High Quality Ultrasound B-mode Image Generation Using 2-D ...

High Quality Ultrasound B-mode Image

Generation Using 2-D Multichannel-based

Deconvolution and Multiframe-based

Adaptive Despeckling Algorithms

A thesis submitted to the Department of Electrical and Electronic Engineering

of

Bangladesh University of Engineering and Technology

in partial fulfillment of the requirement for the degree of

MASTER OF SCIENCE IN ELECTRICAL AND ELECTRONIC ENGINEERING

byJayanta Dey

Student ID: 0417062214 P

DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING

BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY

May 2019

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Dedication

To my parents.

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Acknowledgements

First and foremost, I am grateful to the Almighty for giving me the opportunity andstrength to carry out this thesis work.

I would like to extend my heart-felt gratitude to my supervisor, Dr. Md. KamrulHasan for giving me the opportunity to work on this topic. I am grateful for his un-conditional support, constant guidance and supervision throughout this research work.Working under his supervision is a reminder of the discipline, patience, and persever-ance required to carry out high-level research.

I am also grateful to the Head of the Department of Electrical and Electronic Engi-neering, BUET, for the research lab facilities. I would also like to thank the presentand past members of the DSP research lab, without whose help this thesis may nothave been possible, especially, Dr. Sharmin Rowshan Ara, Mr. Nabid Ibtehaz Nizam,Mr. Md. Hadiur Rahman Khan and Mr. Md. Shifat-E-Rabbi. Moreover, I would liketo thank my colleagues at the Department of Electrical Engineering, AUST, who werea constant source of support and inspiration and my friends, respected seniors, andbeloved juniors at BUET, who are all shining examples to follow.

This work has been supported by Higher Education Quality Enhancement Project, Uni-versity Grants Commission (CPSF#96/BUET/Win-2/ST(EEE)/2017), Bangladesh.The in vivo breast data were acquired at BUET Medical Center by Dr. Farzana Alam,Assitant Professor, Department of Radiology and Imaging, Bangabandhu Sheikh MujibMedical University, Dhaka-1000, Bangladesh.

Contents

Approval Certificate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iCandidate’s Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Abstract xii

1 Introduction 1

1.1 Motivation of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Deconvolution of Ultrasound Images 5

2.1 Deconvolution of RF Echo Data: Literature Review . . . . . . . . . . . 62.2 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Fundamentals and Frequency Domain Approach . . . . . . . . . 92.3.2 bMCFLMS Algorithm for TRF estimation . . . . . . . . . . . . 142.3.3 Effect of Noise on the Convergence of the Algorithm . . . . . . . 20

2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.1 Simulation Phantom Results . . . . . . . . . . . . . . . . . . . . 272.4.2 In-Vivo Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 B-mode Image Generation Framework 35

3.1 Despeckling of the Envelope of Deconvolved Data: Literature Review . 363.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Deconvolution of RF Echo Data . . . . . . . . . . . . . . . . . . 373.2.2 Proposed Despeckling Algorithm . . . . . . . . . . . . . . . . . 40

3.3 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

v

CONTENTS vi

4 Results 51

4.0.1 Simulation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.0.2 In-Vivo Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Discussion and Conclusion 62

5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

List of Publications 65

List of Figures

2.1 Illustration of ultrasonic data acquisition system as SIMO model, show-ing the relationship between the backscattered RF data xi(n) and thepoint spread function, s(n). . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Block diagram of the proposed bMCFLMS algorithm. The upper partshows the modeling of the RF data and RF image from the transducersegmented into B blocks. The lower part shows the sequential estimationof the TRF from the RF segments using the proposed algorithm. . . . . 11

2.3 Effect of noise on the convergence of the bMCFLMS algorithm (a) Be-havior of the NPM curve around the misconvergence point for the firstblock with SNR = 30 dB. (b) Behavior of the correlation cost functionaround the misconvergence point for the first block. (c) Behavior of theNPM curve around the misconvergence point for the first block withcorrelation constraint (⇠ = 3e-7, ⇢ = 2.55, � = 2.3). (d) Misconvergencephenomenon of the second block of TRFs for B = 2 for with no addi-tive white noise in the data. (e) Misconvergence Problem Solved for theSecond Block of TRFs for B = 2 using correlation constraint. . . . . . . 21

2.4 Convergence of the bMCFLMS algorithm with the proposed correlationconstraint for the simulation phantom data at different SNR. . . . . . . 23

2.5 Deconvolution performed on a simulation phantom with 20 scatterersper resolution cell and SNR = 30 dB. The RF data size is 1038 ⇥ 128.The darker circular inclusion with radius 5 mm is created by placingscatterers with relatively lower strength than the surroundings. Log-envelope image of the (a) true TRF, (b) backscattered standard RF data,and (c) deconvolved TRF by the bMCFLMS algorithm, (d) spectra ofthe true and estimated RF data of a single scan-line marked by verticalgreen line in the log envelope images of (b) and (c). (e) The NPM curvebetween the deconvolved TRF and true TRF of the first block. . . . . . 26

vii

LIST OF FIGURES viii

2.6 Performance analysis of the bMCFLMS algorithm for the in-vivo backscat-tered RF data of a breast cyst. Spectrum of the (a) deconvolved TRF,(b) R-MINT estimated PSF, and (c) spectra of the true and estimatedRF data of a single scan-line marked by vertical red line in log envelopeimages. Standard log envelope images of (d) the backscattered RF data,(e) deconvolved TRFs, and (f-g) zoomed-in views of (d-e). . . . . . . . 27

2.7 Performance analysis of the bMCFLMS algorithm for the in-vivo backscat-tered RF data for a left carotid artery. Spectrum of the (a) deconvolvedTRF, (b) R-MINT estimated PSF, and (c) spectra of the true and es-timated RF data of a single scan-line marked by vertical red line in logenvelope images. Standard log envelope images of (d) the backscatteredRF data, (e) deconvolved TRFs, and (f-g) zoomed-in views of (d-e). . . 28

2.8 Time required per iteration by the time- and the frequency-domain al-gorithms for the simulation phantom data of size Na ⇥ 128 where Na ismade variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1 A SIMO model for backscattered RF signal. . . . . . . . . . . . . . . . 383.2 A new SIMO model for deconvolved 2-D RF data. . . . . . . . . . . . . 413.3 The block diagram for the estimation procedure of true ultrasound image. 463.4 Block diagram of the proposed ultrasound image reconstruction method

from the raw RF data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 Effect of the number of frames in despeckling the modified Shepp-Loganphantom image using the proposed algorithm: (a) clean phantom, (b)noisy phantom (� = 0.4), (c)-(f) despeckled using 5,10, 15 and 20 frames,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Despeckling of Shepp-Logan phantom image corrupted by synthetic specklenoise. (a) clean phantom, (b) noisy phantom (� = 0.4), (c) true specklenoise in the 5th frame; despeckled image using (d) SRAD, (e) OBNLM,(f) proposed MSNE; (g) extracted noise from the 5th frame using MSNE;(h) NPM measure between the true and estimated noise using MSNEwithout constraint; (i) NPM measure between the true and estimatednoise using MSNE with constraint. . . . . . . . . . . . . . . . . . . . . 53

4.3 Deconvolution of ultrasound images using adaptive bMCFLMS algo-rithm. (a) Raw RF image, (b) 1-D deconvolved image, (c) 2-D de-convolved image, (d)-(f) zoomed-in views of image segments of (a)-(c),respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

LIST OF FIGURES ix

4.4 Despeckling of breast ultrasound image- 1. (a) Deconvolved image, im-ages obtaind using (b) SRAD, (c) OBNLM, (d) proposed algorithm, (e)machine B-mode image, (f) estimated speckle pattern of the 5-th frame. 58

4.5 Despeckling of breast ultrasound image- 2. (a) Deconvolved image, im-ages obtaind using (b) SRAD, (c) OBNLM, (d) proposed algorithm, (e)machine B-mode image, (f) estimated speckle pattern of the 5-th frame. 59

4.6 Estimated B-mode image with the deconvolution step as (a) bMCFLMS,and (b) cepstrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

List of Tables

2.1 Symbols and Description . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 Constrained bMCFLMS Algorithm . . . . . . . . . . . . . . . . . . . . 332.3 Performance of different algorithms on simulation phantom data with

additive noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4 Performance of different algorithms on in-vivo data . . . . . . . . . . . 34

3.1 bMCFLMS algorithm for 2-D deconvolution of ultrasound RF image . . 41

4.1 Simulation results on the estimation accuracy in terms of NPM (dB) ofspeckle pattern using the proposed method for different noise levels . . 55

4.2 Performance measures computed for the simulation study with differentnoise level (�) using diffetrent despeckling approaches . . . . . . . . . . 56

4.3 Axial and lateral correlation energy for raw RF, 1-D and 2-D decon-volved data using the b-MCFLMS algorithm. . . . . . . . . . . . . . . . 57

4.4 NIQE measure for images despeckled with different algorithms . . . . . 60

x

Glossary

VSS Variable step-size

RF Radio-frequency

PSF Point spread function

TRF Tissue reflectivity function

LMS Least mean square

MADS Multiframe-based Adaptive Despeckling

MSNE Multiframe-based Adaptive Speckle Noise Estimation

SNC Speckle noise cancellation

xi

Abstract

Improving resolution and removing speckle noise from medical ultrasound images whilepreserving tissue texture, small details, and edges without introducing artifact and dis-tortion is a major challenge in ultrasound image restoration. The underlying physicalphenomena related with US image acquisition and imperfection of US imaging systemdesign give rise to low resolution and speckle noise that tend to reduce the image con-trast, obscure and blur image details such as inclusion and small structure boundary,tissue texture and thereby, decrease the quality and reliability of medical ultrasound.In this thesis, a complete framework of signal processing approaches comprising of de-convolution to enhance resolution, despeckling, and post-processing for the generationof ultrasound B-mode image with superior edge, details and tissue texture has beenestablished. In the first step, we propose a correlation constrained blind multichan-nel frequency-domain least-mean-squares (bMCFLMS) algorithm to undo the effectof point spread function (PSF) on the ultrasound radio-frequency (RF) data. ThebMCFLMS algorithm, however, shows misconvergence due to both channel noise andpropagation of TRF estimation error from the previous blocks. This phenomenon ismore intense in the case of md-bMCFLMS algorithm because of increased estimationerror. To address this problem, a novel constraint based on the correlation between themeasured RF data and estimated TRF is proposed in this thesis. Then in the secondstep, based on a multiple input single output (MISO) model over the consecutive de-convolved ultrasound image frames, a multiframe-based adaptive despeckling (MADS)algorithm to reconstruct a high-resolution B-mode image from raw radio-frequency(RF) data has been proposed. It utilizes the speckle patterns estimated using a novelmultiframe-based adaptive approach for ultrasonic speckle noise estimation (MSNE)based on a single input multiple output (SIMO) modeling of consecutive deconvolvedultrasound image frames to estimate the despeckled ultrasound image as single outputfrom the deconvolved image frames as multiple input. The elegance of the proposed al-gorithms is that it addresses the deconvolution and despeckling problem by completelyfollowing the signal generation model rather than the existing ad-hoc smoothening orfiltering approach described in the literature, and therefore, it is likely to maximally

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ABSTRACT xiii

preserve the image features. The efficacy of our proposed blind deconvolution algorithmis measured using simulation phantom and in-vivo data. The proposed md-bMCFLMSalgorithm shows normalised projection misalignment (NPM) improvement of about2.12 ⇠ 16 dB and resolution gain (RG) improvement of 1.14 ⇠ 6.4 dB compared toother techniques in the literature. Moreover, because of the frequency-domain im-plementation it is computationally more efficient, fast converging and robust than itstime-domain counterpart l1-bMCLMS algorithm reported in the literature. Again, theefficacy of the proposed despeckling algorithm is evaluated both visually and quantita-tively on the simulation and in-vivo data. The results show 8.55�15.91 dB, 8.24�14.94

dB, 0.57�7.03 improvement in terms of SNR, PSNR, and NIQE, respectively, for sim-ulation data and 2.22� 3.17 improvement in terms of NIQE for in-vivo data comparedto the traditional despeckling algorithms.

Chapter 1

Introduction

In this Chapter, we discuss the motivation behind the development of a complete

framework of novel signal processing approaches, including deconvolution, despeckling,

and post-processing to generate a high quality B-mode image from raw radio-frequency

(RF) image. Next, the primary objectives of this thesis are highlighted. Finally, the

organization of the thesis is described.

1.1 Motivation of the Thesis

Ultrasound (US) imaging system being non-invasive, non-ionizing, portable, and cost

effective has become the most prevalent diagnostic tool among all the currently avail-

able imaging modalities, e.g., X-ray, magnetic resonance imaging, and computed to-

mography. However, imperfection of US imaging system design and the underlying

physical phenomena related with US image acquisition give rise to low resolution and

speckle noise that tend to reduce the image contrast, obscure and blur image details

such as inclusion and small structure boundary, tissue texture and thereby, decrease

the quality and reliability of medical ultrasound [1]. The low resolution of US image

can be modelled as the convolution of point spread function (PSF) of ultrasound imag-

ing system with the tissue reflectivity function (TRF). The removal of PSF effect from

the measured backscattered RF images can restore the resolution of images and thus

improve the diagnostic quality of ultrasound imaging. On the other hand, speckle noise

1

CHAPTER 1. INTRODUCTION 2

is a granular pattern inherent in any coherent imaging modalities [2], [3] similar to ul-

trasound imaging. It results from the constructive and destructive interferences of the

reflected echos with different phases and amplitudes from the target at the receiving

transducer. Removal of speckle noise from US images is difficult due to its multiplica-

tive nature and the challenge of maintaining the precise texture of the image [4]. As

traditional despeckling filters distort the original image texture and introduce artifacts

like blurring edges, changing the shape of structures present in the image by smoothen-

ing the noise corrupted image, the original noise affected images are sometimes more

preferred than the noise-removed ones in the analysis where the image details have high

importance [4]. Therefore, an effective signal processing approach for removing speckle

noise from the image while preserving the original tissue texture and small details of

the image is vital to increase the diagnostic potential of medical ultrasound.

1.2 Objectives of the Thesis

The objectives of this work are:

1. To develop a new algorithm for the deconvolution of radio frequency (RF) ultra-

sound data.

2. To develop a novel model and estimation algorithm for despeckling of deconvolved

ultrasound RF data.

3. To establish a complete framework for high quality ultrasound B-mode image

generation.

4. To compare the performance of the proposed techniques with some well-known

methods using data from simulation phantom, experimental phantom, and pa-

tient (in vivo).


1.3 Problem Formulation

Low resolution and speckle noise are the major issues related with US imaging. How-

ever, an overall US image enhancement can be achieved by addressing these two issues

in a sequential two-step process as described in [1]. First, the correlation between the

image samples is to be minimized to increase the image resolution, and second, speckle

noise has to be removed from the decorrelated image to improve the image contrast

and better visualize the tissue texture. To achieve the aforementioned two objectives,

a suitable model representing the US imaging system is necessary. With this in view,

considering linear wave propagation through the tissue, and the scattering of the ultra-

sound pulse in the tissue as weak, we can use the first order Born approximation and

consider the tissue scattering system as a linear system [1]. Therefore, the blurring, i.e.,

low resolution of an RF-image can be modeled as the result of convolution between the

point-spread function (PSF) s(m,n) of the imaging system with the tissue reflectivity

function (TRF) h(m,n) [1], [5], [6]. Mathematically, this can be written as

x(m,n) = s(m,n) ⇤ h(m,n) + v(m,n) (1.1)

where x(m,n) is the backscattered ultrasound image data from the n-th A-line at

discrete time m and v(m,n) is the additive noise associated with measurement error

and other physical phenomena not accounted by the convolution model. In US imaging,

h(m,n) is corrupted by speckle noise, and as described in [1], [7], [8], he(m,n), i.e., the

envelope of h(m,n) can be modeled as multiplicative with the true image as

he(m,n) = r(m,n)u(m,n) + ⇣(m,n) (1.2)

where r(m,n), u(m,n), and ⇣(m,n) are true image, speckle noise, and additive noise

resulting from the deconvolution process and the portion not accounted by the mul-

tiplicative process, respectively. Therefore, in ultrasound imaging, we are given the

back-scattered data x(m,n), and our objective is to design algorithms to estimate the

TRF h(m,n) from x(m,n), and thereafter, obtain the despeckled image r(m,n) from

the envelope of h(m,n).


1.4 Organization of the Thesis

This thesis consists of five chapters. Chapter 1 gives a brief discussion about the

motivation of the thesis and the limitations of the existing techniques. Chapter 2 has

the detailed description of the deconvolution algorithm (bMCFLMS) along with its

performance analysis. Chapter 3 establishes and proposes the framework for a high

quality B-mode image generation. Chapter 4 tests our proposed approaches on the

simulation and the in-vivo breast data. It also represents RF image processing and

the information of the presets of the ultrasound instrument. Finally, in Chapter 5, a

conclusion has drawn showing the outcomes and drawbacks of the thesis and indicating

the thoughts and topics which require further research for improvements.

Chapter 2

Deconvolution of Ultrasound Images

In this chapter, we propose a blind deconvolution algorithm in the frequency-domain

with correlation constraint for noise-corrupted ultrasound RF data. Due to smaller

eigenvalue spread, a frequency-domain approach facilitates faster convergence of the

adaptive algorithms compared to the time-domain ones [9], [10]. Moreover, the variable

step-size (VSS) MCFLMS algorithm is in general known to be more noise robust com-

pared to the time-domain ones at the same noise level [11], [12]. Unlike the blocking

procedure described in [13], in this thesis, we introduce a new blocking technique that

facilitates the use of FFT and make the algorithm computationally efficient. However,

as with the other reported cross-relation based blind adaptive algorithms [14], [15], the

proposed algorithm also suffers from misconvergence in the presence of channel noise

and estimation error from the previous estimated blocks of the TRFs. To overcome this

problem, we propose a novel constraint based on the correlation between the measured

RF data and estimated TRFs that can compensate the effect of noise and estimation

error stated above. The performance of the proposed blind deconvolution techniques

is evaluated on the simulation phantom and in-vivo data.

5

CHAPTER 2. DECONVOLUTION OF ULTRASOUND IMAGES 6

2.1 Deconvolution of RF Echo Data: Literature Re-

view

A number of algorithms have been proposed in the literature to deconvolve the PSF

from the received RF data and thereby restore the image resolution. These algorithms

are mainly of two groups. The first group estimates the PSF first and then a classical

deconvolution algorithm is applied to estimate the TRFs. These methods are based on

homomorphic filtering [16], [17], [18], [19] which involve filtering out the PSF either in

the cepstrum domain [16], [18] or in the log magnitude domain [19]. These algorithms

are elegant in the sense that they are simple and can be implemented in real-time.

Filtering the wavelet coefficients of the log magnitude spectrum gives better result in

terms of mean square error (MSE) than the cepstrum based methods [18]. However,

tuning the length of the filter in cepstrum domain and selecting the decomposition

level of wavlet decomposition determines the smoothness of the estimated PSF [20]

and hence the overall deconvolution accuracy. In addition, this category of techniques

assume that the spectrum of the PSF and the TRF lie in separate spectral band

which is not completely true [16]. Moreover, inaccuracy in phase unwrapping poses

another problem for these algorithms [18]. To solve the problem of phase unwrapping,

a recent hybrid parametric inverse filtering (HYPIF) algorithm [18] has been proposed

which estimates the PSF in two steps. First, partial information of the PSF, i.e.,

power spectrum is estimated using the homomorphic filtering. This partial information

obtained is used to constraint the shape of the inverse filter. Then linearity of the

inverse filter is exploited to recover the phase of the PSF. However, in this method the

energy of the inverse filter should be regularized to avoid instability where the PSF has

zero or very low magnitude.

The second group of algorithms estimates the PSF and the TRFs simultaneously.

Among them the blind deconvolution method described in [21] improves the conver-

gence speed and reduces the computational load by projecting the TRF into the null

space of the correlation matrix and the PSF onto the space spanned by the third-order

B-spline wavelet basis. However, in the presence of noise thresholding is necessary to


determine the null space bases. Another classical algorithm of iterative blind deconvo-

lution described in [22] suffers from poor convergence property. Again, the parametric

methods [23], [24] of this group models the imaging process using a low-order autore-

gressive moving average (ARMA) system. However, these methods are only applicable

for fairly smooth RF images only. In addition, due to extremely complex composition

of most biological tissues, derivation of a convenient and accurate parametric model

for the imaging system is not possible [25].

The algorithms discussed so far assume spatially invariant PSF. However, while

propagating through the tissue the PSF suffers from attenuation dependent on the

depth of penetration. In order to address this problem, the spatially variant meth-

ods reported in the literature divides the RF data into several overlapping or non-

overlapping blocks so that the PSF may be considered stationary for that block. The

algorithm proposed in [25] assumes cyclic convolution of the PSF with the TRF blocks

for each of these RF data blocks instead of linear convolution between the PSF and the

total TRF. Lately, a block-based blind deconvolution method in the time-domain using

the multichannel LMS (MCLMS) algorithm has been presented in [13] which estimates

the TRFs block by block to account for the nonstationarity of the PSF. The blocking

procedure described in [13] uses convolution matrix segmentation and thereby restricts

the use of fast Fourier transform (FFT) for implementing convolution. Therefore, the

algorithm is computationally inefficient due to the implementation of time-domain con-

volution using matrix multiplication. In addition, the size of the matrix increases as

the number of blocks increases. Furthermore, it suffers from misconvergence in the

presence of additive white noise and propagation of estimation error from block to

block. An attempt was made to solve this problem using a damped variable step-

size [26], gradient averaging and l1-norm constraint. In order to apply the l1-norm

regularization in the algorithms described in both [25] and [13], the data is assumed

to be sparse which is not generally true for the case of in-vivo data. The noise effect

minimizing methods described in these algorithms are adopted on ad hoc basis and the

misconvergence may not be completely stopped even after applying all the aforesaid

noise effect minimization techniques.


2.2 Signal Model

In the ultrasound imaging system, an array of piezoelectric elements sequentially emit

the same ultrasound pulse in the tissue and receive echo signals from multiple A-lines.

This system can be modeled as a SIMO model with the ultrasound pulse as the single

s(n)

h1(n)

h2(n)

hM

(n)

x1(n)

x2(n)

xM

(n)

v1(n)

v2(n)

vM

(n)

PSF

TRFBackscattered

RF Data

Additive

Noise

Figure 2.1: Illustration of ultrasonic data acquisition system as SIMO model, show-

ing the relationship between the backscattered RF data xi(n) and the point spread

function, s(n).

input, the measured echo signal lines as multiple outputs, and the TRFs along the

axial direction as multiple system channels [21]. Figure 2.1 shows the SIMO model of

the backscattered ultrasound RF data, where the ultrasound pulse or the PSF s(n)

convolves with the i-th channel transfer function or TRF hi(n). With additive noise

vi(n), the measured RF data are given by

xi(n) = s(n) ⇤ hi(n) + vi(n), i = 1, 2, · · · ,M (2.1)

where xi(n) denotes the backscattered RF data of the i-th scan line and M is the

number of total scan lines. If the length of xi(n) is L and that of s(n) is Ls with

Ls << L, we can write (2.1) in matrix form as

xi(n) = S(n)hi + vi(n) (2.2)


where S(n) is the (L+ Ls � 1)⇥ L convolution matrix constructed from s(n), and

hi =hhi(n) hi(n� 1) · · · hi(n� L+ 1)

iT

xi(n) =hxi(n) xi(n� 1) · · · xi(n� L� Ls + 2)

iT

vi(n) =hvi(n) vi(n� 1) · · · vi(n� L� Ls + 2)

iT

However, as each data sample of the ultrasound RF signal results from the reflection

of the ultrasound pulse from a scatterer of the tissue, in reality, we have RF echo data

equal to the TRF length L instead of L + Ls � 1. Moreover, (2.1) assumes that

the PSF is stationary, i.e., remains constant while penetrating the tissue. But, if

the acquired RF data is long in the axial direction, the PSF suffers from the depth

dependent attenuation while traveling through the tissue. Then this nonstationary PSF

restricts the direct use of the cross-relation based MCFLMS algorithm for blind SIMO

model identification in the deconvolution of ultrasound images [13]. Therefore, the

main challenge in ultrasound deconvolution process is to estimate the TRFs, hi(n), i =

1, 2, · · · ,M , using the truncated and nonstationary RF data corrupted by additive

white noise. Here the deconvolution process is carried out in the frequency-domain to

obtain more robustness to noise than in the time-domain along with faster convergence

facilitated by smaller eigenvalue spread and reduced computational complexity.

2.3 Method

2.3.1 Fundamentals and Frequency Domain Approach

For noise-free case, the cross-relation error defined in the following can be exploited to

estimate the TRFs:

eij(n) = xi(n) ⇤ hj(n)� xj(n) ⇤ hi(n), i, j = 1, 2, · · · ,M,

i 6= j (2.3)

Note that the error function in (2.3) becomes zero when xi(n) in (2.1) for noiseless

condition is substituted into it. However, as shown in [13], due to incomplete data


acquisition in ultrasound imaging only the first L samples of the error function will be

equal to zero. Taking this into account, the truncated error function in matrix form is

written as

eij = D(Cxihj �Cxjhi) (2.4)

where,

xi(n) = xi(n), n = 0, 1, 2, · · · , L� 1,

Cxi is the convolution matrix formed with the truncated RF data xi(n) of length L

and

D =hIL⇥L 0L⇥(L�1)

i

is the truncation matrix, I is the identity matrix of size L⇥ L, 0 is the null matrix of

size L⇥ (L� 1),

hi =hhi(n) hi(n� 1) · · · hi(n� L+ 1)

iT

Due to nonstationarity of the PSF in ultrasound imaging, (2.4) cannot be directly

used to estimate the TRFs. An appropriate solution to this problem is to estimate

the TRFs block by block from the error blocks formed from (2.4) as shown in [13].

Then we can consider the PSF as quasi-stationary within that particular block. How-

ever, the blocking approach described in [13] requires matrix multiplication between

segments of a convolution matrix formed from the RF data and estimated TRF. Its

frequency-domain implementation will again require multiplication of an error block

with a DFT matrix making the algorithm computationally inefficient. Unlike in [13],

here we propose a different blocking approach that uses smaller and fixed size vector

operations with FFT applicability and thus a faster approach.

The cross-relation error in (2.3) is basically the difference between two convolution

operations. Therefore, if we can implement a full convolution as a summation of

convolutions between smaller blocks of signals, (2.4) can be implemented efficiently by

eliminating the need of using the truncation operator D as well as the unnecessary

computation of the full convolution (the bracketed part in (2.4)). Before we go into

details, we first show how a convolution can be implemented block by block. In the


con

volu

tion

PSF

TRF

Block #1

. . .

Block #B

RF

seg

men

ts

L

Ls

=1

...

B

Not

ava

ilab

le

from

tra

nsd

uce

r

}

LL

L -1s

bMCFLMS

RF image from transducer

b = 1,2,..,B

TR

F s

egm

ents

RF

seg

men

ts

convoluted RF image

TR

F s

egm

ent

))))))))

Block #1

. . .

Block #b-1

Block #1

. . .

Block #b

Figure 2.2: Block diagram of the proposed bMCFLMS algorithm. The upper part

shows the modeling of the RF data and RF image from the transducer segmented into

B blocks. The lower part shows the sequential estimation of the TRF from the RF

segments using the proposed algorithm.

subsequent discussion, ‘˜ ’ is used to denote truncated data, channel number is placed

as subscript and block number is presented in the superscript. Now consider the

convolution between the truncated RF data xi(n) of an arbitrary channel i and the

estimated TRF data hj(n) of the j-th channel given by

yij(n) = xi(n) ⇤ hj(n) (2.5)

The z-transform of hj(n), denoted as Hj(z) can be expressed as

Hj(z) = hj(0) + hj(1)z�1 + · · ·+ hj(Lb � 1)z�(Lb�1)

+ hj(Lb)z�Lb + · · ·+ hj(BLb)z

�(BLb�1)

= H1j(z) + z

�LbH2j(z) + · · ·+ z

�(B�1)LbHB

j(z) (2.6)


where Hb

j(z) can be written as

Hb

j(z) = hj

�(b� 1)Lb) + hj

�(b� 1)Lb + 1)z�1

+ · · ·+ hj

�(bLb � 1))z�(Lb�1)

, b = 1, 2, · · · , B (2.7)

Here B is the total number of blocks and Lb = floor(L/B).

Similarly, for the i-th channel RF data xi(n), we can write

Xi(z) = X1i(z) + z

�LbX2i(z) + · · ·+ z

�(B�1)LbXB

i(z) (2.8)

where

Xb

i(z) = xi

�(b� 1)Lb) + xi

�(b� 1)Lb + 1)z�1

+ · · ·+ xi

�(bLb � 1))z�(Lb�1)

, b = 1, 2, · · · (2.9)

Using (2.7) and (2.8) , (2.5) can be written in the z-domain as

Yij(z) = Xi(z)Hj(z)

= X1i(z)H1

j(z) + z

�LbX1i(z)H2

j(z)+

z�LbX

2i(z)H1

j(z) + z

�2LbX2i(z)H2

j(z)

+ · · ·+ z�2(B�1)LbX

B

i(z)HB

j(z) (2.10)

Taking the inverse z-transform of (2.10), we get

yij(n) = x1i(n) ⇤ h1

j(n) + z

�Lbx1i(n) ⇤ h2

j(n)

+ z�Lbx

2i(n) ⇤ h1

j(n) + z

�2Lbx2i(n) ⇤ h2

j(n)

+ · · ·+ z�2(B�1)Lbx

B

i(n) ⇤ hB

j(n) (2.11)

Here multiplication by z�1 refers to unit sample delay, and x

b

i(n) and h

b

j(n) represent

the b-th block of the i-th channel RF and j-th channel TRF, respectively. As evident

from (2.11), a convolution operation can be splitted into a sum of smaller convolution

blocks of identical length. For reasons explained in (2.4), we will consider the first L

samples of total 2L� 1 samples of the convolution in (2.11). Now modifying (2.11) we


can write:

yij(n) = x1i(n) ⇤ h1

j(n) + z

�Lbx1i(n) ⇤ h2

j(n)

+ z�Lbx

2i(n) ⇤ h1

j(n) + z

�2Lbx2i(n) ⇤ h2

j(n)+

· · ·+ z(B�1)Lbx

B

i(n)h1

j(n) + z

(B�1)Lbx1i(n)hB

j(n)

= y11ij(n) + z

�Lby21ij(n) + z

�Lby12ij(n) + · · ·

+ z(B�1)Lby

1Bij(n) + z

(B�1)LbyB1ij(n) (2.12)

where

ypq

ij(n) = x

p

i(n) ⇤ hq

j(n)

To account for the nonstationarity problem of ultrasound PSF, the convolution must

be evaluated in smaller blocks of Lb samples. Here each of the smaller convolutions

in (2.12) is of length 2Lb � 1. Now if we observe (2.12), it is evident that only the

first Lb samples of y11ij(n) contributes to the first convolution block y

1ij

and its last

Lb � 1 samples contributes to the next convolution block y2ij. Adding to this the first

Lb samples of y12ij(n) and y

21ij(n), we get the second convolution block y

2ij

and so on.

As the first block does not represent the general idea behind the blocking technique,

we explain the mathematical operations on the second convolution block of length Lb.

Here two truncation matrices A1 and A2 are used to take the last Lb � 1 and the first

Lb samples of a convolution, respectively. Now,

1. y11ij

= A1Cx1ih1j= A1y

11ij

where A1 =

2

40(Lb�1)⇥LbI(Lb�1)⇥(Lb�1)

01⇥Lb01⇥(Lb�1)

3

5

2. y12ij

= A2Cx1ih2j= A2y

12ij

where A2 =hILb⇥Lb

0Lb⇥(Lb�1)

i

3. y21ij

= A2Cx2ih1j= A2y

21ij


Therefore, the second block of yij described in (2.12) is given by

y2ij= y

11ij+ y

12ij+ y

21ij

= A1y11ij+A2y

12ij+A2y

21ij

=1X

p=1

A1yp(2�p)ij

+2X

p=1

A2yp(2�p+1)ij

In general, for any block b, the last Lb�1 length of the following convolutions contribute

to the b-th block convolution yb

ij:

yp(b�p)ij

= Cxpih(b�p)j

, p = 1, 2, · · · , b� 1 (2.13)

And the first Lb length of the following convolutions contribute to yb

ij:

yp(b�p+1)ij

= Cxpih(b�p+1)j

, p = 1, 2, · · · , b (2.14)

The convolution between xi(n) and hj(n) for the b-th block is then given by

yb

ij=

b�1X

p=1

A1yp(b�p)ij

+bX

p=1

A2yp(b�p+1)ij

(2.15)

2.3.2 bMCFLMS Algorithm for TRF estimation

As the truncated cross-relation error described in (2.4) is the difference between two

convolutions, a similar approach as described in (2.13), (2.14) and (2.15) can be adopted

to evaluate the cross-relation error block by block. In a similar way to (2.13), the last

Lb�1 samples of the following error function contributes to the b-th block cross-relation

error:

ep(b�p)ij

= Cxpih(b�p)j

�Cxpjh(b�p)i

, p = 1, 2, · · · , b� 1 (2.16)

where ep(b�p)ij

is the cross-relation error considering the p-th block of RF data and

(b� p)-th block of TRF data. Notice that now the true TRF has been replaced by the

estimated TRF. In the same manner, according to (2.14), the first Lb samples of the

following error function contributes to the b-th block cross-relation error:

ep(b�p+1)ij

= Cxpih(b�p+1)j

�Cxpjh(b�p+1)i

, p = 1, 2, · · · , b (2.17)


Therefore, the cross-relation error for the b-th block is

eb

ij=

b�1X

p=1

A1ep(b�p)ij

+bX

p=1

A2ep(b�p+1)ij

=b�1X

p=1

A1ep(b�p)ij

+bX

p=2

A2ep(b�p+1)ij

+A2e1bij

=b�1X

p=1

ep(b�p)ij

+bX

p=2

ep(b�p+1)ij

+ e1bij

(2.18)

Notice that the error components ep(b�p)ij

and ep(b�p+1)ij

that constitute the b-th block

error function eb

ij, can be computed parallely. Here only the third term of the right

side of (2.18) depends on the b-th block TRF, hb and the first two terms depend on

h1, h

2, · · · , and h

b�1. While estimating the TRF of the b-th block, all the TRF hq

for q = 1, 2, · · · , b � 1 are already known and therefore, may be treated as constant.

Therefore, we can write (2.18) as

eb

ij= e

1bij+ c (2.19)

where c is a constant defined as

c =b�1X

p=1

ep(b�p)ij

+bX

p=2

ep(b�p+1)ij

Taking Fourier transform of (2.19), we get the Fourier transformed error ebij

as

eb

ij= F1e

1bij+ F1c

= F1.A2(Cx1ihb

j�Cx

1jhb

i) + c

= F1A2F�12 F2(Cx

1ihb

j�Cx

1jhb

i) + c (2.20)

where F1 and F2 denote the DFT matrix of size Lb ⇥ Lb and (2Lb � 1) ⇥ (2Lb � 1),

respectively. Rewriting (2.20), we get

eb

ij= F1A2F

�12 (x1

i. ⇤ h

b

j� x

1j. ⇤ h

b

i) + c

= Be1bij+ c (2.21)


where B = F1A2F�12 , x1

iis the Fourier transform of the first block of the i-th channel

TRF data x1i

and hb

idenotes the Fourier transform of hb

iof length 2Lb�1. As convolu-

tion in the time-domain becomes multiplication in the frequency-domain, we can write,

for example, F2(Cx1ihb

j) as x

1i. ⇤ h

b

j, where ‘.⇤’ denotes the element-wise multiplication

operation. Subsequently, underbar with any quantity will define its Fourier transform.

Now, the cost function for the b-th block for estimating hb

can be defined as

Jb =

M�1X

i=1

MX

j=i+1

ebH

ijeb

ij(2.22)

Here ‘H’ denotes the hermitian operation. An estimate of the b-th block TRF, hb

can

be obtained by minimizing the cost function Jb as

hb

= arghb min J b, subject to ||h|| = 1 (2.23)

where ‘|| · ||’ denotes the l2-norm and

h(m) =hh1T(m) h

2T(m) · · · h

bT

(m)iT

(2.24)

with

hb

(m) =hhb

1(m) hb

2(m) · · · hb

M(m)

i(2.25)

In what follows, we derive a variable step-size multichannel FLMS algorithm for the

solution of (3.6).

Substituting (2.21) into (2.22), we get

Jb =

M�1X

i=1

MX

j=i+1

�Be

1bij+ c�H �

Be1bij+ c�

=M�1X

i=1

MX

j=i+1

�e1bHij

BHBe

1bij+ e

1bHij

BHc+ c

HBe

1bij+ a�

(2.26)

where a = cHc is a constant. Taking the gradient of (2.26) with respect to the b-th


block TRF of the k-th channel, we get

@Jb

@hb⇤k

=@

@hb⇤k

M�1X

i=1

MX

j=i+1

(e1bHij

BHBe

1bij+ e

1bHij

BHc+

cHBe

1bij+ a)

�

=k�1X

i=1

�x1⇤i. ⇤BH

Be1bik+ x

1⇤i. ⇤BH

c�

�MX

j=k+1

�x1⇤j. ⇤BH

Be1bkj+ x

1⇤j. ⇤BH

c�

=MX

i=1

x1⇤i. ⇤BH(Be

1bik+ c)

=MX

i=1

x1⇤i. ⇤BH

eb

ik, k = 1, 2, · · · ,M (2.27)

Here ‘*’ denotes the conjugate operation. The update equation of the bMCFLMS

algorithm for the b-th block of the RF data is given by

hb

(m+ 1) = hb

(m)� µb(m)rbJ

b(m)|h=h(m), b = 1, 2, · · · , B (2.28)

where,

rbJb(m) =

@Jb(m)

@hb⇤(m)

=h

@Jb(m)

@hb⇤1 (m)

@Jb(m)

@hb⇤2 (m)

· · · @Jb(m)

@hb⇤M (m)

i(2.29)

Here, hb

(m) denotes the m-th iteration estimate of hb and µb(m) is the variable step-

size (VSS) for the b-th block. The step-size is adapted so that the distance between

hb

(m + 1) and hb

(m) becomes minimum at each iteration and for noise-free case it is

given by (see [12] for details)

µb(m) =

hbT

(m)

||rbJb(m)||2rbJ

b(m)� hbT (m)

||rbJb(m)||2rbJ

b(m) (2.30)

Here ‘T ’ denotes the transpose operation. The problem with this equation is that we

need to know the true TRF of the b-th block to calculate µb(m) which is not known


beforehand. Unlike in [26], hb and rbJb(m) are not orthogonal because rbJ

b(m) is

not only a function of hb, but also of other blocks up to the b-th block. Therefore, we

use the TRFs and the gradient of the cost function up to the b-th block to calculate

the step-size as

µb(m) =

hT (m)

||rJ b(m)||2rJb(m)� h

T (m)

||rJ b(m)||2rJb(m) (2.31)

where,

rJb(m) =

@Jb(m)

@h⇤(m)

=h( @J

b(m)

@h1⇤

(m))T ( @J

b(m)

@h2⇤

(m))T · · · ( @J

b(m)

@hb⇤(m)

)TiT

(2.32)

Now, the second term of (2.31) becomes zero as the true TRFs vector, h formed by

concatenating the true TRFs of all the blocks up to the b-th block is orthogonal to

rJb(m). The step-size in (2.31) then becomes

µb(m) =

hT

(m)

||rJ b(m)||2rJb(m) (2.33)

To evaluate (2.32) and (2.33), we also need to derive the gradients of (2.22) with respect

to other blocks q, where q 6= b. To this end, as we are taking gradient with respect to

hq

, we can consider other parts of (2.18) which do not depend on hq

as constant. From

(2.18), we can write

eb

ij= e

(b�q)q + e(b�q+1)q +

b�1X

p=1,p 6=b�q

ep(b�p)

+bX

p=1,p 6=b�q+1

ep(b�p+1)

= e(b�q)q + e

(b�q+1)q + c1 (2.34)

where the constant c1 is defined as

c1 =b�1X

p=1,p 6=b�q

ep(b�p) +

bX

p=1,p 6=b�q+1

ep(b�p+1)


Taking the Fourier transform of (2.34), we get

eb

ij= B1e

(b�q)q +Be(b�q+1)q + c1 (2.35)

where B1 = F1A1F�12 .

Now the cost function in (2.22) becomes

Jb =

M�1X

i=1

MX

j=i+1

ebH

ijeb

ij

=M�1X

i=1

MX

j=i+1

�B1e

(b�q)qij

+Be(b�q+1)qij

+ c1)H

�B1e

(b�q)qij

+Be(b�q+1)qij

+ c1)

Taking gradient with respect to the conjugate of hq

k, where q 6= b, we get

@Jb

@hq⇤k

=MX

i=1

(x(b�q)⇤i

. ⇤BH

1 eb

ik+ x

(b�q+1)⇤i

. ⇤BHeb

ik) (2.36)

Using (2.36) with (2.32), the VSS µb(m) in (2.33) can now be calculated and the TRFs

are updated using (2.28). The TRF up to the b-th block is normalized to avoid the

trivial zero solution after each update, i.e.,

h(m+ 1) =h(m+ 1)

||h(m+ 1)||(2.37)

Here unity norm constraint is applied on all the blocks of TRFs up to the b-th block

among the total B blocks of TRFs, because the cost function Jb contains all the TRFs

up to hb

. Finally, the inverse Fourier transform of the estimate obtained using (2.28)

will result in an estimate of the b-th block of the TRF. Executing (2.28) for all the

blocks b = 1, 2, · · · , B, an estimate of the full TRF can be obtained from (3.7).

The overall idea of the algorithm is depicted in Fig. 2.2 where the RF image is

divided into B segments. As reported in the literature [25], the relative scaling factor

between two adjacent estimated TRF segments may result in visible discontinuities at

the segment boundaries known as blocking artifacts. However, as shown in (2.26), the

cost function for the b-th block TRF includes all the TRFs up to the (b� 1)-th block.

Therefore, optimization of (2.26) to update the b-th block TRF ensures the continuity

between the blocks of TRFs up to the b-th block.


2.3.3 Effect of Noise on the Convergence of the Algorithm

So far we have assumed a noise-free case for the adaptive algorithm. However, the

presence of noise may not be avoided in practice. It is well-known that noise has a

significant impact on the convergence of the cross-relation based adaptive algorithms

[14], [15], [27], [28], [29]. To stop misconvergence of this type of algorithms due to

the effect of noise, a spectral constraint is proposed in [14] where it is assumed that

the acoustic channel impulse response is spectrally flat. This assumption, therefore, is

not valid for TRFs. The time-domain approach described in [13] uses damped variable

step-size as described in [26], gradient averaging, and l1-norm constraint to prevent

misconvergence. The l1-norm constraint gives some sort of robustness against noise

when the RF data is sparse. But unfortunately sparsity of in-vivo data cannot be

guaranteed. In [15] a modified cost function has been proposed where it is assumed that

the additive noise in different channels have the same variance, a condition that may

not be satisfied in practice. Thus, none of these techniques are generalized to address

the problem of misconvergence. In this work, we develop a generalized approach to get

rid of this problem and thereby make the adaptive algorithm robust to noise.

As described in the previous section, we attempt to make the error function de-

scribed in (2.3) zero block by block in the proposed algorithm. In noisy case, (2.3)

becomes

eij(n) = [xi(n) + ni(n)] ⇤ hj(n)� [xj(n) + nj(n)] ⇤ hi(n)

= [xi(n) ⇤ hj(n)� xj(n) ⇤ hi(n)]+

[ni(n) ⇤ hj(n)� nj(n) ⇤ hi(n)]

= es

ij(n) + e

n

ij(n) (2.38)

where ni(n) denotes the additive noise in the i-th channel, es

ij(n) is the error due

to noiseless data and en

ij(n) is the error due to noise. Therefore, in noisy case the

gradient in (2.27) will also have two components – one due to es

ij(n) and the other

is for en

ij(n). However, as the noise power is generally lower than the desired signal

power, in the beginning, the signal gradient will be higher than the noise gradient.

Thus esij(n) will reduce to a lower value faster than e

n

ij(n). And at a certain instant of


iteration, the two error values will be comparable. After this point, the noise gradient

dominates and causes the solution to misconverge. In order to prevent the algorithm

from misconverging, we have to introduce a constraint in the estimation process that

somehow restrains the noisy gradient.

-17.2

-17.6

-18

-18.4

-17

-17.2

-17.4

-17.6

-17.8

-18

-18.2

-18.4

-18.6

-18.8

500 1000 1500 150010005000

a0

-2

-4

-6

-8

-100 1 2 3 4 5 6 7 8 9 10

-2

-4

-6

-8

-10

-120 1 2 3 4 5 6 7 8

x 104

x 104

No. of iterations No. of iterations No. of iterations

0

320

318

316

314

312

310

Jc

orr

-17

-17.2

-17.4

-17.6

-17.8

-18

-18.2

-18.4

-18.6

-18.8

0 500 1000 1500N

PM

(dB

)

NP

M(d

B)

NP

M(d

B)

NP

M(d

B)

a

b

c d

e

Figure 2.3: Effect of noise on the convergence of the bMCFLMS algorithm (a) Behavior

of the NPM curve around the misconvergence point for the first block with SNR = 30

dB. (b) Behavior of the correlation cost function around the misconvergence point for

the first block. (c) Behavior of the NPM curve around the misconvergence point for the

first block with correlation constraint (⇠ = 3e-7, ⇢ = 2.55, � = 2.3). (d) Misconvergence

phenomenon of the second block of TRFs for B = 2 for with no additive white noise in

the data. (e) Misconvergence Problem Solved for the Second Block of TRFs for B = 2

using correlation constraint.

Consider the following correlation between the RF data xi(n) and estimated TRF

hi(n):

r0i(n) = xi(n) ⇤ hi(�n) (2.39)


Using (2.1), (2.43) can be expressed as

r0i(n) = [s(n) ⇤ hi(n) + vi(n)] ⇤ hi(�n)

= s(n) ⇤ hi(n) ⇤ hi(�n) + vi(n) ⇤ hi(�n)

= s(n) ⇤ hi(n) ⇤ hi(�n)

= s(n) ⇤ rhi(n) (2.40)

where the correlation of noise with TRF, i.e., vi(n) ⇤ hi(�n) is assumed zero and

rhi(n) = hi(n) ⇤ hi(�n) (2.41)

From filtering point of view, (2.40) can be viewed as if hi(�n) is filtered by hi(n)

and the filter output is further filtered by s(n). Here only hi(n) is changing with

iteration. Up to the misconvergence point hi(n) is getting closer in shape to the true

TRF hi(n) and after the misconvergence point, it deviates from the shape of hi(n).

From the concept of matched filter [30], we know that a filter passes maximum energy

at its output if the input signal is of the same shape of the filter impulse response.

Therefore, the energy in rhi(n) increases as hi(n) gets closer to the misconvergence

point, but it decreases after the misconvergence point. As a result, the energy in r0i(n)

reaches its maximum value at the misconvergence point. In this thesis, we exploit this

phenomenon to prevent misconvergence. With a little modification of (2.39), consider

the following convolution:

ri(n) = xi(n) ⇤ hi(n) (2.42)

It is obvious that r0i(n) and ri(n) both have the same power spectrum but different

phase spectrum. Therefore, we can use (2.42) instead of (2.39) while using its energy

as constraint to prevent misconvergence. Since we estimate TRFs block by block, we

cannot use the full length convolution as described in (2.42), rather we will follow the

block-based convolution as described in the previous section. In a similar manner to

(2.15), for the b-th block of the total convolution length defined in (2.42), we can write

rb

i=

b�1X

p=1

A1Cxpih(b�p)i

+bX

p=1

A2Cxpih(b�p+1)i

(2.43)


0 500 1000 1500-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

30 dB25 dB20 dB10 dB

No. of iterations

NP

M(d

B)

30 dB25 dB

20 dB

10 dB

Figure 2.4: Convergence of the bMCFLMS algorithm with the proposed correlation

constraint for the simulation phantom data at different SNR.

To use the energy in rb

ias constraint, consider the following cost function for the

b-th block of data for i = 1, 2, · · · ,M :

Jb

corr=

MX

i=1

rbH

irb

i(2.44)

In order to show that J b

corrbecomes maximum at the misconvergence point graphically,

the simulation phantom data as described later in the result section was used. In Fig.

2.3(a) the misconvergence phenomenon is shown on the simulation data with 30 dB

SNR. As shown in Fig. 2.3(b), the first block correlation cost function J1corr

is maximum

at the misconvergence point and then it decreases. Therefore, the misconvergence due

to noise can be avoided if we minimize the b-th block cost function Jb in (2.22) while

at the same time maximize Jb

corror equivalently minimize �J

b

corr. Adding (2.44) as

constraint to our previous cost function in (2.22) gives

Jbt(m) = J

b(m)� (m)J b

corr(m) (2.45)

where (m) is the Lagrange multiplier, also known as the coupling factor. In general,

for any block b, the gradient of the cost function with respect to hb⇤k

can be written as

@Jbt

@hb⇤k

=MX

i=1

x1⇤i. ⇤BH

eb

ik� (m)x1⇤

k. ⇤BH

rb

k(2.46)


Considering the correlation constraint with (2.36), we get

@Jbt

@hq⇤k

=MX

i=1

(x(b�q)⇤i

. ⇤BH

1 eb

ik+ x

(b�q+1)⇤i

. ⇤BHeb

ik)

� (m)(x(b�q)⇤k

. ⇤BH

1 rb

k+ x

(b�q+1)⇤k

. ⇤BHrb

k) (2.47)

Here the update process of the TRFs is the same as described in the previous section

with a change in the gradient. Now using the gradient from (2.46) and (2.47) in (2.32),

(3.7) and (2.33), we can calculate µb(m). Next, we update the b-th block TRFs using

(2.28).

The coupling factor (m) in (2.45) should be so chosen that it gives a smaller value

for a higher value of Jb(m) and a larger value for a lower value of J

b(m). This is

due to the fact that initially the value of J b(m) will be high and the gradient due to

noiseless data is dominant. Therefore, initially the coupling factor should be of a small

value to facilitate unconstrained update of the TRF due to the dominant gradient for

the noiseless data. The update equation described in (2.28) makes the cost function

decrease even after the misconvergence point and at this point the gradient due to noise

becomes comparable to signal gradient. For this reason, (m) should be so chosen that

it increases with the decrease of J b(m) for a small value of J b(m). As it crosses the

misconvergence point, the higher value of (m) makes the noise effect compensation

stronger. A suitable expression for the coupling factor is empirically obtained as

(m) = ⇠(|⇢ log10(Jb(m))|)� (2.48)

Here � determines the sensitivity of (m) to decrease in the value of J b(m). A higher

value of � means that (m) will increase highly for a small decrease of J b(m). Fig.

2.3(c) shows that the misconvergence problem is solved after adding the constraint

described in (2.45). Misconvergence phenomenon can also emerge from estimation

noise. For example, as we are using the estimated TRFs of the first block instead

of true TRFs while estimating the TRFs of the second block, actually we are adding

estimation noise to the process. This phenomenon is shown in Fig. 2.3(d). The

misconvergence becomes stronger as the number of blocks increases. Fig. 2.3(e) shows


that our proposed constraint also works against the estimation noise. Figs. 2.3(c) and

(e) also show that the proposed constraint helps to achieve better NPM than that of

the misconverging point. Fig. 2.4 shows the efficacy of the proposed constraint against

the misconvergence of the bMCFLMS algorithm at different noise level. However, the

final NPM level degrades for decreasing SNR.

The proposed bMCFLMS algorithm is summarized in Table II.

2.4 Results

In this section, the performance of the proposed algorithms is measured on simu-

lation phantom and in-vivo RF data. The results obtained are compared with the

time-domain method described in [13], cepstrum method [16] and CR based method

[21]. Here the performance is measured using the normalized projection misalignment

(NPM) and the resolution gain (RG) [31]. The NPM is defined as

NPM(m) = 20log10

✓k⇣(m)kkhk

◆(2.49)

⇣(m) = h� hTh(m)

hT (m)h(m)h(m) (2.50)

where h and h(m) represent the true and estimated TRFs, respectively. Measurement

of NPM requires the true TRF and hence it can be calculated only for the simulation

phantom data where the true TRF is known. The RG is defined as

Gd =R

o

d

Rd

d

, d = 5 dB, 10 dB (2.51)

where Ro

dand R

d

drepresent resolutions before and after the deconvolution, respectively.

To calculate Ro

dand R

d

d, the normalized 2-D autocovariance function of the RF and

the TRF data are calculated. Then the axial slice through the peak is considered and

the width of the slice at a level d dB is measured which represents Rd

d.

In all the subsequent figures the reference images were generated from the RF data

which have the blurring effect in it introduced by the PSF. To produce the ultrasound

images the envelope of the data was taken from the absolute value of the Hilbert


Width (mm)

De

pth

(m

m)

0 10 20 30 40

0

10

20

30

40

True TRF Envelope Deconvolved TRF Envelope

De

pth

(m

m)

Width (mm)

0 10 20 30 40

0

10

20

30

40

De

pth

(m

m)

Width (mm)

0 10 20 30 40

0

10

20

30

40

-6

−10

−14

−18

−22

0 3 6 9 12

x 13

0

NP

M (

dB

)

No. of iterations

True

Deconvolved with constraint

d

x 107

-2 e

Frequency (MHz)

No

rma

lize

d M

ag

nitu

de

1

0.8

0.6

0.4

0.2

00 0.40.2 0.6 0.8 1.41.21 1.6 1.8 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Standard RF Envelope

b ca

Figure 2.5: Deconvolution performed on a simulation phantom with 20 scatterers per

resolution cell and SNR = 30 dB. The RF data size is 1038 ⇥ 128. The darker circu-

lar inclusion with radius 5 mm is created by placing scatterers with relatively lower

strength than the surroundings. Log-envelope image of the (a) true TRF, (b) backscat-

tered standard RF data, and (c) deconvolved TRF by the bMCFLMS algorithm, (d)

spectra of the true and estimated RF data of a single scan-line marked by vertical

green line in the log envelope images of (b) and (c). (e) The NPM curve between the

deconvolved TRF and true TRF of the first block.

transform of the data. Then a log compression was performed on the normalized

envelope data where the dynamic range was set as 35 dB for all the images. The

data used in the result section have axial depth of 40 mm and for this depth a total

block number of 2 was found reasonable. However, the number of total blocks may be

increased which will increase the propagation of estimation errors from the previous

block to the subsequent blocks. Therefore, for all data the total number of block was

set at B = 2 and the parameters of noise effect compensating constraint were set to


0 5 10 15

x 106

0

0.5

1

Frequency (Hz)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

7

0

0.5

Frequency (Hz)

No

rm.

ma

gn

itu

de

No

rma

lize

d

ma

gn

itu

de

Zoomed-in view

0 0.5 1 1.5 2

x 107

0

0.5

1

Frequency (Hz)

No

rm.

ma

gn

itu

de

0 10 20 30

0

5

10

15

20

25

30

35

40

Standard Log Envelope

Dep

th, m

m

Deconvolved Log Enveope

Lateral distance, mm

g

300

Lateral distance, mm

10 20

20

x 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

d

TRF spectruma b

Original RF spectrum

Estimated RF spectrum

c

40 40

e

f

g

Figure 2.6: Performance analysis of the bMCFLMS algorithm for the in-vivo backscat-

tered RF data of a breast cyst. Spectrum of the (a) deconvolved TRF, (b) R-MINT

estimated PSF, and (c) spectra of the true and estimated RF data of a single scan-line

marked by vertical red line in log envelope images. Standard log envelope images of

(d) the backscattered RF data, (e) deconvolved TRFs, and (f-g) zoomed-in views of

(d-e).

⇠ = 1e� 4, ⇢ = 2.55 and � = 2.4.

2.4.1 Simulation Phantom Results

The ultrasound simulation was done using the FIELD-II [32] where the transducer

element height was chosen to be 5 mm. Ultrasound simulation was done on a 3-D

simulation phantom (40 mm ⇥10 mm ⇥40 mm) with a scatterer density of 20 scatterers


0 10 20 30

0

5

10

15

20

25

30

35

40

0 5 10 15

x 106

0

0.5

Frequency (Hz)

No

rm.

ma

gn

itu

de

0 0.5 1 1.5 2

x 107

0

0.5

1

Frequency (Hz)

No

rm.

ma

gn

itu

de

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 107

0

0.5

1

Frequency (Hz)

No

rma

lize

d

ma

gn

itu

de

Zoomed-in view

g

i

0 10 20 30

b PSF Spectrum

a TRF spectrum

Original RF spectrum

Estimated RF spectrum

c

Standard Log Envelope

Lateral distance, mm Lateral distance, mm

Deconvolved Log Enveope

1

20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

4040

d e

g

f

Figure 2.7: Performance analysis of the bMCFLMS algorithm for the in-vivo backscat-

tered RF data for a left carotid artery. Spectrum of the (a) deconvolved TRF, (b)

R-MINT estimated PSF, and (c) spectra of the true and estimated RF data of a single

scan-line marked by vertical red line in log envelope images. Standard log envelope

images of (d) the backscattered RF data, (e) deconvolved TRFs, and (f-g) zoomed-in

views of (d-e).

per resolution cell. A circular inclusion with radius 5 mm was simulated in a slice along

the transverse plane of the phantom by reducing the magnitudes of scatterers in that

region. The focus of the ultrasound beam was set at 30 mm depth from the phantom

surface. The transducer center frequency was selected as 10 MHz and the sampling

frequency as 40 MHz. The total number of scan lines (M) was set to 128 to match with

that of available commercial ultrasound scanners. To simulate noisy data zero-mean


additive white Gaussian noise was added to the data so as to obtain an SNR of 30 dB.

The performance of the proposed bMCFLMS algorithm on the simulation data is

shown in Fig. 2.5. From visual comparison of the images provided in Figs. 2.5(a)-

(c) we see that our estimated TRF (Fig. 2.5(c)) has finer texture compared to the

standard RF image (Fig. 2.5(b)) and the sharpness of the inclusion boundary has been

restored in the deconvolved TRF image. Fig. 2.5(d) shows that the estimated RF

spectrum, derived from the estimated TRF along the green marked line of the image

and estimated PSF, matches closely with that of the true RF. Again, since we have

the true TRF for the simulation data, we can evaluate NPM for our estimated TRF

which is shown in Fig. 2.5(e). Fig. 2.5(e) proves the effectiveness of our proposed

correlation-based constraint to prevent misconvergence of the proposed algorithm as

the NPM remains stable at �22.1 dB.

Comparative results of different algorithms on simulation phantom data at SNR

= 30 dB are presented in Table 2.3. The quantitative performance measures used for

comparison are NPM (see (2.49)) and Gd (see (2.51)). It is obvious from Table 2.3 that

our proposed bMCFLMS algorithm gives better image quality in terms of resolution

gain (5 and 10 dB level) and NPM than that of the other methods. In order to show

the improvement after using the missing data in our algorithm, results on the final

deconvolved image from the bMCFLMS are presented.

2.4.2 In-Vivo Results

The performance of the proposed algorithms was tested on two in-vivo data. One of

these data is of fibrocystic breast with the largest cyst size of 4.75 mm⇥4.26 mm and

the other one is of a carotid artery. These data were collected from the patients who

appeared for medical examination at the Medical Centre of Bangladesh University of

Engineering and Technology (BUET), Dhaka, Bangladesh. This study was approved

by the Institutional Review Board (IRB) and prior patient consent was taken.

Ultrasound image quality is measured in terms of texture, tissue structure boundary

and size in the reconstructed image [17], [18], [25]. To demonstrate that these features

improve, we present in Fig. 2.6 the performance of our proposed bMCFLMS algorithm


on the fibrocystic breast data. Fig. 2.6(a) shows the wideband TRF spectrum where

the TRF line is chosen along the red line of Fig. 2.6(e), and Fig. 2.6(b) shows the

narrowband, smooth PSF spectrum. The estimated RF spectrum evaluted from the

multiplication of TRF spectrum and PSF spectrum, matches closely with true RF

spectrum which is shown in Fig. 2.6(c). This ensures the deconvolution efficacy of the

proposed algorithm. From visual perspective, Fig. 2.6(e) shows a significant increase

in texture quality compared to Fig. 2.6(d). For better visualization zoomed-in view

of the two images are provided in Figs. 2.6(f)-(g). It is evident from Figs. 2.6(d)-(e)

that the deconvolved image offers finer texture compared to the blurred RF image.

Again, from the zoomed-in view it is apprent that the partially obscured cyst in the

standard RF image is more prominently visible in the deconvolved image with better

boudary, shape and size. In addition to that, some tiny structures are also visible in the

deconvolved image which are marked by a ellipse in Figs. 2.6(d)-(e). From the visual

inspection of the marked region we can infer that the tiny structures have emerged as

a result of deconvolution that were obscured in the standard RF image.

As in the case of in-vivo data in Fig. 2.6, the overall image quality of the carotid

artery depicted in Fig. 2.7 has also improved. From the zoomed-in view of the carotid

artery in Figs. 2.7(f)-(g), we see that the deconvolved image has finer texture and

comparatively sharp arterial boundary than the RF image. Again, Fig. 2.7(c) shows the

close match between our estimated RF spectrum and the original RF spectrum which

proves the estimation accuracy of the estimated TRF using our proposed algorithm,

and of the PSF estimated using the MINT algorithm.

As shown in Table 2.4, our proposed bMCFLMS algorithm gives significantly better

resolution gain at 10 dB level compared to its time-domain counterpart, CR-based

method and cepstrum method for the in-vivo data. The latter methods also show

data dependency as they give resolution gain improvement in different range for the

breast cyst and the carotid artery data. As discussed in the introduction section, the

effectiveness of cepstrum method depends on the separability of the TRF and the PSF

spectrum. Again, the CR-based method requires thresholding to find the null space

bases of the correlation matrix. However, l1-bMCLMS and our proposed bMCFLMS


Data Length N in the Axial Direction

Tim

e P

er

Itera

tion (

logari

thm

ic s

cale

)

a

100 200 800700600500400300 900 1000

100

101

102

l1-bMCLMS

md-bMCFLMS

Figure 2.8: Time required per iteration by the time- and the frequency-domain algo-

rithms for the simulation phantom data of size Na ⇥ 128 where Na is made variable.

algorithm do not impose any stringent requirements on the data. Therefore, they

show similar resolution improvement for both type of data which justifies their less

dependence on parameter tuning. The parameters for correlation constraint once set

remain effective for all types of data presented in the result section. Again to investigate

the speckle noise level before and after deconvolution signal-to-noise ratio (sSNR) was

calculated in Table 2.4 as [31]. sSNR is defined as the ratio of µ and �, where µ and �

are the mean and standard deviation of the absolute value of the RF data, respectively.

Insignificant changes in the value of sSNR justify the claim that deconvolution can

significantly improve the resolution though has negligible effect on speckle noise as

reported in [31].


Table 2.1: Symbols and Description

Symbols Description

Lb Axial length of an RF data block

xbi (n) RF data of the b-th block of the i-th A-line

hbi TRF of the b-th block of the i-th A-line

Cxpi

Convolution matrix of length 2Lb � 1 for the

p-th block of the i-th A-line RF data

epqij Cross-relation error between the p-th block of

the i-th and q-th block of the j-th A-line

epqij Truncated epqij in the frequency domain

m Iteration index

rbJb(m) Gradient of the b-th block cost function with

respect to the estimated b-th block TRF at the

m-th iteration

rJb(m) Gradient of the b-th block cost function with

respect to the estimated total TRF at the m-

th iteration

µb(m) Variable step-size for the update of the b-th

block TRF at the m-th iteration

Jbcorr(m) Cost function for the correlation constraint of

the b-th block at the m-th iteration

Jbt(m) Total cost function for the b-th block including

correlation constraint

(m) Coupling factor for the correlation constraint

at the m-th iteration

⇠, ⇢ and � Empirical constants for the correlation cou-

pling factor (m)

s(n) Estimated PSF using the MINT algorithm

⌫i Scaling factor for the estimated i-th A-line

missing RF data

Jb0(m) Total cost function for the b-th block including

estimated missing data and correlation con-

straint at the m-th iteration


Table 2.2: Constrained bMCFLMS Algorithm

Step 1 . Set total block number B and appropriate value for ⇠, ⇢ and �

. Initialize the i-th channel TRF, hi = [1 01⇥(L�1)]T for i = 1, 2, · · · ,M

Step 2 . Set current block number b = 1

Step 3 . Set iteration index m = 1

Step 4 . Calculate the error functions for b-th block using (3.6), (3.6), (2.18) and

(2.21)

. Calculate correlation of estimated TRF with RF data using (2.43)

Step 5 . Calculate @Jbt

@hb⇤k

according to (2.46)

. Calculate @Jbt

@hq⇤k

according to (2.47)

. Calculate step-size for b-th block and m-th iteration µb(m) using J

bt instead

of J b in (2.32), (3.7) and (2.33)

Step 6 . Update h using (2.28)

. Normalize h according to (2.37)

Step 7 . If m is less than required iterations, set m = m+ 1 and go to step 4

. Else set b = b+ 1, m = 1 and go to step 3


Table 2.3: Performance of different algorithms on simulation phantom data with addi-

tive noise

Gd(RF)

Data Method NPM (dB) 5 dB 10 dB

CR-based Method �15.13 3.82 4.23

Cepstrum �6.11 1.77 1.78

Simulation data l1-bMCLMS �19.99 4.11 5.14

Proposed bMCFLMS �21.20 4.42 6.03

Table 2.4: Performance of different algorithms on in-vivo data

Gd(RF) sSNR

Data Method 5 dB 10 dB Original Deconv.

Fibrocystic Breast CR-based Method 2.45 1.32 0.7835 0.6788

Cepstrum 1.13 0.72 0.7835 0.7722

l1-bMCLMS 3.45 4.84 0.7835 0.6405

Proposed bMCFLMS 4.00 6.17 0.7835 0.8366

Carotid Artery CR-based Method 3.44 5.03 0.7451 0.7223

Cepstrum 2.44 2.16 0.7451 0.6743

l1-bMCLMS 3.72 5.89 0.7451 0.7564

Proposed bMCFLMS 4.01 7.05 0.7451 0.7129

Chapter 3

B-mode Image Generation Framework

In this chapter, we propose a multiframe-based adaptive despeckling (MADS) algo-

rithm that treats the speckle noise in its multiplicative form and utilizes the speckle

patterns estimated using the multiframe-based adaptive ultrasonic speckle noise esti-

mation (MSNE) algorithm proposed in this thesis. The MSNE algorithm is based on

formulating the true image as single input and the envelope of deconvolved consecutive

US image frames with multiplicative speckle noise pattern in each frame as multiple

outputs. The despeckling algorithm, on the contrary, treats the envelope of deconvolved

consecutive US image frames as multiple inputs and the true image as single output.

According to the mathematical model representing the US imaging system, deconvolu-

tion is necessary prior to despeckling for resolution enhancement of the raw RF data.

Hence, a 2-D deconvolution approach as an extension of our previously published 1-D

deconvolution algorithm, i.e., bMCFLMS has been also described in this chapter. To

prevent misconvergence of the MSNE algorithm in the presence of additive noise and

estimation error resulting from the deconvolution step, a zero-lag correlation contraint

derived from the deconvolved image and the estimated speckle pattern is attached with

the original cost function. As the overall despeckling approach has been derived by

completely following the signal generation models, it is likely to maximally preserve

the diagnostically important details and tissue texture present in the image. Finally, a

complete framework including deconvolution, despeckling, and post-processing for ul-

35

CHAPTER 3. B-MODE IMAGE GENERATION FRAMEWORK 36

trasound B-mode image generation with superior quality in terms of resolution, edges,

small details and texture is also established in this chapter for greater interest of the

researchers.

3.1 Despeckling of the Envelope of Deconvolved Data:

Literature Review

A number of algorithms have been reported in the literature for despeckling US images

which attempts to remove speckle noise either in their mulplicative form or converting

the noise in the additive form. Among the first group of algorithms, the spatial averag-

ing based approaches exploit the repetitive nature of the US image, and among them,

linear filtering methods, such as Gaussian filter and mean filter are effective in reducing

speckle noise [2]. However, they tend to oversmooth the texture and blur edges present

in the image. To overcome this problem, nonlinear approaches based on local [33], [34]

and non-local statistics [35], [36] of the image have been proposed. These algorithms

are mainly weighted filters, in which the weights depend on the similarity between the

intensity values of the patches surrounding the pixels [37]. The main difference between

local and non-local means methods is that the non-local means method employs the

most similar pixels in the image to denoise the current pixel regardless of their Eu-

clidean distance. Although these approaches tend to preserve textures and edges, their

performance is dependent on tuning parameters, such as filter and patch size. Again,

the nonlinear approaches based on the diffusion equation [38] not only preserves edges

but also enhances edges by inhibiting diffusion across edges and allowing diffusion on

either side of the edge. However, selection of the parameter-values is a major issue

in this method, as a value of parameter that is smaller than the optimum one leads

to unsatisfactory noise suppression whereas a higher parameter-value results in poor

structure preservation [39].

The second group of algorithms converts the multiplicative speckle noise into an

additive one by using logarithmic transformation. Now, as shown in [1], the additive

noise can be handled using any traditional denoising scheme, and the performance


of this scheme determines the overall efficacy of despeckling. Finally, the denoised

image is exponentially transformed back to give the despeckled image. The overall

process is termed as homomorphic filtering approach [1], and among the denoising

schemes used in this process, the transfrom domain or multi-resolutional based ap-

praoches [40–43] are of higher efficacy. In [44], an advanced ultrasound despeckling

algorithm is proposed based on the intra-scale correlation between the wavelet coeffi-

cients. Among the multi-resolutional approaches, as shown in [4], despeckling based

on non-subsampled contourlet gives superior performance. However, in wavelet-based

despeckling, a threshold is a critical parameter that is to be determined based on the a

priori knowledge of the distribution of the speckle pattern. In addition, the threshold-

based filtering of wavelet coefficients implies texture smoothening, and it gives rise to

artifact such as Gibbs phenomenon near the edges [45]. All of the approaches discussed

so far relies on ad-hoc filtering or smoothening technique without addressing mathe-

matically the speckle noise generation model. Therefore, it cannot be guaranteed that

these algorithms only operate on the speckle noise without significantly distorting the

true image.

3.2 Method

In this chapter, our main concern is to derive a novel despeckling algorithm that is

likely to preserve maximum features present in the image. However, according to (1.1)

and (1.2), deconvolution is a necessary pre-processing step to despeckling. Therefore,

for the completeness of a high-resolution B-mode image generation, we also consider

it important to include a 2-D extension of our previously proposed 1-D deconvolution

(bMCFLMS) algorithm.

3.2.1 Deconvolution of RF Echo Data

In this section, we attempt to estimate the TRF, h(m,n), in (1.1) with increased

resolution by removing the effect of the PSF, s(m,n), from the raw RF data, x(m,n).

However, to estimate h(m,n), similar to the approach as described in [31], we consider


s (m)

h1(m)

h2(m)

hN

(m)

x1(m)

x2(m)

xN

(m)

v1(m)

v2(m)

vN

(m)

PSF

TRFBackscattered

RF Data

Additive

Noise

a

a

a

a

‘

‘

‘

Figure 3.1: A SIMO model for backscattered RF signal.

the 2-D distortion kernal, i.e., PSF s(m,n) in (1.1) decomposible into two 1-D distortion

kernals (PSFs): one along the axial direction and the other along the lateral direction.

Following this assumption, (1.1) can be modified as

x(m,n) = sa(m) ⇤a sl(n) ⇤l h(m,n) + v(m,n) (3.1)

where sl(n) and sa(m) are the lateral and axial PSFs, respectively, and ‘⇤l’ and ‘⇤a’represent convolution along the lateral and the axial directions, respectively. A novel

technique for removing the effect of axial PSF sa(m) from the measured RF image

x(m,n) was reported in the previous chapter using a single input multiple output

(SIMO) model as shown in Fig. 3.1, where sa(m) convolves with the i-th A-line of the

axial TRF denoted as ha

i(m) and with additive noise vi(m) gives the i-th A-line RF

data xi(m):

xi(m) = sa(m) ⇤a ha

i(m) + vi(m) (3.2)

where

ha

i(m) = sl(n) ⇤l h(m,n) (3.3)

In matrix form, (3.2) can be written as

X = Saha + v (3.4)


where Sa is the convolution matrix formed using the axial PSF sa(m) and

X =hX1 X2 · · · XN 0

i,

ha =

hha

1 ha

2 · · · ha

N 0

i.

Here, N 0 is the total number of A-lines, and Xi and ha

iare the i-th A-line with M

0

samples taken from xi(m) and ha

i(m), respectively where m = 1, 2, · · · ,M 0. To account

for the non-stationarity of the axial PSF, the RF data were divided into B blocks with

equal length Lb, and a block-based cost function Jb for the b-th block was formulated

(for details see [46]) to estimate the axial TRF block-by-block in the frequency-domain

as

Jb =

n�1X

i=1

nX

j=i+1

ebH

ijeb

ij(3.5)

where, ‘H’ denotes the Hermitian operation, any variable with ‘ ’ represents the vari-

able in the frequency-domain, and eb

ijis the Fourier transform of eb

ijdefined as

eb

ij=

b�1X

p=1

A1ep(b�p)ij

+bX

p=1

A2ep(b�p+1)ij

Here,

ep(b�p)ij

= Cxpiha(b�p)j

�Cxpjha(b�p)i

, p = 1, 2, · · · , b� 1,

ep(b�p+1)ij

= Cxpiha(b�p+1)j

�Cxpjha(b�p+1)i

, p = 1, 2, · · · , b,

and A1, A2 are the truncation matrices truncating the last (Lb � 1) and the first Lb

samples of the error function, respectively. Cxpi

is the convolution matrix formed using

the RF data, x(m,n) along the i-th A-line and the p-th block. Now, the b-th block

axial TRF hab was estimated as

hab

= arghab min J b, subject to ||h

a

|| = 1 (3.6)

where ‘|| · ||’ denotes the l2-norm and

ha

=hha1T

ha2T

· · · haBTiT

(3.7)


with ‘T ’ denoting matrix transpose operation and

hab

=hhab

1 hab

2 · · · hab

N 0

i(3.8)

Here, hab

idenotes the estimated axial TRF along the i-th A-line and the b-th block.

In sample-domain, the estimated axial TRF along the i-th A-line can be written as

ha

i(m). Now, the estimated axial TRF hi(m) along the i-th A-line at discrete time m

can be modeled as

ha

i(m) = sl(n) ⇤l h(m,n) + v

0(m,n) (3.9)

where v0(m,n) is the noise resulting from the estimation error of the axial TRF. There-

fore, from (3.2) and (3.9), it is apparent that an attempt, similar to the approach

adopted in the axial direction, can be made in the lateral direction to undo the effect

of the lateral distortion kernal (PSF) from the estimated axial TRF ha

i(m) to estimate

the 2-D deconvolved TRF h(m,n). In this approach, the PSF is considered to be later-

ally stationary as described in [25], and therefore, no blocking is required in the lateral

direction. The method is summarized in Table 3.1. The estimated TRF after lateral

deconvolution is given by

h =hh1 h2 · · · h

M 0

iT(3.10)

where, hiis the estimated lateral TRF along the i-th sample line, i.e., samples along

the i-th row of h. In sample-domain, the m-th row and n-th column sample of the

estimated TRF, h can be denoted as h(m,n).

3.2.2 Proposed Despeckling Algorithm

The envelope of the estimated TRF h(m,n) in the previous subsection is corrupted

with speckle noise as given by (1.2). It is apparent from (1.2) that knowledge of the

speckle pattern u(m,n) in an image frame can help despeckling that frame. However, in

the absence of additive noise, direct division of he(m,n) by u(m,n) may amplify noise

and/or give rise to division by zero problem. Hence, in this subsection, we attempt to

formulate an energy constrained iterative approach to find an equalization multiplying

factor to despeckle the frame. In what follows, we attempt to formulate a novel SIMO


Table 3.1: bMCFLMS algorithm for 2-D deconvolution of ultrasound RF image

Step 1 . Select the bMCFLMS method reported in [46]

Step 2 . Set X = raw radio-frequency (RF) data, where X is the data to be decon-

volved

. Set the data length, L = axial length, and the number of channels, M =

lateral length of the raw RF data. Execute the bMCFLMS algorithm with

block number, B = 2 along the axial direction.

Step 3 . Set X = axially deconvolved data in step 2

. Set the data length, L = lateral length, and the number of channels, M =

axial length of the raw radio-frequency (RF) data. Execute the bMCFLMS

algorithm with block number, B = 1 along the lateral direction.

model for the deconvolved image frames and thereby, estimate the speckle noise in

the respective deconvolved frames using an adaptive filtering technique. Then a novel

MISO model will be proposed to despeckle the decconvolved image frames using the

estimated speckle pattern of the respective frames.

w (m,n)

Additive

Noise

Multiplicative

Speckle Noise

u

u

u

1

2

p

(m,n)

(m,n)

(m,n)

r(m,n)

h (m,n)

h (m,n)

h (m,n)

e1

e2

ep

Decon

volv

ed

data

1

2

p

w (m,n)

w

(m,n)

True Image

Figure 3.2: A new SIMO model for deconvolved 2-D RF data.


In an ultrasound imaging system, images are generally acquired at a frame rate

ranging from 10�60 frames per second (FPS) with speckle patterns generated randomly

in each image frame from the interference of the US pulse at the receiving transducer.

In our work, we attempt to use p consecutive deconvolved frames to formulate a single

input multiple output (SIMO) model as shown in Fig. 3.2. Here, the true ultrasound

image r(m,n) that is considered stationary throughout the p frames, multiplies with

the speckle noise ui(m,n) of the i-th frame, and with an additive noise wi(m,n) gives

the envelope, hei(m,n), of the estimated deconvolved image hi(m,n) of the i-th frame:

hei(m,n) = r(m,n)ui(m,n) + wi(m,n) (3.11)

In matrix form, (3.11) can written as

Hei = R · ⇤Ui +Wi (3.12)

where ‘·⇤’ denotes elementwise multiplication. Here, hei(m,n), r(m,n), ui(m,n) and

wi(m,n) represents the m-th row and n-th column elements of Hei, R, Ui and Wi,

respectively. Therefore, here the challenge is to estimate the speckle noise ui(m,n)

from each frame in the presence of additive noise wi(m,n) and then remove the speckle

noise from the deconvolved envelope image hei(m,n). The assumptions behind the

SIMO model formulation and the identifiability condition [47] for the speckle noise

pattern ui(m,n) of the i-th frame are

1. The true image r(m,n) is stationary throughout the p consecutive frames.

2. The speckle patterns of each frame do not share common zeroes with the rest

p� 1 consecutive frames.

These assumptions are realistic because for an ultrasound video recording with 30 FPS,

consecutive 5�10 frames take around 0.17�0.33 second during which the hand motion

can be ignored. Then we can consider the true ultrasound image r(m,n) as stationary

throughout these frames. Again, formation of speckle noise in consecutive frames is

a completely random phenomena, and hence, they are unlikely to contain common

zeros. However, the probablity of sharing common zeros between the speckle patterns


of the consecutive frames can be further reduced by increasing the number of fames

into consideration; doing this will improve the identification accuracy of the patterns

provided that the stationarity assumption remains valid as shown in the result section

later. In what follows, we derive a multiframe-based adaptive speckle noise estimation

algorithm using the proposed SIMO model.

Speckle Noise Estimation

In the absence of additive noise, the following error function eij(m,n) can be used to

estimate the speckle noise:

eij(m,n) = hei(m,n)uj(m,n)� hej(m,n)ui(m,n) (3.13)

where ui(m,n) is the estimated speckle noise of the i-th frame. Notice that for additive

noiseless case if we can estimate the speckle pattern accurately, the error function

defined in (3.13) becomes zero. Using this fact, we can build the following cost function

to iteratively estimate the speckle noise:

J =p�1X

i=1

pX

j=i+1

||Eij · ⇤Eij||2F (3.14)

where

Eij = Hei · ⇤Uj � Hej · ⇤Ui (3.15)

and ‘|| · ||F ’ indicates the Frobenius norm. An estimate of the speckle noise U can be

obtained by minimizing the cost function J as

U = argU min J, subject to ||U||F = 1 (3.16)

where

U =hU1 U2 · · · Up

i(3.17)

Taking the gradient of J in (3.14), we get

rkJ =@J

@Uk

= 2pX

i=1

Hei. ⇤ Eik (3.18)


The update equation for the MSNE algorithm at the q-th iteration is

U(q + 1) = U(q)� µ(q)rJ(q)��U=U(q)

(3.19)

where,

rJ(q) =@J(q)

@U

=hr1J(q) r2J(q) · · ·r pJ(q)

i(3.20)

and µ(q) is the variable step-size (VSS) which is such that the misalignment of U(q+1)

with the true noise pattern U is minimum at every iteration, given the current estimate

U(q):

Jµ(q) = (||U� ↵U(q + 1)||F )2|U(q)

=��U · ⇤U� 2↵U · ⇤U(q) + 2↵µ(q)U ·⇤r J(q)

+ ↵2U(q) · ⇤U(q)� 2↵2

µ(q)U ·⇤r J(q)

+ ↵2µ2(q)rJ(q) ·⇤r J(q)

��S|U(q) (3.21)

where ↵ is a scaling constant inherent in any blind channel identification approach

based on the cross-relation, and we define an operator ‘|| · ||S’ which evaluates the

sum of the matrix elements. Minimizing (3.21), i.e., setting the gradient of Jµ(q) with

respect to µ(q) to zero, we get µ(q) as

µ(q) =

��U(q) ·⇤r J(q)��S��rJ(q) ·⇤r J(q)��

S

(3.22)

Equation (3.22) can be considered as a variant of VSS derived in [12].

So far additive noise has been ignored in the derivation of the proposed MSNE

algorithm. However, it has a similar effect on the convergence of the MSNE algorithm

as described in [46]. To solve the problem, we need to impose a constraint on (3.14)

so as to prevent the deviation of the estimated speckle pattern from the true speckle

pattern. To this end, consider the following model of speckle corrupted image for the

k-th frame as described in [35]:

hek(m,n) = r(m,n) + r⇣(m,n) Vk(m,n) (3.23)


where Vk(m,n) ⇠ N (0, �2). Ignoring the additive noise in (1.2) and comparing with

(3.23), our proposed algorithm is basically estimating the true speckle pattern as

uk(m,n) = 1 + r(⇣�1)(m,n) Vk(m,n) (3.24)

Then the speckle pattern uk(m,n), estimated using the proposed method, can be ex-

presses as

uk(m,n) = 1 + r(⇣�1)(m,n) Vk(m,n) (3.25)

Since r(m,n) is not a variable here, to make the estimated speckle pattern uk(m,n)

in (3.25) close to the true speckle pattern uk(m,n) in (3.24), we need to maximize

the zero-lag correlation between Vk(m,n) and Vk(m,n). However, to attach this as

a constraint on (3.14), estimates of the true image r(m,n) and the parameter ⇣ are

necessary. Alternatively, consider the zero-lag correlation between the deconvolved

image hek(m,n) and the estimated speckle pattern uk(m,n):

Jcorr =��Hek · ⇤Uk

��S

=��(Hek +Wk) · ⇤Uk

��S

=��Hek · ⇤Uk

��S

(3.26)

where the zero-lag correlation between the additive noise Wk and speckle noise Uk is

considered zero. Using (3.23) and (3.25) in (3.26), we get

Jcorr =��R+R

⇣ ·⇤V k +R⇣ · ⇤Vk + Vk · ⇤Vk

��S

=��Hek +R

⇣ · ⇤Vk + Vk · ⇤Vk

��S

= c+��Vk · ⇤Vk

��S

(3.27)

where at the point of misconvergence when the estimated speckle pattern is close to

the true speckle pattern, the zero-lag correlation between r⇣(m,n) and Vk(m,n) can be

considered zero, and c is a constant defined as c =��Hek

��S. From (3.27), it is apparent

that maximizing Jcorr or equivalently minimizing �Jcorr is analogous to maximizing the

zero-lag correlation between Vk(m,n) and Vk(m,n). To prevent the misconvergence of

the proposed algorithm in the noisy case, we propose to use the zero-lag correlation


constraint in (3.27) with the MSNE cost function in (3.14). Then modifying (3.14) for

noisy case, we obtain

Jc(q) = J(q)� �1Jcorr(q) (3.28)

where �1 is the Lagrange multiplier or also known as the coupling factor. Taking

gradient of (3.28) with respect to Uk, we get

rkJc(q) = rkJ(q)� �1Hek (3.29)

Replacing rkJ(q) in (3.18) by rkJc(q), we can estimate the speckle noise pattern in

each of the p frames.

Estimation of the True Ultrasound Image

So far, a novel algorithm for estimating the speckle pattern has been explained with

a view to estimating the true ultrasound image r(m,n) in (1.2) using U from (3.16).

Now, we describe a novel MISO model as shown in Fig. 3.3, where the speckle noise

cancellation (SNC) factors for each of the i-th frame gi(m,n) multiplies with hei(m,n),

to obtain an estimate of the true US image ri(m,n) for that frame. In the absense of

additive noise in (3.11) and estimation error in the estimated speckle pattern Ui, the

estimated ultrasound image of the i-th frame, Ri, can be obtained in matrix form as

g (m,n)

g (m,n)

g (m,n)

h (m,n)

r(m,n)

e1

1

2

p

Envelope ofDeconvolved

RF data

Speckle Noise Cancellation

EstimatedImage

h (m,n)e2

h (m,n)ep

Figure 3.3: The block diagram for the estimation procedure of true ultrasound image.


Ri = Hei · ⇤Gi (3.30)

= Ri · ⇤Ui · ⇤Gi (3.31)

where Gi is the elementwise multiplying SNC factor to equalize the speckle pattern

for the i-th frame. Here for an accurate estimation of Ri, the elements of the matrix

Ui. ⇤Gi in (3.31) should be equal to 1. Therefore, we can formulate the following cost

function to estimate the SNC factors:

Jeq = ||U · ⇤G�D||2F

(3.32)

where

G =hG1 G2 · · · Gp

i(3.33)

and D is a matrix with all entries equal to 1. Taking gradient of (3.32) with respect

to G, we get

rJeq =@Jeq

@G= 2(U. ⇤G�D). ⇤U (3.34)

The update equation for estimating G at the q0-th iteration is given by

G(q0 + 1) = G(q0)� µ(q0)rJeq(q0) (3.35)

where µ(q0) is the VSS for the q0-th iteration, and can be obtained following (3.22) as

µ(q0) =

��G(q0) ·⇤r Jeq(q0)��S�� rJeq(q0) ·⇤r Jeq(q0)��S

(3.36)

Up to now, we have ignored the effect of estimation error in U. If we assume that Es

be the estimation error in U, then we can write

U = U+ Es (3.37)

Now, replacing the true speckle pattern U with the estimated speckle pattern U, the

cost function in (3.32) becomes

Jeq = ||U. ⇤G�D||2F

(3.38)

= ||(U+ Es). ⇤G�D||2F

= ||U. ⇤G�D||2F+ E (3.39)


where E represents the terms including the estimation error Es. Therefore, the gradient

in (3.34) will have two components– one from the desired part of the cost function and

the other from the estimation error part, i.e.,

rJeq(q0) = rJ

desired

eq(q0) +rJ

error

eq(q0) (3.40)

From (3.38), we observe that U and G have inverse relation so as to make their elemen-

twise product equal to 1. Therefore, in (3.37), the term associated with the estimation

error in U gives rise to an equalization factor G in which the small estimation error

is magnified. To make G less sensitive to such phenomenon, we impose an energy

regularization constraint on G in (3.32):

J0eq= ||U · ⇤G�D||2

F+ �2||G||2

F(3.41)

where �2 is the Lagrange multiplier. Now, the gradient in (3.34) becomes

rJ0eq= 2(U · ⇤G�D) · ⇤U+ 2�2G (3.42)

In addition to the energy regularization constraint, we propose the following gradi-

ent averaging technique to average out or at least reduce the detrimental effect of

rJerror

eq(q0):

rJ00eq(q0) = ↵rJ

0eq(q0) + (1� ↵)J 0

eq(q0 � 1) (3.43)

where ↵ is the weighting factor given on the current gradient rJeq(q0). Now, using the

average gradient rJ00eq(q0) in (3.35) and (3.36) in place of rJeq(q0), we can estimate

the equalization factor G which can then be used to get an estimate of the estimated

true image of the i-th frame, Ri, using (3.30). Finally, averaging the estimates for

i = 1, 2, · · · , p for SNR improvement, the true image is reconstructed as

R =pX

i=1

Ri (3.44)

The additive noise in (3.11) has been ignored so far. However, considering the

additive noise in (3.30), we get

Ri = (R · ⇤Ui +Wi) · ⇤Gi

= R0i+W

0i

(3.45)


where W0iis the modified additive noise in the i-th frame. From (3.44) and (3.45), we

can write

R = R0 +W

0 (3.46)

Due to superior performance of non-subsampled shearlet transform (NSST) to capture

the geometric and mathematical properties of an image such as scales, directionality,

elongated shapes and oscillations as described in [48], we attempt to denoise the im-

age R with a hard-thresholding on the NSST co-efficients following a similar method

as described in [49]. Here, W0 is somewhat minimized due to averaging. However,

according to [1], [8], [50], the effect of speckle noise is more pronounced compared to

the additive noise and hence, the SNR for the additive noise can be considered high in

the despeckled image. In this approach, the estimated true image is decomposed into

4 levels with each having 3, 3, 4 and 4 directions. The coarse scales of the NSST co-

efficients are not thresholded, but the finest scale is hard-thresholded using a tunable

low threshold value.

3.3 Post-processing

To match with the characteristics of the display monitor and control the overall bright-

ness of the image, further post-processing like gamma correction [51] is necessary. The

gamma correction [52] of the estimated true image R is done using

I(m,n) =

r(m,n)

max(R)

!�

(3.47)

Finally, to control the image contrast, gray level transformation [53] of the image

I(m,n) is done as

G(m,n) =

8>>>>>>><

>>>>>>>:

0, if G(m,n) < Wlow

(I(m,n)/max(I)�Wlow)(Whigh�Wlow) ,

if Wlow G(m,n) < Whigh

1, otherwise

(3.48)


where Wlow and Whigh are tunable parameters (intended for tuning the contrast of the

image) such that

Wlow < Whigh < 1

The complete framework of the proposed ultrasound image enhancement and recon-

struction process is depicted in Fig. 3.4.

Raw Image

Data

bMCFLMS

along Axial

Direction

bMCFLMS

along Lateral

Direction

Deconvolution

Despeckling

N

ois

e

Est

imati

on

Speckle Noise

Removal

Gamma Correction

&

Gray Level

Transformation

Image

Post-processings

B-mode

Image

True

US Image

Estimation

MS

NE

Figure 3.4: Block diagram of the proposed ultrasound image reconstruction method

from the raw RF data.

Chapter 4

Results

In this section, the efficacy of our proposed framework for high-resolution B-mode im-

age generation is evaluated on both the simulation and in-vivo data. The contents of

the thesis cover: deconvolution, despecking via MSNE, and post-processing for a com-

plete B-mode image generation. However, as the main contribution is the despeckling

algorithm, the simulation study is designed to show the effectiveness of our proposed

method for speckle noise estimation and despeckling only. The convergence of the

algorithm in the presence of additive noise is also shown to justify the use of the pro-

posed constraint. On the other hand, the in-vivo images suffer from low resolution and

speckle noise arising from the physical phenomena related with the US image acquisi-

tion system. Therefore, according to the signal generation models, the in-vivo study

includes the 2-D deconvolution to enhance the resolution as the first phase for all the

methods involved for comparing despeckling performance. Two types of deconvolu-

tion methods, namely– bMCFLMS and cepstrum [54] are investigated. Finaly, post

processing stage is included to make a complete investigation of the high-resolution

B-mode image generation pipeline in a single thesis.

The quality of the despeckled image is compared with those of SRAD (speckle

reducing anisotropic diffusion filter) [38] and OBNLM (optimized Bayesian non-local

means-based filtering) [35] methods. The performance matrices, used in this case, are

SNR (signal-to-noise ratio), PSNR (peak signal-to-noise ratio), SSIM (structural sim-

51

CHAPTER 4. RESULTS 52

ilarity index measure) [55], and NIQE (natural image quality evaluator) [56]. Among

these indices SNR, PSNR, SSIM require reference image and hence, cannot be used

for in-vivo data. In the absence of a reference image, NIQE and visual evaluation are

the only ways to evaluate the performance of the proposed algorithm. On the other

hand, in case of simulation data with the original noiseless image at hand, we attempt

to build the intuition behind different aspects, i.e., the number of image frames to

be chosen, runtime, efficacy in preserving small details, of the proposed despeckling

algorithm.

(f )(e)(d)

(a) (b) (c)

Figure 4.1: Effect of the number of frames in despeckling the modified Shepp-Logan

phantom image using the proposed algorithm: (a) clean phantom, (b) noisy phantom

(� = 0.4), (c)-(f) despeckled using 5,10, 15 and 20 frames, respectively.

4.0.1 Simulation Data

Simuation data were generated using the ‘Modified Shepp-Logan’ phantom available

in MATLAB with size 256 ⇥ 256. In the simulation study, we have investigated the

despeckling efficacy of the proposed framework and have not considered the PSF effect

on the image. Hence, the image was corrupted with speckle noise only as described

in [35] following (3.23) where V(m,n) ⇠ N (0, �2) and as in [57], ⇣ = 0.5 was used.


(a) (b)

(d) (e) (f )

(g)

(c)

(i)

NP

M(d

B)

-22

-20

-18

-16

-14

20 40 60 800Iteration

NP

M(d

B)

-15

-14

-13

-11

-12

20 40 60 800

(h) Iteration

Figure 4.2: Despeckling of Shepp-Logan phantom image corrupted by synthetic speckle

noise. (a) clean phantom, (b) noisy phantom (� = 0.4), (c) true speckle noise in the

5th frame; despeckled image using (d) SRAD, (e) OBNLM, (f) proposed MSNE; (g)

extracted noise from the 5th frame using MSNE; (h) NPM measure between the true

and estimated noise using MSNE without constraint; (i) NPM measure between the

true and estimated noise using MSNE with constraint.

The level of noise was varied by setting � = {0.2; 0.4; 0.8}. At a particular noise level,

the speckle pattern was varied using different ⌫(m,n) patterns with the same distribu-

tion for different frames. Among the frames used for despeckling using our proposed

method, the last frame was despeckled using SRAD and OBNLM for comparison with

the proposed algorithm. Here, the implementation platform used were: CPU: IntelR�

CoreTM i7-8700K, RAM: 32 GB, software: MATLAB R�, The MathWorks, Natick, MA.

Comparing (1.2) and (3.23) in additive noiseless case, we can write the speckle pattern


of the k-th frame as

uk(m,n) = 1 + r�0.5(m,n)⌫k(m,n) (4.1)

Therefore, (4.1) can be used to calculate the true speckle pattern for the simulation data

that can be used to quantify the performance of the proposed despeckling algorithm.

As claimed in Section 3.2.2, increasing the number of image frames in the proposed

speckle pattern estimation algorithm has an impact on the accuracy of estimation. In

what follows, we attempt to establish a suitable frame number that optimally meets

all the assumptions made in Section 3.2.2, consumes less runtime, and produces a

visually pleasant despeckled image. The performance index used in this case is NPM

(normalized projection misalignment) defined as

NPM(q) = 20log10✓k⇢(q)kkUk

◆dB (4.2)

⇢(q) = U� UTU(q)

UT (q)U(q)U(q) (4.3)

A lower value of NPM indicates better estimation of U. From Table 4.1, it is apparent

that increasing the level of noise deteriorates the estimation accuracy of the speckle

pattern. However, it can be improved by around 6 dB for the three different noise lev-

els as mentioned above by increasing the number of image frames from 5 to 20 in the

proposed algorithm. As described in [47], for an accurate estimation using the blind

multichannel algorithm, the channels should not have common zeros. As we introduce

more image frames in the estimation process, the probability of having common zeros

decreases. This in turn improves the speckle estimation accuracy. However, increasing

the number of image frames imply more computational complexity leading to higher

runtime, and at the same time, it causes the violation of the quasi-stationarity assump-

tion for the true image as described earlier in Section 3.2.2. The despeckled images

using different number of frames for noise level � = 0.4 are shown in Fig. 4.1. It is

apparent from this figure that consideration of frame number greater than 10 results

in visually imperceptible change in the despeckled images. Therefore, in Fig. 4.2,

we have used 10 image frames to compare our simulation phantom results with other

algorithms.


Table 4.1: Simulation results on the estimation accuracy in terms of NPM (dB) of

speckle pattern using the proposed method for different noise levels

NPM (dB)

Number of frames Noise level Runtime (sec)

� =0.2 � =0.4 � =0.8

5 �31.65 �25.69 �19.96 0.61

10 �34.69 �28.72 �22.96 2.01

15 �36.41 �30.45 �24.68 3.95

20 �37.64 �31.64 �25.83 6.63

Width (mm)

De

pth

(m

m)

0 10 20 30 40

0

10

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30

40

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pth

(m

m)

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0 10 20 30 40

0

10

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(m

m)

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0 10 20 30 40

0

10

20

30

40

(a) (c)(b)

(d) (e) (f )

Figure 4.3: Deconvolution of ultrasound images using adaptive bMCFLMS algorithm.

(a) Raw RF image, (b) 1-D deconvolved image, (c) 2-D deconvolved image, (d)-(f)

zoomed-in views of image segments of (a)-(c), respectively.

The performnce of the proposed algorithm in comparison to others is illustrated

in Fig. 4.2. From Fig. 4.2(d), we can deduce that SRAD distorts the texture in the

homogeneous region and blurs the small details in the phantom. Again, as evident

from Fig. 4.2(e), although OBNLM is superior in performance compared to SRAD

in preserving texture and edges, it fails to remove speckle noise completely when the


Table 4.2: Performance measures computed for the simulation study with different

noise level (�) using diffetrent despeckling approaches

Methods Noise level SNR PSNR SSIM NIQE

(�) (dB) (dB)

0.2 20.33 24.98 0.9996 7.55

OBNLM 0.4 14.24 19.35 0.9985 11.03

0.8 10.45 15.94 0.9969 13.17

0.2 12.97 18.28 0.9980 7.45

SRAD 0.4 9.32 15.04 0.9954 10.06

0.8 7.89 13.62 0.9935 12.97

0.2 28.88 33.22 0.9999 6.88

Proposed 0.4 24.30 28.79 0.9998 6.21

0.8 21.63 26.23 0.9997 6.10

noise level is high, e.g., � = 0.4. On the other hand, our proposed algorithm (see

Fig. 4.2(f)) shows significant visual improvement in terms of maintaing original tex-

ture, edges and small details compared to the SRAD and OBNLM approaches, and

the despeckled image is visually close to the clean phantom image. Again, the quanti-

tative metrics as presented in Table 4.2 also demonstrate that our proposed algorithm

performs significantly better in terms of quantitative indices SNR, PSNR, SSIM, and

NIQE at different noise levels compared to SRAD and OBNLM. To show the effect of

the zero-lag correlation constraint on the convergence profile of the proposed MSNE

algorithm, we depict in Fig. 4.2(h)-(i) the NPM curve with 20 dB SNR. As can be

seen in 4.2(h), the algorithm misconverges near �16 dB, whereas in 4.2(i), there is no

sign of misconvergence and the constrained MSNE algorithm smoothly convergences to

around NMP= �22.96 dB implying that the zero-lag correlation constraint is effective

in preventing misconvergence.


Table 4.3: Axial and lateral correlation energy for raw RF, 1-D and 2-D deconvolved

data using the b-MCFLMS algorithm.

Data Correlation energy

Axial Lateral

RF 0.0379 0.0501

1-D deconvolved 0.0287 0.0497

2-D deconvolved 0.0279 0.0329

4.0.2 In-Vivo Data

Performance of the proposed complete framework for US B-mode image generation

comprising of deconvolution, despeckling and post-processing, respectively, is evalu-

ated on the in-vivo data, collected using a commercial SonixTOUCH Research (Ultra-

sonix Medical Corporation, Richmond BC, Canada) scanner integrated with a linear

array transducer, L14-5/38, operating at 10 MHz with sampling frequency of 40 MHz.

These data were collected from the patients who appeared for medical examination at

the Medical Centre of Bangladesh University of Engineering and Technology (BUET),

Dhaka, Bangladesh. This study was approved by the Institutional Review Board (IRB),

and prior patient consent was taken.

The performance evaluation of the deconvolution step is done subjectively as elab-

orate performance evaluation of the 1-D bMCFLMS algorithm is already done in our

published work [46]. However, the performance of the complete framework is eval-

uated using two approaches. First, we keep the deconvolution step fixed for all the

despeckling algorithms to be compared with the proposed MADS algorithm and eval-

uate their comparative performance both from visual and quantitative perspectives.

Second, we compare subjectively the image generated using our complete framework

with the data acquiring machine B-mode image. All the in-vivo images shown here

were log compressed and dynamic range was set to 35 dB as described in [46] for display

purpose.


Width (mm)

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(m

m)

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40

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10

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(a) (b) (c)

(d) (e) (f )

Figure 4.4: Despeckling of breast ultrasound image- 1. (a) Deconvolved image, images

obtaind using (b) SRAD, (c) OBNLM, (d) proposed algorithm, (e) machine B-mode

image, (f) estimated speckle pattern of the 5-th frame.

2-D Deconvolution Performance Evaluation on In-Vivo Data

The successive stages of deconvolution offer images with improved and finer texture

as shown in Figs. 4.3(a)-(c) and the zoomed-in views of their marked portion in Figs.

4.3(d)-(f), respectively. The speckle pattern in Fig. 4.3(d) is blurry and highly auto-

correlated in the spatial domain as convolution of point speckle with the US PSF results

in the spreading of the point spatially and thereby, reduces the resolution of the image.

This large spatial coverage of speckle leads to considerable correlation between speckle

noise not only in the same frame but also in the consecutive frames leading to a greater

number of common zeros. To justify our claim, we select an axial and a lateral line

from two consecutive frames along the marked lines in Figs. 4.3(a)-(c) and calculate

the energy of the normalized correlation among two axial lines as well as lateral lines

of two consecutive frames as presented in Table 4.3. Higher value of correlation energy


Width (mm)

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pth

(m

m)

0 10 20 30 40

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10

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30

40

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0 10 20 30 40

0

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40

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0 10 20 30 40

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0 10 20 30 40

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40

(a) (b) (c)

(d) (e) (f )

Figure 4.5: Despeckling of breast ultrasound image- 2. (a) Deconvolved image, images

obtaind using (b) SRAD, (c) OBNLM, (d) proposed algorithm, (e) machine B-mode

image, (f) estimated speckle pattern of the 5-th frame.

indicates higher correlation among the two frames along the axial or the lateral direc-

tion. From Table 4.3, it is evident that the raw RF data of a particular frame is more

correlated with the next frame in both the axial and the lateral directions compared to

those of the 1-D and the 2-D deconvolved data. However, the 1-D deconvolved data has

higher lateral correlation compared to that of the 2-D deconvolved data with nearly

the same axial correlation (see Table 4.3). And in Fig. 4.3(e), the speckle pattern

becomes fiber-like with greater spatial coverage along the lateral direction than the ax-

ial direction leading to higher correlation with the next frame in the lateral direction.

Finally, in Fig. 4.3(f), the lateral correlation is sufficiently removed, and the speckle

pattern becomes randomly distributed and uncorrelated. As the speckles in the final

2-D deconvolved image occupies lesser space, this has additional advantage of reducing

common zeros between the speckle patterns in consecutive frames along with increasing


resolution of the image. Hence, the speckle pattern estimation can be done efficiently

with 5 � 10 number of frames without violating the quasi-stationarity assumption of

the true image.

Despeckling Performance on in-vivo Data

Table 4.4: NIQE measure for images despeckled with different algorithms

OBNLM SRAD Proposed

Image- 1 7.38 7.69 5.16

Image- 2 7.61 8.49 5.32

We have used two in-vivo breast images with one containing a solid mass (RF

image-1) and the other containing a cyst (RF image-2) to validate the performance

of our proposed algorithm. Similar type of post-processing with � = 0.97 in (3.47)

and Wlow = 1e � 2,Whigh = 0.98 in (3.48) were set for all the images despeckled

with different algorithms for illustration purpose. The performance index used here

is NIQE that relies on the deviation from the statistical regularities of distortionless

images to rate an image as defined in [37], [56]. The lower the value of the NIQE

metric, the better the quality of the despeckled image. However, the NIQE index was

measured on the despeckled image without post-processing. As shown in Table 4.4,

our proposed algorithm gives the lowest NIQE score and hence, the best quality image

compared to that of SRAD and OBNLM. Figs. 4.4 and 4.5 are provided for subjective

evaluation of the proposed algorithm. As shown in Figs. 4.4(b) and 4.5(b), SRAD

algorithm succeeds in preserving edges although it blurs the texture and degrades

the contrast of the image. On the other hand, according to Figs. 4.4(c) and 4.5(c)

OBNLM shows superior performance compared to SRAD in preserving undistorted

texture and contrast. However, it fails to remove the speckle noise completely from

the image. To visually compare the performance of our proposed framework to that of

the commercial US image acquiring machine used in this experiment, Figs. 4.4(d)-(e)

and 4.5(d)-(e) are portrayed. From these figures, observe that tissue texture is more


(a) (b)

Figure 4.6: Estimated B-mode image with the deconvolution step as (a) bMCFLMS,

and (b) cepstrum.

prominent in the images provided by our proposed framework compared to those in

the machine B-mode images. To facilitate the observations, significant structures of

the images are marked with arrow and circle in Figs. 4.4(d)-(e) and 4.5(d)-(e) which

show the machine B-mode images have blurred and distorted the tissue structures.

Again, the cyst boundary in Fig. 4.5(d) is sharper and well-defined comapred to that

of Fig. 4.5(e). The estimated speckle noise patterns of the 5-th image frame as shown

in Figs. 4.4(f) and 4.5(f) contain tissue structures that justify the relevance of true

image dependent modeling [35] of speckle pattern as shown in (3.23).

In spite of offering an elegant solution to the speckle removal problem, the proposed

framework has a flaw in its complete pipeline as the deconvolution step is not realtime

implementable requiring 76 minutes in total for a single image of 128 A-lines with each

line having 1040 samples. However, as an alternate approach, the deconvolution step

can be replaced by a time-efficient cepstrum-based deconvolution as described in [54]

with a little cost paid in image quality as evident from higher NIQE score of 5.59 (see

Fig. 4.6 (b)) compared to that of 5.32 (see Fig. 4.6(a)) using the bMCFLMS algorithm,

and this brings down the total runtime to 6.3 seconds. A graphical processing unit

(GPU) based deconvolution technique to be investigatd in future may bring down the

overall B-mode image generation framework into real-time.

Chapter 5

Discussion and Conclusion

5.1 Discussion

In this study, we present a complete framework of signal processing approaches com-

prising of deconvolution, despeckling, gamma correction, and gray level transformation

to produce a high-resolution B-mode image with superior edge and texture from the

raw RF image. The parameters for SRAD and OBNLM algorithms were tuned for

the lowest NIQE score of the despeckled image. While deriving SRAD and OBNLM

algorithms in [38] and [35], respectively, deconvolution of raw RF image to enhance

resolution was not addressed. Hence, introducing deconvolution prior to SRAD and

OBNLM may have resulted in their poor performance. However, although their speckle

removal efficacy may be good without deconvolution, the image resolution will be poor.

In the proposed framework, the performance of the despeckling algorithm (MADS) is

dependent on the number of consecutive image frames to be considered in the MSNE

algorithm. As mentioned earlier, there is a tradeoff between the number of frames that

can be used without violating the quasi-stationarity assumption of the true US image

and the speckle noise estimation accuracy. In our experiment, we observed that five

consecutive image frames are good enough for a visually pleasant B-mode image gener-

ation. Again, the Lagrange multipliers– �1 in the constraint preventing misconvergence

of the MSNE algorithm (see (3.28)) and �2 in the energy constraint of the iterative

62

CHAPTER 5. DISCUSSION AND CONCLUSION 63

despeckling algorithm (see (3.41)) remain effective once set at an optimum level for

a particular US imaging set-up. To make the framework independent of the display

monitor, gamma correction as a post-processing step has been introduced. Again, to

offer the user a tunable contrast adjustment, two parameters Wlow and Whigh have been

used in the gray level transformation step.

In addition to providing a guideline for high-resolution B-mode image generation,

the thesis introduces a method to extract the speckle pattern inherent in a US image.

Despite of being a random process, speckle noise is not devoid of information. Since the

statistics of the speckle depends on the microstructure of the tissue parenchyma, it can

be useful for differentiating between different tissue compositions or types [58], [59].

5.2 Conclusion

This thesis has dealt with a complete framework for high-resolution ultrasound image

reconstruction from raw RF data. The proposed method relies on SIMO models for

both deconvolution and speckle noise estimation, and MISO model for despeckling. The

proposed framework completely follows the signal generation model and sequentially

addresses the issues with US imaging such as low resolution, speckle noise, and additive

noise. In the first step, to enhance the resolution, a 2-D deconvolution technique has

been introduced as an extension of our previously proposed 1-D bMCFLMS algorithm

which is necessary prior to despeckling according to the mathematical model of US

imaging. In the next step, a novel multiframe-based adaptive speckle noise estimation

(MSNE) algorithm estimates the speckle pattern without any a priori information on

the statistics of the image or the noise pattern. Using the estimated speckle pattern, an

energy constrained iterative algorithm estimates the true US image following a MISO

model. As the despeckling procedure is completely based on signal generation model

and does not involve any kind of ad-hoc filtering operation as reported in the literature,

it has resulted in a high quality tissue texture and edges in the image. Finally, gamma

correction and gray level transformation have been done as post-processing to produce

a complete B-mode image. The efficacy of the proposed algorithm has been tested

CHAPTER 5. DISCUSSION AND CONCLUSION 64

both quantitatively and qualitatively on simulation and in-vivo data. The results

have demonstrated the superiority of our proposed despeckling algorithm compared to

SRAD and OBNLM methods. Again, the proposed framework offers B-mode image

with superior texture and image details compared to those provided by a commercial

ultrasound scanner.

As our proposed framework preserves original image features such as texture, details

and edges, it may have a far reaching impact on medical imaging for diagnostic purpose.

At the same time, the proposed despeckling algorithm may be efficacious in dealing with

the speckle noise problem in other imaging such as synthetic aperture radar (SAR) [60]

and optical coherence tomography (OCT) [61].

List of Publications

Journal

1. Jayanta Dey, and M. K. Hasan, “Ultrasonic tissue reflectivity function esti-

mation using correlation constrained multichannel flms algorithm with missing

RF data”, Biomed. Phys. Eng. Express, vol. 4, no. 4, pp. 045024, 2018.

2. Jayanta Dey, Sharmin R. Ara, and M. K. Hasan, “Multiframe-based Adaptive

Despeckling Algorithm for Ultrasound B-mode Imaging with Superior Edge and

Texture,” submitted to IEEE Transactions on Ultrasonics, Ferroelectrics, and

Frequency Control, 2019.

65

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