DENOISING AND ENHANCEMENT OF MEDICAL

IMAGES

By

Ayesha Saadia

A thesis submitted to the faculty of Department of Computer Software Engineering,Military College of Signals, National University of Sciences and Technology,Islamabad, Pakistan, in partial fulfillment of the requirements for the degree of

Doctor of Philosphy in Computer Software Engineering

April 2017

ABSTRACT

Medical imaging captures visual representation of human body’s structural and functional

aspects like tissues, bones, blood flow etc. for clinical analysis and medical intervention.

Optical Imaging, X-ray, Computer tomography, Magnetic Resonance Imaging (MRI), Ul-

trasound etc. are common medical imaging techniques used by physicians. Among them ul-

trasound is the most widely used imaging technique due to its cost effectiveness and human

health friendly characteristic. But ultrasound images are inherently corrupted with speckle

noise and thus makes physician’s interpretation complex and time-consuming. Therefore, in

medical image analysis image denoising has more clinical value since it helps the physicians

to reach correct, reliable and speedy diagnosis by mitigating noise from the image. Im-

age denoising also facilitate image segmentation, image fusion, object detection and target

recognition. Computer-aided image denoising and image enhancement techniques helps to

improve efficiency and accuracy of physician’s interpretation.

This research work focused on the development of reliable image denoising and enhance-

ment techniques for echocardiographic images. It aimed to denoise an echocardiographic

image without introducing noise distortion and loss of information. Fractional calculus has

been used to efficiently mitigate noise of various levels from the echocardiographic image.

Also rough set theory and fuzzy logic have been used to draw boundaries between image

regions. These concepts helped to handle uncertainty caused by the speckle noise. Three

image denoising methods have been proposed in thesis. First proposed denoising method-

ology performs image denoising in two stages. Stage-1 applies weighted fuzzy mean filter

and stage-2 convolves every pixel of the image with a fractional integration filter. Second

proposed approach intelligently selects appropriate filter for every image region. Fractional

order differintegral filter is proposed in third image denoising methodology. All three pro-

posed denoising schemes not only preserve details in the denoised image but also efficiently

reduce noise. Image Enhancement further improves the visual quality of an image. This

research also proposes two echocardiographic image enhancement schemes to effectively

utilize gradient magnitude and eigenvalue hessian matrix calculations and fractional order

derivative concept.

Real echocardiographic b-mode images and standard images artificially corrupted with

speckle noise have been considered in simulations for this research. Visual and quantitative

analysis of simulation results presents significant improvement by the proposed schemes as

compared to state-of-the art image denoising and image enhancement techniques.

ii

DEDICATION

This thesis is dedicated to

MY BELOVED PARENTS

for their love, endless support and encouragement

iii

ACKNOWLEDGEMENTS

In the name of Allah, the Most Gracious and the Most Merciful.

All praises are due to Allah Almighty who has bestowed me with the strength and the passion

to accomplish this thesis and I am thankful to Him for His mercy and benevolence. Without

His consent I could not have indulged myself in this task.

I would like to express my sincere gratitude to my advisor Dr. Adnan Rashdi for his

continuous support throughout my degree, for his patience, motivation, enthusiasm, and

immense research knowledge. He engaged me in new ideas and demanded a high quality

work in all my endeavors. His guidance helped me in all the time of research and writing of

this thesis. May Allah richly bless him with a lot of happiness, success and good health.

I extend my profound thanks to my thesis guidance and evaluation committee members

including Dr. Adil Masood Siddiqui, Dr. Naveed Iqbal, and Dr. Nadeem Abbas Zaidi for

their constant supervision and support.

I am also thankful to Dr. Amir Gillani and Dr. Farhan Tayyab for their guidance and

help. The work presented here would certainly not have been accomplished without their

influence and support.

My special and heartily thanks to my parents, sister and brothers for the endless support

they provided me through my entire life and in particular during my studies. They prayed for

me throughout the time of my research. I must acknowledge their love and encouragement.

May Allah bless them all in this world and the world hereafter.

iv

PUBLICATIONS

The following relevant publications have been produced during PhD period.

1. A. Saadia, A. Rashdi. “Fractional Order Integration and Fuzzy Logic Based Filterfor Denoising of Echocardiographic Image.” Computer Methods and Programs inBiomedicine, vol. 137, pp. 65-75, 2016. Impact Factor 1.867.

2. A. Saadia, A. Rashdi. “A Method for Speckle Noise Removal” accepted with minorrevision in Circuit Systems and Signal Processing, 2017. Impact Factor 1.17

3. A. Saadia, A. Rashdi. “Incorporating Fractional Calculus in Echocardiographic ImageDenoising” accepted with minor revision in Computers and Electrical Engineering,2017. Impact Factor 1.08

4. A. Saadia, A. Rashdi.“Adaptive Image Enhancement Using Fractional Order Deriva-tives” under review in Signal Processing Journal, 2017. Impact Factor 2.00

5. A. Saadia, A. Rashdi.“Echocardiography Image Enhancement Using Adaptive Frac-tional Order Derivatives” published in the proceedings of International Conference ofSignal and Image Processing (ICSIP), Beijing, China, Aug 13-15, pp. 764-769, IEEE,2016.

v

Table of Contents

ABSTRACT ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

PUBLICATIONS v

TABLE OF CONTENTS vi

LIST OF FIGURES x

LIST OF TABLES xiii

ACRONYMS xiv

NOTATIONS xv

1 INTRODUCTION 1

1.1 Motivation and Problem Statement . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Research Objectives and Contributions . . . . . . . . . . . . . . . . . . . . 3

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 REVIEW OF ECHOCARDIOGRAPHIC IMAGING AND FRACTIONAL

CALCULUS 6

2.1 Anatomy of Cardiovascular System . . . . . . . . . . . . . . . . . . . . . 6

2.2 Echocardiographic Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Ultrasound Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Ultrasound Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Denoising of Echocardiographic Images . . . . . . . . . . . . . . . . . . . 11

2.3.1 Adaptive Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 Nonlocal Means Filter . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.3 Diffusion Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.4 Multiscale methods . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.5 Bilateral Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

vi

2.4 Quantitative Measures for Image Denoising . . . . . . . . . . . . . . . . . 14

2.4.1 Speckle Suppression Index (SSI) . . . . . . . . . . . . . . . . . . . 14

2.4.2 Structural Similarity (SSIM) . . . . . . . . . . . . . . . . . . . . . 14

2.4.3 Peak Signal to Noise (PSNR) . . . . . . . . . . . . . . . . . . . . 15

2.4.4 Edge Preservation Index . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.5 Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.1 Definition and Origin of Fractional Calculus . . . . . . . . . . . . . 15

2.5.2 2-D Fractional Integration and Differentiation Filters . . . . . . . . 17

2.5.3 Effect of Fractional Differentials/Integrals On Signals . . . . . . . 18

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 FUZZY LOGIC AND FRACTIONAL INTEGRATION BASED IMAGE DE-

NOISING 24

3.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.1 Fuzzy Weighted Mean Filter . . . . . . . . . . . . . . . . . . . . . 25

3.2.2 Fixed Order Fractional Integration Filter . . . . . . . . . . . . . . . 28

3.3 Simulation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 INTELLIGENT ADAPTIVE IMAGE DENOISING 38

4.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Adaptive Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Simulation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 ROUGHSET THEORY AND FRACTIONAL INTEGRATION BASED IM-

AGE DENOISING 48

5.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2 Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2.1 Image Region Classifier . . . . . . . . . . . . . . . . . . . . . . . 48

5.2.2 Adaptive Fractional Differintegral Filter . . . . . . . . . . . . . . . 50

5.3 Simulations and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

vii

6 ECHOCARDIOGRAPHIC IMAGE ENHANCEMENT USING ADAPTIVE

FRACTIONAL ORDER DERIVATIVE 58

6.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.2 Proposed Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.2.1 Gradient Magnitude Based Adaptive Image Enhancement . . . . . 59

6.2.2 Hessian Matrix Based Adaptive Image Enhancement . . . . . . . . 60

6.3 Simulation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7 CONCLUSION AND FUTURE WORK 68

7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

REFERENCES 69

viii

List of Figures

2.1 Anatomy of a Heart (a) External View (b) Internal View [106] . . . . . . . 7

2.2 Ultrasound Imaging System [107] . . . . . . . . . . . . . . . . . . . . . . 8

2.3 An example of B-Mode imaging [108] . . . . . . . . . . . . . . . . . . . . 9

2.4 An example of M-Mode [109] . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Example of Colour Doppler and Power Doppler [110] . . . . . . . . . . . . 10

2.6 Fractional differential mask in different directions (a) 180◦ (b) 90◦ (c) 135◦

(d) 45◦ (e) 0◦ (f) 315◦ (g) 225◦ (h) 270◦ . . . . . . . . . . . . . . . . . . . 18

2.7 Superposition of 3× 3 fractional integral mask in eight directions . . . . . 18

2.8 Fractional order differential based on G-L definition (a) Original image (b)

Order 0.1(c) Order 0.3 (d) Order 0.5 (e) Order 0.7 (f) Order 0.9 . . . . . . . 19

2.9 Fractional order differential based on R-L definition (a) Original image (b)

Order 0.1(c) Order 0.3 (d) Order 0.5 (e) Order 0.7 (f) Order 0.9 . . . . . . . 20

2.10 Fractional order integration based on G-L definition (a) Noisy image (b)

Order -0.1(c) Order -0.3 (d) Order -0.5 (e) Order -0.7 (f) Order -0.9 . . . . 21

2.11 Fractional order integration based on R-L definition (a) Original image (b)

Order -0.1(c) Order -0.3 (d) Order -0.5 (e) Order -0.7 (f) Order -0.9 . . . . 22

3.1 Fuzzy Membership Function . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Various denoising methods applied on Cameraman image with speckle noise

variance = 1 (a) TV [66] (b) Wavelet [65] (c) Perona et al. [62] (d) GAFIA

[86] (e) IFD [85] (f) Lee [58] (g) Wiener [59] (h) Geometric [64] (i) Nl

means [31] (j) Bilateral [49] (k) Proposed Methodology (l) Noisy Image (m)

Original Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Various denoising methods applied on Lena image with speckle noise vari-

ance = 0.1 (a) TV [66] (b) Wavelet [65] (c) Perona et al. [62] (d) GAFIA

[86] (e) IFD [85] (f) Lee [58] (g) Wiener [59] (h) Geometric [64] (i) Nl

means [31] (j) Bilateral [49] (k) Proposed Methodology (l) Noisy Image (m)

Original Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Residual Images (Noise) of Cameraman obtained after applying (a) TV [66]

(b) Wavelet [65] (c) Perona et al. [62] (d) GAFIA [86] (e) IFD [85] (f) Lee

[58] (g) Wiener [59] (h) Geometric [64] (i) Nl means [31] (j) Bilateral [49]

(k) Proposed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 31

ix

3.5 Residual Images (Noise) of Lena obtained after applying (a) TV [66] (b)

Wavelet [65] (c) Perona et al. [62] (d) GAFIA [86] (e) IFD [85] (f) Lee [58]

(g) Wiener [59] (h) Geometric [64] (i) Nl means [31] (j) Bilateral [49] (k)

Proposed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.6 Various denoising methods applied on an Echocardiographic image (short

axis view of left ventricular) (a) TV [66] (b) Wavelet [65] (c) Perona et

al. [62] (d) GAFIA [86] (e) IFD [85] (f) Lee [58] (g) Wiener [59] (h) Ge-

ometric [64] (i) Nl means [31] (j) Bilateral [49] (k) Proposed Methodology

(l) Original Echocardiographyic Image . . . . . . . . . . . . . . . . . . . . 34

3.7 Various denoising methods applied on an Echocardiographic image (apical

four chamber view) (a) TV [66] (b) Wavelet [65] (c) Perona et al. [62] (d)

GAFIA [86] (e) IFD [85] (f) Lee [58] (g) Wiener [59] (h) Geometric [64]

(i) Nl means [31] (j) Bilateral [49] (k) Proposed Methodology (l) Original

Echocardiographic Image . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.8 Residual Echocardiographic (short axis view of left ventricular) Images

(Noise) obtained after applying (a) TV [66] (b) Wavelet [65] (c) Perona et

al. [62] (d) GAFIA [86] (e) IFD [85] (f) Lee [58] (g) Wiener [59] (h) Geo-

metric [64] (i) Nl means [31] (j) Bilateral [49] (k) Proposed Methodology . 36

3.9 Residual Echocardiographic (apical four chamber view) Images (Noise) ob-

tained after applying (a) TV [66] (b) Wavelet [65] (c) Perona et al. [62] (d)

GAFIA [86] (e) IFD [85] (f) Lee [58] (g) Wiener [59] (h) Geometric [64] (i)

Nl means [31] (j) Bilateral [49] (k) Proposed Methodology . . . . . . . . . 37

4.1 Fuzzy Membership Function . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Improved Fractional Integral Mask [85] . . . . . . . . . . . . . . . . . . . 41

4.3 Denoising results of checkerboard image (a) Hu et al. [83] (b) Huang et

al. [84] (c) IFD [85] (d) GAFIA [86] (e) Lee (f) Kuan (g) Proposed (h)

Noisy image (i) Original Image . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Denoising results cameraman image with noise variance 0.02 (a) Hu et al.

[83] (b) Huang et al. [84] (c) IFD [85] (d) GAFIA [86] (e) Lee (f) Kuan (g)

Proposed (h) Noisy image (i) Original cameraman image . . . . . . . . . . 43

4.5 Denoising results echocardiographic image (a) Hu et al. [83] (b) Huang et

al. [84] (c) IFD [85] (d) GAFIA [86] (e)Lee [58] (f) Kuan [60] (g) Proposed

(h) Noisy image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

x

4.6 Denoising results echocardiography image (a) Hu et al. [83] (b) Huang et

al. [84] (c) IFD [85] (d) GAFIA [86] (e)Lee [58] (f) Kuan [60] (g) Proposed

(h) Noisy image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.1 Rough Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 Various denoising methods applied on Checkerboard image with noise var =

0.1 (a) IFD [85] (b) GAFIA [86] (c) Hu et al. [83] (d) Huang et al. [84] (e)

Lee [58] (f) Kuan [60] (g) Proposed (h) Noisy (i) Original Image . . . . . . 52

5.3 Various denoising methods applied on Lena image with noise var = 0.05

(a) Noisy image (b) Hu et al. [83] (c) Huang et al. [84] (d) IFD [85] (e)

GAFIA [86] (f) Lee [58] (g) Kuan [60](h) Proposed (i) Original Image . . . 53

5.4 Various denoising methods applied on real echo image (short-axis view of

left ventricular) (a) Hu et al. [83] (b) Huang et al. [84] (c) IFD [85](d)

GAFIA [86] (e) Lee [58] (f) Kuan [60](g) Proposed (h) Original Image . . . 55

5.5 Various denoising methods applied on real echo image (Four chamber view

of left ventricular) (a) Hu et al. [83] (b) Huang et al. [84] (c) IFD [85](d)

GAFIA [86] (e) Lee [58] (f) Kuan [60](g) Proposed (h) Original Image . . . 56

6.1 Comparison among different methods for ultrasound image of a heart (Image

1) (a) Original image (b) Proposed method (c) Histogram Equalization (d)

0.1-order (e) 0.5-order (f) 0.9-order fractional differential . . . . . . . . . . 61

6.2 Comparison among different methods for ultrasound image of a heart (Image

2) (a) Original image (b) Proposed method (c) Histogram Equalization (d)

0.1-order (e) 0.5-order (f) 0.9-order fractional differential . . . . . . . . . . 62

6.3 Comparison among different methods for ultrasound image of a heart (a)

Original image (b) Proposed method (c) Histogram Equalization (d) 0.1-

order (e) 0.5-order (f) 0.9-order fractional differential . . . . . . . . . . . . 64

6.4 Comparison among different methods for ultrasound image of an ovary (a)

Original image (b) Proposed method (c) Histogram Equalization (d) 0.1-

order (e) 0.5-order (f) 0.9-order fractional differential . . . . . . . . . . . . 65

xi

List of Tables

3.1 Comparison of Edge Preservation Index (β) for various methods . . . . . . 30

3.2 Comparison of Correlation Coefficient (ρ) for various methods . . . . . . . 31

3.3 Comparison of various methods for Cameraman image with different noise

variances σn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Comparison of various methods for Lena image with noise variance σn . . . 33

3.5 Comparison of SSI for various denoising methods applied on Echocardio-

graphic image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 Quantitative Analysis of Cameraman Image . . . . . . . . . . . . . . . . . 45

4.2 Quantitative Analysis of Checkerboard Image . . . . . . . . . . . . . . . . 47

5.1 Comparison of different methods for Lena image with noise var = 0.05 . . . 53

5.2 Comparison of different methods for Lena image with noise var = 0.1 . . . 54

5.3 Comparison of different methods for Lena image with noise var = 0.5 . . . 54

5.4 Comparison of different methods for Checkerboard image with noise var =

0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.5 Comparison of different methods for Checkerboard image with noise var = 0.1 55

5.6 Comparison of different methods for Checkerboard image with noise var = 0.5 55

6.1 Comparison of Different Methods for Ultrasound of a Heart Image 1 . . . . 66

6.2 Comparison Of Different Methods for Ultrasound of a Heart Image 2 . . . . 66

6.3 Comparison Of Different Methods For Ultrasound of a Heart . . . . . . . . 67

6.4 Comparison of Different Methods for Ultrasound of an Ovary . . . . . . . 67

xii

ACRONYMS

Magnetic Resonance Image MRI

Computer Tomography CT

Non Local Means NL Means

Principal Component Analysis PCA

Mean Square Error MSE

Root Mean Square Error RMSE

Peak Signal to Noise Ratio PSNR

Speckle Suppression Index SSI

Structural Similarity SSIM

Coefficient of Variation CV

Standard Deviation SD

Improved Fractional Differentials IFD

Global Adaptive Fractional Integral Algorithm GAFIA

Total Variation TV

Gradient Magnitude Gmag

xiii

NOTATIONS

ρ Correlation Coefficient

Γ Gamma Function

I Mean of an image I

β Edge Preservation Index

∆(.) High pass filtered version of an image

η Additive Noise

xiv

Chapter 1

INTRODUCTION

Medical imaging is the process to capture visual representations of the structural and func-

tional aspects of the human body like tissues, bones, blood flow etc. for clinical analysis

and medical intervention. Over the years, various medical imaging methods have been de-

veloped including Optical Imaging, X-ray, Computer Tomography (CT or computer-assisted

tomography), Magnetic Resonance Imaging (MRI), Ultrasound, etc.

In X-ray imaging, X-ray tube throw X-ray beam on the human body. While passing

through the body, this beam is attenuated as portions of the X-ray beam get absorbed. On

the other side of the body, an image is formed on a film by capturing these attenuated rays.

X-ray is helpful in diagnosis of breast cancer and inspecting blood vessels and organs. A

contrast agent is injected in the blood which improves visibility of the X-ray image. This

X-ray technique is called Angiography and is usually used to view blood vessels of a heart.

To view real time X-ray image of the patient’s internal organs on a monitor, Fluoroscopy

is used. Fluoroscopy provides real-time examination of the patient’s body. X-ray beam

after attenuation, creates an image on the fluorescent screen. Our body organs and tissues

are sensitive to radiations. While producing X-ray image, special care is taken to use low

radiation dose. Patients who are more frequently exposed to ionized radiations, develop

cancer risk.

CT helps to acquire multiple high resolution images. During CT examination, a tube and

a detector both revolves around the body. CT scanners provide 3D visualization of the area

under examination and give more information than conventional radiography.

MRI is a sophisticated medical imaging technique that produces high quality image. It

uses magnetic fields and radiofrequency to capture detailed image of the body’s internal

organs as cross-sectional images or slices, specially in the examination of brain tumors to

slipped discs. MRI scanners are used to capture image of joints injuries, the blood vessels,

the breast, as well as abdominal and pelvic organs such as the liver or reproductive organs. It

gives details about complete brain structure, excellent contrast detail of soft tissue, gray and

1

white matter in the brain. MRI imaging is not produced using ionizing radiations however

patients with pacemakers and implants might get affected by the strong magnetic field there-

fore they should not undergo an MRI examination. Some patients also have allergic reaction

of contrast agents, if used.

Ultrasound, sometimes also known as sonography, is a useful way of examining many of

the body’s internal organs, including liver, gallbladder, spleen, kidneys etc. Using ultrasound

modality it is possible to visualize underlying anatomy in real time and capture image of

dynamically varying structures within the body which is not possible using conventional

imaging techniques. Unlike X-ray, it captures image of soft tissues. Ultrasound addresses

the limitations of other imaging modalities. It is noninvasive, portable, accurate and low

cost. It offers significant advantages over the other imaging modalities.

1.1 Motivation and Problem Statement

Advantages of ultrasound makes it an ideal imaging tool to be used in many situations.

Ultrasound is gaining more popularity particularly in cardiac imaging. Specific form of

ultrasound that is used to examine the heart is known as echocardiography [1] [2]. It provides

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

Echocardiographic images provide information about the heart size, shape, blood flow rate,

etc. Echocardiography video shows different stages of contraction and relaxation of the

heart. With the increase in application of mobile and portable telemedicine, requirement of

ultrasound scanning instruments for better image processing techniques also increases.

During acquisition and transmission ultrasound images are affected with noise [3]. Noise

is unwanted information present in an image. It alters original information of the image and

disturbs the visual quality of image and brings variation in image intensity. Principle source

of noise in ultrasound images are reflected rays/waves. In ultrasound imaging technique high

frequency sound waves are transmitted through the patient’s body and reflected waves are

gathered to make an image. When ultrasound waves strike body structures back-scatter of

the ultrasound beam occurs. This gives rise to noise. Different types of noise have their own

characteristics, distinguishable from others. Although advancements have been made in the

ultrasound imaging technology but noise is a problem which still needs to be addressed [30].

Denoising techniques are applied on the image to remove such noise without affecting im-

2

portant features of an image. Noise that affects the quality of of ultrasound images is speckle

noise. It gives granular appearance to the image. Speckle noise is modeled as signal depen-

dent noise hence is an inherent property of the ultrasound image itself. It is multiplicative

noise. Besides ultrasound images, speckle noise is also present in other coherent imaging

modalities like synthetic aperture radar (SAR), laser images etc. It not only reduces the res-

olution and contrast of image but also reduces the diagnostic value of ultrasound imaging

modality. For medical ultrasound images, it is important to suppress noise while preserv-

ing important details and structure. Therefore before performing different image processing

tasks like segmentation, registration and fusion etc. speckle should be removed. It improves

diagnostic value of ultrasound imaging modality. Hence denoising becomes a critical pre-

processing step for ultrasound image processing. Aim of this thesis is to develop denoising

and enhancement techniques that aid correct diagnosis of echocardiographic images i.e., to

provide information that helps doctors to evaluate patient’s health. This can be done by

suppressing maximum amount of speckle noise and preserving important details and struc-

ture. A denoised image will provide more accurate information about different structures

and helps in performing other image processing tasks like segmentation, registration, fusion

etc. [4] [5] [6] with more confidence and speed.

1.2 Research Objectives and Contributions

In the presence of speckle noise, filtering process becomes essential preprocessing step for

echocardiographic images. It increases quality and diagnostic value of image. This put

emphasis on efficient despeckle filtering which is to be applied as a preprocessing step of

medical image processing. This study gives overview about different speckle reduction tech-

niques and presents new speckle suppression methods, which are effective in suppressing

noise from an echocardiographic image.

The objective of this thesis is to design algorithms for echocardiograhic image denoising

and enhancement, with improved performance when compared with existing schemes in

terms of:

i) minimising information loss and artifacts

ii) removing maximum amount of noise

iii) enhancing contrast and structure of already denoised image.

3

The contributions are summarized as follows:

1) Development of image denoising techniques for echocardiographic images based on fuzzy

logic and fractional calculus. Fuzzy Logic is used to handle the uncertainty caused by the

speckle noise and fractional integration filter is further incorporated to get a denoised image

[54].

2) An intelligent technique is proposed which selects different filters for denoising every

region of an image. Image is divided into different regions using coefficient of variation.

3) Roughset theory and fractional integration based denoising methodology is developed

which adaptively selects order of fraction integration for each pixel of the image. This

results in good denoising performance and preservation of important structures of echocar-

diographic image.

4) Proposed echocardiographic image enhancement methodologies are based on fractional

order derivatives proved effective. Fractional order derivatives have been proven good for

enhancing image details and contrast. It brings clarity to different structures of an echocar-

diographic image as compared to existing state of the art techniques [55].

Proposed methods are compared with other methods using quantitative and qualitative

evaluation metrics. For evaluation, real echocardiographic images and standard images arti-

ficially corrupted with speckle noise have been considered. It has been shown that the pro-

pose methodologies are superior in quantitative and quantitative terms by removing speckle

and preserving details and edges of the image. Furthermore, image enhancement method-

ologies for echocardiographic images are also a part of this thesis. Image enhancement

techniques are applied on echocardiographic images. Visual and quantitative comparison

proves the effectiveness of proposed algorithms.

1.3 Thesis Outline

This thesis is divided into seven chapters. A brief review of each chapter is given below.

The first chapter gives introduction about the problem definition, thesis objective and its

organization. It describes the motivation behind this research and outcomes of the study.

Second chapter presents anatomy of a cardiovascular system, basics of echocardiographic

imaging and covers review of speckle suppression methodologies. Different quantitative

measures for performance evaluation of image denoising and enhancement are discussed in

4

this chapter. Chapter 2 also explains the theory of fractional calculus and how this theory

has revolutionized the area of image processing.

Third chapter describes two stage denoising methodology [54] which applies fuzzy filter

in the first stage and fractional integration filter in the second stage. This technique preserves

structure and edges during denoising process.

In forth chapter adaptive denoising algorithm based on improved fractional integration

has been included. This intelligent scheme divides pixels of input noisy image in different

image regions and process every pixel of an image according to the properties of that pixel.

This scheme shows good results.

Fifth chapter entitled “Adaptive Denoising Approach Using Roughset Theory and Fuzzy

Logic with Fractional Calculus” discusses denoising algorithms which select suitable frac-

tional order for each image pixel and then convolve the pixel with the fractional order in-

tegration mask. Roughset theory is good for handling uncertainty brought in the image by

speckle noise. This technique has been proved effective to remove noise of various levels

from the image.

Sixth chapter presents two enhancement methodologies for echocardiographic images.

Enhancement brings further improvement in denoised image. First image enhancement al-

gorithm [55] uses gradient magnitude and second algorithm utilizes the concept of hessian

matrix along with fractional order derivatives. These image enhancement methodologies

bring sharpness and clarity in the echcardiographic image.

Seventh chapter concludes the work presented in this thesis and also presents future direc-

tions in the area of despeckling and enhancement of echocardiographic images.

5

Chapter 2

REVIEW OF ECHOCARDIOGRAPHIC IMAGING AND

FRACTIONAL CALCULUS

2.1 Anatomy of Cardiovascular System

Human cadiovascular system or heart is a muscular organ located behind the ribs, between

lungs [9]. It is of fist size. Heart a pumping organ that supplies oxiginated blood to the

whole body through arteries and receives deoxiginated blood through veins. These phases

of heart cycle are known as diastole and systole [10]. Systole is the phase when the muscle

shifts from its totally relaxed state to the completely contract form. Diastole is opposite to

systole. Diastole starts when systole ends. Heart is surrounded by a fluid to protects friction

between heart and other organs surrounding it. This fluid is filled in the pericardial cavity

and its walls are known as pericardium.

The wall of a heart has three layers. Epicardium is the outermost layer of the heart wall,

myocardium is the middle layer and endocardium is the innermost layer. Most of the heart

is made of myocardium.

As illustrated in Fig 2.1, the heart is divided into four parts or chambers namely the left and

right ventricles, and the left and right atria [11]. The left and right ventricles are separated by

the wall and the atria also have an the interatrial septum. Left atrium (LA) and left ventricle

(LV) carries oxiginated blood to the whole body. Right atrium (RA) and right ventricle (RV)

carries deoxiginated blood. Wall of a heart is not evenly thick from all sides. The atria of the

heart pump blood to ventricles so its wall is thin. Contrary to this, the ventricles have a very

thick myocardium to supply blood to far off body organs.

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

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

atrium from its associated ventricle. The left AV valve or mitral valve has two flaps, and

the right AV valve or tricuspid valve has three flaps. Papillary muscles arise from ventric-

ular walls and connect with AV valves via the chordae tendinae to papillary muscles which

6

Figure 2.1: Anatomy of a Heart (a) External View (b) Internal View [106]

emerge from ventricular walls. The operation of valves is determined by the pressure gradi-

ent between the atrium and the ventricle. The semilunar valves, positioned on the pulmonary

artery and the aorta, separate each ventricles from its connected artery and prevent the back-

ward flow from the artery.

2.2 Echocardiographic Imaging

Figure 2.2 shows a block diagram of an ultrasound imaging system. It consist of hand held

transducer and a processing device with display unit. Ultrasound gel placed on patient’s

skin and transducer is brought in contact with the skin to transmit high frequency sound

waves into the body. Waves are reflected back after striking different body structures. The

transducer collects these sound waves which are bounced back, convert them into electric

signals and display an image on the display unit. This process is repeated until whole desired

body area has been covered. Ultrasound images are captured in real-time and so they also

show blood flow and movement of different organs [13].

2.2.1 Ultrasound Physics

An ultrasound uses high frequency sound waves that propagate through a medium by com-

pression and rarefaction [12]. Wavelength of ultrasound waves is dependent on the acoustic

7

Figure 2.2: Ultrasound Imaging System [107]

properties of medium. High frequency sound waves penetrate less than low frequency sound

waves due to attenuation. Ultrasound pulse is the impulse function and received echo pulse

is the impulse response of the system. When ultrasonic waves propagate through a medium,

they either reflect, refract, scatter or attenuate. Scattering refers to the interaction of the ultra-

sound wave with microstructures that are much smaller than its wavelength. Arising from the

spatial arrangement of the scatterers, there are two types of scattering; periodic and diffuse.

Periodic arrangement of scatters results in the coherent scattering, producing periodicity in

the echo spectrum. Scattered waves with random phase result in diffuse scattering. Diffuse

scattering give birth to speckle noise in the image whereas coherent scattering creates clear

light and dark spot [14].

2.2.2 Ultrasound Modes

Various scanning modes are available including A-mode, B-mode, M-mode, spectral ultra-

sound, color-coded ultrasound, and power doppler ultrasound. These modes are described

below:

A-mode

A-mode denotes amplitude mode scanning. In this mode, reflected echo is plotted as a con-

tinuous 1-D signal. In A-mode, strong reflections represents increase in the signal amplitude.

Limitation of A-mode is that it is 1-D and displays limited information. Nowadays it is only

8

Figure 2.3: An example of B-Mode imaging [108]

used in ophthalmology to perform very accurate measurements of distance.

B-mode

B-mode is the electronic conversion of the A-mode. It refers to the brightness mode. In

the field of biomedical imaging, the ultrasound B-Scan images are used for diagnosis of

different diseases. In B-mode, a line of brightness-modulated dots is displayed, and the

line represents the orientation of the transducer. The ultrasound transducer transmit sound

pulses into the body and receive the reflected sound waves to form an image. Reflected

echoes are displayed as a 2D grayscale image as shown in Fig. 2.3. Intensity value of each

pixel is determined by the amplitude of the returning echoes. Pixels of ultrasound image are

represented with different gray shades [8]. The brightness of the pixel represents the strength

of the returning echo. White shade of a pixel represent strong reflections, e.g., the reflection

caused by diaphragm, gallstones and bones; grey shade denotes weaker reflections, e.g.,

solid organs and thick fluid; and black shade indicates no reflection, e.g., fluid within a cyst.

The image is constructed vertically one line at a time. In this study we will consider B-Mode

images for simulations.

M-mode

M-mode is valuable for studying the movement of structures within the heart, such as valves

and the ventricular walls. M-mode is basically an A-scan plotted against time. It is used to

9

Figure 2.4: An example of M-Mode [109]

record accurate timing of vascular motion. A line is drawn over a structure of interest (Fig.

2.4) to move toward or away from the probe over time.

Doppler Ultrasound

Doppler ultrasound shows blood flow in relation to the anatomy. There are three types of

doppler ultrasound:

Figure 2.5: Example of Colour Doppler and Power Doppler [110]

10

Color Doppler

In color doppler, a color map is superimposed on grey-scale ultrasound image. It combines

anatomical information with velocity information to generate color-coded maps. These color

maps show the speed and direction of blood flow. Computer converts Doppler measurements

into an array of colors which is displayed on the screen. Color Doppler image is helpful in

assessing abnormalities of blood flows qualitatively and quantitatively.

Power Doppler

Power Doppler provides greater detail as compared to color doppler by showing direction of

blood flow. It is capable of capturing minimal or little blood flow.

Spectral Doppler

Unlike color doppler it displays flow velocities recorded over time. It also produce sound

for blood flow information. This sound is heard with every heartbeat.

2.3 Denoising of Echocardiographic Images

Scattering of reflected waves produces speckle noise in the echocardiographic image [7].

Speckle noise covers the underlying structure and disturbs its clearity [25] [13]. Speckle

is a multiplicative noise. Instead of being added to the intensity value of image pixels, it

is being multiplied. It follows gamma distribution [14]. It is difficult to remove speckle

and retain important information present in an image [29]. A number of techniques have

been presented so far for reducing or eliminating speckle [29] [30]. Different approaches to

reduce speckle noise are discussed below:

2.3.1 Adaptive Filters

Adaptive filters rely on local statistics of image region [57] [56]. These filters performs

different level of filtering at each image location. The Lee filter [58], Kuan filter [60], Frost

filter [59] and median filter [61] are adaptive filters initially found applications in synthetic

aperture radar (SAR). Lee, Kuan and Frost filters are designed to utilize first order statistics

such as the variance and mean of the neighborhood. Median filter is one of the simplest

nonlinear filter. It replaces every pixel of the image by the median of intensity values of

pixels in a specific neighborhood. Enhanced versions of these filters are also proposed [26]

11

[27] [28]. Adaptive methods are based on local statistics therefore these filters suppress noise

from smooth regions only and fail to give good performance in areas near edge regions [29].

2.3.2 Nonlocal Means Filter

Ultrasound image is composed of different regions which have same properties. Nonlocal

means filter [31] takes advantage of high degree of redundancy present in present in ultra-

sound images. In other words, any small group of pixels has many similar groups in the

same image. Nonlocal means (Nl means) filter estimates the value of each pixel as the

average of all pixels having similar intensity as of central pixel in a window [32]. Similar-

ity can be defined in intensity domain or in frequency domain. Different variations of this

techniques have been proposed by incorporating weights before taking average. Weight is

calculated according to the Euclidean distance between similar pixels [31] or using using

principal components analysis (PCA) [33]. Weight defined in this way improves the perfor-

mance of filter. Nl means based technique can also give good results if speckle parameters

are estimated first and then apply nl means method to retrieve true value of a pixel [34].

Nl means has also been extended to denoise series of image frames [35]. It denoise many

frames simultaneously hence reducing time complexity. Main drawback of nl means is noise

variance. It is highly sensitive to noise. This method rely on similarity of pixels, so noise

can effect its performance. As the noise variance increases, it brings down the performance

of the method.

2.3.3 Diffusion Filtering

Anisotropic diffusion is an efficient, nonlinear denoising technique which not only reduces

noise but also performs contrast enhancement at the same time [38]- [41]. Anisotropic diffu-

sion [62] is heat equation based speckle reduction technique. A diffusion function controls

the level of denoising in various image areas. It allows smoothing in homogeneous areas

and prohibit across image edges. This method is applied directly to logarithmic-compressed

images as it does not require any information from image power spectrum [39]. It performs

smoothing in the direction of highest gradient values i.e. perpendicular to the edges and

and ignores smoothing in edge areas [40]. The choice of diffusion coefficient can affect the

performance of the filter [63]. Ref. [37] proposed detail preserving anisotropic diffusion.

This approach is computationally complex and it fails to remove noise from the edges by

12

performing diffusion in smooth region only and ignoring filtering from edges.

2.3.4 Multiscale methods

Multiscale methods include both pyramid and wavelets techniques. Image pyramids are

formed reducing the resolution and thus size of image in each level [42] [43]. The origi-

nal image is decomposed into a pyramid structure such that the finer details or low scale

information is contained within the high resolution pyramid levels while the coarse reso-

lution pyramid levels encompass the large scale information. The advantage of denoising

images in their pyramid representation is that the image features of different sizes are con-

sidered separately during the denoising process. Wavelet techniques uses soft thresholding

or hard thresholding methods [45]. Wavelet decomposition process is used to transform

the input signals into the wavelet representation [46]. The image is broke up into different

sub-band signals [44]. However, as opposed to pyramid based methods, the orientation as

well as scale is considered during the wavelet decomposition process. In every new level

of a pyramid, the image is reduced by a factor of 2 in both horizontal and vertical direc-

tions. The horizontal, vertical and diagonal components are calculated at each level in the

wavelet pyramid representation. The saliency of information is represented by the size of

the pyramid coefficients which corresponds to contrast at that particular scale in the original

signal. Moreover, wavelet representation has the added advantage of being compact i.e. all

sub-bands are same in size as that of original image. Noise removal in the wavelet domain

is also performed by modeling wavelet coefficients using Bayesian distribution. However,

these schemes often introduce blocking effects in the denoised image. Wavelet does not

possess the shift-invariance property. The transform coefficients may completely change by

a slight shift of the input signal, which is undesirable and produces distortion in the output

image in case of noise. Computational complexity is another major issue. [46] [47] [48]

2.3.5 Bilateral Filter

Bilateral filtering was developed by Tomasi and Manduchi as a new denoising technique

[49]. It is a non-linear filter and replaces every pixel of the image by weighted average of

the input [50]. In bilateral filter weight is adjusted by two functions defined as spatial and

intensity domain functions. It not only takes into account the close relationship between

space, but also takes into account the gray similar relationship. Weight of spatial function

13

increases with decrease in the spacial distance between pixels and the weight of a pixel

depends also on another function, which decreases the weight of pixels with large intensity

differences and vice versa [51]- [53].

2.4 Quantitative Measures for Image Denoising

Quantitative analysis is used to evaluate the performance of different schemes in terms of

some quantitative statistics. These measures are discussed below.

2.4.1 Speckle Suppression Index (SSI)

Speckle Suppression Index (SSI) is defined as:

SSI =σUµU

.µIσI

(2.1)

where µI and µU are the mean of noisy and filtered image respectively. σI and σU are stan-

dard deviation of noisy and filtered image. SSI measures the speckle suppression capability

of a filter. Value of SSI is usually less than 1. Less value of SSI shows more noise removing

capability of a filter.

2.4.2 Structural Similarity (SSIM)

Structure Similarity (SSIM) index is defined as:

SSIM =(2µOµU + c1)(2σOU + c2)

(µ2Oµ

2U + c1)(σ2

O + σ2U + c2)

(2.2)

where µO and µU are mean of original noise free image and filtered image respectively, σ2O

and σ2U are variance of original image and filtered image respectively, σOU is the co-variance

of two images. c1 and c2 are constants defined as:

c1 = Lk1 (2.3)

c2 = Lk2 (2.4)

where L = 255, k1 and k2 are 0.01 and 0.03 respectively. SSIM shows how much structure

is preserved in the noise removal process. Its value is between 0 and 1, 0 shows no similarity

and 1 shows full similarity between the original reference image and the denoised image.

14

2.4.3 Peak Signal to Noise (PSNR)

PSNR is defined as:

PSNR = 10log10(2552

MSE) (2.5)

where MSE is the mean square error between original image and filtered image.

2.4.4 Edge Preservation Index

Edge Preservation Index (β) is defined as:

β =Γ(∆O −∆O,∆U −∆U)√

Γ(∆O −∆O,∆O −∆O).Γ(∆U −∆U,∆U −∆U)(2.6)

where ∆O and ∆U are the high pass filtered version of original noise free reference image

O and denoised image U , ∆O and ∆U are mean of images ∆O and ∆U and

Γ(I1, I2) = Σi,j(I1(i, j).I2(i, j)) (2.7)

β is equal to 1 for perfect edge preservation. Edge preservation is very important in medical

image denoising.

2.4.5 Correlation Coefficient

Correlation Coefficient (ρ) is defined as:

ρ =

∑M

∑N

(OMN −O)(UMN − U)√(∑M

∑N

OMN −O)2(∑M

∑N

UMN − U)2(2.8)

where O and U are mean of original and denoised image respectively.

2.5 Fractional Calculus

In this study various algorithms for denoising and enhancement of images have been pre-

sented. These algorithms are based on fractional calculus. This section will give an insight

into the theory of fractional calculus.

2.5.1 Definition and Origin of Fractional Calculus

Fractional Calculus was emerged almost 300 years ago. On September 30th, 1695,

L′Hopital wrote a letter to Leibniz and asked what the result would be if Leibniz′s no-

15

tation is used to find derivative of fractional order. Leibniz replied that “It would be an

apparent paradox, from which one day useful consequences will be drawn”. After this dis-

cussion, many mathematicians start working on the concept of fractional calculus and de-

rived equations for it. Concept of fractional calculus has been applied to different real world

applications [15] [16]. Fractional calculus (integral and differential operators) has recently

found its applications in signal and image possessing. Image processing applications include

denoising, super resolution and enhancement applications. Fractional calculus is capable of

improving image quality by enhancing and denoising an image [19] [82] .

Many definitions of fractional calculus have been proposed by different mathematicians

so far [20]- [22]. These mathematicians include Euler, Laplace, Fourier, Lacroix, Abel,

Riemann and Liouville., they presented their own notation and approach that describe the

idea of a fractional order integral or derivative. For any square integrable signal s(t) ∈

L2(R), its v−order fractional differential is:

Dvsn(t) =dvs(t)

dtv(2.9)

Fractional differential is inverse operation of fractional integral. Based on fractional order

operator theory, I = D′, I denotes fractional integral operator.

Well known definition of fractional order differential and integral are given below. Rie-

mann Liouvile definition is:

aDνb s(t) =

1

Γ(n− ν)(dn

dtn

∫ b

a

(t− τ)(m− v − 1)s(τ)dt, (n− 1 ≤ ν < n) (2.10)

Caputo proposed a definition for fractional calculus in 1967:

aDνb s(t) =

1

Γ(n− ν)

∫ b

a

sn(τ)dτ

(t− τ)ν−a+1, (n− 1 ≤ ν < n) (2.11)

Grunwald Letnikov (GL) definition is one of the basic definitions of the fractional order

differentiation. It involves no direct use of the ordinary derivative or integrals which gives it

an advantage over the other definitions of fractional calculus.

The first, second and third order derivatives of function s(t) are obtained by L′Hospital′s

16

rule:

s′(t) = limh→0

s(t+ h)− s(t)h

s′′(t) = [s′(t)]′ limh→0

s(t+ 2h)− 2s(t+ h) + s(t)

h

s′′′(t) = [s′′(t)]′ limh→0

s(t+ 3h)− 3s(t+ 2h) + 3s(t+ h)− s(t)h

(2.12)

The v− order derivative (v ∈ N) of function s(t) is obtained by mathematical induction:

sν(t) = limh→0

h−νν∑j=0

(−1)j(νj

)s(t− jh) (2.13)

The fractional order is generated with the help of a gamma function. Considering function

s(t) has (v+ 1)− order derivatives on the interval [a, b], the v− order fractional differential

of function f(t) is defined as:

aDνb s(t) = lim

h→0h−v

[(b− a)

h

]∑j=0

(−1)j(νj

)s(t− jh) (2.14)

where the integer part of(b− a)

his[(b− a)

h

]and

(vj

)= ν!

j!(v−j)! is binomial coefficient.

2.5.2 2-D Fractional Integration and Differentiation Filters

Suppose the duration of a unitary signal s(t) is t ∈ [a, b], divide this signal into equal intervals

by taking interval size h = 1, then n = [(b − a)/h] = [b − a], the difference of s(t) is

expressed as:

dvs(t)

dtv≈ s(t) + (−v)s(t− 1) +

(−v)(v + 1)

2

s(t− 2) + . . .+Γ(−v + 1)

m!Γ(−ν +m+ 1)s(t−m)

(2.15)

Equation (2.15) can be extended for a two dimensional function and the fractional differ-

ential on x and y axis is given by:

dvs(x, y)

dxv≈ s(x, y) + (−v)s(x− 1, y) +

(−v)(v + 1)

2

f(x− 2, y) + . . .+Γ(−v + 1)

m!Γ(−v +m+ 1)s(x−m, y)

(2.16)

17

dvs(x, y)

dyv≈ s(x, y) + (−ν)s(x, y − 1) +

(−ν)(ν + 1)

2

s(x, y − 2) + . . .+Γ(−ν + 1)

m!Γ(−ν +m+ 1)s(x, y −m)

(2.17)

3 × 3 mask based on first three non-zero terms of (2.15) is obtained (Fig. 2.6) as explained

in [82].

Figure 2.6: Fractional differential mask in different directions (a) 180◦ (b) 90◦ (c) 135◦ (d)45◦ (e) 0◦ (f) 315◦ (g) 225◦ (h) 270◦

ν2−ν2 0 ν2−ν

2 0 ν2−ν2

0 -ν -ν -ν 0ν2−ν2 -ν 8 -ν ν2−ν

2

0 -ν -ν -ν 0ν2−ν2 0 ν2−ν

2 0 ν2−ν2

Figure 2.7: Superposition of 3× 3 fractional integral mask in eight directions

Apply this fractional integration mask in eight directions (0◦, 45◦, 90◦, 135◦, 180◦, 225◦,

270◦ and 315◦) and add them to obtain a 5× 5 mask as shown in Fig 2.7.

2.5.3 Effect of Fractional Differentials/Integrals On Signals

Frequency response of the fractional differential operation describes that with the increasing

order, fractional differential enhances high frequency part of a signal and preserves the low

18

(a) (b)

(c) (d)

(e) (f)

Figure 2.8: Fractional order differential based on G-L definition (a) Original image (b) Order0.1(c) Order 0.3 (d) Order 0.5 (e) Order 0.7 (f) Order 0.9

frequency portion to some extent. On the other hand, fractional integral operator of higher

order attenuates the high frequency portion of a signal [86]. Therefore, fractional integral

act as denoising filter as it weakens high frequency component such as noise, whereas frac-

tional differential enhances high frequency component i.e. image edges and preserves low

frequency part such as texture [86].

19

Results obtained by applying R − L and G − L differential masks on Lena image are

shown in Fig. 2.8 and Fig. 2.9 From these results we can see that fractional order differ-

entials enhance images and with the increase in order high frequency components increases

more. Comparing results of R − L and G − L, R − L gives low contrast images whereas

image enhancement capability of G − L is appreciateable. Comparison of fractional order

(a) (b)

(c) (d)

(e) (f)

Figure 2.9: Fractional order differential based on R-L definition (a) Original image (b) Order0.1(c) Order 0.3 (d) Order 0.5 (e) Order 0.7 (f) Order 0.9

20

(a) (b)

(c) (d)

(e) (f)

Figure 2.10: Fractional order integration based on G-L definition (a) Noisy image (b) Order-0.1(c) Order -0.3 (d) Order -0.5 (e) Order -0.7 (f) Order -0.9

integration results for R − L and G − L is given in Fig. 2.10 and Fig. 2.11. Fractional

order integration mask when applied to an image gives image denoising effect. Results of

Fig. 2.10 shows that G − L fractional integration gives better performance as compared to

R − L (Fig 2.11). In this thesis we will use G − L fractional order differentials to develop

image enhancement algorithms and G − L fractional order integrals will be used in image

21

(a) (b)

(c) (d)

(e) (f)

Figure 2.11: Fractional order integration based on R-L definition (a) Original image (b)Order -0.1(c) Order -0.3 (d) Order -0.5 (e) Order -0.7 (f) Order -0.9

denoising applications.

2.6 Conclusion

In this chapter, basic anatomy of a heart has been discussed. Heart is the central organ of hu-

man circulatory system. Ultrasound imagery is used to capture image of a heart. This image

is helpful for cardiologists to diagnose various diseases. Their decision is greatly affected

22

by the quality of echocardiographic image. A noisy or unclear image may lead to wrong

diagnosis. Therefore image denoising and enhancement is an essential preprocessing step

for echocardiographic images. Various image denoising filters have been proposed so far

including adaptive, nonlocal means, diffusion, multiscale and bilateral filters. Performance

of a filter is assessed on qualitative and quantitative basis. Different quantitative measures

used during this study have been discussed as a pet of this chapter. All algorithms presented

in this thesis are based on fractional calculus. Therefore, history and definition of fraction

calculus is included in this chapter to provide necessary background for understanding this

emerging concept.

23

Chapter 3

FUZZY LOGIC AND FRACTIONAL INTEGRATION BASED

IMAGE DENOISING

3.1 Preliminary

Presence of speckle noise in ultrasound images, makes it difficult for a physician to analyze

the image correctly. Hence despeckling is an essential pre-processing step. In this chapter

a denoising methodology has been proposed and results are presented to show its effective-

ness. Many denoising techniques have been proposed so far. Sometimes of them combine

two methodologies to form a single methodology. This resultant methodology is called hy-

brid techniques. In hybrid technique, shortcoming of one technique is addressed by the use

of second technique. So combination of techniques helps in achieving better results. One

such technique is proposed in [69], which combines Principal Component Analysis (PCA)

and non-local means based method to obtain good denoising results. Another hybrid de-

noising algorithm is presented in [70] which is a combination of wiener filter and Bayesian

estimator. An iterative method based on fuzzy sub pixel fractional partial difference has

been explored in [89]. Application of this technique on noise corrupted image brings en-

hancement in image contrast but it is computationally complex method. Multiresolution

speckle reduction methodology has been proposed in [72]. This method makes an image

pyramid and apply generalized principal component analysis on each level of pyramid to

get a denoised image. Pyramid construction increases complexity of this method. Fuzzy

logic is good to address the problem of speckle noise. Fuzzy average filter [74] perform

significant denoising by replacing each pixel by average of its neighboring pixels but this

method results in poor edge preservation. Another fuzzy filter is proposed in [94], it works

by dividing an input image into different regions and then search similar image patches

to denoise them simultaneously. Denoising method for ultrasound images based on fuzzy

logic [76] is a good contribution towards medical image denoising but high noise variance

can effect the performance of a filter.

24

Considering the effectiveness of hybrid techniques we present a hybrid denoising method-

ology [54]. Proposed denoising filter combines properties of fuzzy logic filter and fractional

integration filter. Details of the methodology has been discussed in next section.

3.2 Proposed Algorithm

Speckle noise affects the quality of ultrasound image. Speckle noise changes intensity value

of pixels and the variation appears as granular noise pattern in echocardiographic images.

Speckle noise is a multiplicative noise. Speckle noise and signal are statistically indepen-

dent. Speckle noise follows Gamma distribution. To improve diagnostic value of ultrasound

image speckle noise must be removed. If I(x, y) is the noisy ultrasound image with size

A×B, we can write it as:

I(x, y) = O(x, y).N(x, y) + η(x, y) (3.1)

where O(x, y) is the noise-free image, N(x, y) and η(x, y) represent the multiplicative

speckle noise and additive noise respectively. Effect of multiplicative noise is more dom-

inant, therefore (3.1) can be written as:

I(x, y) = O(x, y).N(x, y) (3.2)

Purpose of denoising methodology is to retrieve imageO when given image I by eliminating

N fromO. The proposed algorithm takes image I as input image and denoise it in two stages.

In stage-1, it utilizes fuzzy logic [78] [79] to assign weights to eight nearest neighbors of a

pixel and then computes weighted mean of all pixels in a window. This weighted mean

value is assigned to the center pixel of a window. In stage-2, to further improve speckle

suppression fractional integration filter is applied on the outcome of stage-1. Proposed two

stages based method has the capability to remove noise and retain structure of an image.

3.2.1 Fuzzy Weighted Mean Filter

An ultrasound image has different regions namely homogeneous, detail and edge region.

Each region has its distinguished characteristics. These characteristics are represented by

different intensity values. Therefore, we can utilize intensity value or more precisely, inten-

sity difference between adjacent pixels to distinguish pixels of different regions and noise.

25

Algorithm 1 Proposed Fuzzy Logic and Fractional Integration Based Image DenoisingRequire: Input image I .

1: Start of Stage:1 (Fuzzy Weighted Mean Filter)2: Define fuzzy set T

T = {LESS,MORE}3: Let J be the set describing intensity value difference in the input image I .4: Calculate fuzzy membership of J as:

µT =

µLESS = 1 J ≤ 40

µLESS = 170−J170−40 40 < J < 170

µMORE = J−40170−40 40 < J < 170

µMORE = 1 170 ≤ J ≤ 255

5: Calculate weight w(i, j) for each pixel of I

w(i, j) =

µLESS µLESS ≥ µMORE

1− µMORE µLESS ≤ µMORE

0.1 µMORE = 1

6: Calculate resultant image R of stage:1

R(x, y) =∑Q

i=−Q

∑Qj=−Q w(i,j)×I(x+i,y+j)∑Q

i=−Q

∑Qj=−Q w(i,j)

7: End of Stage:18: Start of Stage:2 (Fractional Order Integration Filter)9: Apply fractional integration mask with v = −0.9 on R

U = R ∗mask10: End of Stage:211: Output denoised image U

Pixels belonging to the same image region has less intensity difference whereas more inten-

sity difference between adjacent pixels occur if both pixels belong to different regions such

as edge and smooth area. This large difference may also be due to the presence of noise.

Intensity variation is a subjective term and is difficult to describe therefore we have used the

concept of fuzzy sets to address vague and overlapped concepts.

When multiplicative noise affects a pixel, the effect of noise is also transferred to the neigh-

boring pixels. Noise either decrease or increase the intensity value of a pixel. Fuzziness

caused by the presence of speckle noise can be dealt using fuzzy logic theory which helps to

find the most appropriate output in a situation where information is ambiguous and impre-

cise.

In the proposed technique, each pixel is processed by taking into account 3 × 3 pixels win-

dow around it. Instead of applying simple mean filter, if we assign weights by considering

intensity difference between pixel being processed and all of its neighboring pixels, we can

achieve good results. If large difference occurs between intensity values of neighborhood

26

Figure 3.1: Fuzzy Membership Function

pixel and center pixel then the neighboring pixel belongs to different image region otherwise

we can say that center pixel and neighborhood pixel, both belong to the same image region.

In the proposed weighted mean filter, weights are calculated with the help of fuzzy function

which automatically adjust weights by looking at the intensity value of each pixel in a win-

dow. As intensity difference is a subjective term and it cannot be described with the help

of a crisp set so fuzzy functions are very useful here. A fuzzy set which describes intensity

variation is composed of two functions namely LESS and MORE.

In an image with intensity values 0− 255, minimum absolute difference between two pixels

will be 0 and maximum possible difference will be 255. Let J be the set describing intensity

value difference in the input image I and T is the set of fuzzy terms defining different in-

tensity levels, T = {LESS,MORE}. From Fig. 3.1, it is clear that fuzzy function LESS

assigns highest membership value to pixels with intensity difference less than 40 and then

gradually reduces the membership value with the increase in intensity difference. Similarly,

fuzzy functionMORE assigns highest membership value to pixels with more difference and

it gradually keep on decreasing membership value as the difference decreases. Membership

degree can be expressed by a mathematical function that assigns each element of input set J

a membership degree between 0 and 1. 0 shows no membership, 1 shows full membership

and fuzzy values in between 0 and 1 show the strength of membership to a particular fuzzy

function. Each pixel of a window is then assigned weight according to its membership de-

gree with the fuzzy functions.

27

In Fig. 3.1, it has shown that intensity values between 0 - 40 show full membership to fuzzy

function LESS and therefore have membership value µLESS = 1 and values from 170 -

255 have µMORE = 1. In the overlapped portion, a pixel belongs to both fuzzy functions

simultaneously. To break this tie we used mathematical Maximum function. Membership

value for each pixel is calculated using (3.3).

µT =

µLESS = 1 J ≤ 40

µLESS = 170−J170−40 40 < J < 170

µMORE = J−40170−40 40 < J < 170

µMORE = 1 170 ≤ J ≤ 255

(3.3)

where µLESS and µMORE show membership value of a pixel to fuzzy functions LESS and

MORE respectively.

The corresponding weight w(i, j) of a pixel is calculated as defined in (3.4)

w(i, j) =

µLESS µLESS ≥ µMORE

1− µMORE µLESS ≤ µMORE

0.1 µMORE = 1

(3.4)

After assigning weights to all pixels present in a window, we replace the intensity value of a

central pixel with the weighted mean value calculated as follows:

R(x, y) =

∑Qi=−Q

∑Qj=−Qw(i, j)× I(x+ i, y + j)∑Qi=−Q

∑Qj=−Qw(i, j)

(3.5)

whereR(x, y) is the output image of stage-1 andw(i, j) is the weight coefficient, I(x+i, y+

j) is the corresponding pixel in a local window of size (2Q+ 1)× (2Q+ 1), i, j ∈ [−Q,Q],

in this paper we have considered Q = 1.

3.2.2 Fixed Order Fractional Integration Filter

In second stage of the proposed algorithm, to further suppress speckle noise, fractional filter

is applied on image R. Fractional order integral filter shown in Fig. 2.7 has been used here.

Convolve the resultant image R of stage-1 with mask given in Fig. 2.7 by putting ν = −0.9.

This value of ν has been tested on different speckle noise corrupted images and proved

effective in noise suppression.

U = R ∗mask (3.6)

28

where * is the convolution operator and U is the denoised image.

3.3 Simulation and Results

The proposed method is applied on different images and simulation results are presented.

Comparison is done with different state-of-the-art methods including Lee [58], Wiener [59],

Geometric filter [64], Bilateral [49], Nl means [31], Perona et al. [62], Total variation

(TV) [66], Wavelet [65], GAFIA [86] and IFD [85]. For quantitative analysis, performance

of various filters is compared in terms of Peak Signal to Noise Ratio (PSNR), Speckle Sup-

pression Index (SSI), Structural Similarity (SSIM), Edge Preservation Index (β) and Corre-

lation Coefficient (ρ).

To test effectiveness of the proposed methodology, a noisy Lena image and a Cameraman

image has been considered. These images have been corrupted with artificial speckle noise.

Figure 3.2: Various denoising methods applied on Cameraman image with speckle noisevariance = 1 (a) TV [66] (b) Wavelet [65] (c) Perona et al. [62] (d) GAFIA [86] (e) IFD [85](f) Lee [58] (g) Wiener [59] (h) Geometric [64] (i) Nl means [31] (j) Bilateral [49] (k)Proposed Methodology (l) Noisy Image (m) Original Image

29

Figure 3.3: Various denoising methods applied on Lena image with speckle noise variance= 0.1 (a) TV [66] (b) Wavelet [65] (c) Perona et al. [62] (d) GAFIA [86] (e) IFD [85] (f)Lee [58] (g) Wiener [59] (h) Geometric [64] (i) Nl means [31] (j) Bilateral [49] (k) ProposedMethodology (l) Noisy Image (m) Original Image

Simulations have also been performed considering real echocardiographic images.

Various echocardiographic images acquired in multiple views have been considered.

These echocardiographic images have been obtained from [90]. Results for left ventricu-

lar (short axis view) and apical (four chambers) view of a heart have been presented in this

chapter.

Fig. 3.2 and Fig. 3.3 shows qualitative results of Cameraman and Lena images. Quali-

tative results of real echocardiographic images have been shown in Fig. 3.6 and Fig. 3.7.

Residual images (Noise) obtained after applying different denoising methodologies for Cam-

Table 3.1: Comparison of Edge Preservation Index (β) for various methods

Image Lee Wiener Geometric Nl means Bilateral Wavelet Perona et al. TV GAFIA IFD ProposedCameraman 0.3986 0.3847 0.5288 0.4315 0.4220 0.3856 0.4204 0.5203 0.4461 0.4266 0.5719

Lena 0.4739 0.3325 0.4268 0.5045 0.4499 0.4314 0.4612 0.5786 0.4713 0.4609 0.6579

30

Figure 3.4: Residual Images (Noise) of Cameraman obtained after applying (a) TV [66] (b)Wavelet [65] (c) Perona et al. [62] (d) GAFIA [86] (e) IFD [85] (f) Lee [58] (g) Wiener [59](h) Geometric [64] (i) Nl means [31] (j) Bilateral [49] (k) Proposed Methodology

Table 3.2: Comparison of Correlation Coefficient (ρ) for various methods

Image Lee Wiener Geometric Nl means Bilateral Wavelet Perona et al. TV GAFIA IFD ProposedCameraman 0.8098 0.8988 0.8843 0.8911 0.7379 0.8745 0.8103 0.8990 0.7578 0.7674 0.9392

Lena 0.8193 0.9128 0.8795 0.9367 0.7730 0.9066 0.8879 0.9247 0.7586 0.7895 0.9464

31

Table 3.3: Comparison of various methods for Cameraman image with different noise vari-ances σn

σn = 0.2 σn = 0.5 σn = 1Methods SSI SSIM PSNR SSI SSIM PSNR SSI SSIM PSNRLee [58] 0.8015 0.2834 16.2186 0.7342 0.2307 13.6769 0.7099 0.1982 12.5215

Wiener [59] 0.7706 0.4259 19.1536 0.6659 0.3365 16.2934 0.6097 0.3010 15.1515Geometric [64] 0.7335 0.3141 12.3719 0.6116 0.2531 10.5838 0.5523 0.2128 9.6917Nl means [31] 0.8014 0.4068 18.5889 0.8116 0.3034 13.0705 0.8502 0.2649 10.9128Bilateral [49] 0.9800 0.3312 13.1227 0.9908 0.2697 10.1614 0.9983 0.2280 8.9457Wavelet [65] 0.7181 0.3546 18.2237 0.6335 0.2979 15.6232 0.5385 0.2515 15.0764Perona et al. 0.8747 0.3707 15.6308 0.8995 0.2942 11.4945 0.9154 0.2574 9.8560

TV [66] 0.7991 0.4161 18.9066 0.7330 0.3314 15.0214 0.7145 0.2972 13.3013GAFIA [86] 0.9094 0.3181 14.2430 0.8692 0.2329 11.9557 0.8437 0.1909 11.0696

IFD [85] 0.8856 0.3198 14.6882 0.8362 0.2319 11.8865 0.8146 0.1837 10.7419Proposed 0.7037 0.5019 21.5696 0.5884 0.3926 19.1762 0.5422 0.3445 17.3767

Figure 3.5: Residual Images (Noise) of Lena obtained after applying (a) TV [66] (b) Wavelet[65] (c) Perona et al. [62] (d) GAFIA [86] (e) IFD [85] (f) Lee [58] (g) Wiener [59] (h)Geometric [64] (i) Nl means [31] (j) Bilateral [49] (k) Proposed Methodology

32

Table 3.4: Comparison of various methods for Lena image with noise variance σn

σn = 0.1 σn = 0.5 σn = 1Methods SSI SSIM PSNR SSI SSIM PSNR SSI SSIM PSNRLee [58] 0.8168 0.3508 10.8841 0.6844 0.1514 13.8275 0.6541 0.1127 12.5409

Wiener [59] 0.7856 0.5209 22.1568 0.5777 0.2845 16.5931 0.5045 0.2407 15.4500Geometric [64] 0.7792 0.3786 14.8673 0.5080 0.2142 10.4483 0.4407 0.1638 9.0621Nl means [31] 0.7529 0.5578 23.6833 0.7436 0.2238 13.5082 0.7946 0.1397 11.0611Bilateral [49] 0.9518 0.3284 16.0356 0.9819 0.1220 10.6669 0.9936 0.0821 8.7700Wavelet [65] 0.8916 0.5035 20.5239 0.5762 0.2913 16.1122 0.5266 0.2561 15.5867

Perona et al. [62] 0.8095 0.4967 20.8992 0.8615 0.1844 11.6399 0.8864 0.1130 9.8344TV [66] 0.7921 0.5425 22.7172 0.6601 0.2502 15.2568 0.6412 0.1802 13.2370

GAFIA [86] 0.8794 0.3369 15.8586 0.8442 0.1500 11.8838 0.8228 0.1142 10.9568IFD [85] 1.0067 0.3268 16.8989 0.8187 0.1440 11.8228 0.8052 0.1001 10.5512Proposed 0.7049 0.6500 24.4041 0.4737 0.3951 19.6252 0.4207 0.3082 17.5878

eraman, Lena and echocardiographic images are shown in Figs 3.4 - 3.5 and Figs 3.8 - 3.9.

It is evident from these figures that TV preserves edges of the images but fails to suppress

maximum amount of noise. Wavelet produces lots of artifacts. Perona et. al removes fine

texture during denoising process. GAFIA outputs a low contrast denoised image. No sig-

nificant performance has been shown by IFD both in terms of noise suppression and edge

preservation. Lee shows poor edge preservation and little noise removal whereas Wiener

filter produces blur image. Geometric filter has introduced artifacts. Nl means rely of self

similarity present in an image therefore we can see that its performance severely degrades

with the increase in noise variance. Bilateral filter has failed to remove sufficient amount

of noise along with structure preservation. Quantitative results for these images are shown

in Tables 3.1-3.5. In case of echocardiographic images, no reference noise free image is

available therefore its quantitative analysis is done on the basis of SSI. Lowest value of SSI

indicates good noise suppression capability of the proposed method. Proposed method pos-

sess good noise removal characteristics in addition to structure preservation quality. It can

be clearly seen that proposed method outperforms in all situations.

Visual assessment of echocardiographic images was done by an experienced echocardio-

grapher. The cardiologist stressed that the proposed technique outperforms other methods in

noise suppression and clarity of fine structure.

3.4 Conclusion

Methodology presented in this chapter is for denoising of echocardiographic images. Pro-

posed filter is based on fuzzy logic and fractional order integration. Filtering has been done

33

Figure 3.6: Various denoising methods applied on an Echocardiographic image (short axisview of left ventricular) (a) TV [66] (b) Wavelet [65] (c) Perona et al. [62] (d) GAFIA [86] (e)IFD [85] (f) Lee [58] (g) Wiener [59] (h) Geometric [64] (i) Nl means [31] (j) Bilateral [49](k) Proposed Methodology (l) Original Echocardiographyic Image

Table 3.5: Comparison of SSI for various denoising methods applied on Echocardiographicimage

Method/Image 4-Chamber View Short-Axis ViewLee [58] 0.9861 0.9891

Wiener [59] 0.9896 0.9914Geometric [64] 0.9931 0.9953Nl means [31] 0.9742 0.9787Bilateral [49] 0.9873 0.9805Wavelet [65] 0.9855 0.9867

Perona et al. [62] 0.9734 0.9707TV [66] 0.9909 0.9858

GAFIA [86] 0.9890 0.9750IFD [85] 0.9817 0.9772Proposed 0.9647 0.9679

34

Figure 3.7: Various denoising methods applied on an Echocardiographic image (apical fourchamber view) (a) TV [66] (b) Wavelet [65] (c) Perona et al. [62] (d) GAFIA [86] (e) IFD[85] (f) Lee [58] (g) Wiener [59] (h) Geometric [64] (i) Nl means [31] (j) Bilateral [49] (k)Proposed Methodology (l) Original Echocardiographic Image

35

Figure 3.8: Residual Echocardiographic (short axis view of left ventricular) Images (Noise)obtained after applying (a) TV [66] (b) Wavelet [65] (c) Perona et al. [62] (d) GAFIA [86] (e)IFD [85] (f) Lee [58] (g) Wiener [59] (h) Geometric [64] (i) Nl means [31] (j) Bilateral [49](k) Proposed Methodology

36

Figure 3.9: Residual Echocardiographic (apical four chamber view) Images (Noise) obtainedafter applying (a) TV [66] (b) Wavelet [65] (c) Perona et al. [62] (d) GAFIA [86] (e) IFD [85](f) Lee [58] (g) Wiener [59] (h) Geometric [64] (i) Nl means [31] (j) Bilateral [49] (k)Proposed Methodology

in two stages, in stage-1 adaptive fuzzy weighted mean filter has been applied and in stage-2

fractional integration filter has been used. Both stages are important as stage-1 is addressing

the intensity variation by assigning corresponding weight to each pixel and stage-2 further

improves the denoised image. Results have been proved both qualitatively and quantitatively.

So we conclude that proposed technique can efficiently despeckle an echocardiographic im-

age and it also improves diagnostic results of ultrasound imaging modality.

37

Chapter 4

INTELLIGENT ADAPTIVE IMAGE DENOISING

4.1 Preliminary

In chapter 3, we have seen that fuzzy logic is good to handle uncertainty caused by speckle

noise and at stage-2 of the algorithm a fixed order fractional differential was applied. It

improved denoising results. But using fixed order on the entire image is not a good ap-

proach. Pixels of smooth, texture and edge regions have different properties so these should

be treated differently. This chapter proposes an algorithm that selects appropriate order of

fractional integration for every pixel of the image based on the characteristics of that pixel.

4.2 Proposed Algorithm

Fractional calculus is an emerging research area in image processing and its applications

have brought out effective results. Keeping this in mind we have used fractional integral

mask proposed in [85]. Difference between [85] and proposed technique is that in [85]

whole image is convolved by a fixed order filter but proposed algorithm adaptively selects

appropriate order of filter for each pixel of the image. A digital image is composed of

different regions, namely smooth, texture and edge. Treating pixels of different regions with

same filter is not a good approach. Let I be the ultrasound image corrupted with speckle

noise:

I(x, y) = O(x, y).N(x, y) (4.1)

where O(x, y) is the noise-free image, N(x, y) represents multiplicative gamma noise. Ad-

ditive noise is also present but its effect is negligible therefore efforts are made to remove

multiplicative noise only. Image I has dimensionsM×Qwhere x = 1.....M and y = 1.....Q.

In the proposed technique, an image is divided into three distinct regions and fuzzy function

is used to draw boundary between these regions. Coefficient of variation (CV ) is used as

image region classifier. CV is represented as:

CV =sd

mean(4.2)

38

Figure 4.1: Fuzzy Membership Function

where sd is standard deviation and mean is arithmetic mean of 3× 3 neighborhood around

a pixel. A pixel with small value of CV is a smooth region pixel, high value of CV is

for edge pixels and medium value belongs to the texture region. A simple fuzzy triangular

membership function, has been deployed to deal with the fuzziness of speckle noise. It is

used to define fuzzy values of various image regions. Triangular fuzzy membership function

is defined as:

f(w; q, e, t) = max(min(w − qe− q

,t− wt− e

), 0) (4.3)

Every pixel of image H will have fuzzy membership value defined by the following equa-

tion:

µI(x,y) =

µSmooth = c−CV

c−a a ≤ CV ≤ m1

µTexture = max(min(CV−m1c−m1

, m2−Cvm2−c ), 0) m1 < CV ≤ m2

µEdge = CV−cb−c otherwise

(4.4)

Fuzzy function for the whole image will look as shown in Figure 4.1. Thresholds a, b, c,

m1 and m2 are calculated as:

a = min(CVi) (4.5)

b = max(CVi) (4.6)

c = median(unique(CVi)) (4.7)

m1 = average(CV[a,c]) (4.8)

39

Algorithm 2 Proposed Intelligent Image DenoisingRequire: Input image I with dimensions M ×Q.

1: Calculate CV using 3× 3 window around each pixel of I(x, y)

CV (x, y) = sdmean

2: Calculate thresholds for triangular fuzzy functions usinga = min(CVi)b = max(CVi)

c = median(unique(CVi))m1 = average(CV[a,c])m2 = average(CV[c,b])

where i = M ×Q3: Segregate all pixels of image H into various regions using fuzzy function

µI(x,y) =

µSmooth = c−CV

c−a a ≤ CV ≤ m1

µTexture = max(min(CV−m1c−m1

, m2−Cvm2−c ), 0) m1 < CV ≤ m2

µEdge = CV−cb−c otherwise

4: Each pixel of I has membership as defined

I(x, y) =

Smooth a ≤ µI(x,y) ≤ m1

Texture m1 < µI(x,y) ≤ m2

Edge m2 < µI(x,y) ≤ b

5: Denoise Smooth region pixel usingU(x, y) = 1

9Σ1i=−1Σ

1j=−1I(x+ i, y + j)

6: Denoise texture region pixels with fractional integration mask with v = −0.7 and edgeregion pixels using

U(x, y) = I(x, y) ∗mask7: Denoise texture region pixels with fractional integration mask with v = −0.5

U(x, y) = I(x, y) ∗mask8: Resultant image U is denoised image.

m2 = average(CV[c,b]) (4.9)

where i = M ×Q

Fuzzy function given in (4.4) is used to classify each pixel based on CV . As lowest CV

corresponds to smooth region, medium shows texture region and highest belongs to the edge

region. So by using (4.10), we can decide membership (µ) of each pixel of I to one of the

regions. Each pixel of an image is classified as smooth, texture or edge region:

I(x, y) =

Smooth a ≤ µI(x,y) ≤ m1

Texture m1 < µI(x,y) ≤ m2

Edge m2 < µI(x,y) ≤ b

(4.10)

40

v2−v12

v2−v3

v2−v12

−v5 1 −v

5

−v6

−2v3

−v6

v2−v12

−v5

−v6

v2−v3 1 −2v

3

v2−v12

−v5

−v6

Figure 4.2: Improved Fractional Integral Mask [85]

4.2.1 Adaptive Denoising

After classifying every pixel of the image I as smooth, medium or edge region pixels; a filter

will be applied on each pixel to get a denoised image. As we mentioned earlier, in this paper

we have proposed adaptive version of [85]. So we will apply fractional integral mask to

denoise an image. But as smooth region has no high frequency component like texture and

edge region therefore applying fractional integral mask on smooth region pixel is useless and

it will only add complexity. To make algorithm less complex, we will process smooth region

pixels with average filter and for texture and edge region pixels fractional integral mask will

be used as fractional integral operation has the ability to preserve texture and edges with

removing noise.

Smooth Region

Noisy pixel of smooth region will be replaced by the average of all pixels in a 3× 3 window

around it. Pixels which are classified as smooth region are processed using the equation

below:

U(x, y) =1

9Σ1i=−1Σ

1j=−1I(x+ i, y + j) (4.11)

Texture Region and Edge Region

Average filter tends to remove texture and fine details so this filter is not appropriate for

texture and edge region. Therefore we have treated texture and edge region with fractional

order integration filter. This filter has the capability to remove noise while preserving edges

and details [86]. Frequency response of fractional order integration shows that it has different

response at different frequencies. Texture and edge region pixels are denoised using (4.11),

for texture pixels v = −0.7 and for edge region v = −0.5.

41

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 4.3: Denoising results of checkerboard image (a) Hu et al. [83] (b) Huang et al. [84](c) IFD [85] (d) GAFIA [86] (e) Lee (f) Kuan (g) Proposed (h) Noisy image (i) OriginalImage

U(x, y) = I(x, y) ∗mask (4.12)

where mask is improved fractional integral mask given in Fig. 2.7 and U is the denoised

image.

4.3 Simulation and Results

Simulations of proposed and state-of-the-art existing techniques ( [83], [84], [85], [86], [58]

and [60]) are performed on standard cameraman image and checkerboard image. For

echocardiographic images analysis, various images have been considered. Results of two

echocardiographic images have been included in this chapter. Cameraman image and

checkerboard image is artificially corrupted with speckle noise of different variance (σn)

and results are captured visually and qualitatively. Quantitative analysis is performed using

42

Figure 4.4: Denoising results cameraman image with noise variance 0.02 (a) Hu et al. [83](b) Huang et al. [84] (c) IFD [85] (d) GAFIA [86] (e) Lee (f) Kuan (g) Proposed (h) Noisyimage (i) Original cameraman image

43

Figure 4.5: Denoising results echocardiographic image (a) Hu et al. [83] (b) Huang et al. [84](c) IFD [85] (d) GAFIA [86] (e)Lee [58] (f) Kuan [60] (g) Proposed (h) Noisy image

44

Table 4.1: Quantitative Analysis of Cameraman Image

σn Methods MSE PSNR β SSIM ρ0.02 Proposed 193.5129 25.2637 0.8236 0.6905 0.9747

Hu et al. [83] 2480.1 14.1860 0.7058 0.6556 0.9731Huang et al. [84] 187.8942 25.3917 0.6744 0.6330 0.9751

GAFIA [86] 519.6713 20.9735 0.6304 0.5470 0.9544IFD [85] 272.0033 23.7851 0.6631 0.5511 0.9639

Lee 771.1000 19.2597 0.6575 0.5597 0.8979Kuan 622.8425 20.1870 0.5596 0.5910 0.9164

0.04 Proposed 249.1679 24.1659 0.7957 0.6127 0.9677Hu et al. [83] 2497.21 14.1562 0.6121 0.5926 0.9661

Huang et al. [84] 295.4343 23.4262 0.5772 0.5517 0.9612GAFIA [86] 845.7879 18.8582 0.5479 0.4778 0.9213

IFD [85] 514.3926 21.0179 0.5675 0.4795 0.9332Lee [58] 830.4824 18.9375 0.6047 0.4630 0.8906

Kuan [60] 664.8754 19.9034 0.5348 0.5212 0.91090.06 Proposed 298.6975 23.3785 0.7678 0.5698 0.9611

Hu et al. [83] 2507.9 14.1376 0.5557 0.5480 0.9593Huang et al. [84] 398.4863 22.1267 0.5299 0.5055 0.9478

GAFIA [86] 1111.0 17.6736 0.5091 0.4361 0.8915IFD [85] 761.1245 19.3162 0.5166 0.4366 0.9062Lee [58] 903.5175 18.5714 0.5556 0.4088 0.8820

Kuan [60] 706.2845 19.6410 0.5157 0.4767 0.9056

mean square error (MSE), peak-signal-to-noise ratio (PSNR), structural similarity (SSIM)

index, edge preservation index (β), correlation coefficient(ρ). Quantitative comparison of

existing and proposed techniques is given in Table 4.1 and 4.2.

From Figure 4.3 and Figure 4.4, we can evaluate performance of different denoising fil-

ters for artificially corrupted checkerboard and cameraman image. We see that Hu et al. pro-

duced a low contrast image, Huang et al., IFD and GAFIA removed less amount of noise. As

proposed algorithm is adaptive version of [85] therefore it is evident from the results that the

proposed filter shows improvement in performance. In the presence of different noise vari-

ances, proposed technique outperforms other existing techniques. Quantitative comparison

for various denoising methodologies has been shown in Table 4.1 and 4.2. Besides evaluating

performance of the proposed filter for standard images, denoising results are also produced

for echocardiographic images. As no ground-truth is present in case of echocardiography im-

ages so results are discussed only on visual basis. Visually best image is noise-free, contain

important image structures and minimum amount artifacts. Figures 4.3− 4.6 shows qualita-

tive comparison for cameraman and real echocardiography images. These results have been

45

Figure 4.6: Denoising results echocardiography image (a) Hu et al. [83] (b) Huang et al. [84](c) IFD [85] (d) GAFIA [86] (e)Lee [58] (f) Kuan [60] (g) Proposed (h) Noisy image

46

Table 4.2: Quantitative Analysis of Checkerboard Image

σn Methods MSE PSNR β SSIM ρ0.02 Proposed 330.2662 22.9422 0.8668 0.8418 0.9888

Hu et al. [83] 833.1047 22.9050 0.7575 0.7610 0.9921Huang et al. [84] 5136.2 11.0244 0.7897 0.7490 0.9895

GAFIA [86] 1916.0 15.3069 0.6556 0.6012 0.9724IFD [85] 582.0887 20.4809 0.7328 0.6240 0.9906Lee [58] 1438.4 16.5519 0.7128 0.7183 0.9398

Kuan [60] 1285.7 17.0733 0.6017 0.7275 0.94720.04 Proposed 468.7163 21.4217 0.8404 0.7871 0.9864

Hu et al. [83] 547.6487 20.7458 0.6476 0.6865 0.9880Huang et al. [84] 5514.4 10.7158 0.7089 0.7018 0.9878

GAFIA [86] 2421.7 14.2896 0.5847 0.5412 0.9649IFD [85] 937.9402 18.4091 0.6273 0.5567 0.9814Lee [58] 1562.8 16.1918 0.6976 0.6592 0.9376

Kuan [60] 1422.3 16.6010 0.5938 0.6710 0.94560.06 Proposed 616.4312 20.2320 0.8234 0.7454 0.9834

Hu et al. [83] 717.2748 19.5731 0.5954 0.6406 0.9839Huang et al. [84] 5841.7 10.4654 0.6492 0.6660 0.9857

GAFIA [86] 2810.5 13.6429 0.5477 0.4988 0.9556IFD [85] 1251.3 17.1572 0.5707 0.5096 0.9715Lee [58] 1690.2 15.8515 0.6736 0.6181 0.9358

Kuan [60] 1546.1 16.2385 0.5819 0.6391 0.9442

evaluated by an experienced echocardiographer who confirmed superiority of the proposed

methodology over other state-of-the art techniques. Proposed technique produced denoised

images with less noise and more structural clarity.

4.4 Conclusion

In this chapter, we have proposed adaptive method for denoising of echocardiographic im-

age. Proposed methodology is based on improved adaptive fractional integration filter. Every

pixel of an image is processed by adaptively selecting appropriate filter according to the char-

acteristics of the pixel’s region. Fuzzy logic is incorporated to deal with the vague concept

of speckle noise. Each region of an image is treated differently in order to remove maximum

noise and preserve characteristics of that region. This method significantly removes noise

from an image and preserves image structure which is easily lost in non-adaptive filters.

Proposed methods produce high-quality image with better quality and no artifacts. Simula-

tion results compared qualitatively and quantitatively with state-of-the-art existing schemes

proves the robustness of proposed techniques.

47

Chapter 5

ROUGHSET THEORY AND FRACTIONAL INTEGRATION BASED

IMAGE DENOISING

5.1 Preliminary

Rough set theory is a new mathematical tool for imperfect data analysis [99]. It is being

applied in different disciplines like decision support, engineering, environment, banking,

medicine and others. Theory of rough sets is good at handling vague and overlapped con-

cepts, therefore handles speckle noise well [100]. This chapter puts forward speckle remov-

ing techniques for echocardiographic images which adaptively selects order of the filter for

each pixel in an image. Order of fractional integration filter is selected according to the

properties of every pixel. Pixels belonging to different image regions are treated differently.

Therefore image texture and details are well preserved in the denoising process.

5.2 Proposed Algorithm

Ultrasound images are corrupted with speckle noise. As this noise is multiplicative therefore

its components get multiplied with the components of an image. Speckle noise follows

gamma distribution. Suppose input noisy image is I then

I = O ×N (5.1)

where O is the noise free image and N is speckle noise. Image I has dimensions Z × R.

Take log of (1) to convert multiplicative noise into additive noise. Before applying denois-

ing filter, we will first classify each pixel of I into different image regions namely smooth,

texture and edge. Each pixel is then convolved with fractional integral denoising mask to

remove noise.

5.2.1 Image Region Classifier

Three different regions of an image (smooth, texture and edge) have distinct characteristics.

Pixels of these regions can be differentiated on the basis of change in intensity value. This

48

change is well reflected with the help of derivative. Hessian matrix is a 2 × 2 matrix based

on second order derivative. Hessian matrix on each pixel of I will be defined as:

Hp =

pxx pxy

pyx pyy

(5.2)

where each element of matrixH is second order derivative of current pixel with its neighbor-

ing pixels, x and y represent x-direction and y-direction. p εZ ×R which shows that Hessian

matrix will be calculated for each pixel of I . If two neighbouring image pixels belong to the

same image region, their second order derivative will be a small value. Pixels of different

regions will have large value for second order derivative.

Hessian matrix has two eigenvalues:

λ1 =1

2[(pxx + pyy) +

√(pxx + pyy)2 + 4p2xy] (5.3)

λ2 =1

2[(pxx + pyy)−

√(pxx + pyy)2 + 4p2xy] (5.4)

where λ1 and λ2 are large and small eigenvalues respectively.

When a pixel belongs to an edge, value of λ1 is large and λ2 is small. In smooth region,

both λ1 and λ2 have very small values. So keeping this in mind, we can divide an image into

different regions by considering only large eigenvalue (λ1) of every pixel. But the critical

Figure 5.1: Rough Set

49

thing is defining boundaries between different image regions. Apply the concept of rough

set theory [99] on the values of λ1 to classify each pixel into different image regions. The

roughset theory is good to handle speckle noise [100]. The rough sets are defined by the

construction of lower and upper approximations for each class present in a rough set. Lower

approximation corresponds to those elements of a universe which are certainly classified as

a member of that class, Upper approximations is the set of all those elements which can

possibly belong to a specific class [100]. Here, input image is the universe and λ1 calculated

for every pixel are the elements of this universe. These elements are divided into two classes

namely Small and Large as shown in Fig. 5.1. Each rough class is further divided into upper

and lower approximations. Pixels belonging to lower approximation of Small class will be

those pixels which belong to homogeneous region, pixels in lower approximation of Large

class belong to edge region and pixels which donot belong to any lower approximation are

those pixels which belong to texture region. So in this way we can classify a pixel by

considering large eigenvalue of its hessian matrix.

Threshold values defining the range of upper and lower approximation of both classes is

important. Thresholds τa and τb have been defined through empirically developed formulas

as described in (5.5) and (5.6). In this way threshold values can be calculated for any image.

Threshold values defining the range of class upper and lower approximation of both

classes is important. Thresholds τa and τb have been defined through empirically devel-

oped formulas as described in (5.5) and (5.6). In this way, threshold values can be calculated

for any image.

τa = max(λ1)− λ (5.5)

where λ is the mean of λ1 for all pixels in an image.

τb =max(λ1) + τa

2(5.6)

5.2.2 Adaptive Fractional Differintegral Filter

Denoising of ultrasound image is performed using fractional calculus. We have utilized a

5 × 5 mask given in Fig 2.7. When we will convolve a digital image with this mask, the

resultant image will be the image processed by v − order fractional differential/integral.

50

Algorithm 3 Rough Set Theory and Fractional Integration Based Image Denoising Algo-rithmRequire: Input images I with dimensions Z ×R.

1: Take logarithm of input image I .2: Calculate Hessian matrix at each pixel of I using

Hp =

[pxx pxypyx pyy

]3: Calculate eigenvalues of hessian matrix using

λ1 = 12[(pxx + pyy) +

√(pxx + pyy)2 + 4p2xy]

λ1 = 12[(pxx + pyy)−

√(pxx + pyy)2 + 4p2xy]

4: Define roughset (Fig 5.1) asC = {Small, Large}

5: Calculate thresholds for roughset usingτa = max(λ1)− λτb = max(λ1)+a

2

6: Decompose all pixels of image I into smooth, texture and edge region using rough settheory.

7: Calculate order of fractional filter v using

v =

−0.5 λ1εSmallLow approx.

−0.2 λ1εSmallUpper approx.,λ1εLargeUpper approx.

0.5 λ1εLargeLow approx.

8: Convolve every pixel of image I with fractional ordermask by putting appropriate valueof v as calculated in step 7

U = I ∗mask9: Resultant image U is denoised image.

Calculate the order v for every pixel of I according to the following equation:

v =

−0.5 λ1εSmallLow approx.

−0.2 λ1εSmallUpper approx.,λ1εLargeUpper approx.

0.5 λ1εLargeLow approx.

(5.7)

Selection of ν plays very important role in filter performance. When the order ν will be

negative, mask in Fig. 2.7 will act as fractional integral mask and when ν will be positive,

it will be a fractional differential mask. So, we will apply mask with negative orders on

smooth areas fine texture whereas positive values of ν will be for edges. Fractional differen-

tial operation increases the high-frequency components of signal whereas fractional integral

operation on a signal will attenuate its high frequency components. Therefore, fractional in-

tegral is appropriate for texture and homogeneous areas and fractional differential of higher

51

Figure 5.2: Various denoising methods applied on Checkerboard image with noise var = 0.1(a) IFD [85] (b) GAFIA [86] (c) Hu et al. [83] (d) Huang et al. [84] (e) Lee [58] (f) Kuan [60](g) Proposed (h) Noisy (i) Original Image

order is good for edges. Final denoised image is obtained as described in the following

equation:

U = I ∗mask (5.8)

where ∗ is the convolution operator and mask is a 5× 5 Gaussian filter with weights based

on fractional calculus. These weights are adjusted by putting (5.7) into 5× 5 mask shown in

Fig. 2.7. Proposed algorithm is simple yet robust for high noise variance. Anti-logarithm is

applied on image U to revert the changes.

5.3 Simulations and Results

We compared performance of the proposed method with different state-of-the-art techniques

and latest fractional order integration based techniques like IFD [85], GAFIA [86], Lee [58],

Kuan [60], Hu et al. [83] and Huang et al. [84]. All simulations were carried out in Matlab.

We considered standard Lena image and checkerboard image, corrupted them with speckle

noise of different variances. As no reference noise free image of real echocardiographic

52

Figure 5.3: Various denoising methods applied on Lena image with noise var = 0.05 (a)Noisy image (b) Hu et al. [83] (c) Huang et al. [84] (d) IFD [85] (e) GAFIA [86] (f) Lee [58](g) Kuan [60](h) Proposed (i) Original Image

Table 5.1: Comparison of different methods for Lena image with noise var = 0.05

Method SSIM r MSE PSNR β

IFD [85] 0.4276 0.8775 745.3162 19.4074 0.5027GAFIA [86] 0.4285 0.8912 1204.0 17.3247 0.5028

Huang et al. [84] 0.6706 0.9480 423.6916 21.3880 0.6511Hu et al. [83] 0.6013 0.9533 2398.6 14.3313 0.5423

Lee [58] 0.1550 0.5600 2669.6 13.8663 0.3881Kuan [60] 0.6671 0.9065 412.3031 21.9786 0.6271Proposed 0.6968 0.9593 369.1518 22.4588 0.7021

53

Table 5.2: Comparison of different methods for Lena image with noise var = 0.1

Method SSIM r MSE PSNR βIFD [85] 0.3259 0.7919 1353.5 16.8162 0.4602

GAFIA [86] 0.3409 0.7618 1708.6 15.8045 0.4754Huang et al. [84] 0.4360 0.8841 544.4330 20.7714 0.4608

Hu et al. [83] 0.5256 0.9283 2503.8 14.1449 0.4932Lee [58] 0.3525 0.8221 844.1718 18.8665 0.4699

Kuan [60] 0.5387 0.8924 475.62 21.3581 0.5643Proposed 0.6115 0.9459 456.4168 21.5372 0.6783

Table 5.3: Comparison of different methods for Lena image with noise var = 0.5

Method SSIM r MSE PSNR βIFD [85] 0.1437 0.4712 4279.5 11.8169 0.4315

GAFIA [86] 0.1527 0.4611 4161.5 11.9383 0.4488Huang et al. [84] 0.2164 0.6333 2104.8 14.8987 0.4305

Hu et al. [83] 0.2164 0.6333 2104.8 14.8987 0.4305Lee [58] 0.2618 0.7056 1356.7 16.2590 0.4578

Kuan [60] 0.2580 0.7211 1240.9 17.1935 0.4521Proposed 0.3163 0.7672 1147.0 17.5351 0.5695

images is available therefore results of standard images will be considered to verify the

performance of proposed technique. Results are compared both qualitatively and quantita-

tively. Quantitative comparison metrics used for comparison are peak signal to noise ratio

(PSNR), mean square error (MSE), correlation coefficient (r), edge preservation index (β)

and structural similarity index measure (SSIM).

From Table 5.1-5.6, it can be seen that proposed algorithm shows highest values for PSNR

as compared to other techniques, which indicates its best noise suppression capability. High

value of correlation coefficient shows that output image of proposed method correlates with

Table 5.4: Comparison of different methods for Checkerboard image with noise var = 0.05

Method SSIM r MSE PSNR β

IFD [85] 0.5519 0.9772 1011.8 18.0799 0.5975GAFIA [86] 0.5176 0.9594 2734.6 13.7618 0.5624Hu et al. [83] 0.6816 0.9867 5673.0 10.5927 0.6760

Huang et al. [84] 0.6616 0.9860 6236.2 20.1816 0.6108Lee [58] 0.6385 0.9368 1622.3 16.0294 0.6860

Kuan [60] 0.7663 0.9475 1239.6 17.1981 0.6057Proposed 0.7893 0.9882 444.5405 21.6516 0.7066

54

Table 5.5: Comparison of different methods for Checkerboard image with noise var = 0.1

Method SSIM r MSE PSNR β

IFD [85] 0.4601 0.9524 1774.6 15.6397 0.5130GAFIA [86] 0.4368 0.9336 3747.5 12.3934 0.5011Hu et al. [83] 0.6141 0.9812 6357.2 10.0981 0.5816

Huang et al. [84] 0.5769 0.9747 1094.3 17.7395 0.5245Lee [58] 0.5620 0.9300 1982.1 15.1596 0.6233

Kuan [60] 0.5912 0.9408 1796.8 15.5858 0.5557Proposed 0.6924 0.9856 716.428 19.5791 0.6856

Table 5.6: Comparison of different methods for Checkerboard image with noise var = 0.5

Method SSIM r MSE PSNR β

IFD [85] 0.2652 0.7603 6039.6 10.3207 0.3978GAFIA [86] 0.2379 0.7379 8602.5 8.7864 0.4125Hu et al. [83] 0.4205 0.9279 9567.9 8.3226 0.4156

Huang et al. [84] 0.3484 0.8686 4357.3 11.7387 0.3919Lee [58] 0.3528 0.8148 5219.8 10.9543 0.3634

Kuan [60] 0.4030 0.8925 4212.0 11.8859 0.4365Proposed 0.3713 0.9368 3195.1 13.0860 0.5575

Figure 5.4: Various denoising methods applied on real echo image (short-axis view of leftventricular) (a) Hu et al. [83] (b) Huang et al. [84] (c) IFD [85](d) GAFIA [86] (e) Lee [58](f) Kuan [60](g) Proposed (h) Original Image

55

Figure 5.5: Various denoising methods applied on real echo image (Four chamber view ofleft ventricular) (a) Hu et al. [83] (b) Huang et al. [84] (c) IFD [85](d) GAFIA [86] (e)Lee [58] (f) Kuan [60](g) Proposed (h) Original Image

the original image. SSIM is a proof of structural similarity among denoised and original

image, which is high for our proposed method. Edge preservation is important for medical

image denoising. Edges are important in medical image processing. Value of β shows that

the proposed method preserves edges well in the denoising process, even in the presence of

high noise variance. Fig. 5.2 and Fig. 5.3 shows comparison among different techniques in

terms of visual quality. In Fig. 5.2 and Fig. 5.3 Hu et al. produced a low contrast image, IFD

and GAFIA fail to remove considerable amount of noise from the image, Lee and Kuan also

do not show considerably good performance. The proposed method works well in removing

noise as we can see in Fig. 5.2(g) and Fig. 5.3(h). In case of real echocardiography image,

no noise free reference image is available. So we have tested two echocardiography images

on qualitative basis only. For real echo image shown in Fig. 5.4, our technique not only

removed noise from outside the heart chamber but also from inside portion. Fig. 5.5 shows

4-chamber view of a heart, proposed method removed noise from the tissue and making its

texture more clear whereas other methods does not produce attractive results. Visual results

of the proposed technique once again out performs other techniques. Again, it is obvious

56

that the proposed method gives more clear image by removing maximum noise and keeping

small structure visible.

Results of real echo images were assessed by an experienced cardiologist who confirmed

the ability of the proposed method in suppressing noise and making small structure visible.

5.4 Conclusion

In this chapter, roughset theory has been used to denoise echocardiographic images. Rough-

set theory is good to deal with the uncertainty of speckle noise. Every pixel of the image is

convolved with the fractional differintegral mask. Edge pixels are enhanced using differen-

tial mask and texture and smooth pixels are denoised using integral mask. Results showed

effectiveness of the proposed algorithm.

57

Chapter 6

ECHOCARDIOGRAPHIC IMAGE ENHANCEMENT USING

ADAPTIVE FRACTIONAL ORDER DERIVATIVE

6.1 Preliminary

Low quality of medical images is a great hurdle in making correct diagnosis. It is there-

fore necessary to enhance medical images. Enhanced image will influence the speed and

accuracy of doctor’s diagnosis. It highlights fine details and edges of an image. Enhanced

image reveals signs of illness more correctly. Image enhancement is necessary to get full ad-

vantage from ultrasound imaging modality. This chapter proposes enhancement algorithms

for ultrasound images and more specifically for echocardiographic images. Enhancement

brings improvement in the visual quality of an image [101]- [105]. So an enhanced image is

more helpful for a physician to do analysis of patient’s health. Most commonly used method

for image enhancement is histogram equalization. Fractional calculus is an old topic but

its contribution in the field of image processing is not very old [80] [97]. Concept of frac-

tional calculus when applied to images brings remarkable improvement. That’s why it is the

center of recent research. In this chapter we have utilized this innovative idea to enhance

medical images and proposed two image enhancement algorithms [54]. Both algorithms are

adaptive and apply different order of fractional differential masks on each pixel of the image

separately. Both algorithms use different pixel classification rules.

6.2 Proposed Algorithms

Algorithms proposed in this section are image enhancement algorithms. First algorithm seg-

regate image pixels into different regions namely smooth, texture and edge, on the basis of

gradient magnitude whereas second algorithm uses eigenvalue of Hessian matrix. After the

segregation step, enhancement is performed on each pixel separately. Details of algorithms

are given below. Fractional calculus mask act as fractional integral filter when the order of

filter is negative. It acts as fractional differential mask when the order is positive and for or-

der equal to 0, mask has no effect on an image. Fractional order integral operator attenuates

58

Algorithm 4 Proposed Echocardiographic Image Enhancement Using Gradient Magnitudeand Fractional Order DerivativesRequire: Input image I .

1: Calculate gradient magnitude Gmag of every pixel of I .2: Divide gradient magnitude into different regions using thresholds

a = peak+min(Gmag)2

b = peak+max(Gmag)2

3: Calculate order of fractional filter using

v =

0.1 = i(x, y) < aξ = a ≤ i(x, y) ≤ b

0.9 = i(x, y) > b

where ξ =[i(x,y)−min(I)max(I)−min(I) × 7

]+ 0.1

4: Convolve enhancement mask with order as derived in step 3 with the input image I .E = I ∗mask

5: Resultant image E is enhanced image.

the high frequency components of a signal so it is appropriate for denoising applications as

we have seen in previous chapters. Fractional differential operator is appropriate for high

frequency components of signal. As the order of fractional differential increases, the high

frequency components are enhanced in a non-linear way and with small order of fractional

differential, low frequency components are preserved. If we apply same order to the whole

image than it will bring improvement in one region but at the same time other regions will be

affected e.g. apply higher order on the entire image, edges will be enhanced but at the same

time it will have poor effect on the texture of an image. Similarly, higher order fractional

differentials fail to preserve texture therefore texture on the image will be lost. This shows

that fractional differential is more suitable for image enhancement.

6.2.1 Gradient Magnitude Based Adaptive Image Enhancement

In order to get an enhanced image via application of fractional differential mask, we should

first divide input image into different regions. An image is comprised of different regions,

namely smooth, texture and detail. Higher order of fractional differentials is suitable for edge

region, lower orders are appropriate for smooth region and middle orders are for texture

region. Let I be the input image having intensities [0-255]. To classify a pixel as edge,

texture or smooth, we have utilized the concept of gradient magnitude. If gradient magnitude

of a pixel is large then it is edge pixel, if it lies within texture region range then it is texture

region pixel otherwise it is a pixel belonging to smooth region. But the critical thing is to

59

adjust the thresholds for all these regions. Least gradient magnitude will represent smooth

region, maximum gradient represent edge region and the region in between the thresholds of

these two regions is texture region. Thresholds are computed by the following eqs:

a =peak +min(Gmag)

2(6.1)

b =peak +max(Gmag)

2(6.2)

where peak is the peak value of gradient magnitude calculated for every pixel of I ,

min(Gmag) and max(Gmag) are minimum and maximum value of gradient magnitude

of all the pixels of input image I . If gradient magnitude of a pixel is below threshold a then

it is a smooth region pixel, gradient magnitude above threshold b, it is an edge pixel and all

the pixels with gradient magnitude in between thresholds a and b are texture pixels. Next,

order v of fractional differential mask is computed by the following equation:

v =

0.1 i(x, y) < a

ξ a ≤ i(x, y) ≤ b

0.9 i(x, y) > b

(6.3)

ξ =

[i(x, y)−min(I)

max(I)−min(I)× 7

]+ 0.1 (6.4)

Now, convolve each pixel of the input image I with the fractional differential mask (Fig

2.7) by putting the order v as derived above. Enhanced image E will be calculated as:

E = I ∗mask (6.5)

6.2.2 Hessian Matrix Based Adaptive Image Enhancement

In the last section, gradient magnitude based image enhancement algorithm was proposed.

We now discuss another image enhancement algorithm which first divides all pixels of the

image into three regions using eigenvalue of hessian matrix and then perform enhancement

using fractional differential mask. Let I be the input image with dimensionsM×N . Hessian

60

Figure 6.1: Comparison among different methods for ultrasound image of a heart (Image 1)(a) Original image (b) Proposed method (c) Histogram Equalization (d) 0.1-order (e) 0.5-order (f) 0.9-order fractional differential

matrix on each pixel of I will be defined as:

H =

Ixx Ixy

Iyx Iyy

(6.6)

where each element of matrix H is second order derivative in different directions. Hessian

matrix has two eigenvalues:

λ1 =1

2[(Ixx + Iyy) +

√(Ixx + Iyy)2 + 4I2xy] (6.7)

61

Figure 6.2: Comparison among different methods for ultrasound image of a heart (Image 2)(a) Original image (b) Proposed method (c) Histogram Equalization (d) 0.1-order (e) 0.5-order (f) 0.9-order fractional differential

λ2 =1

2[(Ixx + Iyy)−

√(Ixx + Iyy)2 + 4I2xy] (6.8)

where λ1 and λ2 are large and small eigenvalues respectively.

If a pixel is edge pixel, its λ1 would be large and λ2 is small. For smooth region pixel,

both λ1 and λ2 would be small. In this way, we can discriminate between pixels of different

regions by considering the difference between both eigenvalues.

62

Algorithm 5 Proposed Echocardiographic Image Enhancement Hessian Matrix FractionalOrder DerivativesRequire: Input image I .

1: Calculate Hessian matrix at each pixel of I using

H =

[Ixx IxyIyx Iyy

]2: Calculate eigenvalues of hessian matrix using

λ1 = 12[(Ixx + Iyy) +

√(Ixx + Iyy)2 + 4I2xy]

λ2 = 12[(Ixx + Iyy)−

√(Ixx + Iyy)2 + 4I2xy]

3: Calculate D asD = λ1 − λ2

4: Select the appropriate order of image enhancement mask usingv = D + 0.1

5: Convolve I with the fractional differential mask of order v as calculated in step 4.E = I ∗mask

6: Resultant image E is desired enhanced image.

D = λ1 − λ2 (6.9)

Small values of D belongs to smooth region pixels and large values are for edge region

pixels. In order to enhance image, smooth region pixels are convolved with fractional order

differential mask of low order and as the D will increase the order will also increase. Larger

value of v will then enhance the edge pixels and mediocre values will preserve texture and

small value will give slight enhancement to the smooth region. The order of fractional

differential is adjusted with the help of following equation:

v = D + 0.1 (6.10)

After selecting the order, a pixel is processed by convolving 5 × 5 neighborhood window

around it with the fractional differential mask given in Fig. 2.7.

E = I ∗mask (6.11)

where E is the desired enhanced image.

63

Figure 6.3: Comparison among different methods for ultrasound image of a heart (a) Orig-inal image (b) Proposed method (c) Histogram Equalization (d) 0.1-order (e) 0.5-order (f)0.9-order fractional differential

6.3 Simulation and Results

To test effectiveness of proposed algorithms, we have applied these algorithms on various

ultrasound images. Results of two images for each algorithm have been presented here. Im-

ages have been tested both quantitatively and qualitatively. For quantitative analysis average

gradient and information entropy are considered. Average gradient reflects the degree of im-

provement in quality of image. Large value of average gradient shows more enhanced and

clear image. Entropy shows how much information is present in an image. Table 6.1 and

Table 6.2 show results of the gradient magnitude based algorithm. We can see that state-of-

the-art histogram equalization method did not enhance images well. Fig. 6.1 and Fig. 6.2

show comparison among proposed method and other methods. Different fixed orders are

applied on images to show comparison between fixed and adaptive order fractional differen-

tial algorithms. Here it is important to note that when fixed order is applied on both images,

results are not very attractive. Fixed order-0.9 over enhanced the image with very high value

64

Figure 6.4: Comparison among different methods for ultrasound image of an ovary (a) Orig-inal image (b) Proposed method (c) Histogram Equalization (d) 0.1-order (e) 0.5-order (f)0.9-order fractional differential

of average gradient but at the same time it reduces the entropy value which shows that it

damaged the fine texture of an image. Fixed order-0.1 failed to give considerable enhance-

ment to the input images with very small value of average gradient. Form these results, it is

verified that applying same order on the whole image does not bring good results.

Proposed method brought enhancement in images by increasing average gradient and it

also preserved information contents as its entropy value is very much near the original image.

It is verified in Table 6.1 and Table 6.2. Proposed method increases the average gradient 2-4

times as compared to the original image. So fractional differential enhances the edges of

image and brings clarity in image.

We considered one echocardiographic image and one ovary ultrasound image to see results

of hessian matrix based method. The effect of image enhancement of proposed method is

65

Table 6.1: Comparison of Different Methods for Ultrasound of a Heart Image 1

Method Average Gradient Entropy0.1-Order 5.9439 3.73950.4-Order 10.2290 3.72000.9-Order 85.7105 2.6030

Histogram Equalization 4.1767 2.7715Proposed 11.2226 3.7235Original 5.1523 3.7122

Table 6.2: Comparison Of Different Methods for Ultrasound of a Heart Image 2

Method Average Gradient Entropy0.1-Order 6.6689 4.61710.5-Order 14.1393 4.79550.9-Order 89.6691 3.3095

Histogram Equalization 6.2965 3.7964Proposed 26.3746 4.6736Original 5.8154 4.6178

compared with fixed fractional differential methods as well as with state-of-the-art histogram

equalization technique. Although image enhancement is best evaluated by visual analysis

but we have also evaluated it with the help of average gradient. In Fig. 6.3 we can see

that the proposed method shows good enhancement. Histogram equalization method did not

give good enhancement results, in fixed order methods 0.1 order didn′t enhance edges well,

0.5 order showed good results, 0.9 order over enhanced image and also produced noise in

the image. Similarly in Fig. 6.4, 0.1-order again did not give satisfactory result, 0.4-order

showed good enhancement. Quantitative comparison is given in Table 6.3 and Table 6.4.

From the experiment results we can clearly see that fractional order differentials are good at

enhancing images but finding the right order for an image is a tedious and time consuming

task. So, proposed method gives a clear image by enhancing the details of image. Proposed

methods select appropriate order of the fractional differential dynamically in a time effective

manner. Main purpose of these algorithms is to select order of fractional differential. It will

save time and effort in adjusting appropriate fractional order for an image manually.

6.4 Conclusion

Enhancement of medical images helps a physician in making accurate and speed decision.

Enhancement brings sharpness in the image. Medical image enhancement based on frac-

66

Table 6.3: Comparison Of Different Methods For Ultrasound of a Heart

Method Average Gradient0.1-Order 8.68950.4-Order 12.60100.9-Order 94.1148

Histogram Equalization 10.9186Proposed 12.0714Original 7.8850

Table 6.4: Comparison of Different Methods for Ultrasound of an Ovary

Method Average Gradient0.1-Order 7.25170.5-Order 13.50530.9-Order 46.8151

Histogram Equalization 7.9505Proposed 12.1233Original 6.5125

tional differential is a developing research area. This chapter proposed adaptive image en-

hancement algorithms which automatically selects fractional order for each pixel. Proposed

methodologies selects higher order of fractional calculus for edge region and smaller order

for texture and smooth region of the image. Proposed adaptive fractional order differential

methods enhance image and also saves time for selecting appropriate order for the image. It

speeds up medical image processing task. When compared to existing image enhancement

methods, proposed methods show improved results. Simulation results show that adaptive

fractional differential is perfect for medical image enhancement.

67

Chapter 7

CONCLUSION AND FUTURE WORK

7.1 Conclusion

This chapter summarizes and concludes the investigation of denoising and enhancement

methodologies presented in this thesis. It emphasizes the main points of the theoretic and

practical work performed during this project and presents conclusions that can be drawn

from the results. General recommendations for further work and development in the area

of denoising of echocardiographic image have been proposed. The main subject of this the-

sis was the design of denoising and enhancement algorithms and five different approaches

were proposed. In chapter 3, echcocardiographic image denoising based on fractional cal-

culus and fuzzy logic was proposed. The focus of this chapter was on the effectiveness of

fractional calculus in the area of image denoising. Knowledge of fractional integration was

fused with the concept fuzzy logic with the aim to develop a hybrid methodology to mitigate

speckle noise from ultrasound image. The main advantage of this technique was that it pre-

served edges and performed well even in high noise variance. Noise removal process was

performed in two stages which assured maximum noise removal. Simulation results showed

the significance of proposed scheme compared to existing state of art schemes. Chapter 4

presented another algorithm for speckle noise mitigation. Algorithm based on fuzzy logic

theory intelligently selected an appropriate filter for each pixel of the image. This technique

was designed to convolve every pixel of the image with different order of fractional integra-

tion filter. Chapter 5 discussed a denoising scheme, which used latest concepts of roughset

theory and fractional differintegral. Benefit of this technique was that it selected level of

filtering considering the properties of a pixel. In this way edge, texture and smooth region

pixels were treated with different filters. Simulation results based on visual and quantitative

analysis showed the significance of proposed scheme. All proposed denoising algorithms

works well for every image. First algorithm is robust in removing noise of high variance

whereas the other two work well for comparatively low noise variances. The focus of chapter

68

6 of this thesis was on the importance of image enhancement for echocardiographic images.

The aim was to design enhancement techniques to bring clarity in small structure of a heart

ultrasound image. Enhanced image aid correct diagnosis and also reduce time required by

the physician to analyze an image. Two enhancement methodologies were presented and

tested for different ultrasound images. Simulation results proved their effectiveness.

7.2 Future Work

Investigation into the field of echocardiographic image denoising and enhancement pre-

sented in this thesis was thorough, however, it was not exhaustive and some natural exten-

sions to the research presented in it are recommended in this section. Fractional calculus is

emerging topic in the area of image processing. But its applications in the field of denoising

and enhancement of echocardiographic image are very rare. Main drawback of fractional

calculus filter is that if it is applied alone on a noisy image, it will work only for low vari-

ance of noise. We have tried to overcome this weakness of fractional integration filter but

still improvement is required. Adaptive denoising algorithms bring more improvement in the

denoised. In these adaptive methods improvement can be brought by focusing on two points.

First, divide pixels into different image regions by using more robust region classifier. More

accurate is the division of pixels, more good will be the performance of filter. Secondly,

selection of filters for various image regions also affect the results of denoising. We applied

fractional calculus based filters in all proposed adaptive algorithms. Improvement can be

brought by incorporating other filters as well. Finally, instead of denoising single frame of

echocardiographic video, multiple frames should be processed at a time. Heart cycle repeats

itself. Therefore, we can extract similar frames from consecutive cycles and process these

frames all together to save time. Now a days 3D ultrasound is becoming more popular. We

used 2D b-mode images for this study but our future research will be extended towards 3D

ultrasound images.

69

REFERENCES

[1] Q. Ciampi, B. Villari, “Role of echocardiography in diagnosis and risk stratification inheart failure with left ventricular systolic dysfunction”, Cardiovascular Ultrasound,pp. 5-34, 2007.

[2] S. Kaddoura, “Echo Made Easy”, Elsevier Health Sciences, Churchill Livingstone,2012.

[3] N. J. Alison, N. Navab, H. Becher, “Ultrasonic image analysis and image-guidedinterventions.” Interface focus, vol. 1, no. 4, pp. 673-685. 2011

[4] N. J. Alison, and D. Boukerroui, “Ultrasound image segmentation: a survey”, IEEETransactions on Medical Imaging, vol. 25, no. 8, pp. 987-1010, 2006.

[5] T. Dietenbeck, M. Alessandrini, D. Barbosa, J. Dhooge, D. Friboulet, O. Bernard,“Detection of the whole myocardium in 2d-echocardiography for multiple orienta-tions using a geometrically constrained level-set”, Medical Image Analysis, vol. 16,no. 2, pp. 386401, 2012.

[6] D. Lamont, L. Parker, M. White, N. Unwin, S. M. A. Bennett, M. Cohen, D. Richard-son, H. O. Dickinson, A. Adamson, K. G. M. M. Alberti, A. W. Graft, “Risk ofcardiovascular disease measured by carotid intima-media thickness at age 49-51: Lifecourse study”, Br. Med. J., vol. 320, pp. 273-278, 2000.

[7] R. F. Wagner, S. W. Smith, J. M. Sandrik, H. Lopez, “Statistics of Speckle in Ultra-sound B-Scans”, IEEE Transactions on Sonics and Ultrasonics, vol. 30, no. 3, pp.156-163, 1983.

[8] D. J. Dowsett, P. A. Kenny, and R. E. Johnston, “The physics of diagnostic imaging”,Chapman and Hall Medical, 1998.

[9] D. Silverthorn, “Human Physiology: An Integrated Approach”, Benjamin Cummings,2006.

[10] R. E. Klabunde, “Cardiovascular Physiology Concepts”, Lippincott Williams Wilkins,2004

[11] A. Guyton and J. Hall, “Textbook of Medical Physiology”, W.B. Saunders Company,2000.

[12] J. Bushberg, J. Seibert, E. L. Jr, J. Boone, “The Essential Physics of Medical Imag-ing”, Lippincott Williams Wilkins, the second ed., 2001.

[13] J. G. Abbott, F. L. Thurstone, “Acoustic speckle: Theory and experimental analysis”,Ultrasound Imag., vol. 1, pp. 303-324, 1979.

[14] C. B. Burckhardt, “Speckle in ultrasound b-mode scans”, IEEE Transactions on Son-ics and Ultrasonics, vol. SU-25, no. 1, pp. 1-6, 1978.

[15] K. B. Oldham, J. Spanier, “The fractional calculus”, New York: Academic Press,1974.

70

[16] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, “Theory and applications of fractionaldifferential equations”, Amsterdam, Netherlands: Elsevier Science, 2006.

[17] Y. F. Pu, J. L. Zhou, Y. Xiao, “Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement”, IEEE Transactions on ImageProcessing, vol. 19, pp. 491-511. 2010.

[18] C.B. Gao, J. L. Zhou, J. R. Hu, F.N. Lang, “Edge detection of colour image basedon quaternion fractional differential” IET Image Processing, vol. 5, no. 3, pp 261-72,2011.

[19] A. Nakib, Y. Schulze, E. Petit, “Image thresholding framework based on two-dimensional digital fractional integration and Legendre moments”, IET Image Pro-cess, 2012, vol. 6, no. 6, pp 717-27.

[20] A. C. McBride, “Fractional calculus”, New York: Halsted Press, 1986.

[21] P.W. Huang, S, K, Dai, P.L. Lin, “Texture image retrieval and image segmentationusing composite sub-band gradient vectors”, Journal of Visual Communication ImageRepresentation, vol. 17, no. 5, pp. 947-57, 2006.

[22] E. R. Love, “Fractional derivatives of imaginary order”, Lond Math Soc, 1971, vol. 3,pp 241-59.

[23] D. Cafagna, “Fractional calculus: a mathematical tool from the past for present engi-neers”, IEEE Transactions on Industrial Electronics, vol 1, no. 2, pp. 35-40, 2007.

[24] K. S. Miller, B. Ross, “An introduction to the fractional calculus and fractional differ-ential equations”, New York: Wiley, 1993.

[25] S. Singh, A. Jain, V. Bhateja, “A comparative evaluation of various de-speckling algo-rithms for medical images”, In Proceedings of the CUBE International InformationTechnology Conference, Sep 3, pp. 32-37, ACM, 2012.

[26] A. Sharma, V. Bhateja, A. Tripathi, “An Improved Kuan Algorithm for Despeckling ofSAR Images”, Information Systems Design and Intelligent Applications, pp. 663-672,Springer, 2016.

[27] A. Lopes, R. Touzi, E. Nezry, “Adaptive speckle filters and scene heterogeneity”,IEEE Transactions on geoscience and remote sensing, vol. 28, no. 6, pp. 992-1000,1990.

[28] J. C. Bamber, C. Daft, “Adaptive filtering for reduction of speckle in ultrasonic pulse-echo images”, Ultrasonics, vol. 24, no.1, pp.41-44, 1986.

[29] T. Joel, R. Sivakumar, “Despeckling of ultrasound medical images: A survey”, Jour-nal of Image and Graphics, vol.1, no. 3, pp. 161-65, 2013.

[30] S. Finn, M. Glavin, E. Jones, “Echocardiographic speckle reduction comparison”,IEEE Transactions on Ultrasonics Ferroelectric Frequency Control, vol. 58, pp. 82-101, 2011.

71

[31] P. Coupe, P. Hellier, C. Kervrann, C. Barillot, “Nonlocal means-based speckle filteringfor ultrasound images”, IEEE transactions on image processing, vol. 18, no. 10, pp.2221-2229, 2009.

[32] B. Wilson, J. P. Dhas, “A Survey of Non-Local Means based Filters for Image De-noising”, International Journal of Engineering Research and Technology, vol. 2, no.10, 2013.

[33] Y. Zhan, M. Ding, L. Wu, X. Zhang, “Nonlocal means method using weight refin-ing for despeckling of ultrasound images”, Signal Processing, vol. 103, pp. 201-213,2014.

[34] P. V. Sudeep, P. Palanisamy, J. Rajan, H. Baradaran, L. Saba, A. Gupta, J. S. Suri,“Speckle reduction in medical ultrasound images using an unbiased non-local meansmethod”, Biomedical Signal Processing and Control, vol. 31, no. 28, pp. 1-8, 2016.

[35] Y. Gu, Z. Cui, C. Xiu, L. Wang, “Ultrasound echocardiography despeckling with non-local means time series filter”, Neurocomputing, vol. 124, pp. 120-30, 2014.

[36] V. Bhateja, G. Singh, A. Srivastava, “A novel weighted diffusion filtering approachfor speckle suppression in ultrasound images”, In Proceedings of the InternationalConference on Frontiers of Intelligent Computing: Theory and Applications (FICTA),pp. 459-466, 2013.

[37] S. A. Fernandez, C. A. Lopez, “On the estimation of the coefficient of variation foranisotropic diffusion speckle filtering”, IEEE Transactions on Image Processing, vol.15, no. 9, pp. 2694-2701, 2006.

[38] K. Krissian, C. F. Westin, R. Kikinis, K. G. Vosburgh, “Oriented speckle reducinganisotropic diffusion”, IEEE Transactions on Image Processing, vol. 16 no. 5, pp.1412-1424, 2007.

[39] Q. Sun, J. A Hossack, J. Tang, S. T. Acton, “Speckle reducing anisotropic diffusionfor 3D ultrasound images”, Computerized Medical Imaging and Graphics, vol. 28,no. 8, pp. 461-470, 2004.

[40] D. Mittal, V. Kumar, S. C. Saxena, N. Khandelwal, N. Kalra, “Enhancement of theultrasound images by modified anisotropic diffusion method”, Medical biologicalengineering computing, vol. 48, no. 12, pp. 1281-1291, 2010.

[41] C. P. Loizou, C. S. Pattichis, C. I. Christodoulou, R. S. Istepanian, M. Pantziaris, A.Nicolaides, “Comparative evaluation of despeckle filtering in ultrasound imaging ofthe carotid artery”, IEEE transactions on ultrasonics, ferroelectrics, and frequencycontrol, vol. 52, no. 10, pp. 1653-1669, 2005.

[42] T. Liu, N. Feng, Y. Shen, W. Pan, “Pyramid based multiscale anisotropic diffusionfilter for ultrasound image despeckling”, In the proceedings of IEEE InternationalInstrumentation and Measurement Technology Conference (I2MTC), pp. 1345-1349,2014.

[43] D. Ai, J. Yang, Y. Chen, W. Cong, J, Fan, Y. Wang. “Multiresolution generalized Ndimension PCA for ultrasound image denoising”, Biomedical engineering online, vol.13, no. 1, pp. 1, 2014.

72

[44] S. Gupta, R. C. Chauhan, S. C Sexana, “Wavelet-based statistical approach for specklereduction in medical ultrasound images”, Medical and Biological Engineering andcomputing. vol.42, no. 2, pp. 189-192, 2004.

[45] S. Sudha, G. R. Suresh, R. Sukanesh, “Speckle noise reduction in ultrasound imagesusing context-based adaptive wavelet thresholding”, IETE Journal of Research, vol.55, no. 3, pp. 135-143, 2009.

[46] M. S. Uddin, M. Tahtali, A. J. Lambert, M. R. Pickering, “Speckle reduction for ultra-sound images using nonlinear multi-scale complex wavelet diffusion”, In the proceed-ings of IEEE International Conference On Signal and Image Processing Applications(ICSIPA), pp. 31-36, 2013.

[47] H. H. Choi, J. H. Lee, S. M. Kim, S. Y. Park, “Speckle noise reduction in ultra-sound images using a discrete wavelet transform-based image fusion technique”, Bio-Medical Materials and Engineering, vol. 26, pp. 1587-97, 2015.

[48] X. Fu, Y. Wang, L. Chen, J. Tian, “An image despeckling approach using quantum-inspired statistics in dual-tree complex wavelet domain”, Biomedical Signal Process-ing and Control, vol. 18, pp. 30-35, 2015.

[49] C. Tomasi, R. Manduchi, “Bilateral filtering for gray and color images”, In the pro-ceedings of IEEE International Conference on Computer Vision, pp. 839-846, 1998.

[50] S. V. Sagheer, S. N. George, “A novel approach for de-speckling of ultrasound imagesusing bilateral filter”, In the proceedings of IEEE 3rd International Conference onRecent Advances in Information Technology (RAIT), pp. 453-459, 2016.

[51] T. Loganayagi, K. R. Kashwan, “A robust edge preserving bilateral filter for ultra-sound kidney image”, Indian Journal of Science and Technology, vol. 8, no. 23, 2015.

[52] D. Bhonsle, V. Chandra, G. R. Sinha, “Medical image denoising using bilateral filter”,International Journal of Image, Graphics and Signal Processing, vol. 4, no. 6, pp. 36,2012.

[53] D. Shao, P. Liu, D. C. Liu. “Characteristic matching-based adaptive fast bilateral filterfor ultrasound speckle reduction”, Pattern Recognition Letters, vol. 34, no. 5, pp. 463-469, 2013.

[54] A. Saadia, A. Rashdi, ”Fractional order integration and fuzzy logic based filterfor denoising of echocardiographic image.” Computer Methods and Programs inBiomedicine, vol. 137, pp. 65-75, Dec 2016.

[55] A. Saadia, A. Rashdi,“Echocardiography image enhancement using adaptive frac-tional order derivatives”, In the proceedings of IEEE International Conference of Sig-nal and Image Processing (ICSIP), Beijing, China, pp. 764-769, 2016.

[56] S. Khachira, F. Kallel, O. Tankyevych, O and A. B. Hamida, “Speckle reduction andsegmentation in echocardiographic images: A comparative study”, In the proceedingsof IEEE First International Image Processing, Applications and Systems Conference(IPAS), pp. 1-6, 2014.

73

[57] V. Dutt, F. James, “Adaptive speckle reduction filter for log-compressed B-scan im-ages”, IEEE Transactions on Medical Imaging, vol 15, no. 6, pp. 802-813, 1996.

[58] J. S. Lee, “Digital image enhancement and noise filtering by use of local statistics”,IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 2, pp. 165-168,1980.

[59] V. S. Frost, J, A. Stiles, K. S. Shanmuqan, “A model for radar images and its ap-plication to adaptive digital filtering of multiplicative noise”, IEEE Transactions onPattern Analysis and Machine Intelligence, vol.2, pp. 157-166, 1982.

[60] D. T. Kuan, A. Sawchuk, T. C. Strand, “Adaptive restoration of images with speckle”,IEEE Transactions on Speech and Signal Processing, vol. 35, no. 3, pp. 373-383,1987.

[61] T. Loupas, W. N. McDicken, P. L. Allan, “An adaptive weighted median filter forspeckle suppression in medical ultrasonic images”, IEEE transactions on Circuitsand Systems, vol. 36, no. 1, pp. 129-135, 1989.

[62] P. Perona, J. Malik, “Scale-space and edge detection using anisotropic diffusion”,IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.12, no. 7, pp.629-639, 1990.

[63] Y. Yu, T. A. Scott, “Speckle reducing anisotropic diffusion”, IEEE Transactions onImage Processing, vol.11, no. 11, pp.1260-1270, 2002.

[64] T. R. Crimmins, “Geometric filter for speckle reduction” Applied optics, vol. 24, no.10, pp. 1438-1443, 1985.

[65] D. Zhou, W. Cheng, “Image denoising with an optimal threshold and neighbouringwindow” Pattern Recognition Letters, pp.1694-1697, 2008.

[66] Y. Wang, W. Chen, S. Zhou, T. YU, Y. Zhang, “MTV: modified total variation modelfor image noise removal”, Electronics Letters, pp. 592-594, 2011.

[67] G. Aubertt, J. F. Aujol, “A variational approach to removing multiplicative noise”SIAM Journal of Applied Mathematics, vol. 68, no. 4, pp. 925-946, 2008.

[68] V. Bhateja, K. Rastoqi, A. Verma, C. Malhotra, “A Non-Iterative Adaptive MedianFilter for Image Denoising” IEEE International Conference on Signal Processing andIntegrated Networks (SPIN), Noida (UP). pp. 113-118, 2014.

[69] J. V . Manjon, C. Pierrick, B. Antonio, “MRI Noise Estimation and Denoising UsingNon-Local PCA”, Medical image analysis, vol. 22, no. 1, pp. 35-47, 2015.

[70] N. Biradar, M. L. Dewal, M. K. Rohit, “Speckle Noise Reduction in Echocardio-graphic Images of Aortic Valve and Cardiac Chambers”, Optik-International Journalfor Light and Electron Optics, vol. 126 , no.2, pp.153-163, 2015.

[71] Y. Zhang, H. D. Cheng, J. Tian, J. Huang , X, Tang, “ Fractional Subpixel Diffusionand Fuzzy Logic Approach for Ultrasound Speckle Reduction”. Pattern Recognition,2010, vol. 43, no. 8, pp. 2962-2970, 2010.

74

[72] D. Ai, J. Yang, Y. Chen, W. Cong, J. Fan, Y. Wang, “Multiresolution Generalized NDimension PCA for Ultrasound Image Denoising”, Biomedical engineering online,vol. 13, no. 1, pp. 112, 2014.

[73] H. K. Kwan, “Fuzzy filters for noise reduction in images”, Fuzzy Filters for ImageProcessing, Springer Berlin Heidelberg, pp. 25-53, 2003.

[74] H. K. Kwan, Y. Cai, “Fuzzy filters for image filtering”, IEEE 45th Midwest Sympo-sium on circuits and systems, vol. 3, pp. III-672, 2002.

[75] K. Binaee, P. H. Reza, “An ultrasound image enhancement method using local gradi-ent based fuzzy similarity”, Biomedical Signal Processing and Control, vol. 13, pp:89-101, 2014.

[76] J. Babu, F. S. Gnanou, “Adaptive speckle reduction in ultrasound images using fuzzylogic on Coefficient of Variation”, Biomedical Signal Processing and Control, vol.23, pp. 93-103, 2016.

[77] H. A. Jalab, W. I. Rabha, “Denoising algorithm based on generalized fractional inte-gral operator with two parameters”, Discrete Dynamics in Nature and Society, 2012.

[78] L. A. Zadeh, “Fuzzy sets, Information and Control, vol. 8, no. 3, pp. 338-353, 1965.

[79] L. A. Zadeh, “Toward a theory of fuzzy information granulation and its centrality inhuman reasoning and fuzzy logic Fuzzy Sets System, vol. 90, pp. 111-127, 1970.

[80] V. Garg and K. Singh, “An improved Grunwald-Letnikov fractional differential maskfor image texture enhancement”, International Journal of Advanced Computer Sci-ence and Applications, vol.3, no. 3, pp. 130-135, 2012.

[81] L. Cangpin, Q. Deliang and C. YangQuan, “On Riemann Liouville and Caputo deriva-tives”, Discret. Dyn. Nat. Soc, 2011. DOI: 10.1155/2011/ 563494. (article ID 562494).

[82] Y. F. Pu, J. L. Zhou and X. Yuan, Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement, IEEE Trans. Image Process, vol.19, no. 2, pp. 491-511, Feb 2010.

[83] J. Hu, Y. Pu, J. Zhou, A novel image denoising algorithm based on Riemann-Liouvilledefinition, Journal of Computers, vol. 6, no. 7, pp. 1332-1338, Jan 2011 DOI:10.4304/jcp.6.7.1332-1338.

[84] G. Huang, Y. F. Pu, Q. L. Chen and J. L. Zhou, “Research on image denoising basedon fractional order integral”, Systems Engineering and Electronics, vol. 33. no.4, pp.925-932, Apr 2011.

[85] N. He, J. B. Wang, L. L. Zhang and K. Lu, “An improved fractional-order differenti-ation model for image denoising”, Signal Processing, pp. 180-188, 2015.

[86] B. Li and W. Xie, “Image denoising and enhancement based on adaptive fractionalcalculus of small probability strategy”, Neurocomputing, 2015.

[87] B. Li and W. Xie, “Adaptive fractional differential algorithm based on otsu standard”,In Proceedings of IEEE Conference on Control and Decision, pp. 2020-2025, 2014.

75

[88] D. Gragnaniello, G. Poggi, G. Scarpa and L. Verdoliva, “SAR despeckling based onsoft classification”, In Proceedings of IEEE Conference on Geoscience and RemoteSensing Symposium (IGARSS), 2015, pp. 2378-2381.

[89] Y. Zhang, H. D. Cheng, J. Tian, J. Huang and X. Tang, “Fractional subpixel diffusionand fuzzy logic approach for ultrasound speckle reduction”, Pattern Recognition, vol.43, no. 8, pp. 2962-70, 2010.

[90] www.medison.ru

[91] G. Parisa, H. Behnam, and Z. Sani, “Noise reduction of echocardiographic imagesbased on temporal information”, IEEE Transactions Ultrasonics, Ferroelectrics, andFrequency Control. 2014. p. 620-630.

[92] E. Farzana, M. Tanzid, K. M. Mohsin, “Adaptive bilateral filtering for despeckling ofmedical ultrasound images”. In the Proceedings of IEEE Conference TENCON Japan,pp. 1728-1733, 2010.

[93] H. Cheng, J. Tian, “Speckle reduction of synthetic aperture radar images based onfuzzy logic”, First International Workshop on Education Technology and ComputerScience IEEE Computer Society. pp. 933-937, 2009.

[94] K. Binaee, R.P. R. Hansanzadeh, “An ultrasound image enhancement method usinglocal gradient based fuzzy similarity”, Biomedical Signal Processing Control, vol. 13,pp. 89-101, 2014.

[95] S. Kempfle, “Fractional calculus via functional calculus: theory and applications”Nonlinear Dyn, vol. 29, pp. 99-127, 2002.

[96] D. Cafagna, “Fractional calculus: a mathematical tool from the past for present engi-neers” IEEE Ind. Electron. Mag, vol. 1, no. 2, pp. 35-40, 2007.

[97] J. Manchado, V. Kiryakova, F. Mainardi, “Recent history of fractional calculus” Com-munications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140-1153, 2011.

[98] H.A. Jalab, R. W. Ibrahim, “Fractional Alexander polynomials for image denoising”Signal Process, vol. 107. pp. 340-354, 2015.

[99] Z. Pawlak, “Rough sets and intelligent data analysis”, Information sciences, vol. 147,no. 1, pp. 1-12, 2002.

[100] D. Pra, A. Lucia,“Temporal and Spatial Fuzzy Granularity of Laser and UltrasoundFrames”, VI Latin American Congress on Biomedical Engineering CLAIB Parana.Argentina, Springer International Publishing. 2015.

[101] S. Kuldeep, and R. Kapoor, “Image enhancement using exposure based sub imagehistogram equalization”, Pattern Recognition Letters, vol. 36, pp. 10-14, 2014.

[102] S . Huang, and C. Yeh, “Image contrast enhancement for preserving mean brightnesswithout losing image features”, Engineering Applications of Artificial Intelligence,vol. 26, pp. 1487-1492, 2013.

76

[103] E. Kaur, E . Jain, and E . Lather, “Study of Image Enhancement Techniques: A Re-view”, International journal of advanced research in computer science and softwareengineering, vol. 3, 2013.

[104] G. Artyom , J. Jenkinson, and S. Agaian, “Quaternion Fourier transform based alpha-rooting method for color image measurement and enhancement”, Signal Processing,vol. 109, pp. 269-289, 2015.

[105] S. Khan. W. Yin Chai, W. Soo See, and A. Khan, “X-Ray image enhancement using aboundary division wiener Filter and wavelet- based Image Fusion Approach”, Journalof Information Processing Systems, vol. 12, 2016.

[106] https://en.wikipedia.org/wiki/Medicalultrasound.htm

[107] https://en.wikipedia.org/wiki/ultrasound.htm

[108] http://www.medison.ru/uzi/eho408.htm

[109] http://www.medison.ru/uzi/eho36.htm

[110] http://www.rainbowhospitals.org

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