximization of the similarity measure as an energy minimization. Within

the framework of deformable registration, tissue motion is estimated via optimization

of a speckle-featured energy function. Accuracy of the method was initially evaluated

by using a computer model that synthesized echocardiographic image sequences, and

subsequently by an animal model that provided continuous intracardiac echocardio-

graphic images as well as reference measurements for myocardial deformation. In

conclusion, accurate estimation of regional myocardial deformation from intracardiac

echocardiography by novel speckle tracking is feasible. This approach has important

clinical implications for multimodal imaging during catheterization.

In Chapter 8, we draw conclusions for our work.

5

Chapter 2

Ultrasound Principles

Diagnostic ultrasound employs high frequency ultrasound waves to image tissue

structures and their motion. Ultrasound imaging is fundamentally a non-reconstructive

imaging process wherein image information is obtained by localizing an ultrasound

echo signal reflected from a scattering medium. In this chapter, we review the ultra-

sound physics and basic principles of image formation.

2.1 Ultrasound Physics

An ultrasound wave is a form of mechanical energy that propagates through a

medium by compression and rarefaction [1, 2]. Ultrasonic frequency is unaffected by

changes in speed as the acoustic wave propagates through various media, therefore

the ultrasound wavelength is dependent on the acoustic properties of medium. Ultra-

sound wavelength determines the spatial resolution achievable along the direction of

the beam. A high-frequency ultrasound beam (small wavelength) provides superior

resolution and image detail compared with a low-frequency beam. However, the depth

of beam penetration is reduced at high frequency and increased at low frequencies

due to attenuation (discussed in 2.1.3).

6

Figure 2.1: The resolution components in 3-D space [1].

2.1.1 Ultrasound Pulse

The emitted ultrasound pulse is the impulse function of the system. Correspond-

ingly, the received echo pulse can be considered as the impulse response of the bi-

ological medium. When it represents the output of the ultrasound system during

interrogation of an ideal point target, the echo pulse is also known as the system’s

point spread function (PSF). The character of the PSF in the axial dimension is de-

termined predominantly by the center frequency and bandwidth of the acoustic signal

generated at each transducer element, whereas its character in the lateral and eleva-

tion dimensions is determined predominantly by the aperture and element geometries

and the beamforming applied.

2.1.2 Resolution

In ultrasound system, the axial, lateral and elevational (slice thickness) dimensions

determine the spatial resolution and visibility of the system. The definitions of three

components are illustrated in Fig. 2.1.

Axial resolution defines the ability of the ultrasound pulse to differentiate between

7

two closely spaced objects that lie along the axis of an ultrasound beam. Axial

resolution is determined by the spatial pulse length (SPL), which depends on the

number of cycles within a pulse and on the length of each cycle. Achieving good

axial resolution requires that the returning echoes can be distinct without overlap.

Typical axial resolution is 0.5mm. Higher frequencies reduce SPL and improve axial

resolution. However, this is at the expense of increased signal attenuation.

Lateral (azimuthal) resolution defines the ability to resolve adjacent objects per-

pendicular to the beam direction. Lateral resolution is determined by the beam width

(diameter), and is also depth-dependent. Typical lateral resolution is 2-5 mm.

Elevational resolution is dependent on the transducer element height. Use of

a fixed focal length lens across the entire surface of the array provides improved

elevational resolution at the focal distance, however partial volume effects may appear

before and after focal zone.

2.1.3 Interactions with Matter

When ultrasonic waves propagate through a medium, some effects may occur, in-

cluding reflection, refraction, scattering and attenuation [2]. A particular interaction

is determined by the acoustic properties of matter and the PSF of the system.

Reflection

Sound reflection occurs at tissue boundaries with differences in acoustic impedance

Z = ρc , where ρ is the density of the tissue, and c is the sound speed. When a sound

wave is directly incident on a boundary between two media with acoustic impedances

Z1 and Z2, the ratio of incident to reflected pressure is predicted by the reflection

8

coefficient R, defined as:

R =Z2 − Z1

Z2 + Z1

.

The sound reflected back toward the source is called an echo. The magnitude of a

surface reflection is dependent on the relative impedances between tissues, not the

absolute value of the individual impedances. For example, air/tissue interfaces can

reflect most of incident ultrasound beam. To minimize the large reflections, a regular

pre-examination procedure is to apply gel between the surfaces of the scanning area

and transducer.

Scattering

Scattering refers to the interaction of the ultrasound wave with microstructures

that are much smaller than its wavelength. Arising from the spatial arrangement of

the scatterers, there are two types of scattering. If the scatterers have a periodic

arrangement, it results in the coherent scattering, producing periodicity in the echo

spectrum. If the scatterers are spatially random distributed, it leads to the diffuse

scattering. The diffuse scattering further gives rise to speckle in the ultrasound

image [2].

Attenuation

As a sound wave passes through the tissue, it progressively loses energy, and is

transformed into other energy forms, such as heat. Amplitude attenuation is primar-

ily caused by the inner friction or viscosity of the tissue. Signal attenuation depends

highly on the carrier frequency. Higher frequencies allow a better spatial resolution,

but are more attenuated than lower ones and thus have less penetrating ability. Con-

9

versely, a lower frequency transducer has a greater depth of penetration but poorer

resolution. The optimal carrier frequency is a trade-off between the requirements

of penetration depth and image resolution. Frequencies generally used in diagnostic

ultrasound range from 3.5 to 10 MHz.

2.2 Ultrasound Imaging

A typical ultrasound system (Fig. 2.2) includes the transducer, the signal pro-

cessing device, and the display device. The ultrasound transducer uses an array of

piezoelectric elements to transmit a sound pulse into the body and to receive the

echoes that return from scattering structures within.

Basic principle of ultrasound imaging can be generalized as: to emit pulses, and

to collect reflected echoes. In the imaging system, the strength or amplitude of each

reflected wave is represented by a dot. These dots are combined to form a complete

image. The brightness of the dot represents the strength of the returning echo. The

position of the dot represents the depth from which the returning echo was received.

2.2.1 Display Modes

The detected echoes may be displayed in one-dimensional formats such as ampli-

tude mode (A-mode), brightness mode (B-mode) or motion mode (M-mode) formats.

These different display modes are briefly described in the following [2].

A-mode displays echo amplitude versus time (depth), and is used when accurate

distance measurements are required (e.g. in ophthalmology).

B-mode is the electronic conversion of the A-mode. In this mode, a line of

brightness-modulated dots is displayed, and the line represents the orientation of

10

Figure 2.2: B-mode ultrasound imaging system [2].

the transducer.

M-mode is B-mode set in motion, and is used to display time evolution versus

depth. Sequential B-mode lines are displayed adjacent to each other, allowing vi-

sualization of interface motion. M-mode is valuable for studying the movement of

structures within the heart, such as valves and the ventricular walls.

An ultrasonic scanner generally operates in B-mode, and presents a gray-scale

image that represents a spatial map of echo amplitude. In the B-mode image, white

dots represent strong reflections, e.g., the reflection caused by diaphragm, gallstones

and bones; grey dots denote weaker reflections, e.g., solid organs and thick fluid; and

black dots indicate no reflection, e.g., fluid within a cyst.

11

2.2.2 Image Artifacts

Ultrasound image artifacts arise from an incorrect display of anatomy or noise

during imaging. Incorrect anatomical imaging can cause shadowing, reverberation,

and speed displacement artifacts. However, in this study, we are mainly concerned

with the system noise artifact, speckle. Speckle is a textured appearance that results

from small, closely-spaced structures that are too small to be resolved by the PSF.

Speckle therefore is the result of diffuse scattering, and it can considered as an inherent

property of the ultrasound image. Speckle generally does not reflect the structure of

the underlying tissue. The regional mean brightness of texture pattern, however,

reflects the regional echogenicity of tissue. Therefore, speckle can be considered as

noise, since it may obscure structures in medium under observation.

12

Chapter 3

Heart and Echocardiography

For a better understanding of the remainder of this dissertation, a brief review of

the anatomy and function of the human cardiovascular system as well as echocardio-

graphy is given in this chapter.

3.1 Heart

The heart is a muscular organ that is located between the lungs in the middle of

the chest, behind and slightly to the left of the sternum [6]. The heart is enveloped

in two layered fluid-filled sac, called the pericardium. The outer layer is the parietal

pericardium, and the inner layer is the visceral pericardium or epicardium. Both lay-

ers secrete the pericardial fluid, which lubricates the heart during motion. The heart

itself is composed of an interior surface, called the endocardium. The endocardium

consists of a layer of endothelial cells and an underlying layer of connective tissue.

Most of the heart is made of cardiac muscle tissue, called the myocardium.

As illustrated in Fig. 3.1, the heart is divided into four muscular chambers:

the left and right ventricles, and the left and right atria. The left ventricle is an

axis-symmetric conical shaped chamber, and the right ventricle is a roughly crescent

shaped chamber. The left and right ventricles are separated by the interventricular

13

(a) (b)

Figure 3.1: Anatomy of the human heart [3]. (a) External View. (b) Internal View.

septum, and the atria also have an the interatrial septum. Functionally, the heart is

separated as left and right heart pumps. The left heart, composed of the left atrium

(LA) and left ventricle (LV), pumps blood from the pulmonary veins to the aorta.

The right heart, composed of the right atrium (RA) and right ventricle (RV), moves

blood from the vena cavae to the pulmonary arteries. The ventricle are the major

pumping chambers that deliver blood to pulmonary and systemic circulations. The

atria receive venous blood, and also function as small pumps to assist ventricular

filling.

Four pressure-operated valves control the direction of blood flow by preventing

backward flow during the contraction of ventricles. The atrioventricular (AV) valves

separate each atrium from its associated ventricle. The left AV valve or mitral valve

has two flaps, and the right AV valve or tricuspid valve has three flaps. The free

ends of AV valves attach via the chordae tendinae to papillary muscles which emerge

14

(a) (b)

Figure 3.2: (a) Conduction system of the heart, and (b) typical ECG during heart cycle. [3]

from ventricular walls. The operation of valves is determined by the pressure gradient

between the atrium and the ventricle. The papillary muscles contract synchronously

with the myocardium to help preventing backward flow during the contraction. The

semilunar valves, positioned on the pulmonary artery and the aorta, separate each

ventricles from its connected artery and prevent the backward flow from the artery.

3.1.1 Cardiac Conduction System and Electrocardiogram

The heart beat originates in a cardiac conduction system and spread via this sys-

tem to all parts of the myocardium. As shown in Fig. 3.2(a), the structure that make

up the conduction system are the sinoatrial node (SA node), the internodal atrial

pathways, the atrioventricular node (AV node), the bundle of His and the Purkinje

system. The electrocardiogram (ECG) is a recording of the electrical fluctuations of

the myocardium during the cardiac cycle. A typical ECG is shown in Fig. 3.2(b). In

the cardiac cycle, the SA node normally discharges most rapidly. Impulses generated

in the SA node rapidly spreads across the right and the left atria causing them to

contract. The onset of this atrial activity generates the P wave of the ECG. The SA

15

node is also connected to a set of special internodal fibers that convey an activity

signal from the SA node to the AV node, which is located in the septal wall of the

right atrium. This system of fibers is part of the specialized conduction system within

the heart. The AV node thus receives input from three such internodal pathways (su-

perior, middle, inferior), as well as the right atrial musculature. Additional special

conduction fibers (purkinje fibers) connect the AV node to the inner surface of the

ventricular walls. The output of the AV node conveys to the Bundle of His, which

branches downstream into a right and a left bundle branch (RBB and LBB). The

bundle branches convey the activation signal to the ventricular walls. The Purkinje

fibers terminate on the ventricular walls of the myocardium and consequently spread

activation through the wall. This depolarization generates the QRS complex of the

ECG. The ventricular repolarization produce T wave of ECG, which is longer than

QRS complex but smaller in amplitude since the ventricular repolarization is less well

synchronized than the ventricular depolarization.

3.1.2 Cardiac Cycle

The heart cycle is a pumping action that is divided into two major alternating

phases: systole and diastole. Systole is the period of time during which the mus-

cle transforms from its totally relaxed state to the instant of maximal mechanical

activation. Diastole is the period of time during which the muscle relaxes from the

end-systolic state back towards its resting state. Figure 3.3 demonstrates the relation-

ship between the ECG signal, the cardiac pressure and ventricular volume during the

cardiac cycle. The cycle can be further characterized into four subphases by tracking

the pressures and volumes in the ventricle as a function of time. Systole includes the

16

Figure 3.3: Cardiac cycle. Phase 1. the isovolumetric contraction; 2. the ejection; 3. theisovolumetric relaxation; 4. the ventricular filling. [4]

isovolumic contraction and ejection phases, and diastole includes isovolumic relax-

ation and filling phases. The details of four phases are depicted as following:

1. The isovolumic contraction (Phase 1 in Fig. 3.3) is initiated by the ventricular

depolarization. During this period, all four valves are closed and the volume of

the ventricles remains constant. However, due to myocardial activation, there

is a rapid increase in the ventricular pressure.

2. The ventricular ejection happens when the ventricular pressures exceed aortic

17

and pulmonary artery pressures. The increase of pressures opens outlet valves,

then blood is ejected from the ventricles. As the contraction process of the

cardiac muscle reaches its maximal effort, the ventricular tension is reduced

and ejection slows down. Blood however continues to flow until ventricular

pressure falls below arterial pressure.

3. In the isovolumic relaxation, the ventricular pressures decrease. However, atrial

pressures continue to rise due to venous return. Ventricular volumes remains

constant, since all four valves are closed in this interval.

4. The ventricular filling occurs when the ventricular pressures fall bellow the atrial

pressures. The reduced pressure opens the AV valves. The rapid flow of blood

causes a rapid fall in the atrial pressure. As filling proceeds, the ventricular

pressures rise as the ventricles fill with blood. This reduces the pressure across

the AV valves, and the rate of filling decreases.

3.2 Echocardiography

Echocardiography is a procedure using ultrasonic compression waves applied to

the chest wall to obtain a graphic record of the heart’s position, or the motion of

parts such as ventricular walls and valves. There are several modes of data acqui-

sition distinguished from different anatomic location, such as: transthoracic (TTE),

transesophageal (TPE) and intracardiac echocardiography (ICE) [7]. Identified by

their spatial imaging capability, echocardiographic techniques can be also recognized

by terms, such as 2-D and 3-D echocardiography. Among them, 2-D transthoracic

echocardiography is a widely used ultrasound imaging modality in clinical diagnosis,

18

Figure 3.4: Parasternal long axis view of a human heart.

often referred as the echocardiography standard. Generally, TTE is limited by the

acquisition positions (windows) of the transducer. These problems of windows for

data acquisition are not presented with TPE and ICE, since imaging is done within

the body either at the esophagus or within the left ventricle via catheter.

3.2.1 Transthoracic Echocardiography

In the scanning process of TTE, the ultrasound probe is placed on the chest wall

of the patient, and images are displayed on the image console. In this mode, there

are four standard acoustic windows through which the heart can be interrogated

transthoracically, including parasternal, apical, subcostal and suprasternal windows.

The suprasternal window is well suited for imaging the aorta while the subcostal win-

dow allows to image the interatrial septum and the inferior vena cava. The parasternal

and apical windows permit the heart to be scanned along its long or short axis, respec-

tively. Therefore, these two windows are primarily used for the analysis of ventricular

function.

The parasternal window is a series of small apertures that lie to the the immediate

19

Figure 3.5: Parasternal short axial view of a human heart recording at the papillary musclelevel.

left of the sternum at levels of the third, fourth, and fifth intercostal spaces. The long

axis planes of parasternal window can provides clear views of long axis of the left

heart, the right ventricular inflow/outflow tract, the main pulmonary artery, and

the cardiac apex. Among them, the parasternal long axis of the left heart is the

most important and most frequently scanned image planes. As shown in Fig. 3.4, it

includes most of the cardiac structures of left heart: the left ventricle, left atrium,

mitral vale, aortic valve, and interventricular septum. There are four standard short

axis planes of parasternal windows. These planes are the short axis of the aorta and

left atrium, the left ventricle at the mitral valve level, the left ventricle at the papillary

muscle level, and the left ventricular apex. Figure 3.5 is an example of a parasternal

short axial recording at the papillary muscle level.

Apical acquisition position varies from patient to patient, and can be located

by palpation. There are four primary apical view: the apical four-chamber, the

apical five-chamber, the apical two-chamber, and the apical long-axis views of the

left ventricle. The apical four chamber view is acquired with the transducer located

20

Figure 3.6: Apical four-chamber view of a human heart.

Figure 3.7: An ICE image of a dog heart.

directly over the anatomic cardiac apex. As shown in Fig. 3.6, it includes primary

cardiac structures of four chambers and permits the evaluation of their relative sizes,

orientation, and structural integrity.

3.2.2 Intracardiac Echocardiography

Intracardiac echocardiography (ICE) has been accepted as a high spatial resolution

imaging modality for the diagnosis of cardiac structure and function [8]. It was

21

first developed on the technical basis of intravascular ultrasound (IVUS) [7], which

visualizes the structure of vessel walls using a catheter-based image system. The ICE

catheter has a distal transducer that emits and receives ultrasound pulses. To acquire

images, a ICE catheter has to be guided into the ventricle through the great vessel.

Tomographic views of the cavity are acquired by attaching the ICE catheter to a motor

drive unit that enables automatic and continuous rotation of the transducer at a fixed

speed. The motor unit is fitted with a custom computer-controlled pullback device

with an optical sensor that enabled external automatic and accurate withdrawal of the

ICE catheter in desired increments. The ICE catheter and motor unit are connected

to an imaging console to acquire continuous echocardiographic images. Figure 3.7

shows an ICE image of the LV of a dog.

3.2.3 Myocardial Segmentation and Nomenclature

The Cardiac Imaging Committee of the Council on Clinical Cardiology of the

American Heart Association has published the standard myocardial segmentation

and nomenclature for tomographic imaging of the heart [5]. In this standard, the

heart is divided into 17 segments for assessment of the myocardium and the left ven-

tricular cavity (shown in Fig. 3.8). With the definition of the apex segment, this

standard is different with 16-segment model of the American Society of Echocardiog-

raphy (ASE) [9]. Since the myocardial apex segment or apical cap becomes pertinent

in the assessment of myocardial perfusion, a 17-segment model is applicable for both

the assessment of wall motion and myocardial perfusion using echocardiography. For

regional analysis of left ventricular function or myocardial perfusion, the left ventricle

is divided into three circular short axis slices: basal, mid-cavity and apical slices. The

22

Figure 3.8: Standard definition of the left ventricular 17 segments by American HeartAssociation. Left column: basal, mid, and apical-cavity of heart; right column top: longaxis view; bottom: a circumferential polar plot of the 17-myocardial segments and therecommended nomenclature for tomographic imaging of the heart (adapted from [5]).

names for the myocardial segments are defined based the location relative to the long

axis of the heart and the circumferential location (Fig. 3.8). The circumferential seg-

ments of the basal and mid-cavity are anterior (front wall), anteroseptal, inferoseptal,

inferior, inferolateral, and anterolateral. The apical cavity is circumferentially divided

into anterior, septal, lateral, inferior segments.

23

Chapter 4

Ultrasound Image Model

A diagnostic ultrasound B-mode image is a collection of ultrasound echoes, result-

ing from the interference between the ultrasound pulses and the scanned tissue. An

ultrasound image generally features with the granular texture, called speckle. Speckle

patterns are a characteristic feature of coherent imaging system, and carry informa-

tion about the unresolvable scattering structure. Therefore, speckle is considered as

an inherent property of the ultrasound image. Speckle analysis has been a major

subject in the ultrasound image processing, interpretation, and simulation. In this

chapter, we review the statistical model of speckle, and introduce several ultrasound

image models.

4.1 Statistical Model of Speckle

A number of groups have investigated the envelope of the backscattered echo

as a source of information about the underlying scattering characteristics in tissue.

Early work on ultrasound speckle research adopted the approach taken in Goodman’s

work on coherent laser imaging [10,11], where tissue was considered as a collection of

randomly located scatterers, and speckle was modeled as having a Rayleigh or Rice

distribution [12]. For the non-Rayleigh distribution cases, more recent studies show

24

that general speckle models, e.g. K distribution [13, 14], Nakagami distribution [15]

are more suited for the limited scattered density or phase shift.

4.1.1 Analytic Ultrasound Signal

Given a carrier signal (ultrasound pulse) with the phasor exp(jωct) and ωc is the

center frequency of transmission, the ultrasound backscattering echo s(x, y, z; t) can

be expressed in the analytic form [11,16]:

s(x, y, z; t) = a(x, y, z)ejωct. (4.1)

Here, a(x, y, z) is the complex phasor amplitude, such as,

a(x, y, z) = ar(x, y, z) + jai(x, y, z) (4.2)

where ar(x, y, z) is the real sequence, and ai(x, y, z) is the Hilbert transform of

ar(x, y, z). Alternatively, a(x, y, z) can also be represented in terms of magnitude

and phase, i.e.,

a(x, y, z) = A(x, y, z) exp jφ(x, y, z) (4.3)

where A(x, y, z) =√

a2r(x, y, z) + a2

i (x, y, z) and φ(x, y, z) = arctan ai(x,y,z)ar(x,y,z)

. Equa-

tion (4.3) can be expressed in the terms of the contribution of each scatterer in the

resolution cell

a(x, y, z) =1√N

N∑

k=1

ak(x, y, z) =1√N

N∑

k=1

Ak(x, y, z)ejφk(x,y,z) (4.4)

25

where ak(x, y, z) is the complex phasor of kth scatterer, N is the number of scatterers

in the resolution cell, and Ak(x, y, z) is its corresponding amplitude. Combining (4.2)

and (4.4), we have

ar(x, y, z) =1√N

N∑

k=1

Ak(x, y, z) cos(φk(x, y, z)) (4.5)

and

ai(x, y, z) =1√N

N∑

k=1

Ak(x, y, z) sin(φk(x, y, z)). (4.6)

The ultrasound backscattering echo s(x, y, z; t) also can be expressed as

s(x, y, z; t) = sr(x, y, z; t) + jsi(x, y, z; t) (4.7)

where sr(x, y, z; t) is the real sequence, and si(x, y, z; t) is the Hilbert transform of

sr(x, y, z; t). From equations (4.1),(4.2) and (4.7), we obtain

sr(x, y, z; t) = ar(x, y, z) cos(ωct)− ai(x, y, z) sin(ωct). (4.8)

After expanding using (4.5) and (4.6), we obtain

sr(x, y, z; t) =N∑

k=1

Ak(x, y, z)√N

cos(ωct + φk(x, y, z)). (4.9)

Similarly, we have

si(x, y, z; t) =N∑

k=1

Ak(x, y, z)√N

sin(ωct + φk(x, y, z)). (4.10)

26

The signal sr is called the backscattered echo, or radio-frequency (RF) signal.

4.1.2 Speckle Statistics

The time varying component of the phasor, ωct, does not affect the statistics of

the signal, hence we drop it in the statistical study of sr in (4.9) and si in (4.10).

Rather, the real and imaginary parts of the complex phasor, ar and ai in (4.5) and

(4.6), determine the statistics of backscattered echo s(x, y, z; t). If φk is uniformly

distributed over [−π, π], the mean values 〈ar〉 = 〈ai〉 = 0 due to 〈cos φk〉=〈sin φk〉 = 0.

Similarly, the second moments of real and imaginary parts is given by

〈|ar|2〉 = 〈|ai|2〉 =1

N

N∑

k=1

|Ak|22

.

It is fair to assume that the real and imaginary parts are uncorrelated, such as 〈arai〉 =

0. If the number of scatterers are sufficiently large (N → ∞) to satisfy the central

limit theorem, the joint distribution of ar and ai is Gaussian distributed

pr,i(x, y) =1

2πσ2exp

(x2 + y2

2σ2

)

where σ2 = limN→∞

1N

∑Nk=1

|Ak|22

.

The amplitude of the complex phasor is given by

A =√

a2r + a2

i .

27

The probability density function (PDF) of A is Rayleigh distributed,

pA(x) =x

σ2exp

(− x2

2σ2

).

The signal-to-noise ratio (SNR) equals 1.91. This model assumes a large number of

spatially uniformly distributed scatterers.

When a coherent component is introduced to speckle, it adds a constant phasor

A0 to the scattered echoes (4.4)

a(x, y, z) = A0 +1√N

N∑

k=1

Ak(x, y, z)ejφk(x,y,z). (4.11)

As a consequence, it leads to unresolved periodically varying scattering. Upon detec-

tion, this has the effect of changing the Rayleigh into a Rice distribution:

pA(x) =x

σ2exp

(−x2 + A2

0

2σ2

)I0

(A0x

σ2

), x ≥ 0 (4.12)

where I0(.) is zero order modified Bessel function of the first kind.

The Rayleigh density function, and its extension, the Rice density function, pro-

vide a general model for the backscattered echo signals when the scatterer density

is very large. If the scatterer density is limited, both the Rayleigh and Rice mod-

els are no longer valid. Due to the similarity of the speckle generation mechanism,

most ultrasound speckle noise models originated from models of the radar system,

where the target signal arrives at the receiver after being reflected, scattered and

diffracted from objects in the intervening medium. Several general noise models (e.g.

K and Nakagami distributions) have been developed to model speckle in radar sys-

28

tems [17, 18]. Recently, these models have been utilized to study ultrasound speckle

noise [13, 15, 19, 20]. Within this class of models, the Nakagami and K distributions

more closely approximate actual image noise.

When the number of scatterers N is small (compared to Rayleigh model), speckle

can be modeled as a K distribution [13]

pA(x) =2b

Γ(α)

(bx

2

)α

Kα−1(bx), x ≥ 0, α > 0, b > 0 (4.13)

where b =√

4α/E(x2), α is the shape parameter and Kα−1 is the modified Bessel

function of the second kind and order α−1. The computation of parameters of the K

distribution is much complicated, and K distribution can’t incorporate post-Rayleigh

(Rician) statistics.

Recently, another general envelope statistical model, the Nakagami model, has

been developed [15]:

pA(x) =2mm

Γ(m)Ωmx(2m−1) exp

(−mx2

Ω

), x ≥ 0,m ≥ 1/2, Ω > 0 (4.14)

where Γ denotes the gamma function, and m is the Nakagami parameter, which de-

notes the effective scatterer density. We note that speckle is pre-Rayleigh distributed

for m < 1; for m = 1, it has a Rayleigh distribution; and for m > 1, it is similar to

a Rician distribution [15]. Compared with other models, the Nakagami distribution

has the advantage of computational simplicity and accuracy [21]. Therefore, we use

the Nakagami model to evaluate the statistical reliability of speckle in our study.

29

4.2 Ultrasound Image Model

In general medical applications, speckle is considered as noise that severely de-

grades the image quality. A well-founded ultrasound image model is an important

tool to help evaluate the performance of the signal-processing algorithms for the ul-

trasound images. In this section, we introduce several ultrasound image models.

4.2.1 Constructing the Ultrasound Image from the RF Signal

The RF echographic signal usually is modeled as the convolution of ultrasound

point spread function (PSF) h(x, y, z) and the tissue scattering function t(x, y, z) in

the real plane [22]:

sr(x, y, z) = h(x, y, z) ∗ t(x, y, z) (4.15)

where ∗ denotes the convolution operation. The tissue scattering function represents

the tissue properties along the propagation direction of ultrasound pulse. A simple

tissue model is given by

t(x, y, z) =∑

n

gnδ(x− xn, y − yn, z − zn) (4.16)

where δ(·) is the Dirac function, (xn, yn, zn) is the center of each inhomogeneity, and

gn is the echogenicity of the each scatterer. To model a 2-D ultrasound image, PSF

is considered to be separable, h(x, y, z) = h(x, y) ∗ hz(z). Therefore, 2-D slice of RF

is obtained as [22]

sr(x, y) = h(x, y) ∗ t(x, y) (4.17)

30

where t(x, y) =∫

t(x, y, z)hz(−z)dz. If the scatterers in the resolution cell are ran-

domly distributed, and their number is sufficiently large to satisfy the central limit

theorem, the scattering function t(x, y) can be modeled as Gaussian distributed. The

PSF h(·) can be obtained from the experimental measurement or simulation. A simple

PSF can be a low-pass filter, or Gabor function

h(x, y) =1

2πσxσy

exp

(− x2

2σ2x

− y2

2σ2y

+ iωcx

).

The envelope-detected ultrasound image is expressed as the magnitude of the

backscattered signal

f(x, y) = |sr(x, y) + jsi(x, y)| (4.18)

where the imaginary part si(x, y) is the Hilbert transform of sr.

4.2.2 Constructing from Complex Tissue Scattering Function

If the scattering function in (4.15) is defined in 2-D complex plane, the ultrasound

analytic signal can be expressed as [12]

s(x, y) = h(x, y) ∗ T (x, y). (4.19)

Here, T (x, y) is a hypothetical 2-D scattering function defined in the complex plane,

T (x, y) = t(x, y) + jt(x, y) (4.20)

31

where t(x, y) is the Hilbert transform of the tissue scattering function t(x, y). Similar

to (4.18), the envelope-detected ultrasound image can be generated by the magnitude

of s(x, y).

4.2.3 Multiplicative Model

Let’s expand the problem further. The complex tissue scattering T (x, y) in (4.20)

can also be expressed in terms of the amplitude G(x, y) and phase ψ(x, y), i.e.,

T (x, y) = G(x, y) exp (jψ(x, y)) (4.21)

where ψ(x, y) = arctan t(x,y)t(x,y)

and ψ(x, y) is uniform distributed over [−π, π]. Consid-

ering the scattering within the resolution cell, with (4.18) and (4.21), the envelop-

detected image can be express as

f(x, y) = |h(x, y) ∗ (G(x, y) exp jψ(x, y))|

=

∣∣∣∣∫ ∫

Rcell

h(x− x′, y − y′)G(x′, y′) exp (jψ(x′, y′)dx′dy′∣∣∣∣

≈ |G(x, y)| ·∣∣∣∣∫ ∫

Rcell

h(x− x′, y − y′) exp (jψ(x′, y′)dx′dy′∣∣∣∣. (4.22)

In (4.22), the amplitude of scattering function G(x, y) is assumed not to change

appreciably within the resolution cell, i.e., the impulse response decays rapidly outside

the resolution cell. We further define g(x, y) ≡ |G(x, y)| and

n(x, y) ≡∣∣∣∣∫ ∫

Rcell

h(x− x′, y − y′) exp (jψ(x′, y′)dx′dy′∣∣∣∣ (4.23)

32

Then, (4.22) can be reformulated as

f(x, y) = g(x, y)n(x, y), (4.24)

where g(x, y) represents the echogenicity or ground truth of the image, and n(x, y)

denotes the noise term. Equation (4.24) is in the multiplicative form, which has been

generally used to model the speckle image [10,23].

For a given resolution cell, speckle is generated by the discrete form of (4.23)

n[m,n] =

∣∣∣∣N1∑

k=1

N2∑

l=1

h(m− k, n− l) exp (jψ(k, l))

∣∣∣∣ (4.25)

where N = N1×N2 represents the number of the scatterers in the resolution cell. To

simulate speckle in a given scanned area, we assume that the PSF h(x, y) is spatially

invariant, and scatterers are uniformly dispersed in the given area. Therefore, speckle

can be generated from the discrete convolution between PSF and dispersed phasors

in the scanned area, where the number of phasors is the product of scatterer density

and the number of resolution cells in the given area. The discrete form of (4.22) is

expressed as:

f [m,n] = g[m,n]n[m,n]. (4.26)

For a given size of resolution cell, different distributed-speckle can be simulated by

simply varying the number of scatterers N in the resolution cell. For examples, the

large number of scatterers could lead to Rayleigh distributed speckle [10,11], whereas

limited scatterer density could produce pre-Rayleigh speckle.

In the references [24–26], n(x, y) is generated by lowpass filtering a complex Gaus-

33

sian random field and taking the magnitude of the filtered output. Equation (4.23)

explains why these methods generally work to model reality: the lowpass filter is used

as h(x, y), and complex Gaussian field is used as the scattering function. Therefore,

these methods are efficient for the general purpose of speckle image simulation.

4.2.4 Dynamic Compressed Ultrasonic Image

The envelope-detected image could have a dynamic range on the order of 50-70

dB, whereas a typical display would have dynamic range of 20-30 dB. Consequently, a

clinical ultrasound imaging system usually uses logarithmic amplification to compress

the input signal to fit the display device. The compressed ultrasonic image is given

by

I(x, y) = D ln f(x, y) + E (4.27)

where f(x, y) is the envelope-detected ultrasound image, D is a parameter of the am-

plifier, and E is the linear gain of the amplifier. Logarithmic compression increases the

amplitude of the smaller input signals at the expense of the larger signals. Although

the linear gain E does not affect the statistics of the output signal, the logarithmic

amplification totally changes the statistics of the input envelope signal [27].

letting the random variable x ∈ f Rayleigh distributed, we have y = D ln x + E,

34

x = ey−E

D , and y′(x) = D/x. The pdf of y ∈ I is given by

PI(y) =

∣∣∣∣Px(x)

y′(x)

∣∣∣∣x=e

y−ED

=1

Dσ2exp

(2(y − E)

D

)exp

(−exp(2(y−E)

D)

2σ2

)

=2

Dexp

(2(y − E)

D− ln(2σ2)− exp

[2(y − E)

D− ln(2σ2)

])

=2

Dexp

(y − (E + D

2ln(2σ2))

D/2− exp

[y − (E + D

2ln(2σ2))

D/2

])(4.28)

This is the Fisher-Tippet or log-Weibull distribution [28].

Another practical log-compressed ultrasound image model has been proposed by

Lopes et al. [29]. Experimental measurements show that displayed ultrasonic images

can be modeled as [29]:

I(x, y) = µg(x, y) +√

µg(x, y)n(x, y) (4.29)

where n(x, y) is a zero-mean Gaussian noise with mean one. Although this model

doesn’t involve logarithmic transformation, it is still referred to as a “log-compressed”

model by convention.

4.3 Experiments

In this section, we demonstrate the performance of the ultrasound image model

introduced in the previous section. In our study, the PSF was generated using the

Field II ultrasound simulator [30], wherein a linear array transducer was used, and

the center frequency is chosen as 3 MHz.

35

Figure 4.1: Simulation an ultrasound image of the short-axial view of a left ventricle.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1histogramRayleigh

Figure 4.2: The histogram and Rayleigh fitting of the background region in Fig. 4.1.

4.3.1 Simulation with RF Signals

The RF signals were considered the interaction of the system response with the

tissue scattering function (suggested by (4.15)). We assumed that the samples of the

appropriate tissue scattering function are uncorrelated, zero-mean, Gaussian random

36

(a) (b) (c)

Figure 4.3: Simulated ultrasound images with the multiplicative model. Speckle wassynthesized by (a) lowpass filtering method, and speckle model with the number of scatterers(b) N=100, (c) N=3 per resolution cell

variables. The simulation procedures are illustrated in Fig. 4.1. The echogenicity

map was designed to mimic the short-axis scanning view of the left ventricle (LV),

which contained a blood cavity, ventricle wall and background tissue. Then, the 2-D

RF signals were generated by (4.15), and the envelope of the RF data was generated

by (4.18). The histogram of an homogenous region of the envelope data was fitted

with the Rayleigh model, and the result (shown in Fig. 4.2) indicates a good matching

between the simulation and theoretical predication.

4.3.2 Simulation with the Multiplicative Model

Similar to the previous RF signal method, we used the same echogencity map

for the ultrasound phantoms . First, speckle was generated by lowpass filtering a

complex Gaussian random field and taking the magnitude of the filter output. The

synthesized ultrasound image is shown in Fig. 6.3(a).

With (4.25), we also simulated speckle with the scatterer density 100 and 3, shown

in Figs.6.3 (b) and (c), respectively. We evaluated the accuracy of our simulation

37

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) (b)

Figure 4.4: The histograms (solid, −), Rayleigh (dash, −−) and Nakagami (dash-dot, −.)of speckle homogenous regions, generated with the scatterer density (a) 100 and (b) 3.

101

102

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Number of scatterers per resolution cell

Nak

agam

i par

amet

er m

Figure 4.5: The Nakagami parameter m for the different distributed speckle, generatedby varying the scatterer density.

method with both Rayleigh and Nakagami fittings. As shown in Fig. 4.4(a), the his-

togram of speckle generated using a large scatterer number (N=100) can be accurately

fitted by both Rayleigh and Nakagami (m=1.0) models. However, Fig. 4.4(b) shows

that the low scatterer density (N=3) leads to a pre-Rayleigh (m=0.5678) rather than

Rayleigh distribution. In the additional tests, we examined speckle with the various

scatterer densities (3 ≤ N ≤ 100) using the Nakagami model. The results (Fig. 4.5)

accurately matched the experimental phantom studies of Shankar et al. [15].

38

(a) (b)

Figure 4.6: Dynamic compressed ultrasound images. (a) the logarithmic compressed imageof Fig. 4.1, (b) the result of (4.29)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4.7: The histogram and Fisher-Tippet fitting curves for the log-compressed imageFig. 4.6(a).

4.3.3 Logarithmic Compressed Ultrasound Image

The simulation of log-compressed image was straightforward. For example, Fig.

4.6(a) was generated by applying the logarithmic transform on the enveloped detected

image Fig. 4.1 with D = 60 and E = 0. The histogram of an homogenous region has

a good matching with the Fisher-Tippet distribution, shown in Fig 4.7. The second

approach of log-compressed image simulation is based on the empirical model (4.29).

We used the square root of the envelope-detected speckle to approximate Gaussian

39

distributed noise and mimic speckle pattern. The result is shown in Fig. 4.6(b).

4.4 Conclusion

In this chapter, we have introduced several ultrasound image models, which were

developed on the basis of speckle statistics. A robust ultrasound image model should

be featured with accuracy and low computational cost. The multiplicative model

provides a very close approximation to the ultrasound image, both in image features

and statistical characteristics. In practice, the selection of the ultrasound image model

depends on the specific application. In the rest of this thesis, we used serval different

image models in the applications of denoising and motion estimation.

40

Chapter 5

Speckle Suppression in Ultrasound Images

5.1 Introduction

Ultrasound speckle is the result of the diffuse scattering, which occurs when an

ultrasound pulse randomly interferes with the small particles or objects on a scale

comparable to the sound wavelength. Speckle is an inherent property of an ultrasound

image, and is modeled as spatial correlated multiplicative noise. In most cases, it is

considered a contaminating factor that severely degrades image quality.

To improve clinical diagnosis, speckle reduction is generally used for two applica-

tions: visualization enhancement and auto-segmentation improvement. Most speckle

filters are developed for enhancing visualization of speckle images [25, 31, 32]. For

these approaches, texture recovery is a desired feature of filtering, and needs to be

addressed. Another goal of ultrasonic speckle suppression is to improve image sim-

plification, which is in turn very beneficial in automated object detection (e.g. in

segmentation and motion tracking). In this sense, texture removal significantly im-

proves the speed and accuracy of automated object detection. Consequently, speckle

filters that are designed for texture recovery have rather limited application in auto-

segmentation improvement.

We therefore consider the design of a speckle suppression algorithm, which mainly

41

focuses on producing the denoising result for auto-segmentation improvement, but

is also able to provide visualization enhancement. For the main goal (segmentation

improvement), image texture does not improve boundary tracking. Rather, texture

recovery is better ignored. In fact, a simplified image with piecewise smoothing

regions and the essential edges of objects, often improves segmentation performance.

Such simplification can be described by use of a “cartoon model”, which has been

elaborated by the Mumford-Shah functional [33].

An image can be simplified using iterative filtering such that the output of each

iteration represents a coarser version of its input. A class of techniques for accomplish-

ing that purpose is, the scale-space denoising methods, called nonlinear anisotropic

diffusion, e.g., Perona-Malik filter [34], Weickert filter [35] and total variation diffu-

sion [36]. These techniques rely on the diffusion flux to iteratively eliminate small

variations due to noise or texture, and to preserve large variations due to edges.

For the multiplicative noisy image, however, the general signal/noise relationship no

longer exists, since the variations due to noise may be larger than those due to sig-

nal. This limits the application of the nonlinear diffusion method in the processing

of ultrasound images. A solution is to integrate the speckle suppression algorithm

into the diffusion technique. For instance, a speckle reducing anisotropic diffusion

(SRAD) method [37] has been derived by casting the typical spatial adaptive filers

(the Lee and Frost filter [31, 38]) into the nonlinear diffusion technique. Although

the SRAD method improves edge detection via the anisotropic filtering, the filter-

ing result with regard to speckle suppression and edge preservation is still preserved

for segmentation purposes. For example, low-contrast edges are often smeared with

speckle, and speckle texture is usually retained in the high-intensity region.

42

The nonlinear diffusion technique relies on the gradient operator to distinguish

signal from noise. Such a method often cannot achieve a precise separation of sig-

nal and noise. Ultrasound image denoising problems are better solved if a powerful

signal/noise separating tool (e.g., wavelet analysis) is incorporated in the speckle-

reducing diffusion process. Moreover, multiscale wavelet despeckling methods have

demonstrated tremendous performance improvement compared to typical spatial speckle

filters [39–41]. Intuitively, integration of the multiresolution and sparsity properties

of the wavelet with anisotropic speckle reduction from nonlinear diffusion should lead

to stronger speckle suppression and edge preservation than that achieved by spatial

domain filtering alone. Recent work [42] has shown that nonlinear anisotropic dif-

fusion can be employed within framework of the dyadic wavelet transform (DWT).

We refer to the integration of nonlinear diffusion and wavelet shrinkage as wavelet

diffusion. Inherited from the wavelet, this technique has more favorable denoising

properties than nonlinear diffusion (namely, via multiscale analysis and more effi-

cient signal/noise separation). It is also distinguished from wavelet-based denoising

methods by its improved edge-enhancement and iterative noise reduction features.

In this chapeter, we present a normalized modulus-based nonlinear multiscale

wavelet diffusion method for speckle suppression and edge enhancement. The pro-

posed approach aims to improve the ultrasound image quality for automated image

interpretation. With a tunable parameter, the algorithm can also preserve texture

for visual enhancement. The proposed algorithm is versatile for both the envelope-

detected speckle image and the log-compressed ultrasonic image. Relying on edge

detection by the normalized wavelet modulus, the algorithm can directly take either

type of image as input without prior compressing via the logarithmic transform or

43

uncompressing via the exponential function. This feature actually solves the perfor-

mance instability problem, which is caused by inaccurate estimation of the compres-

sion coefficient — a tricky problem for most speckle filters.

The chapter is organized as follows. In Section 2.2, we review the theories of

nonlinear diffusion, the dyadic wavelet transform and 2D wavelet diffusion. In Section

2.3, we introduce the new algorithm. In Section 2.4, we quantify the performance of

our algorithm and present results for both synthetic (simulated) and real ultrasonic

images. Conclusions are drawn in Section 2.5.

5.2 Wavelet Diffusion

5.2.1 Nonlinear Diffusion

Perona and Malik proposed a fundamental nonlinear anisotropic diffusion based

on partial differential equation (PDE) for noise smoothing [34, 35]. Given a noisy

image f(x, y, t) at time (scale) t, the nonlinear diffusion equation is expressed as

∂∂t

f(x, y, t) = div[c(x, y, t)∇f(x, y, t)]

f(x, y, 0) = f0(x, y)

(5.1)

where ∇ is the gradient operator, div is the divergence operator, and c(x, y, t) is the

diffusion coefficient. If c(x, y, t) is a constant, (6.1) reduces to the isotropic heat dif-

fusion equation. To avoid the edge-smearing during the diffusion, c(x, y, t) should be

constructed to encourage homogenous-region smoothing and to inhibit the smoothing

across the boundaries. A satisfied c(x, y, t) is determined by two components: the

44

edge map η(x, y, t) and the diffusivity function g(·). The edge map η(x, y, t) is the

estimation of the location of the edges at time t. Ideally, η(x, y, t) should have two

properties:

1. η(x, y, t) equals to zero for the region inside boundaries, and

2. η(x, y, t) has the local contrast at edge point in a direction perpendicular to the

edge.

In the scale space, η(x, y, t) = ∇f(x, y, t) can generally provide an accurate esti-

mation of the edge positions. The diffusivity function g(·) has to be a nonnegative

monotonically decreasing function, with g(0) = 1. As a consequence, c(x, y, t) can be

formulated as

c = g(|η|). (5.2)

A diffusivity function proposed in [34] is given by

g(|η|) =1

1 + (|η|/λ)2(5.3)

where λ is an edge magnitude threshold parameter. The influence of λ on the diffusion

process can be illustrated by the flux, defined as Φ(η) = g(η)η [34]. Given a value

of λ, the maximum flux ΦM occurs at |η| = λ for (5.3) [43]. Below ΦM , the flux is

reduced to zero, indicating that diffusion encourages homogenous region smoothing.

Above ΦM , the flux also goes to zero, suggesting that diffusion inhibits smoothing

across edges. Generally, a large value of λ produces a smoother result in a homogenous

region than a smaller one. In this sense, λ acts as a threshold for the diffusion process.

45

5.2.2 Dyadic Wavelet Transform

Mallat and Zhong [44] have generalized the Canny edge detection approach, and

have presented a multiscale dyadic wavelet transform for the characterization of 1D

and 2D signals. With a wavelet function ψ(x) ∈ L2(R), a continuous wavelet trans-

form of f(x) is given by

Wa,bf(x) =< f, ψa,b >=

∫ +∞

−∞f(x)

1

aψ(

x− b

a)dx (5.4)

where a > 0 is the scale number, b ∈ R is the translation parameter, and ψa,b(x) =

1aψ(x−b

a). With a differentiable smoothing function θ(x), ψ(x) is given by

ψ(x) = ∂θ(x)/∂x.

For the 2D wavelet transform, the wavelet functions ψ1(x, y) and ψ2(x, y) are defined

as:

ψ1(x, y) =∂θ(x, y)

∂xand ψ2(x, y) =

∂θ(x, y)

∂y. (5.5)

The dyadic wavelet transform of f(x, y) ∈ L2(R2) at the scale 2j (or level j) has two

components defined by:

W dj f(x, y) = f ∗ ψd

j (x, y) d = 1, 2. (5.6)

46

Hence, the wavelet coefficients W 1j f(x, y) and W 2

j f(x, y) are proportional to the gra-

dient of f ∗ θ(x, y):

W 1j f(x, y)

W 2j f(x, y)

= 2j

∂∂x

(f ∗ θj)(x, y)

∂∂y

(f ∗ θj)(x, y)

= 2j∇(f ∗ θj)(x, y). (5.7)

The modulus of the wavelet coefficients at scale 2j is defined as:

Mjf(x, y) =√|W 1

j f(x, y)|2 + |W 2j f(x, y)|2 (5.8)

which represents the multiscale edge information obtained by combining the hori-

zontal and vertical wavelet coefficients. With a scaling function φ(x, y), the coarse

approximation of f(x, y) at scale 2j is

Sjf(x, y) = f ∗ φj(x, y). (5.9)

A finite-level discrete dyadic wavelet transform of the 2D discrete function f ∈

l2(Z2) can be represented as:

W =

SJf, (W d

j f)d=1,21≤j≤J

(5.10)

where SJf is a coarse scale approximation of f at final scale 2J , and W dj f represents

the detail image at scale 2j. We refer to this discrete wavelet transform as the MZ-

DWT.

47

A 2D discrete function f can be decomposed by a lowpass filter H and a highpass

filter G, and reconstructed with a lowpass filter H (the conjugate filter of H) and two

highpass filters K and L. In the Fourier domain, the Fourier transform of five filters

are denoted by H, G, H, K and L, respectively. Details about filter construction can

be found in [44] and [42]. The coarse scale approximation of f(u, v) at scale 2j+1 can

be represented in the Fourier domain as:

Sj+1f(u, v) = H(2ju)H(2jv)Sjf(u, v) (5.11)

where j ≥ 0, and S0f(u, v) = f(u, v). Correspondingly, the two detail images are

obtained as:

W 1j+1f(u, v) = G(2ju)Sjf(u, v), (5.12)

W 2j+1f(u, v) = G(2jv)Sjf(u, v). (5.13)

With the reconstruction filters, the signal is represented recursively as:

Sjf(u, v) = Sj+1f(u, v) H(2ju) H(2jv)

+ W 1j+1f(u, v) K(2ju)L(2jv)

+ W 2j+1f(u, v) L(2ju)K(2jv). (5.14)

The time domain representation of (5.11)-(5.14) can be found in [42, 44]. By sub-

stituting (5.11)-(5.13) into (5.14), a necessary and sufficient condition for perfect

48

reconstruction is given as [45]:

H(u) H(v)H(u)H(v) + K(u)L(v)G(u) + L(u)K(v)G(v) = 1. (5.15)

5.2.3 Wavelet Diffusion

Recently, Mrazek et al. [46] have sought to determine the correspondence between

wavelet shrinkage and nonlinear diffusion methods. Shih et. al [42] have shown that

nonlinear diffusion can be approximated by a MZ-DWT shrinkage process, and have

proposed a novel denoising scheme which combines the two techniques. We refer to

the integration of nonlinear diffusion and wavelet shrinkage as wavelet diffusion. This

integrated technique has several favorable denoising properties inherited from the

individual techniques (e.g. multiscale analysis and efficient signal/noise separation

properties from the wavelet, edge-enhancement and iterative noise reduction features

from the nonlinear diffusion). A derivation that proceeds from one-dimensional non-

linear diffusion to dyadic wavelet shrinkage has been shown in [42]. For our applica-

tion, we briefly demonstrate the derivation in 2D. From (6.1), we have

∂

∂tf(x, y, t) =

∂

∂x[c(x, y, t)

∂

∂xf(x, y, t)] +

∂

∂y[c(x, y, t)

∂

∂yf(x, y, t)]. (5.16)

Forward time discretization of the time derivative is approximated as:

∂

∂tf(x, y, t) =

f(x, y, t +4t)− f(x, y, t)

4t+ O(4t).

Neglecting the higher-order terms, and substituting the above equation into (5.16),

49

we obtain

f(x, y, t +4t)− f(x, y, t)

4t=

∂

∂x[c(x, y, t)

∂f(x, y, t)

∂x]

+∂

∂y[c(x, y, t)

∂f(x, y, t)

∂y]. (5.17)

With 4t = 1, we can approximate (5.17) as:

f(x, y, t + 1) ≈ f(x, y, t) +d

dx[c(x, y, t)

df(x, y, t)

dx] +

d

dy[c(x, y, t)

df(x, y, t)

dy] (5.18)

and denote f(x, y, t+1), f(x, y, t) and c(x, y, t) as f(x, y), f(x, y) and c(x, y), respec-

tively, for briefness. Letting p(x, y) = 1− c(x, y), (5.18) can be rewritten as:

f(x, y) = f(x, y) +d2f(x, y)

dx2+

d2f(x, y)

dy2

− d

dx

[p(x, y)

df(x, y)

dx

]− d

dy

[p(x, y)

df(x, y)

dy

]. (5.19)

The Fourier transform of (5.19) is

f(u, v) = (1− u2 − v2)f(u, v)− ju[ 1

2πp(u, v) ∗ (juf(u, v))

]

− jv[ 1

2πp(u, v) ∗ (jvf(u, v))

]. (5.20)

Letting A1 · A2 = 1−u2−v2; B = ju; D = −ju; E = jv; F = −jv and p = 12π

p(u, v),

50

and substituting into (5.20), we have

f(u, v) = A2 · A1 · f(u, v) + D · (p ∗ (B · f(u, v)))

+ F · (p ∗ (E · f(u, v))). (5.21)

We note that (5.21) has the same format as (5.14). In addition,

A1 · A2 + B · D + E · F = 1,

which satisfies the filter requirement expressed in (5.15). Finally, the inverse Fourier

transform of (5.21) is:

f(x, y) = (f(x, y) ∗ A1) ∗ A2 + (p(x, y) · (f(x, y) ∗B)) ∗D

+ (p(x, y) · (f(x, y) ∗ E)) ∗ F. (5.22)

Equation (5.22) indicates that the image f(x, y) is first decomposed with the lowpass

filter A1 and the highpass filters B and E. It is then regularized with p(x, y), and

finally reconstructed with the corresponding lowpass filter A2 and the highpass filters

D and F .

From the derivation (see (5.19)), the diffusion coefficient c(·) has its correspon-

dence with p(·) in the wavelet domain. Similar to c(·) in (6.1), the diffusion behavior of

p(·) is also determined by the edge map η and the diffusivity function g(·). Therefore,

wavelet diffusion coefficient is given by

p(|η|) ≡ 1− g(|η|). (5.23)

51

Figure 5.1: Scheme for 3-level wavelet diffusion. Sjf and W dj f denote the filtered wavelet

coefficients at scale 2j .

To achieve edge-preservation and intra-region smoothing, g(·) in (5.23) also has to

be a nonnegative monotonically decreasing function. In this sense, most diffusivity

functions [47], which have already been developed in the nonlinear diffusion, can

be used in wavelet diffusion. Another important factor controlling the effect of the

diffusion is the selection of the edge map η. For a general denoising problem (e.g.

additive Gaussian noise), either wavelet coefficients or wavelet modulus can be used

as the edge map. However, from (5.7), the similarity of the gradient operator and the

wavelet modulus suggests that the wavelet modulus may be more appropriate.

The advantages of wavelet-based diffusion over spatial nonlinear diffusion are ob-

vious: the edges detected by the wavelet coefficients/modulus are more accurate than

the ones estimated by the gradient operator. Moreover, multiscale analysis provides

powerful denoising scheme for the treatment of complicated noise, including speckle.

Similar to the wavelet shrinkage [48], the denoising scheme of wavelet diffusion is

implemented by three steps: 1) the noisy image f is decomposed into the coarse scale

52

approximation Sjf (j ≥ 1) and detail images W dj f (d = 1, 2) by 2D MZ-DWT; 2)

wavelet coefficients W d1 f are regularized as

W dj f = p(|η|)W d

j f. (5.24)

3) the denoised image is reconstructed by taking the inverse MZ-DWT. To achieve a

satisfactory denoising result, wavelet diffusion is often performed iteratively [42]. For

instance, a three-level wavelet diffusion scheme is shown in Fig. 5.1.

5.3 Speckle Suppression with Wavelet Diffusion

Wavelet diffusion can be considered as a special case of nonlinear diffusion which

is employed within the framework of the dyadic wavelet transform. In denoising

applications, the key issue of wavelet diffusion is to find an accurate edge estimation

method. For the image corrupted with additive Gaussian noise, wavelet coefficients

(or wavelet modulus) can precisely distinguish the edge-related components from

noise-related components relying on the difference of their magnitude. However, when

an image is contaminated with multiplicative noise, use of the wavelet coefficient as

an edge estimator experiences difficulty in efficiently detecting edges, since the noise-

related components may indeed be larger than the edge-related components [49]. A

similar problem occurs when the nonlinear diffusion technique is employed in speckle

suppression. For that problem, Yu and Acton [37] proposed a method which cast

the spatial adaptive filtering technique into the nonlinear diffusion algorithm. The

conceptual similarity between the nonlinear diffusion and wavelet diffusion techniques

encourages us to examine their solution more closely at the beginning of this section.

53

Table 5.1: Correspondence between concepts used in different denoising techniques

Spatial Adaptive Filter Nonlinear Diffusion Wavelet DiffusionDiffusion coefficient c(·) p(·)Additive Gaussian noiseEdge map (η) ∇f W d

j f or Mjf

Threshold (λ) or Constant or histogram- Constantnoise estimation based estimation [34] [42]

Multiplicative speckleEdge map (η) Cs q Mjf

Threshold (λ) or Cu q0 Homogenous region-noise estimation [31] [37] based estimation

Later in this section, we propose our edge-detection scheme and diffusion threshold

estimator in the framework of wavelet diffusion.

5.3.1 Related Work

As a typical spatial adaptive filter, the Lee filter assumes signal reflectivity r as

a stationary random variable, and a linear minimum mean square error (LMMSE)

estimator is used to eliminate speckle, given by [31]

rs = µs + (1− C2u/C

2s )(fs − µs). (5.25)

Here, µs is the mean value of image f for a moving window s, C2s = σ2

s/µ2s is the

normalized noisy signal variance, and C2u = σ2

u/µ2u is the normalized noise variance

for the homogenous region u.

Speckle reducing anisotropic diffusion (SRAD) is derived by casting the spatial

adaptive filter into the variational framework. A SRAD diffusivity function is defined

54

as

g(q) =1

1 + (q2 − q20)/[q

2(1 + q20)]

(5.26)

where q is instantaneous coefficient of variation (ICOV), and q0 is diffusion threshold.

The speckle reduction of SRAD can be understood from the relationship of q and

q0 with their correspondence in the spatial adaptive filter. In fact, q in (5.26) is

a variational expression of Cs of (5.25) in terms of the gradient operator, whereas

q0 is exactly same as Cu [37]. On another hand, the similarity observed between the

nonlinear diffusivity function (5.3) and SRAD diffusivity function (5.26) indicates the

roles of q and q0 in the speckle diffusion: q plays a role as the speckle edge detector

in the same manner as the edge detector η in nonlinear diffusion, whereas q0 acts

as the diffusion threshold λ. The conceptual correspondence of different denoising

techniques is illustrated in Table I.

In the spatial nonlinear diffusion scheme, the solution proposed by SRAD for the

despeckling problem is: to estimate the edge map with the normalized noisy signal

variance, and to compute the diffusion threshold from the homogenous speckle region.

With this strategy, the signal mean is removed during the edge estimation, and the

edge-related components can be easily separated from the noise-related components

by the magnitude difference. Therefore, it suggests that, the wavelet diffusion can be

also successfully employed for speckle suppression as long as one can find an appro-

priate edge detector to represent the intrinsic signal/noise relationship in the wavelet

domain, and identify the homogenous speckle region for the diffusion threshold esti-

mation.

55

5.3.2 Edge Detection with Normalized Modulus

Two types of ultrasound images are generally used. One is the envelope-detected

speckle image, which can be generated from the recorded RF signals; whereas the

other is the displayed ultrasonic image, which is commonly used in medical applica-

tions. The latter is generally considered the logarithmic compressed envelope-detected

image (e.g. f2 = D ln f1 + G, where D is the compression coefficient, and G is the

linear gain). This kind of nonlinear compression totally changes the statistics of the

envelope-detected signals, and a different compression coefficient also leads to differ-

ent statistical distribution of signals [27]. To avoid conversion between image types,

we propose two different edge detectors corresponding to image type. The advan-

tage of such direct processing is the avoidance of performance instability caused by

inaccurate estimation of the compression coefficient.

5.3.2.1 Envelope-detected Speckle Image

Statistical studies show that envelope-detected signal can be generally represented

in terms of a multiplicative noise model [49]:

f(x) = µR(x)n(x) (5.27)

where µ is the average amplitude of the target, and R(x) is the intrinsic signal with

mean one, and n(x) is Rayleigh distributed speckle noise with mean one. By definition

(5.4), the wavelet coefficients are [49]

Wa,bf(x) = µ

∫R(x)n(x)ψa,b(x)dx (5.28)

56

For a homogenous region, R(x) is set to one to analyze the noise contribution, and

the wavelet coefficients are proportional to the mean amplitude µ of the signal. Since

n(x) has finite energy and∫ +∞−∞ ψa,b(x)dx = 0, the integral of (5.28) at scale a will be

a nonvanishing function of translation b. Therefore, noise contribution to the wavelet

coefficients depends on the signal mean.

Generally, the normalized variance on wavelet coefficients σWjf/µs is used to char-

acterize the intrinsic signal variance [39]. Here, µs = (∑s

f)/N is the local mean for a

window s with N pixels, and σ2Wjf = (

∑s

Wjf2)/N is the local variance of the wavelet

coefficients. If considering the variance over all sub-bands, we find that this total

variation equals to the variance of modulus, i.e.

σ2Mjf =

1

N

∑s

(Mjf2) =

1

N

∑s

(W 1j f 2 + W 2

j f 2).

Therefore, the normalized variance on wavelet modulus σMjf/µs can also characterize

the intrinsic signal variation.

Prior to using the normalized modulus as an edge map, two adjustments are made

to improve its denoising performance. First, the size of window for the mean esti-

mation is scale-dependent, specifically, Dj = 2j−1(D0 − 1) + 1, where D0 is original

window. Second, noise variance estimation occurs at the current pixel, rather than by

local window estimation. Although this adjustment sacrifices the spatial-correlation

resistance provided by window estimation, a better edge resolution is achieved. More-

over, the spatial-correlation caused by speckle can be easily solved via the diffusion

process. Finally, we propose the following edge detector for the envelope-detected

57

image diffusion:

Mjf = Mjf/µs, j = 1, 2, ...J. (5.29)

Using the modulus (rather than the wavelet coefficients) to characterize the noisy

signal is well-suited to the purposes of image segmentation. After removal of the

signal mean, the edge-related Mjf has a large value, whereas the noise and texture

have a small value of modulus. Consequently, an edge-enhanced diffusion process

leads to modulus-maximization at edges and piece-wise smoothing within the ho-

mogenous regions. Such a result is suitable for the applications of classification and

segmentation.

5.3.2.2 Displayed Ultrasonic Image

The medical ultrasonic images (B-Scan images) generated from clinical imaging

systems have different properties compared with an envelope-detected image. The

signal processing stages contained within the scanner (logarithmic compression, low-

pass filtering, interpolation) modify the statistics of the original signal. Experimental

measurements [29] show that displayed ultrasonic images can be modeled as :

f(x) = µR(x) +√

µR(x)n(x) (5.30)

where n(x) is a zero-mean Gaussian noise with mean one. Although this model

doesn’t involve logarithmic transformation, it is still referred to as “log-compressed”

model by convention. Assuming that a uniform area is scanned (i.e. R(x) = 1), it

can be easily shown that the mean of log-compressed image is proportional to the

variance rather than the standard deviation of the image.

58

(a) (b)

Figure 5.2: (a) Simulated envelope-detected speckle image. (b) Real echocardiographicimage.

Similar to (5.28), the wavelet coefficient are given as:

Wa,bf(x) = µ

∫R(x)ψa,b(x)dx +

õ

∫ √R(x)n(x)ψa,b(x)dx (5.31)

Considering the noise contribution at homogenous regions (R(x) = 1), we find that the

wavelet coefficients are proportional to√

µ. To characterize the intrinsic signal/noise

variation, the edge detector for the log-compressed ultrasound image is constructed

as:

Mjf = Mjfµ− 1

2s , j = 1, 2, ...J. (5.32)

Comparing (5.29) and (5.32), the effect of normalization is to remove the signal mean

during the edge estimation, and the only difference is the contribution of signal mean

to the signal/noise characterization.

59

1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 HistogramFitted

2 4 6 8 10 12 14 16 180

0.05

0.1

0.15

0.2

HistogramFitted

Figure 5.3: Histograms and the Rayleigh mixture model fitting of the normalized modulusat scale 22 for the simulated envelope-detected speckle image (top) and real echocardio-graphic image (bottom), shown in Figs. 5.2(a,b),respectively.

5.3.2.3 Statistical Model of Normalized Modulus

The distribution of the speckle-related modulus depends on the statistical model of

the wavelet coefficients. Several models have been proposed for characterizing speckle,

including the mixture Gaussian distribution [41] for the uncompressed speckle image,

and the normal inverse Gaussian distribution [50] for the logarithmic compressed

speckle image. Except for the Gaussian distribution, most models are analytically

too complicated to yield a practical model for the normalized wavelet modulus. To

60

simplify the estimation, we assume that both of speckle-related and edge-related

normalized wavelet coefficients are Gaussian distributed. Consequently, the speckle-

related normalized modulus Mjf can be modeled by the Rayleigh distribution:

p(x|noise) =x

σ2n

exp

(− x2

2σ2n

)(5.33)

where x denotes the Rayleigh random variable, σn is the standard deviation of the nor-

malized wavelet coefficients. Similarly, p(x|edge) for edge-related Mjf has the same

form as (5.33) with the edge-related standard deviation σe. Overall, the normalized

wavelet modulus Mjf is given by the Rayleigh mixture model,

p(x) = ωnp(x|noise) + (1− ωn)p(x|edge). (5.34)

We demonstrate the performance of the Rayleigh mixture model in matching the

distribution of normalized modulus for both envelop-detected and log-compressed

ultrasonic images. As shown in Fig. 5.2, the envelope-detected image (Fig. 5.2(a))

is simulated by (5.27) (see Section IV for details), whereas Fig. 5.2(b) is a real

echocardiographic image (four chamber view). Both of images are decomposed by

MZ-DWT. The histograms of normalized modulus and their corresponding Rayleigh

mixture fitting at the resolution of 22 are shown in Fig. 5.3. The results indicate that

the Rayleigh mixture model can well characterize the statistics of the normalized

wavelet modulus for both envelop-detected and log-compressed ultrasonic images.

61

5.3.3 Diffusion Threshold

5.3.3.1 Estimation based on the Homogenous Speckle Region

The diffusion threshold should reflect the noise variation in the multiscale wavelet

modulus. The traditional threshold estimation, such as using a constant value or

histogram-based estimation (90% integral of histogram, suggested in [34]), is usually

difficult to control in producing a satisfactory result. Extended from the concept of

Cu in the spatial case, the diffusion threshold can be estimated by the noise variation

present in the homogenous speckle region of the image. This has been pointed out

by Yu and Acton [37]. However, due to the difficulty of automatic selection of a

homogenous region, they simplified the threshold estimation by using a constant

with the pre-designed exponential decay function. Such an estimation becomes less

flexible with a more complicated image.

We pursue the concept of threshold estimation based on the homogenous region.

First, we study the relationship between the estimated threshold and the resolution

scales, using a manual selection method. When the image is decomposed into multi-

scale, the modulus in the coarser scale (j ≥ 2) tends to be much smoother than that

for the finer scale. Therefore, we reduce the threshold of coarser scale to encourage

edge preservation. The homogenous-region based threshold for the multiscale wavelet

diffusion is proposed as

λj = Mjfu2−j′/2 (5.35)

where Mjfu represents the mean normalized modulus for the homogenous region u,

and j′ is the scale factor. Empirically, we use j′ = 0 for j = 1 and j′ = j for j ≥ 2.

In our experiments, the proposed threshold estimator performs well for vari-

62

0 5 10 15 20 25 300.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Iteration

λ j

j=1 2 3

Figure 5.4: Diffusion thresholds λj(j = 1, 2, 3) estimated from the homogenous region inFig. 5.2(b).

ous speckle images with different noise levels. As an example, Fig. 5.4 shows an

homogenous-region estimated threshold λj, which is estimated by the manually se-

lected homogenous region (e.g. cavity of right ventricle) of Fig. 5.2(b). With this

threshold, we are able to generate a result similar to the one shown in Fig. 5.10(d).

After removal of the signal mean, the noise-related Mjf in the homogenous region

can generally represent the intrinsic noise level for the whole image. With iterative

diffusion, the noise, which is originally Rayleigh distributed, gradually becomes Gaus-

sian distributed. As shown in Fig. 5.4, the estimated threshold decays quickly from

a large initial value to a small constant. Therefore, such a threshold would resist the

boundary oversmoothing associated with the diffusion process. The homogenous re-

gion estimation method can be adapted well to complicated speckle images. However,

manual selection of the homogenous region is always laborious and unstable for prac-

tical application. Hence, a scheme for the automatic determination of homogenous

regions within an image would be very desirable.

63

5.3.3.2 Speckle Image Classification

We use likelihood classification and cross-scale edge consistence to separate the

homogenous speckle regions from others. For the classification model, we assume

that the image consists with three classes: edges, speckle and background. The back-

ground commonly exists in the medical ultrasound image, e.g. the region outside

scanning region. Background removal is necessary to reduce the estimation error and

increase speed. Due to its constant value, the background can be easily removed with

intensity thresholding. The problem then becomes binary classification: specifically,

classification of the edge-related and the speckle-related components in the normalized

modulus. From assumption (5.34), the normalized modulus is modelled as a Rayleigh

mixture distribution. We use the expectation maximization (EM) method [51] to es-

timate the parameters ωn, σn and σe of (5.34). Typically, the number of noise-related

coefficients is much larger than those related to edges, and the peak of the normal-

ized modulus histogram is most likely due to noise-related coefficients. Therefore,

the initial value of σn is estimated by the regression method, σn =√

π/2max(hi),

where hi is the segment of histogram. Involved computation is reduced with such

initialization. With the estimated parameters, the image is segmented by likelihood

classification [52], and the classification threshold is given by

T =

√2(log

σ2n

σ2e

+ωn

1− ωn

)

/∣∣∣∣1/σ2e − 1/σ2

n

∣∣∣∣. (5.36)

To achieve a stable classification, we rely on the persistence of the edge-related

normalized modulus across resolution scale. In particular, for an image with back-

ground removed, a coarse-to-fine classification method [25] can be used to determine

64

the homogenous region Uj:

Uj =

1, [(1− Uj+1)Mj+1f ]Mjf < K2TjTj+1

0, elsewhere

(5.37)

Here, K is a tunable parameter that controls the region of interest, and (1−Uj+1)Mj+1f

represents the edge-related components of normalized modulus at scale j + 1. For

coarsest scale, we assume MJf contains only edges of the image, with UJ = 0. In

Fig. 5.5, we demonstrate the performance of the coarse-to-fine classification for the

two test images (Fig. 5.2). For both test images, K = 1, and the classified homoge-

nous speckle-related Mjf at different resolutions are shown in white. It is clearly

shown that the identified speckle-related components decrease with an increase in

decomposition level, whereas edge-related components increase.

With the detected homogenous regions at different scales, the diffusion threshold

is computed as

λj = Mean(UjMjf)2−j′/2 (5.38)

From (6.17) and (5.38), the parameter K of (5.38) plays a tuning role in determining

the diffusion threshold, and further controls the denoising result. When K = 0, all

coefficients are related to edges. Consequently, λj = 0, and no filtering needs to be

performed. When K is extremely large, all coefficients are related to noise. As a con-

sequence, λj is proportional to the mean of normalized modulus. In general, when K

increases, more coefficients close to edges are contributed to the threshold calculation.

Since these coefficients generally have large values, a large value of K would lead to

a large diffusion threshold. On another hand, a small value of K leads to a small

65

Figure 5.5: Classified homogenous speckle regions (white) at scale 21, 22, 23 (from top tobottom) for the simulated envelop-detected speckle image (left column) and real echocar-diographic image (right column).

threshold. Later in Section IV, we further study the influence of K in controlling

the diffusion performance. Briefly, we show that a reasonable value of K always pro-

duces a stable performance improvement with iteration. In fact, the selection of K is

66

determined by the particular application. As an example of a low-speckle image, ul-

trasonic brain imaging requires tiny structure detection. Consequently, a small value

(e.g. K = 0.5) can produce a satisfactory despeckling result without destroying weak

edges. On another hand, for large boundary detection, such as the cardiac structure

in echocardiographic image, a large value of K (e.g. K = 2) will reduce most speckle

and eliminate the texture of objects. This can reduce the computational cost and

improve the accuracy of a segmentation method.

In summary, we generalize our algorithm as the following:

1. Decompose the noisy image f(x, y) into Sjf and W dj f by 2D MZ-DWT.

2. Compute the normalized modulus Mjf using (5.29) or (5.32) according to the

image type.

3. For a background removed image, estimate the Rayleigh mixture parameters

using EM-estimator, and compute the likelihood classification threshold using

(5.36) for each scale.

4. Determine the homogenous region using coarse-to-fine classification rule (6.17).

5. Compute the diffusion threshold with (5.38).

6. Compute the wavelet diffusion coefficient p(Mjf) using (5.23) with a selected

diffusivity function.

7. Regularize wavelet coefficients W dj f using (5.24).

8. Reconstruct the image by taking the inverse 2D MZ-DWT.

67

The homogenous region classification (steps 3 and 4) is performed on the initial

iteration. As part of an iterative filtering algorithm, the other steps are repeated

until a desired result is produced.

5.4 Experiments and Results

We tested our proposed normalized modulus-based nonlinear multiscale wavelet

diffusion (NMWD) speckle suppression algorithm on both of the synthetic and real

ultrasonic images. With the synthetic envelope-detected and log-compressed images,

despeckling performance in terms of image quality indices is compared with other

established despeckling methods. With real ultrasonic images, performance improve-

ment is demonstrated for both visualization and segmentation purposes. In our ex-

periments, the Weickert filter [35] was used as the diffusivity function g(η) in (5.23)

for its robustness regarding boundary preservation,

g(η) =

1 η ≤ 0

1− exp[−3.315(η/λ)4

] η > 0.

(5.39)

A suitable choice for the smoothing function θ(x, y) in (5.5) was a cubic spline with

compact support [44]. Therefore, in our implementation, quadratic spline wavelet

filters were used for decomposition and reconstruction. A three-level NMWD is em-

ployed on all test images (see Fig. 5.1).

68

(a) (b)

(c) (d)

Figure 5.6: Denoising results for the simulated envelope-detected ultrasonic image (Fig.5.2(a)). (a) Echogeneity map. Results filtered by (b) GenLik, (c) SRAD and (d) NMWD,respectively.

5.4.1 Denoising Results for the Simulated Image

To quantitatively evaluate the despeckling performance of the proposed algorithm,

we first experimented with the synthetic speckle images. We generated spatial corre-

lated speckle noise by lowpass filtering a complex Gaussian random field and taking

the magnitude of the filtered output [24–26]. To better mimic the appearance of the

real image, we controlled the correlation length of speckle by appropriately setting

the size of the kernel. The ground truth image (Fig. 5.6(a)) was constructed by

69

using seven elliptic targets with different intensities on a dark background. As shown

in Fig. 5.2(a), the envelope-detected ultrasound image was simulated by corrupting

the ground truth image with full speckle noise using (5.27). For the log-compressed

image, the noise was generated so as to have both the appearance of speckle and the

norm distribution. The image was simulated using (5.30), and the result is shown in

Fig. 5.7(a).

We compared the performance of our speckle suppression algorithm with that of

other speckle reduction techniques: namely, the speckle reducing anisotropic diffusion

(SRAD) technique [37], and the wavelet generalized likelihood ratio filtering method

(GenLik) [25]. Although both algorithms are designed to reduce speckle and preserve

the edges of objects, the differences are: 1) SRAD emphasizes edge-enhancement

more than visualization improvement, whereas GenLik focuses to a greater extent on

visualization improvement. 2) SRAD is a nonlinear diffusion based method, whereas

GenLik is a multiscale wavelet denoising method. 3) SRAD takes the envelope-

detected image as its input, whereas GenLik prefers the log-compressed image. In

addition, Yu and Acton [37] have demonstrated the performance superiority of SRAD

over Perona-Malik nonlinear diffusion, the Lee and Frost filters; whereas Pizuriaca et

al. [25] have shown that GenLik outperforms the homomorphic Wiener filter. Thus,

we consider that a performance comparison between our algorithm and these two

despeckling filters, represents an adequate demonstration that the proposed algorithm

fulfills the denoising design requirements.

A first comparison was made using the envelope-detected full speckle image (Fig.

5.2(a)). In SRAD implementation, q0 in (5.26) is reduced exponentially with iteration,

such as with q0(t) = q0(0) exp(−t/6). Here, q0(0) equals to√

1/L for intensity images

70

(a) (b)

(c) (d)

Figure 5.7: Denoising results for the simulated log-compressed ultrasonic image. (a)Original image. Results filtered by (b) GenLik, (c) SRAD and (d) NMWD, respectively.

and√

(4/π − 1)/L for amplitude images, and L is the look number. Therefore, for

the envelope detected ultrasound image, we used q0(0) = 0.5227 (L = 1) in the test.

The time step was set as 4t = 0.05, and the number of iterations was 300. The

diffusivity function was chosen as (5.26), and the result is shown in Fig. 5.6(c). The

GenLik method was evaluated using the original implementation, which is available

in the author’s website (http://telin.rug.ac.be/∼sanja/). For best performance, the

test image was first log-transformed prior to being filtered by the GenLik method.

The filtered result was recovered by the exponential function. The edge-detection

71

Table 5.2: Performance comparison for different denoising techniques

Speckle Image Log-Speckle ImageMethod ρ FOM ρ FOM

Noisy image 0.7583 0.2281 0.9113 0.2245GenLik 0.9272 0.4953 0.9741 0.5297SRAD 0.9533 0.6121 0.9773 0.6661NMWD 0.9717 0.7566 0.9886 0.9071

threshold factor was chosen as 5 with a window size 5× 5, and the result is shown in

Fig. 5.6(b). In our algorithm, the parameter K in (6.17) was set to 3. The window

size for estimation of the mean is 3 × 3 at the first scale. The image was processed

with 30 iterations, and the output is shown in Fig. 5.6(d).

We further compared the denoising performance of all three filters on the synthetic

log-compressed image (Fig. 5.7(a)). Since SRAD takes an envelope-detected image

as input, the test image was first decompressed by taking the exponential of the

image divided by a compression coefficient prior to being processed by SRAD. The

compression coefficient, D, is estimated empirically to achieve the best performance.

Specifically, D = 50 for this test image. The other parameters were the same as those

used in the first experiment. For the GenLik method, the test image was directly used

as the input. The edge-detection threshold factor was chosen as 5 with window size

5 × 5. In our algorithm, the parameter K was set to 2, the despeckling process ran

adaptively with 30 iterations. The denoised images recovered by the GenLik, SRAD

and proposed algorithm are shown in Figs. 5.7(b-d).

Since speckle in the ultrasound image is modeled as the multiplicative noise, a

linear image fidelity criterion, such as MSE or SNR, is not always an accurate measure

of speckle suppression in images. In our studies, the denoising algorithm performance

72

is quantified by using two quality indices: a noisy suppression quality index ρ [26,53],

an edge preservation index, called figure of merit (FOM) [37,54]. Speckle suppression

is evaluated by comparing the structure similarity between denoised image and noise-

free image. A correlation-based structure similarity measure is given by [26,53]

ρ =

∑i,j∈w

(x(i, j)− µx)(y(i, j)− µy)

√ ∑i,j∈w

(x(i, j)− µx)2 · ∑i,j∈w

(y(i, j)− µy)2(5.40)

where µx and µy are mean values of interested region w in the noise-free image x and

denoised image y, respectively. The FOM is defined as

FOM =1

max(nd, nr)

nd∑i=1

1

1 + γ d2i

(5.41)

where nd is the number of detected edge pixels in the test noisy image, nr is the

number of reference edge pixels in the noise-free image, di is the Euclidean distance

between the ith detected edge pixel and the nearest reference edge pixel, and γ is a

constant typically set to 0.11. We use the Laplacian of Gaussian method to detect

the edges. If the measured image is close to the reference image, the values of ρ and

FOM should be close to 1.

The performance quality of two experiments, in terms of ρ and FOM, are listed

in Table II. In comparing the denoising results, we found that all of the speckle

reduction methods can eliminate speckle in most homogenous regions. However,

only the proposed method can significantly reduce speckle in both high and low

intensity regions, as well as preserve both high-contrast and low-contrast edges. We

also iteratively applied the GenLik method on the test images, however, no significant

73

0 20 40 60 80 1000.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

ρ

K=0.5 1.0 2.0 3.0 4.0 5.0

0 20 40 60 80 1000.2

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0.5

0.6

0.7

0.8

0.9

FO

M

Iteration

K=0.5 1.0 2.0 3.0 4.0 5.0

Figure 5.8: Image quality indices ρ (top) and FOM (bottom), after the simulated envelope-detected image is filtered by NMWD with different values of K.

performance improvement was observed. For example, after the log-compressed image

was processed by GenLik for 30 iterations, FOM= 0.5816, and ρ = 0.9776. This

indicates that nonlinear diffusion-based methods have a significant advantage in being

able to suppress speckle, while preserving edges.

We also studied the stability of the parameter K in the proposed algorithm. The

test images were processed with different value of K, specifically, 0.5, 1.0, 2.0, 3.0,

4.0 and 5.0. For each value, the image was processed with 100 iterations. Figs.

74

0 20 40 60 80 1000.94

0.95

0.96

0.97

0.98

0.99

1

ρK=0.5 1.0 2.0 3.0 4.0 5.0

0 20 40 60 80 100

0.4

0.5

0.6

0.7

0.8

0.9

1

FO

M

Iteration

K=0.5 1.0 2.0 3.0 4.0 5.0

Figure 5.9: Image quality indices ρ (top) and FOM (bottom), after the simulated log-compressed image is filtered by NMWD with different values of K.

5.8 and 5.9 demonstrate the effect of K on controlling the denoising performance

of the proposed algorithm. When K is within a threshold, (e.g. K = 3 for the

enveloped-detected image, K = 2 for the log-compressed image), both ρ and FOM do

not decrease with iteration. Above this value, however, these quality indices decrease

with iteration. Such variation is within expectation. A large value of K indicates

more coefficients close to edges are counted in the diffusion threshold estimation.

If the diffusion threshold is overestimated, edge smearing occurs, and the quality

75

indices decrease with iteration. This becomes evident, when K = 5. In that case, the

diffusion threshold is equivalent to the mean of normalized modulus at the current

scale. However, for a value below threshold, the role of K always improves the image

quality with iteration in a stable fashion. The experiments also illustrate the effect

of the number of iterations on performance. For a given value of K, NMWD fast

approaches reasonable performance within 20 to 40 iterations. After that, only small

improvements are observed. It suggests that diffusion with 20 to 40 iterations has the

highest computational efficiency.

The computational complexity of proposed algorithm can be analyzed from two

stand points: the main procedures (excluding EM estimation) and the EM algo-

rithm. Given N pixels, the complexity of EM estimation for two-Rayleigh mixture is

O(i × N), where i is the iteration number. In the main procedures, wavelet decom-

position and reconstruction exhibit the largest complexity, O(N log N). Overall, the

computational complexity of the complete algorithm is O(i×N +j×N log N), where

j is the iteration number of wavelet diffusion. In practice, NMWD was implemented

in Matlab (Mathworks, Natick, MA), where the main procedures achieved a process-

ing rate of 0.19 sec/scale/iteration for a 256× 256 image on a PC with a Pentium 4

(2.4 GHz) processor.

5.4.2 Real Image

In the first in vivo image experiment, we examined the image quality improve-

ment of the proposed algorithm for both visualization and auto-segmentation. Figure

5.10(a) (also Fig. 5.2(b)) shows an echocardiographic image of the human heart, in

four-chamber view. The data was acquired using a HDI5000 ultrasound scanner man-

76

ufactured by ATL, a Philips Medical Systems Company. Two experiments with two

different values of K were performed on the test image. Specifically, a small value of

K = 0.5 was used to test visualization improvement, whereas a large value K = 1.5

was used for segmentation improvement. The wavelet diffusion was performed for 30

iterations for both experiments, and the denoising results are shown in Figs. 5.10(c,e).

For clear illustration, the profiles, along the highlight line in the original image, are

also compared. The test image was also filtered by two subject algorithms. We used

the GenLik method for the comparison on visualization improvement. The edge-

detection threshold factor of GenLik was chosen as 5 with window size 5 × 5. To

examine the visual improvements, we focused on speckle reduction within the cavity

and at the wall of right ventricle (indicated by the highlight line). We also focused on

structure enhancement at the moderator band near the apex of the right ventricle.

As indicated by the profiles, our algorithm produces a better result for the purpose

of visualization. For the segmentation-purposed comparison, we used SRAD for its

edge enhancement feature. Specifically, the compression coefficient D = 35 was used,

and the other parameters were identical to those used in the previous experiment. In

this case, we compared speckle suppression and texture removal in the wall region,

and the structure enhancement of all ventricular walls. Comparing Figs. 5.10 (d) and

(e), we found that the proposed algorithm achieved better speckle removal and edge

enhancement than the SRAD method.

For a real ultrasound image, criterion used in evaluating the denoising result may

be quite subjective to the specific objectives of the observers. Consequently, the

proposed algorithm has to be flexible so that it can be readily adapted to the require-

ments of different applications. With a small value of K, the proposed algorithm can

77

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Figure 5.10: Denoising results for the echocardiographic image. (a) Original image. Re-sults filtered by (b) the GenLik method, (c) NMWD (K = 0.5), (d) SRAD and (e) NMWD(K = 1.5), respectively. The profiles along the highlight line of the original image (a) areshown in their following row.

78

preserve the textured region, as well as the formation of uniform area in the filtered

image. In the sense of visualization improvement, such a filtered result would be vi-

sually favored in clinical diagnosis. However, for auto-segmentation applications, the

very same result may cause the active contour to be trapped by the retained textured

region and granular boundaries. To improve auto-segmentation, we recommend using

a large value of K, so as to remove speckle texture in the homogenous region and

enhance the edges of structure.

In general, for a nonlinear diffusion method, the balance between noise suppres-

sion and edge preservation often makes threshold selection difficult. A large diffusion

threshold often leads to the significant tiny structure smearing with noise, whereas a

small threshold will produce unsatisfactory noise suppression for boundary tracking.

In the next example, we demonstrate that the algorithm can achieve speckle suppres-

sion and tiny structure preservation simultaneously. The test image is an ultrasound

scan of human liver and kidney region (Fig. 5.11(a)), which is obtained from public

medical image database, MedPixTM (http://rad.usuhs.mil/medpix/medpix.html). In

this test, we focus on evaluating speckle removal in the uniform region of the liver,

and the edge enhancement of the nodular structure of the liver parenchyma. The

denoised results are shown in Fig. 5.11(b-d). These results were also compared via

the profiles, along the highlight line in the original image. As the results show, the

proposed algorithm outperforms the other two filters by clearly outlining the noduli

on the liver surface, while suppressing most of speckle in the liver and kidney regions.

Our result (Fig. 5.11(d)) suggests that the proposed method could lead to reliable

and efficient nodule detection in the diagnosis of cirrhosis of the liver.

79

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Figure 5.11: Denoising results for a liver image. (a) Original image. Results filtered by(b) the GenLik method, (c) SRAD and (d) NMWD, respectively. The profiles along thehighlight line of the original image (a) are shown in their following row.

80

5.5 Conclusion

This chapter introduces a novel multiscale normalized modulus-based wavelet dif-

fusion method for speckle suppression and edge enhancement in ultrasound images. In

our approach, speckle image is iteratively filtered by the nonlinear diffusivity function

via the framework of the dyadic wavelet transform. In each iteration, the noisy image

is processed with three-step wavelet shrinkage-like procedures: decomposition, regu-

larization and reconstruction. Considering the statistical behavior of speckle, success-

ful employment of nonlinear wavelet diffusion in a speckle suppression task, requires

three appropriately designed components: an edge detector, a diffusion threshold and

a diffusivity function. Since most diffusivity functions developed from spatial nonlin-

ear diffusion have been shown to satisfy the denoising requirement, our work mainly

focuses on the design of the first two components above. We use the normalized

wavelet modulus as the edge detector to characterize the intrinsic signal/noise varia-

tion. The significant feature provided by this edge detector is its versatility for images

of different types. Thus, our algorithm can deal directly with either envelope-detected

speckle image or log-compressed medical ultrasonic image without any pre-transform.

To adapt the noise variation with iteration, the diffusion threshold is estimated from

the normalized modulus in the homogenous speckle regions. The automatic identifi-

cation of homogenous regions is implemented using a two-stage classification. First,

the normalized modulus at each scale is classified using the likelihood method based

on the Rayleigh mixture model. Second, the homogenous speckle region is identified

by a coarse-to-fine classification utilizing the edge persistence across scale. In this

procedure, a tuning parameter (K) is introduced to adjust the diffusion threshold,

81

and it further controls the final denoising result. Relying on this feature, the proposed

algorithm is highly flexible in producing a desired result for a specific application.

Using synthetic envelope-detected images, we have shown that our algorithm is a

versatile speckle reduction technique for both envelope-detected and log-compressed

speckle images. We also have demonstrated the performance superiority of the pro-

posed algorithm over the SRAD and GenLik methods in terms of speckle suppression

and edge preservation indices. With real ultrasonic images, we have shown that our

algorithm is quite robust in producing a desired result either for visualization en-

hancement or for auto-segmentation improvement. In summary, by combining the

sparsity and multi-resolution properties of wavelets, with the edge preservation and

enhancement features of the nonlinear diffusion, our algorithm provides very signifi-

cant speckle suppression and edge enhancement for the purposes of visualization and

automatic structure detection.

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Chapter 6

Speckle Suppression for 3-D Ultrasound Images

6.1 Introduction

Undergoing rapid development, three-dimensional (3-D) ultrasound has been ac-

cepted clinically as an extension of conventional 2-D ultrasound methods used for

visualization of 3-D anatomy and pathology [55]. Compared with other medical imag-

ing modalities, 3-D ultrasound scanning has advantages in that it does not involve

invasive measurement and can display volume information in real time. However,

3-D ultrasound image is severely corrupted by speckle and other artifacts, which

considerably complicate tasks of volume visualization and determination in clinical

applications. Therefore, a speckle reduction process is quite necessary for enhancing

visualization of organ anatomy and improving the accuracy of volume determination.

In previous chapter, we developed a speckle reduction filter for the 2-D ultrasound

images [56], called the nonlinear multiscale wavelet diffusion (NMWD) algorithm.

NMWD integrates the technical advantages of nonlinear anisotropic diffusion and

wavelet denoising, and provides a superior despeckling solution for ultrasound images.

Mathematically, the denoising properties of NMWD are inherited from the nonlin-

ear diffusion technique [34] and the dyadic wavelet transform(DWT) [44]. Therefore,

extension from 2-D to the multi-dimensional case rests on extension of these funda-

83

mental methods. Specifically, 3-D nonlinear diffusion method can be directly extended

by introduction of an additional dimensional variable in the PDE equation, whereas

a fast 3-D DWT can also be implemented as decomposition and reconstruction [57].

The major challenge in 3-D NMWD is that the statistical properties of the normalized

wavelet modulus (the edge map) change. For the 2-D case, the speckle-related nor-

malized wavelet modulus is modeled as Rayleigh distribution [56]. However, for the

3-D case, we demonstrate that it must be modeled as Maxwell distribution. There-

fore, in developing the 3-D NMWD algorithm, we first explore the basic theories of

wavelet diffusion. Subsequently, we re-examine the statistics of wavelet modulus and

propose a solution for 3-D ultrasound speckle suppression. Finally, we validate our

algorithm with synthetic and real ultrasound images.

6.2 3-D nonlinear multiscale wavelet diffusion

Given a noisy image f(x, y, z, t) at time t, the 3-D nonlinear diffusion equation is

expressed as

∂∂t

f(x, y, z, t) = div[c(x, y, z, t)∇f(x, y, z, t)]

f(x, y, z, 0) = f0(x, y, z)

(6.1)

where ∇ is the gradient operator, div is the divergence operator, and c(x, y, z, t) is

the diffusion coefficient. The diffusion coefficient is constructed using an edge map

η(x, y, z, t) and the diffusivity function g(·), specifically

c = g(|η|) (6.2)

84

Neglecting high-order terms and assuming 4t = 1, Equation(6.1) can be approx-

imated by a Taylor series expansion to first terms:

f(x, y, z, t + 1) ≈ f(x, y, z, t) +∂

∂x[c(x, y, z, t)

∂f(x, y, z, t)

∂x]

+∂

∂y[c(x, y, z, t)

∂f(x, y, z, t)

∂y]

+∂

∂z[c(x, y, z, t)

∂f(x, y, z, t)

∂z]. (6.3)

Letting p(x, y, z, t) ≡ 1− c(x, y, z, t), we further modify Equation (6.3) so that it can

be put into a format common to signal decomposition and reconstruction by filters

(Ai, Bi, Di, Ei, where i=1,2; see references [42,56] for details):

f(x, y, z, t + 1) = (f(x, y, z, t) ∗ A1) ∗ A2 + (p(x, y, z, t) · (f(x, y, z, t) ∗B1)) ∗B2

+ (p(x, y, z, t) · (f(x, y, z, t) ∗D1)) ∗D2

+ (p(x, y, z, t) · (f(x, y, z, t) ∗ E1)) ∗ E2. (6.4)

The filters satisfy the condition

A1 · A2 + B1 · B2 + D1 · D2 + E1 · E2 = 1. (6.5)

A finite-level discrete dyadic wavelet transform of the 3-D discrete function f ∈

l2(Z3) can be represented as:

W =

SJf, (W d

j f)d=1,2,31≤j≤J

, (6.6)

where SJf is a coarse scale approximation of f at final scale 2J , and W dj f represents

85

the detail image at scale 2j. A fast 3-D forward DWT and inverse-DWT pair can

be implemented by the combination of filters operation: a discrete function f is

decomposed using a lowpass filter H and a highpass filter G in three spatial directions

(d = 1, 2, 3), and reconstructed with a lowpass filter H and highpass filters K and L.

The necessary and sufficient condition for perfect reconstruction is [57]:

3∏

l=1

|H(ωl)|2 +3∑

l=1

K(ωl)G(ωl)L(ω1, ..., ωl−1, ωl+1..., ω3) = 1 (6.7)

The equivalence of Eqs. (6.5) and (6.7) indicates that nonlinear diffusion can be

approximated by the processes of filter decomposition and reconstruction within the

framework of dyadic wavelet transform. In this sense, we refer this specific type

of diffusion as wavelet diffusion. The wavelet diffusion coefficient p(·) in Equation

(6.4) can be expressed as a function of the nonlinear diffusivity function g(·), such as

p(·) = 1− g(·). Here, p(·) plays a crucial role in wavelet diffusion in the same manner

that the wavelet shrinkage function does in wavelet denoising.

6.3 Despeckling using 3-D NMWD

Most 3-D ultrasonic images generated by clinical imaging systems have been com-

pressed to fit dynamic display range. Empirically, the ultrasonic display image can

be modeled as [29]

f(x) = µR(x) +√

µR(x)n(x) (6.8)

where µ is the average amplitude of the target, and R(x) is the intrinsic signal with

mean one, and n(x) is a zero-mean Gaussian noise with mean one. In this section, we

86

focus our attention on the design of the edge map and the estimation of the diffusion

threshold for the ultrasonic display image.

6.3.1 Normalized modulus

In classical wavelet shrinkage methods, noise-related wavelet coefficients can be

reduced using thresholding techniques. In ultrasonic images, however, thresholding

is limited to the reduction of speckle-related wavelet coefficients, since speckle noise

is signal-dependent and noise-related coefficients can actually be larger than signal-

related coefficients. It has been shown that this signal-dependency can be suppressed

by removing the signal mean [49], thus revealing the intrinsic signal/noise relationship.

Consequently, we implement the normalized wavelet modulus Mjf as an edge map

to characterize this intrinsic signal variation:

Mjf = Mjf/√

µs =

√W 1

j f 2 + W 2j f 2 + W 3

j f 2

µs

(6.9)

where µs = (∑s

f)/N represents the local mean for a window s with N pixels.

To study statistical behavior of the speckle-related normalized modulus, we con-

sider the speckle-related normalized wavelet coefficients to be Gaussian distributed.

In our previous 2-D analysis, we assumed the speckle-related normalized wavelet mod-

ulus to be Rayleigh-distributed [56]. This assumption is not valid for the 3-D case,

however, since the envelope of 3-D Gaussian random variables is better characterized

by a Maxwell distribution [28]. Thus, the speckle-related normalized modulus Mjf

87

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Normalized modulus

Pro

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lity

j =1, Rayleigh mixture

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Normalized modulus

Pro

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lity

j =1, Maxwell mixture

2 4 6 8 10 12 14 160

0.05

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0.35

0.4

0.45

Normalized modulus

Pro

babi

lity

j =2, Rayleigh mixture

2 4 6 8 10 12 14 160

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Normalized modulus

Pro

babi

lity

j =2, Maxwell mixture

Figure 6.1: Histograms and fittings of normalized modulus at the first (top) and second(bottom) scales of a 3-D liver image (shown in the first row of Figure 6.2). The fittings aremodeled by the Rayleigh-mixture (left) and Maxwell-mixture (right), respectively.

for 3-D images is given by:

p1(x) =

√2

π

x2 exp(−x2/(2α21))

α31

(6.10)

where x denotes the Maxwell random variable, α1 is the Maxwell parameter. Simi-

larly, the distribution of edge-related normalized modulus p2(x) has the same form

as Equation (6.10) with parameter α2. The normalized wavelet modulus Mjf is thus

88

well-represented by the Maxwell mixture model

p(x) =2∑

i=1

ωipi(x) (6.11)

where ω1 and ω2 are the weights of noise component and signal component, respec-

tively, and ω1 + ω2 = 1. We use the expectation maximization (EM) [51] method

to estimate the parameters Θ = (ω1, ω2, α1, α2). Given a data set of size N , the ob-

served data X = xjNj=1 and unobserved data Y = yjN

j=1 consist a complete data

set (X,Y). Let Y be i.i.d., and yj = i if yj is generated by mixture component i. In

the expectation step, the expectation of log-likelihood of the complete data is

Q(Θ|Θ∗) = E [log p(X,Y|Θ)|X, Θ∗]

=2∑

i=1

N∑j=1

p(i|xj, Θ∗) log ωi +

2∑i=1

N∑j=1

p(i|xj, Θ∗) log pi(xj|αi) (6.12)

where Θ∗ are the current parameters. The posterior probability of jth sample be-

longing to ith mixture component is given by

p(i|xj, Θ∗) =

ω∗i pi(xj|α∗i )2∑

i=1

ω∗i pi(xj|α∗i ). (6.13)

In the maximization step, the parameters are estimated by the differentiating Equa-

tion (6.12) with respect to every parameter. The weight ω∗∗i is estimated by

ω∗∗i =1

N

N∑j=1

p(i|xj, Θ∗) (6.14)

89

and the density parameter α∗∗ is given as

(α∗∗i )2 =

N∑j=1

p(i|xj, Θ∗)x2

j

3N∑

j=1

p(i|xj, Θ∗). (6.15)

With an iterative update strategy, the noise and signal components of normalized

modulus can be accurately estimated.

Figure 6.1 shows a statistical analysis of the first and second scales of the 3-D

normalized moduli calculated from a 3-D ultrasound liver image (shown in Figure 6.2).

The normalized moduli were fitted by both Rayleigh-mixture and Maxwell-mixture

models. As the figures show, Maxwell-mixture model follows the histograms of 3-D

normalized moduli very well. We also computed the RMS (root mean square) errors,

RMS = 0.011 (j = 1), 0.066 (j = 2) for Maxwell-mixture model, and 0.018 (j = 1),

0.072 (j = 2) for Rayleigh-mixture model. The results indicate that Maxwell-mixture

model is more accurate in characterizing the statics of 3-D normalized modulus.

6.3.2 Diffusion threshold

The diffusion threshold is estimated using the noise variation in the homogenous

speckle region of the image. We employ the likelihood classification and cross-scale

edge consistency to separate the homogenous speckle regions from other regions. Us-

ing the parameters estimated by the EM method, we can easily obtain the likelihood

classification threshold Tj.

Tj =2 log(ω1

ω2)− 6 log(α2

α1)

( 1α2

2− 1

α21)

(6.16)

90

Furthermore, we rely on cross-scale edge consistency to identify the homogenous

speckle region Uj in a 3-D ultrasonic image, that is,

Uj =

1,∏

Mj,j+1f < K2∏

Tj,j+1

0, elsewhere

(6.17)

where K is a tunable parameter that controls the region of interest. Finally, the

diffusion threshold is computed as a function of the mean of the normalized modulus

at the detected homogenous regions.

λj = Mean(UjMjf)2−j′/2 (6.18)

where j′ = 0 for j = 1 and j′ = j for j ≥ 2. As an example, 3-D edges/homogenous

regions of a 3-D ultrasound liver image (top row, Figure 6.2) is classified by the

proposed algorithm with K = 1, and results are shown in the bottom of Figure 6.2.

6.4 Results

Our algorithm was tested on both of synthetic and real ultrasonic images. To

quantitatively evaluate the filter performance, a 3-D phantom was constructed using

three layers with different intensities: the external structure is a cubic box, middle is

a ball, and the internal region is diamond-shaped, as shown in Figure 6.3(a). Speckle

noise was generated by lowpass filtering a complex 3-D Gaussian random field and

taking the magnitude of the filtered output. To mimic the clinical data, the envelope

detected ultrasound image was compressed by the logarithmic transform. Figure

6.3(b) shows the synthesized 3-D ultrasound image with size of [64× 64× 64].

91

Figure 6.2: Top row: the arbitrary slices of a 3-D human liver ultrasound image alongYZ, XZ and XY planes (left to right). Bottom row: the corresponding slices taken from theclassified normalized modulus, where homogenous speckle regions are shown in white andedges in black.

We compared the performance of our speckle suppression algorithm with that

of a 3-D speckle reducing anisotropic diffusion (3-D SRAD) technique [37, 58]. In

the 3D-SRAD implementation, the test image was first uncompressed prior to being

filtered by the algorithm. The time step of the diffusion was set as 4t = 0.05, and

the number of iterations was 300. In the 3-D NMWD implementation, the Weickert

filter [35] was used as the diffusivity function

g(η) =

1 η ≤ 0

1− exp[−3.315(η/λ)4

] η > 0.

A two-level filtering scheme was employed on the test image. The parameter K = 1.5,

and the number of iterations was 15. We have implemented our algorithm using

92

(a) (b)

(c) (d)

Figure 6.3: (a) 3-D phantom, (b) synthetic ultrasound image, and the filtered resultsgenerated by (c) 3-D SRAD and (d) 3-D NMWD.

MATLAB. For the image with size of [64 × 64 × 64], the computational time for 15

iteration was 45 seconds on a 2.0GHz Centrino Duo computer.

The results of two algorithms are compared at the middle slice along the XZ, Y Z

and XY planes, as shown in Figures 6.3(c) and (d). Significant edge enhancement

and speckle suppression were observed in the NMWD result. An important objective

of 3-D filtering technique is to improve the ability to identify structure as part of the

93

Table 6.1: Despeckling Performance Comparison

Method Noisy Image SRAD NMWD

ρ 0.3625 0.5233 0.5866FOM 0.3238 0.6686 0.7983

volume visualization process. Therefore, we constructed the internal diamond shape

for both noisy and despeckled synthetic images, shown in the right corner of each

sub-figure. Similarly, the improvement of volume visualization was also observed

in the NMWD result. We further quantified the denoising performance using two

standard quality indices: a figure of merit (FOM) [37, 54] for edge-preservation, and

a correlation-based quality index ρ [26] for speckle suppression,

ρ =

∑i,j,k∈w

(x(i, j, k)− µx)(y(i, j, k)− µy)

√ ∑i,j,k∈w

(x(i, j, k)− µx)2 · ∑i,j,k∈w

(y(i, j, k)− µy)2

where µx is the mean intensity of volume window w in image x. Results are shown

in TABLE I, where improved performance of our algorithm over SRAD is clearly

demonstrated by both quality indices.

A 3-D ultrasound scan of human liver was used as an in vivo test image. As shown

in the first row of Figure 6.4 (which is same as the top row in Figure 6.2), the volume

slices lay out the noduli, veins and soft tissues. Denoised results of SRAD and NMWD

are shown in middle and bottom rows, respectively. Our algorithm outperforms SRAD

by clearly outlining the noduli and veins in the liver without over-smoothing edges of

these tissue structures. In clinical applications, 3-D despeckling technique is expected

to significantly improve 3-D structure visualization and segmentation. Consequently,

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Figure 6.4: The slices of 3-D human liver ultrasound image along YZ, XZ and XY planes(left to right). Top row: original image, middle: 3-D SRAD, bottom: 3-D NMWD.

our algorithm was further tested by employing volume isosurfacing on the test liver

image (shown in Figure 6.4). Our goal was to extract the 3-D structure of liver vessels

from the given images. In the isosurface method [59, 60], the surface extraction is

determined by the selection of threshold: a large value may lead to retain irrelevant

surfaces, whereas a small value could cause insufficient extraction and discontinued

structures. The role of filtering methods in visualization is to reduce the ambiguity

of the selection of threshold by suppressing speckle and enhancing boundaries. For

95

(a) (b)

(c) (d)

Figure 6.5: Volume visualization of vessels in a 3-D liver image, which is pre-processedwith (a) lowpass, (b) BLTP, (c) SRAD and (d) NMWD.

comparison, two general ultrasound filtering methods, lowpass and BLTP, were also

tested. In the lowpass method, the original image was filtered by a Gaussian kernel

of size [5,5,5] and σ2 = 1. In the BLTP method [61], the image was processed by a

filtering pipeline: binarize, lowpass, threshold and propagate. All processed images

were further filtered by a 3-D median filter to remove tiny structures. Then, the

vessel surface was visualized by isosurface extraction [59,60]. As shown in Figure 6.5,

the results indicate that both lowpass based methods have difficulties in dealing with

96

speckle-related rough surface. Direct lowpass filtering often leads to oversmoothed

surfaces when reducing speckle (Figure 6.5(a)), whereas the BLTP method depends

heavily on parameter selection in its procedures. Both SRAD and NMWD produced

acceptable despeckled images for volume visualization. However, in the SRAD result,

the smearing of weak boundaries leads to the appearance of disconnected vessels

(shown in Figure 6.5(c)). The result generated by our 3-D NMWD (Figure 6.5(d))

proves much more favorable for volume visualization due to its improved performance

with regard to speckle suppression and boundary enhancement.

6.5 Conclusion

In summary, we have developed a speckle suppression algorithm for 3-D ultra-

sound images. In our method, ultrasound images was filtered by nonlinear diffusivity

function within the framework of dyadic wavelet transform. In our design, normal-

ized modulus was used as the edge map to reveal signal/noise variation. The diffusion

threshold was estimated from homogenous speckle regions, which was classified rely-

ing on the statistical analysis of the normalized modulus. We tested our algorithm on

synthetic and real 3-D ultrasound images. The results indicated that our algorithm

was robust for both speckle suppression and edge preservation. The potential clinical

value of our algorithm was demonstrated by visualization of 3-D structure in the test

images.

97

Chapter 7

Speckle Tracking in Intracardiac

Echocardiographic Images

7.1 Introduction

Intracardiac echocardiography (ICE) has provided considerable advantages in

guiding clinical electrophysiology procedures such as imaging anatomical structures,

confirming electrode-tissue contact, monitoring ablation lesions, and providing hemo-

dynamic assessment [62–65]. Clinically, ICE is predominantly implemented on the

basis of a catheter carrying at its distal end a rotating transducer that operates at a

frequency of 9 MHz and provides two-dimensional (2-D) tomographic images of the

heart’s interior, or a catheter with a phased transducer array operating at a lower

ultrasound frequency for deeper penetration (5.5-10 MHz) and provides a 2-D sector

view of the heart (80o -90o opening angle). The need for improving the management

of several types of complex rhythm disorders has demanded a better understanding of

these disorders particularly in relation to underlying anatomy and physiological out-

come. This requisite makes ICE an attractive imaging modality for online anatomical

and functional imaging during catheterization [66,67].

In general, dynamic functional analysis methods rely on motion estimation tech-

98

niques for extracting structural information from image sequences. Several techniques

have been applied to motion detection in traditional echocardiography, including opti-

cal flow [68–70], deformable registration [71,72], and block matching [73–77] methods.

Ultrasound speckle is an inherent property of an ultrasound image (B-mode), and is

the result of the diffuse scattering that occurs when an ultrasound pulse randomly

interferes with the small particles or objects on a scale comparable to the sound wave-

length. Classical optical flow approaches (e.g. Lucas-Kanade [68]) estimate motion by

assuming intensity constancy of detected targets between consecutive frames. These

methods are highly sensitive to noise. Since ultrasound speckle noise severely de-

grades the accuracy of methods employing the intensity-based assumption, the use of

optical flow methods often leads to unsatisfactory results [78]. On another hand, a de-

formable registration technique [71] has been adapted to myocardial motion detection,

resulting in excellent spatial capture ability for non-uniform deformation [72]. Many

motion detection methods consider ultrasound image sequences as general images,

and neglect the statistical features of ultrasound in their estimation schemes. How-

ever, a certain class of block matching techniques [74–77] uses the signal-dependent

nature of speckle as a spatial marker for underlying tissue motion. These methods

achieve a more accurate estimation than those methods that assume intensity con-

stancy between frames. A major drawback of the block matching approach however,

is that its capture ability varies with the size of the block. It is very difficult to select

an optimum block-size that captures both large and small deformations simultane-

ously [73].

In developing a myocardial motion estimation scheme for echocardiographic im-

ages, two factors have to be considered and appropriately managed: non-rigid my-

99

ocardial deformation and ultrasound speckle. Myocardial motion is complex and

includes various motion patterns. During systole, the myocardial wall moves inward

to eject blood and thickens, whereas during diastole the wall moves outward and

thins. Additional deformations include longitudinal and circumferential shortening

of the myocardial muscle fibers. The varieties of myocardial motion suggest that a

non-rigid motion model (deformable model) is necessary. Consideration of speckle

(and other artifacts) is also important in achieving good accuracy. In the ideal case,

speckle patterns of moving tissue are temporally stable under the condition of small

motion. In echocardiographic images, however, such temporal constancy usually is

not valid due to non-uniform myocardial motion and speckle decorrelation. Even

for a myocardium with uniform structural/perfusion properties, its texture in ultra-

sound images may appear different from frame to frame. When Gaussian noise-based

estimation techniques (e.g., least square estimation) are used, they often encounter

false matching when subjected to texture varied echocardiographic images. Although

ICE images have an advantage of high spatial resolution, these same motion analysis

challenges remain in effect due to the signal-dependent nature of speckle.

The objective of the study was to develop and validate an algorithm for regional

myocardial deformation analysis on the basis of ICE imaging. A deformable speckle

tracking approach for motion estimation in ICE images is presented. The method

incorporated statistical features of ultrasound images into a maximum likelihood mo-

tion analysis, and treated maximization of the similarity measure as energy mini-

mization. Thus, within the framework of deformable registration, tissue motion was

estimated via optimization of a speckle-featured energy function. The robustness of

our algorithm was evaluated by studying speckle decorrelation on a series of simu-

100

lated ultrasound image sequences. In addition, measurements of regional myocardial

displacement and strain were obtained from animal experiments by sonomicrometry

and were further used to validate our motion analysis algorithm.

7.2 Ultrasound Image Model

In mathematically describing ultrasound pulses transmitted to the body, the de-

tected backscattered signals (echoes) rf (x, y, z) are often modeled as the convolution

of ultrasound point spread function (PSF) h(x, y, z) with the tissue scattering function

t(x, y, z) [22]:

rf (x, y, z) = h(x, y, z) ∗ t(x, y, z) (7.1)

where ∗ denotes the convolution operation. The tissue scattering function t(x, y, z)

represents the tissue properties along the direction of propagation of ultrasound

pulses, which can be represented [22] as:

t(x, y, z) =∑

n

anδ(x− xn, y − yn, z − zn) (7.2)

where (xn, yn, zn) denotes the spatial position of scatterer n, δ(·) the scatterer impulse

function, and an the echogenicity of the scatterer. The envelope of the radiofrequency

signal is obtained by

f(x, y, z) = |rf (x, y, z) + j rf (x, y, z)| (7.3)

where rf is the Hilbert transform of rf . To model a 2-D ultrasound image, one may

consider PSF to be separable, i.e., h(x, y, z) = h(x, y)hz(z). Hence, a 2-D slice of

101

rf (x, y, z) can be obtained [22] as:

rf (x, y) = h(x, y) ∗ t(x, y) (7.4)

where

t(x, y) =

∫t(x, y, z)hz(z)dz (7.5)

If hz(·) is constant within the thickness (elevational plane) of the ultrasound beam,

and the number of scatterers in the ultrasound resolution cell is sufficiently large to

satisfy the central limit theorem, then t(x, y) can be modeled as a normal process [22].

7.2.1 Tissue Motion

Consider a sequence of envelope-detected ultrasound images acquired from moving

tissue. If the PSF is stationary during the scanning process, then from (7.1) and

(7.3), the texture variation within the image sequence is the result of tissue motion.

Let (x0, y0, z0)T represent the initial spatial position of a given scatterer in the tissue.

The new position of the scatterer (x1, y1, z1)T is obtained by employing tissue position

transformation:

(x1, y1, z1)T = A(x0, y0, z0)

T + b (7.6)

where A represents 3-D deformation and rotation of the scatterer, and b = (b1, b2, b3)T

represents the translation vector. For a 2-D ultrasound image, the underlying tissue

motion is assumed to take place only in the x-y plane. That is, z-components in A

and b of (7.6) are set to zero. This leads to z1 = z0, and the 3-D affine transformation

102

(a) (b)

(c) (d)

Figure 7.1: Two consecutive frames of ICE images acquired in mid left ventricle of a dog.(a) Frame 1, and (b) Frame 2, where the recording stages are indicated by N on ECG signals.(c) A region of myocardium in (a). (d) A region of myocardium in (b) and correspondingto that of (c). Going from (c) to (d), there is a [5,5] pixel shift toward the right-bottomdirection.

represented by (7.6) can be simplified to the 2-D case.

7.2.2 Motion Noise

Following the above assumption of 2-D motion, texture variation in the image

sequence is generally presumed to be the result of underlying 2-D tissue motion [22].

Since most of the tissue is well-structured, the spatial arrangement of diffuse scatterers

is assumed to be relatively stable during tissue motion. Therefore in the ideal case,

the texture of a particular moving tissue structure should hold stable (i.e. constant

speckle pattern), which implies that the local speckle pattern is trackable in terms of

103

intensities and correlation. Accordingly, one may conclude that tracking of speckle

pattern can reveal underlying tissue motion.

Actual myocardial motion however, is more complicated than the ideal description

above. As an example, two consecutive ICE images of a left ventricle are shown in

Fig. 7.1. Two regions (white boxes in Fig. 7.1 (a) and (b)), representing the same

segment of myocardium, are selected from the images. Although these two regions

are closely motion related, their texture is obviously different, as shown in Fig. 7.1

(c) and (d). A low similarity measure (correlation coefficient = 0.16) between the

two regions supports this visual observation. Similar texture variation can also be

observed in standard external 2-D echocardiographic imaging. From our observations

of echocardiographic images, texture variation in moving tissue can be described by

the following: 1) a certain percentage of the speckle pattern is stable, which is gener-

ally characterized by Rayleigh distribution; 2) a significant amount of random texture

variation is present, which can cause a twinkling appearance of tissue structure in the

image sequence; and 3) local mean intensities are generally stable during motion.

In some cases, the twinkling texture effect dominates the image sequence, and tis-

sue structure can only be recognized by relying on the constancy of the local mean

intensity.

Many factors can cause random texture variation, and include speckle decor-

relation, tissue deformation, non-negligible out-of-plane motion, and variation in

PSF [79]. These factors may be related to each other, and together contribute to

random texture generation. Speckle decorrelation has the most profound effect on

texture variation. Temporal decorrelation of speckle pattern generally occurs when

tissue scatterers have axial, lateral or elevational motion. Speckle decorrelation due

104

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ

Cor

rela

tion

coef

ficie

nt

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 18

9

10

11

12

13

14

15

16

17

λ

SN

R(d

B)

(b)

Figure 7.2: Correlation coefficients (a) and image quality as described by signal-to-noiseratio (SNR) (b) in relation to speckle decorrelation index (λ).

to axial and lateral motion can be usually solved by relying on the similarity between

speckle patterns [80], whereas speckle decorrelation due to elevational motion is gen-

erally difficult to track, since tissue may move out of the scanning plane. One may

utilize elevational speckle decorrelation to estimate displacement in the direction of

elevation [81], however it is implausible for complex motion, especially myocardial

motion, which generally has radial, circumferential and longitudinal components.

105

Speckle decorrelation results in loss of echo signal coherence and leads to dis-

placement estimation errors. It has been shown that compensation for decorrelation

in speckle tracking is an ill-possed inverse problem [82]. Therefore rather than try-

ing to extract motion components from decorrelated patterns, we consider speckle

decorrelation as a kind of motion noise, which heavily degrades the accuracy of mo-

tion estimation. We assume that out-of-plane motion plays a major role in causing

unsolvable speckle decorrelation. To describe the degree of corruption in an image

sequence by speckle decorrelation, we define a parameter λ, which relates the portion

of scatterers in out-of-plane motion to the total number of scatterers in the tissue

scattering function. Hence, λ = 0 represents the ideal 2-D motion that is free of

elevational motion, whereas λ = 1 represents severe corruption by elevational speckle

decorrelation. For a small value of λ, speckle decorrelation may be considered as

an outlier in the estimation. However, when λ reaches a sufficiently large value, it

becomes the major noise signal in the estimation. As a demonstration, ultrasound

images were synthesized with the same echogenicity (described below as shown in

Fig. 7.4(a)), but corrupted with different levels of elevational speckle decorrelation,

λ = 0.0-1.0, as portrayed in Fig. 7.2(a). The signal-to-noise ratio (SNR) was com-

puted by comparison with the elevational motion free image (λ = 0), and the result

is shown in Fig. 7.2(b).

Speckle decorrelation complicates any tracking algorithm that relies on texture

constancy. Thus, our goal was to develop a robust noise-resistant tracking method

that accurately detects underlying tissue motion. We have rejected general least-

square based methods due to their vulnerability to outliers. Instead, we favor a

tracking method that relies on constancy of tissue echogenicity. The recent applica-

106

tion of maximum likelihood motion estimation to ultrasound image sequences [74–77]

suggests that a good tracking strategy would be one that incorporates statistics of

the ultrasound image into the estimation process.

7.3 Maximum Likelihood Motion Estimation

Let a reference image (Ir) and test image (It) be two motion related frames in a

given sequence, where I ⊂ Z2. If a pixel x(xx,xy) ∈ Ir has its corresponding pixel

y(yx,yy) ∈ It , then motion displacement u between two pixels is given as

u(x) = y − x, (7.7)

where u(x) = u(x, y) = (u1(x, y),u2(x, y)) = ((yx − xx), (yy − xy)). Let ft = ft(x)

represent the intensity value of image It at pixel x, and u = u(x) the displace-

ment field. According to the maximum likelihood method for parameter estimation,

the estimated displacement vector u is obtained by maximization of the conditional

probability density function (pdf) [74]

u = arg maxu

p(fr|ft,u). (7.8)

Since Ir and It are directly related within the same image acquisition model, the

conditional probability of fr given its homologue ft and displacement u is described

by [83]

p(fr|ft,u) =∏

i∈ I

p

(fr(i)

∣∣fωt (i)

)(7.9)

107

where fr(i) denotes the intensity value of the reference image at pixel i ∈ I, and

fωt (i) is obtained by applying the spatial transform on ft(i), i.e., fω

t (i) ≡ ft(u(i) + i).

7.3.1 Image Sequences with Gaussian Noise

Noise is considered independent of the signal for most imaging systems. Conse-

quently, the intensity variation due to motion can be expressed as

ft(x) = fr(x) + n(x),

where n(x) represents additive system noise. If noise is identical independent dis-

tributed (i.i.d.) with pdf pn, maximization of the conditional probability p(ft|fr,u)

is equivalent to

maxu

∏

i∈ I

pn(fωt (i)− fr(i)). (7.10)

For Gaussian noise, the dissimilarity cost function is given by the normalized negative

log-likelihood of (7.10)

E =1

N

∑

i∈ I

|fwt (i)− fr(i)|2 (7.11)

where N is the number of pixels in the image. We recognize (7.11) as the sum of

squared differences (SSD), which is widely used as a cost function in intensity-based

registration methods [71,72].

7.3.2 Ultrasound Image Sequences

Speckle is considered as signal-dependent noise in ultrasound images [12]. Hence,

the Gaussian noise assumption inherent in motion estimation is no longer valid. The

108

envelope-detected ultrasound image (7.3) can also be modeled as the result of interac-

tion between the ultrasound PSF and a complex field T (x) = t(x) + jt(x), such that

f(x) = |h(x)∗T (x)|. Assuming that the amplitude of scattering function varies slowly

within the resolution cell, the envelope-detected image can be approximated [12] as:

f(x) = g(x)n(x)

where g(x) = |a(x)| is the amplitude of scattering function (representing tissue

echogenicity), and n(x) is the noise term. For fully developed speckle, n(x) is identical

independent Rayleigh distributed [10,12],

pn(x) =x

σ2exp(

−x2

2σ2).

Considering the ratio between two ultrasound images having the same amplitude scat-

tering function (i.e. constant echogencity), fr(x) = g(x)nr(x) and ft(x) = g(x)nt(x),

then

ft(x) = nt(x)/nr(x)fr(x). (7.12)

Let η = nt/nr represent the ratio of two Rayleigh random variables, and assume

that speckle variances are unchanged between the two frames, then the pdf of η is

given [75] as:

pη(x) =2x

(x2 + 1)2, η > 0. (7.13)

In most medical applications, one deals directly with displayed ultrasonic im-

ages, which are logarithm-compressed versions of envelope-detected images. This

kind of nonlinear compression changes the statistics of the envelope-detected signals,

109

and different compression coefficients can lead to different statistical distributions of

signals [27]. Consequently, direct use of displayed ultrasonic images (without prior

uncompression) is highly preferred. To this end, the log-transform is applied to both

sides of (7.12) to obtain ft(x) = fr(x) + η(x). The pdf of η is derived as a function

of random variable η [76]

pη(x) =2 exp(2x)

[exp(2x) + 1]2. (7.14)

The conditional pdf for displayed ultrasound images is given by

p(ft|fr,u) =∏

i∈ I

2 exp(2(fωt (i)− fr(i)))

[exp(2(fωt (i)− fr(i))) + 1]2

(7.15)

where N is the number of pixels in the image. Here, for convenience, we continue to

use the symbol f to denote the input log-compressed image instead of f . The motion

between frames is estimated by maximizing (7.15).

7.4 Ultrasound Elastic Speckle Tracking

To formulate the ultrasound motion estimation problem as an optimization pro-

cess, we apply the normalized negative log likelihood function to (7.15) and define a

maximum likelihood cost function for ultrasound speckle tracking (USST) as:

Λ(u) ≡ 1

N

∑

i∈ I

(ln(exp(2r(i)) + 1)− r(i)

)(7.16)

where r(i) = fωt (i) − fr(i). Maximization of the conditional pdf (7.15) is equivalent

to

u = arg minu

Λ(u). (7.17)

110

r

ρ(r)

r

ψ(r

)

r

ψ(r

)

SSD

USST

Figure 7.3: Comparison of robustness between SSD and USST estimators. Top, objectfunctions of SSD and USST estimators in relation to residual r. Bottom left, SSD influencefunction, Bottom right, USST influence function.

We refer to this algorithm as a USST-based maximum likelihood motion estimator to

distinguish it from the more conventional SSD-based maximum likelihood estimator.

7.4.1 Robust Noise Resistance

Robust estimation is often used to achieve accurate estimation for the case of

missing data or isolated points having high residual errors (outliers) [84]. One class of

robust estimators, called M-estimators, is designed to minimize the sum of residuals.

111

Let r be the residual of the difference between the ith fitted value and the observation.

The objective function of the SSD method is a least-squares estimator, ρ(r) = r2 and

the influence function is ψ(r) = 2r. The least-squares solution is highly sensitive to

outliers as its influence function is unbounded to the residual error. As a consequence,

the accuracy of the SSD method is limited in the presence of speckle decorrelation in

the ultrasound image sequence. This kind of speckle outlier is extremely difficult to

exclude from the estimation due to its coherent nature. However, the USST estimator

has the necessary features that lead to robust estimation [84], whereby the objective

function is given by

ρ(r) = log(exp(2r) + 1)− r

and its influence function is

ψ(r) =exp(2r)− 1

exp(2r) + 1. (7.18)

Figure 7.3 illustrates the difference between the SSD and USST estimators. For the

SSD estimator, the influence of a datum on the estimation increases linearly with its

error. On the other hand, the influence function of the USST estimator is bounded

by ±1, which suggests that the USST estimator has a better outlier resistance than

the SSD estimator. Moreover, the robustness of the USST estimator is fairly obvious

in a linear system with limited parameters, (e.g., affine transform) where one may

show that the objective function has a unique minimum in parameters and is convex

in every transform variable. This explains why the speckle similarity measure is more

accurate than the cross-correlation measure [74,75].

112

7.4.2 Deformable Registration

Myocardial motion is mostly non-rigid, hence tissue displacement u(x) (7.7) in

an ICE image can be described by a 2-D non-rigid transformation based on cubic

B-splines [71]:

u(x) =∑

k∈z2

ckB3(

x

h− k) (7.19)

where ck = (c1k, c

2k) is the deformation parameter, h = (h1, h2) is the space between

nodes, and

B3(x

h− k) = β3(

x

h1− k)β3(

y

h2− l)

is the tensor product of cubic B-splines.

During the warping process, computed u(x) may have a non-integer value. Thus,

we interpolate the image using B-splines as:

f(x) =∑

i∈z2

biB3(x− i) (7.20)

where bi is a set of interpolation coefficients, and B3(x− i) = β3(x− i)β3(y− j) is a

tensor product of cubic B-splines.

The solution for the minimum of the cost function (7.16) is the deformation field,

u(x), which is obtained by using an optimization algorithm that acts upon the pa-

rameters cmk , m = 1, 2. First partial derivative of Λ is calculated explicitly as:

∂Λ

∂cmk

=∑

i∈z2

exp(2r(i))− 1

exp(2r(i)) + 1

∂ft(w)

∂wm

∣∣∣∣w=u(x)+x

∂um

∂cmk

(7.21)

113

where

∂ft(w)

∂w1=

∑

i∈z2

bi∂β3(d)

∂d

∣∣∣∣d=u1+x−i

β3(u2 + y − j) (7.22)

∂um

∂cmk

= B3(x

h− k).

The cubic B-spline interpolation affords convenience by explicit differentiation of the

cubic B-spline window, which reduces to the difference of two shifted quadratic B-

splines [85].

7.4.3 Implementation Details

We use the limited memory optimization algorithm of Broyden-Fletcher-Goldfarb-

Shanno with bound (L-BFGS-B) [86] to minimize the cost function in (7.16). Apply-

ing L-BFGS-B is appropriate and efficient for our large scale problem. In addition, the

bounded search range provides a regularization constraint to variables. Two stopping

criteria are employed to terminate the optimization: the maximum iteration num-

ber, and ‖Λ′(ck)‖ ≤ ε, where ε is tolerance value. The deformation parameter is

iteratively updated during optimization as ck+1 = ck +4c [71].

The objective function of the USST method and its influence function (7.18) sug-

gest that the USST estimator is a robust estimator for a rigid transform. However, for

non-rigid B-spline registration, the robustness of the USST estimator is ambiguous.

The objective function of parametric B-spline deformation is not guaranteed to be

convex in all B-spline coefficients. Consequently, we use multiresolution and itera-

tive refinement techniques to reduce the local minima, wherein the global difference

between two images at a coarse scale is propagated to finer scales. This strategy

114

has desirable features in that it speeds up the convergence process and increases the

ability to capture deformations. Both reference and test images are resized so as to

construct an image pyramid from coarse to fine resolution. When the solution con-

verges at a given pyramid scale, the computed parameters are then used as initial

estimates for the parameters at the next finer resolution. This process is repeated

until the finest (original) scale is reached.

7.4.4 Motion field

To compute the displacement field of an image sequence, we apply the registration

algorithm to a whole image sequence (e.g., a cardiac cycle). A very significant ad-

vantage of using multiresolution deformable registration is its ability to capture large

deformations, thus allowing the employment of a relatively simple update strategy.

Specifically, the first image of an ICE sequence (usually end of diastole) is used as

the reference image, and every subsequent image in the sequence is registered with

respect to the first image. Such a method avoids temporal drift errors (normally quite

significant) when using consecutive image pair registration. To integrate the temporal

coherence of deformation, a cubic smoothing spline function s is fit to the computed

displacement field [87, 88]. For a given motion displacement sequence u(i, t), where

i = (i, j) is the grid position, and ti = 1, ..., n is the frame index, the optimal solution

s is obtained by minimizing

γ

n∑ti=1

(u(i, ti)− s(ti))2 + (1− γ)

∫∂2s(t)

∂t2dt (7.23)

115

where γ is a smoothing parameter that controls the tradeoff between fidelity and

smoothness. The parameter γ can be adjusted as necessary to generate a visually

smooth output. The resultant smoothed displacement s is used to calculate the

displacement field according to

vt = st − st−1. (7.24)

7.4.5 Model Validation

Motion tracking was initially tested on a computer model that simulated ultra-

sound image sequences. The USST method was validated by comparing its perfor-

mance with that of the SSD-based deformable registration [71, 72]. Quality indices

were used as quantitative measures of estimation performance. To obtain statistically

meaningful results, extensive tests were performed on ultrasound image sequences

corrupted by different noise levels.

Subsequently, the feasibility and accuracy of depicting regional myocardial de-

formation were examined using ICE images acquired from an animal model. Four

healthy mongrel dogs (30 - 35 kg) were included in the study, and the protocol ad-

hered to the PHS guidelines for the care and use of laboratory animals. The dogs

were preanesthetized with xylazine, atropin, and propofol, endotracheally intubated,

and ventilated using an external respirator while anesthesia maintained by isoflurane

inhalation. Electrodes were attached to the limbs to record ECG leads I, II, and III

(model ECG100; Biopac Systems, Goleta, CA). Midline thoracotomy was performed

and the heart was suspended in a pericardial cradle. A 9-F sheath was inserted into

the LV through a purse string suture in the apex and positioned along the LV ma-

116

jor axis. A standard 9-F 9-MHz ICE catheter (model Ultra ICE; Boston Scientific,

Boston, MA) was inserted through the sheath and forwarded into the LV. The ICE

catheter had a distal transducer that emitted and received ultrasound pulses. To-

mographic short-axis views of the cavity were derived by attaching the ICE catheter

to a motor drive unit that enabled automatic and continuous rotation of the trans-

ducer at a fixed speed. The ICE catheter was connected to an imaging console

(model iLab; Boston Scientific) to acquire continuous 2-D echocardiographic images

(rate=30 frame/s). A calibrated high-fidelity pressure catheter (5F, model SPC-350;

Millar Instruments, Houston, TX) was also inserted into the LV via the apex. The

echocardiographic images were acquired continuously throughout the cardiac cycle,

along with ECG and LV pressure signals (ICE sampling rate was 30 frame/s; ECG

and pressure sampling rates were each 1000 sample/s). Myocardial regional displace-

ment was measured by standard sonomicrometry (Sonometrics Corporation, London,

Ontario, Canada). Two segment-length ultrasound crystals (diameter=2 mm) were

fixed under the guidance of ICE in mid myocardium of anterior LV wall. Specifi-

cally, the first crystal was placed in mid lateral region of the LV. The second crystal

was placed in mid anterior LV, about 2-3 cm from the first crystal. Circumferential

displacement around the long axis was recorded continuously with a time resolution

of 1.0 ms. To eliminate effects of breathing, the respirator was temporarily turned

off for a brief period during each acquisition. Due to overlapping frequency bands

of operation, sonomicrometry and ICE imaging were performed in sequence. Data

were collected at baseline as well as at two levels of increased contractility induced

by dobutamine (DOB) administration (1-2.5 µg/kg.min).

117

7.5 Experimental Validation by Computer Model

7.5.1 Ultrasonic Image Phantom

Envelope-detected 2-D ultrasound images were generated using (7.3) and (7.4),

and an echogenicity map was constructed to mimic tissue structure in a typical

ICE image of the left ventricle. These images contained a cavity, ventricular my-

ocardium, and background tissues. Speckle in the homogenous regions was verified to

be Rayleigh distributed. To simulate a commercial medical-grade ultrasonic image,

the envelope-detected image was then log-compressed. An example of a phantom

image is shown in Fig. 7.4(a).

A test image was synthesized by simulating the real ultrasound imaging process

[69]. Specifically, the moving tissue scattering function was generated by transforming

the reference tissue scattering function using a predefined motion field. To mimic the

real ICE image to an even greater extent, we also introduced speckle decorrelation

into the test image. Most severe speckle decorrelation occurs in 2-D echocardiographic

images when myocardial scatterers move out of the scanning plane. To reproduce this

situation, some scatterers were randomly selected to have elevational motion (z1 6= z0

in (7.6)). The vacancies thus created were randomly occupied by new scatterers.

The decorrelation degree was indexed by λ, the variation parameter in the scattering

function.

7.5.2 Experiments on a Pair of Images

We first considered the registration of a pair of motion-related ultrasound images.

The reference image and the test image are shown in Fig.7.4(a) and (b), respectively

118

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 7.4: Warped results of a pair of synthesized ultrasound images. (a) reference, and(b) test image. (c) superposition of (a) and (b). (d) true deformation field. (e) and (f):warped results by the SSD method. (g) and (h): warped results by the USST method.

119

0 100 200 300 400 500 600 7000

500

1000

1500

2000

IterationE

0 100 200 300 400 5005

10

15

20

25

Iteration

Λ

Figure 7.5: Optimization processes of two registration methods: (a) SSD method, (b)USST method.

(image size: 256 × 256 pixels). The test image was generated by employing a non-

rigid motion field on the reference scattering function, and corrupting the image with

heavy motion speckle (λ = 1). A “difference” was formed from the superposition of

the two images. Figure 7.4(c), shows the reference image in red, test image in green.

The SSD and USST methods were applied to register the test image with the

reference image using the same computational conditions for both algorithms. Cubic

B-splines were used as basis functions for the image interpolation and deformation

function. The knot spacing h in (7.19) was set as [32,32]. The optimization stop

criteria were set to achieve global energy minimization for both methods: ε was set

at 0.001, and the maximum iteration number being resolution dependent was 200

for the coarsest level (size 32 × 32) and 20 for the original image. The optimization

120

processes of the two algorithms are illustrated in Fig. 7.5(a) and (b) by showing the

evolution of cost function values (E for SSD, and Λ for USST), indicating that both

methods achieve similar global minimization. Resultant images are superimposed on

the reference image in Figs. 7.4(e) and (g). Although visual differences of the two

warped results are nearly indistinguishable from gray scale images, recovery accuracy

can still be demonstrated by comparing the true deformation field (Fig. 7.4(d)) with

recovered deformation fields (Fig. 7.4(f) and (h)). The results show that the USST

method is more accurate than the SSD method in recovering the true motion field.

To quantitatively evaluate registration performance, two different quality indices

were used: an angular error measure,

θ = arccos〈u,uc〉||u|| ||uc|| (7.25)

and a relative magnitude displacement error

ε =

∣∣||u|| − ||uc||∣∣

||uc|| (7.26)

where u is estimated displacement, and uc is true deformation. Both θ and ε have

been used as measures of error in optical flow techniques [69, 70]. For the simulation

in Fig. 7.4, the SSD method resulted in θ = 33.61± 21.86 and ε = 24± 19%, while

the USST method resulted in θ = 17.61± 10.47 and ε = 9± 6%.

7.5.3 Experiments on Image Sequences

We tested both USST and SSD algorithms on synthesized ultrasound image se-

quences that mimicked real echocardiographic images throughout a cardiac cycle. To

121

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

45

50

λ 0.00

mean 2.31 deg

std 3.86 deg

median 1.10 deg

Angular error [degree]

Fre

quen

cy o

f occ

urre

nce

[%]

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

45

50

λ 0.00

mean 2.42 deg

std 3.95 deg

median 1.14 deg

Angular error [degree]

Fre

quen

cy o

f occ

urre

nce

[%]

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

λ 0.05

mean 2.73 deg

std 3.74 deg

median 1.54 deg

Angular error [degree]

Fre

quen

cy o

f occ

urre

nce

[%]

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

λ 0.05

mean 2.63 deg

std 3.62 deg

median 1.50 deg

Angular error [degree]

Fre

quen

cy o

f occ

urre

nce

[%]

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

λ 0.10

mean 3.26 deg

std 4.39 deg

median 1.80 deg

Angular error [degree]

Fre

quen

cy o

f occ

urre

nce

[%]

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

λ 0.10

mean 3.16 deg

std 4.28 deg

median 1.79 deg

Angular error [degree]

Fre

quen

cy o

f occ

urre

nce

[%]

0 10 20 30 40 50 60 70 800

5

10

15

20

25

λ 0.30

mean 5.15 deg

std 5.70 deg

median 3.34 deg

Angular error [degree]

Fre

quen

cy o

f occ

urre

nce

[%]

0 10 20 30 40 50 60 70 800

5

10

15

20

25

λ 0.30

mean 4.63 deg

std 5.29 deg

median 3.02 deg

Angular error [degree]

Fre

quen

cy o

f occ

urre

nce

[%]

122

0 10 20 30 40 50 60 70 800

5

10

15

λ 0.60

mean 9.09 deg

std 10.02 deg

median 5.56 deg

Angular error [degree]

Fre

quen

cy o

f occ

urre

nce

[%]

0 10 20 30 40 50 60 70 800

5

10

15

λ 0.60

mean 8.37 deg

std 8.05 deg

median 6.03 deg

Angular error [degree]

Fre

quen

cy o

f occ

urre

nce

[%]

0 10 20 30 40 50 60 70 800

1

2

3

4

5

6

7

8

9

10

λ 1.00

mean 18.76 deg

std 18.76 deg

median 12.66 deg

Angular error [degree]

Fre

quen

cy o

f occ

urre

nce

[%]

0 10 20 30 40 50 60 70 800

1

2

3

4

5

6

7

8

9

10

λ 1.00

mean 14.99 deg

std 13.40 deg

median 10.99 deg

Angular error [degree]

Fre

quen

cy o

f occ

urre

nce

[%]

Figure 7.6: Histograms of average angular errors for sequences with different elevationalspeckle decorrelation index (λ). From top to bottom, λ = 0.0, 0.05, 0.1, 0.3, 0.6, 1.0. Leftcolumn: results of the SSD method, and right column: results of the USST method.

model cardiac motion, we simulated a periodic displacement field that maintained a

constant cross-sectional area of the myocardium. This was achieved by applying a

radial displacement field with a magnitude decreasing with distance from the center.

The displacement field was cosine modulated in time to simulate myocardial relax-

ation and contraction, and subsequent thinning and thickening of the ventricular wall

during diastole and systole, respectively.

We generated six ultrasound image sequences corrupted with different levels of

motion noise. Each sequence consisted of 22 images representing a complete cardiac

cycle. Elevational speckle decorrelation was controlled by the parameter λ. Specif-

ically, for the first sequence, tissue motion was limited to the scanning plane with

123

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20

λ

Ave

rage

ang

ular

err

or [d

egre

e]

SSDUSST

(a)

0 0.2 0.4 0.6 0.8 1.00

0.05

0.1

0.15

0.2

0.25

0.3

λ

Ave

rage

mag

nitu

de e

rror

[%]

SSDUSST

(b)

Figure 7.7: Average angular error (a) and average magnitude errors (b) associated withSSD and USST methods, displayed as functions of speckle decorrelation index λ.

λ = 0. For the rest of the sequences, out-of-plane motion was gradually introduced

with λ = 0.05 for the second, 0.1 for the third, 0.3 for the fourth, 0.6 for the fifth,

and 1.0 for the last sequence. For the last case, the tissue scattering function was

totally changed for every frame during motion.

Image sequence registration was performed based on a non-temporal update strat-

egy, i.e. every frame in a sequence was registered to the same reference frame in the

124

sequence (the first frame in our experiments). Computational conditions were the

same as those used in the previous single pair image registration. The results of

sequence registration were directly used to compute the quality indices without any

smoothing operation (γ = 1 in (7.23)). Results of the two algorithms were first eval-

uated by calculating average angular error, shown in histograms of Fig. 7.6. The

results indicate that estimation errors for both algorithms increase with increasing λ.

Performance of the USST algorithm was superior to that of the SSD algorithm for all

noisy cases. Due to the merit of the multiresolution strategy, the difference in perfor-

mance was more significant for large decorrelation cases than for small decorrelation.

The average angular error and magnitude error are depicted in Fig. 7.7 as functions

of λ, further illustrating that speckle tracking ability of the USST method is robust

for ultrasound image sequences corrupted by a wide range of motion noise.

7.6 Experimental Validation by Animal Model

Each data set consisted of three consecutive cardiac cycles of ICE recording that

were selected for computation of the deformation field, where a cardiac cycle was

defined as the R-R interval on a synchronously recorded ECG. For a typical baseline

recording, there were about 18-20 frames per cardiac cycle. With the increased heart

rate during DOB infusion, there were about 12 frames at the high dose (DOB1), and

15 frames at the low dose (DOB2). Both the SSD-based and USST-based registra-

tion algorithms were employed to compute the deformation field. We used the same

initial parameters and stop criteria as those employed in the previous phantom image

experiments. In each cardiac cycle, the first frame taken at the peak R wave was used

125

Figure 7.8: Schematic of locations of sonomicrometry crystals (marked by rectangles) inthe left ventricle. Measures of circumferential and radial distances are also illustrated.

as the reference image, whereas the other frames in the same cycle were registered to

the first frame.

Illustrated in Fig. 7.8, the circumferential distances between the two crystals were

determined from ICE by computing the deformation fields. A region of interest (ROI)

of size 5× 5 pixels was selected around each crystal location. An additional ROI was

also selected in the subendocardium, which was radially positioned relative to the

second crystal in mid anterior LV. The center of this ROI, marked by the symbol ‘+’

in Fig. 7.8, was used to compute radial distance. The position of every pixel in the

ROIs was updated as a function of time throughout the pre-computed deformation

field. In every updated ROI, centers of gravity were used to calculate the positions

of crystals. Finally, the circumferential and radial distances were computed, and

updated as functions of the cardiac cycle.

Examples of circumferential and radial displacements at baseline are shown in

126

10

10.5

11

Dis

tanc

e [m

m]

20

21

22

23

24

25

26D

ista

nce

[mm

]

50

100

LVP

[mm

Hg]

1 2 3Cardiac cycles

EC

G

C USSTC SSDC SM

R USSTR SSD

Figure 7.9: Displacement determined by computed deformation field and sonomicrometry,where ‘R’ represents radial displacement, ‘C’ represents circumferential displacement, and‘SM’ represents sonomicrometry. Displacements are shown for both USST and SSD methods

Fig. 7.9. The results depict good agreement between motion displacements com-

puted using the USST method and the reference motion displacements measured by

sonomicrometry. The correlation coefficients were 0.95 for the SSD method, and 0.96

for the USST method. Figure 7.10(a) shows an ICE image at end diastole, and Fig.

7.10(b) shows an ICE image at end systole, both with their respective computed

displacement fields superimposed.

We compared regional myocardial strains using the Lagrangian strain S, defined

127

LVRV

Anterior

100 ms 100 ms

Figure 7.10: ICE images in mid LV at end diastole (left) and end systole (right). Arrowsindicate displacement field of the LV myocardium during systole (left) and diastole (right).Intervals for displacement fields are indicated by vertical bars on corresponding ECG.

as the relative elongation with respect to the initial distance Lt0 , i.e.,

S(t1) =Lt1 − Lt0

Lt0

.

Figure 7.11 illustrates an example of myocardial strains at three different recording

stages (baseline, DOB1 and DOB2). The strains were calculated on the basis of

displacements averaged over three cardiac cycles. The results clearly show that my-

ocardial strain increased relative to baseline with dobutamine infusion. Since dobu-

tamine concentration was higher in DOB1 than DOB2, the strain was also larger

with DOB1. We further validated the two registration methods at every time frame

against sonomicrometric measurement by Bland-Altman analysis on circumferential

strain. Figures7.12 (a) and (b) show good agreement between the calculated strains

and sonomicrometry, with the USST method yielding superior results compared to

128

0 100

−15

−10

−5

0

5

10

15

20

25

30

Perecent of heart cycle [%]

Str

ain

[%]

R DOB1R DOB2R BLC BLC DOB2C DOB1

Figure 7.11: Radial and circumferential strains computed by the USST method at threerecording stages: baseline (BL), DOB1 and DOB2. Dobutamine concentration in DOB1was higher than DOB2. ‘R’ represents radial strain, and ‘C’ represents circumferentialstrain.

the SSD method.

7.7 Discussion and Conclusion

Two-dimensional transthoracic echocardiography is the most widely used tech-

nique for the assessment of regional myocardial function. Visual assessment of echocar-

diographic images leads to a qualitative diagnosis that suffers from inter- and intra-

129

−18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4−8

−6

−4

−2

0

2

4

6

8

Average of circumferential strain [%]

Str

ain

mea

sure

diff

ence

[%]

0.82

−4.15

5.78

+2.0 SD

−2.0 SD

Mean

(a)

−18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4−8

−6

−4

−2

0

2

4

6

8

Average of circumferential strain [%]

Str

ain

mea

sure

diff

ence

[%]

0.73

−2.52

3.98+2.0 SD

−2.0 SD

Mean

(b)

Figure 7.12: Bland-Altman plots comparing circumferential strains as determined by twomethods: (a) SSD and sonomicrometry, and (b) USST and sonomicrometry. Baseline (*),DOB1(4) and DOB2 (¤).

130

observer variability. In addition, echocardiographic border detection algorithms pro-

vide limited information on regional wall deformation [89–91]. Tissue Doppler imag-

ing [92,93] has been used to provide an operator-independent quantitative analysis of

regional myocardial function through the analysis of myocardial velocities and deter-

mination of strain and strain rate [94,95]. However, only deformations in the direction

of the ultrasound beam are measured using tissue Doppler imaging, and comparative

measurements at multiple sites depend on the angle of the ultrasound beam. Speckle

tracking imaging associated with small displacements has recently been employed in

detecting tissue motion in images acquired by external echocardiography [72,96,97].

To further advance the utility of ICE and provide capabilities for multimodal

imaging during catheterization, we developed a novel speckle tracking method for

regional myocardial motion estimation from ICE image sequences. Our method ex-

ploits ultrasound statistics by utilizing maximum likelihood motion estimation, and

treats the maximization of motion probability as the minimization of a derived en-

ergy function. Myocardial displacement was estimated by optimization of this energy

function while relying on the framework of non-rigid registration. We have validated

our speckle tracking method in both computer and animal models.

In developing our method, we first considered random texture variation in ultra-

sound images as the major factor hindering motion estimation. Instead of treating

texture variation as system Gaussian noise, we recognized its speckle-like nature and

considered it as the result of motion-related speckle decorrelation. Our analysis sug-

gested that a tracking method based on the constancy of echogenicity was much

more reliable than one utilizing only constancy of the speckle pattern. Employing

this constant echogenicity assumption, our method provided an optimal solution for

131

myocardial motion by integrating speckle tracking with non-rigid motion estimation.

We further showed that our method was theoretically feasible in the sense of robust

estimation.

A practical concern for our method is the utilization of temporal information in

motion estimation. One spatio-temporal model has been proposed by [72], wherein the

temporal coherence of deformation was incorporated into the registration process. For

our purposes (in vivo experiments), the computational cost of such a spatio-temporal

model would be prohibitive in achieving the desired accuracy. Consequently, we

employed a post-temporal update strategy as suggested in [87]. We first registered

every frame in the sequence to obtain the deformation field. Then, the computed

deformation field was smoothed by spatial-temporal interpolation. A limitation of

this strategy was the lack of a global motion constraint for individual registration.

However, this limitation was reduced by employing strict convergence criteria for

registration on each pair of images. This post-temporal update strategy has the

following advantages: (1) much lower computational cost than the spatial-temporal

update method; (2) not limited by the time-resolution of the ultrasound system; and

(3) avoidance of large individual frame errors that can propagate into the estimation

sequence.

Our method of tracking on the basis of constancy of echogenicity plays a major role

in reducing the effect of speckle decorrelation in motion estimation. Another method

of solving the problem of speckle decorrelation is to introduce a regularization term

in the energy function [79], wherein the regularization parameter is appropriately

selected to achieve a balance between resistance to decorrelation and preservation

of displacement fidelity. In our method, the constraint was implemented using a

132

bounded optimizer (L-BFGS-B).

Our method provides a fundamental platform for motion estimation in ultrasound

image sequences. A methodological improvement can be achieved by incorporating

an invertibility constraint into the cost function [98]. The algorithm can be further

improved by updating the maximum likelihood estimation using a Bayesian maxi-

mum a posteriori framework, wherein a priori information about the periodicity of

myocardial motion would be taken into account.

Radial displacement computed from in vivo experiments by our USST method

was not validated by sonomicrometry in a manner similar to circumferential displace-

ment. However, the similarity of USST results to those of the SSD method, and the

consistency of computed radial displacement with previous studies in its relationship

to circumferential displacement are supportive of the capability of our USST method

in depicting regional radial myocardial deformation. Results from in vivo animal

experiments indicate that our method has significant relevance to the analysis of re-

gional myocardial function, such as in automated detection of ischemia and infarction.

Specifically, speckle tracking of catheter derived ICE images could evolve clinically as

a useful method in diagnosing, monitoring, and guiding applications appropriate for

the cardiac catheterization laboratory.

In conclusion, the assessment of regional myocardial deformation by novel speckle

tracking in intracardiac echocardiographic image sequence is feasible. This method

has important clinical implications for multimodal imaging during cardiac catheteri-

zation.

133

Chapter 8

Conclusion

This dissertation has focused on solving two important issues in medical ultra-

sound imaging: speckle suppression and motion estimation.

We first introduced a novel multiscale normalized modulus-based wavelet diffusion

method for speckle suppression and edge enhancement in ultrasound images. In our

approach, the speckle image is iteratively filtered by the nonlinear diffusivity func-

tion via the framework of the dyadic wavelet transform. In each iteration, the noisy

image is processed with three-step wavelet shrinkage-like procedures: decomposition,

regularization and reconstruction. The normalized wavelet modulus is used as the

edge detector to characterize the intrinsic signal/noise variation. The significance of

this edge detector is its versatility for use with images of different types. Specifically,

our algorithm can deal directly with either envelope-detected speckle image or log-

compressed medical ultrasonic image without any pre-transform. To adapt the noise

variation with iteration, the diffusion threshold is estimated from the normalized mod-

ulus in the homogenous speckle regions. A tuning parameter is introduced to adjust

the diffusion threshold, and it further controls the final denoising result. Relying on

this feature, our algorithm is highly flexible in producing the desired result for a spe-

cific application. We have demonstrated the performance superiority of the proposed

algorithm over other despeckling methods in terms of speckle suppression and edge

134

preservation indices. With real ultrasonic images, we have shown that our algorithm

is robust in producing desired results for the clinical applications. In summary, our

algorithm provides very significant speckle suppression and edge enhancement for the

purposes of visualization and automatic structure detection.

The second part of dissertation deals with the development of a speckle tracking

method for myocardial motion estimation from intracardiac echocardiographic (ICE)

image sequences. ICE images are used for anatomical imaging of the heart, and the

ability to detect myocardial wall motion provides an additional means for functional

imaging. Our approach was to solve two problems in motion estimation from ICE im-

age sequences: non-rigid myocardial deformation and speckle decorrelation. Rather

than compensating for decorrelated components in motion, we considered speckle

decorrelation as motion noise. To achieve robust noise resistance, we employed maxi-

mum likelihood estimation while fully exploiting ultrasound speckle statistics. Maxi-

mization of motion probability was treated as the minimization of an energy function.

Non-rigid myocardial deformation was estimated by optimizing this energy function

within a framework of parametric elastic registration. Accuracy of the method was

initially evaluated by using a computer model that synthesized echocardiographic

image sequences, and subsequently by an animal model that provided continuous

intracardiac echocardiographic images as well as reference measurements for myocar-

dial deformation. In conclusion, estimation of regional myocardial deformation from

intracardiac echocardiography by novel speckle tracking is feasible. This approach

has important clinical implications for multimodal imaging during catheterization.

135

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