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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
0.3
0.4
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
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120
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160
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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.
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40
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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
0 5 10 15 20 25 300
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Normalized modulus
Pro
babi
lity
j =1, Rayleigh mixture
0 5 10 15 20 25 300
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Normalized modulus
Pro
babi
lity
j =1, Maxwell 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, 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
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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
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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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