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In the using or citation of parts of the thesis it’s obliged to indicate the name of the author PHDTHESIS Electromagnetic Models for Ultrasound Image Processing Author: Supervisor: Hugo NAVARRETE Ramón MUJAL, Ph.D A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in the Department of Electrical Engineering November 2015 Declaration of Authorship I, Hugo NAVARRETE, declare that this thesis titled, ’Electromagnetic Models for Ultra- sound Image Processing’ and the work presented in it are my own. I confirm that: • This work was done wholly or mainly while in candidature for a research degree at this University. • Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly at- tributed. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. • I have acknowledged all main sources of help. • Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Date: iii “The same equations have the same solutions” Richard Phillips Feynman The Feynman Lectures on Physics Vol II v Acknowledgements I am deeply grateful to my advisor Dr. Ramón Mujal-Rosas, PhD for his helpful guidance and unconditional support. To AECI (Agencia Española de Cooperación In- ternacional) and COLCIENCIAS (Departamento Administrativo de Ciencia, Tecnología e Innovación, Colombia) for their financial support in the initial stages of the project. To my friend Dr Alejandro César Frery, PhD who first pointed out the basic ideas of this work. To my brother Dr. Solón Navarrete M.D. for his comments and fruitful dis- cussions. And to My Wife, Mónica: you are part of my life, this could not have been done without your love and support. vii Contents Declaration of Authorship.............................. iii Acknowledgements vii Abstract 1 Prologue 3 Motivation.......................................3 Objectives.......................................3 Contributions.....................................3 Thesis outline.....................................4 1 Introduction5 1.1 General Considerations.............................5 1.2 Central Limit Theorem Approach.......................6 1.3 Compound Representation Approach....................7 1.4 Multiplicative Model Approach........................8 1.5 Logarithmic compression...........................8 2 Statistical Models for Ultrasound RF data 11 2.1 Central Limit Theorem Approach....................... 11 2.1.1 Rayleigh distribution.......................... 11 2.1.2 Rice distribution............................ 12 2.1.3 K-Distribution.............................. 16 2.1.4 Homodyned K-Distribution...................... 18 2.2 Compound representation approach..................... 18 2.2.1 Homodyned K-Distribution...................... 18 2.2.2 Generalized K-Distribution...................... 30 2.2.3 Rician Inverse Gaussian Distribution (RiIG)............ 30 2.3 Multiplicative model approach........................ 30 2.3.1 Generalized K-Distribution...................... 31 3 A general model for ultrasound B-scan images 33 3.1 Statistical models for SAR data........................ 33 3.1.1 Speckle Noise Model.......................... 33 3.1.2 The Multiplicative Model....................... 34 3.2 GAistributions for SAR images........................ 35 3.2.1 KA-distribution for SAR image.................... 37 3.2.2 GA0 distribution for SAR image................... 37 3.3 G0 distribution for ultrasound image processing.............. 38 3.4 HG0 distribution for Log-compressed B-scan images............ 39 3.4.1 Logarithmic Compression Model................... 39 3.4.2 A New Statistical Model of Log-compressed B-scan images... 40 3.5 Moments generating function method for HG0 parameter estimation.. 41 ix 3.6 Maximum likelihood method for HG0 parameter estimation....... 42 4 Log-compressed data modeling 43 4.1 Montecarlo simulation of Log-compressed data............... 43 4.2 Homodyned-K distributed data simulation................. 43 4.3 Stable process simulation........................... 44 4.4 Log Compression................................ 44 4.5 HG0 distribution for modeling Log-compressed data........... 44 4.6 Concluding Remarks.............................. 45 5 Central Limit Theorem Revisited 73 5.1 Generalized Central limit theorem...................... 73 5.2 α-Stable distributions.............................. 73 5.2.1 Definition................................ 74 5.3 Infinite divisibility and Alpha-stable distributions............. 74 5.4 Self-decomposable distributions....................... 74 5.5 The Kullback–Leibler divergence....................... 75 5.6 G0 and Alpha-stable distributions...................... 75 5.7 The main theorem................................ 75 5.8 Proof....................................... 75 5.8.1 Multiplicative Model and G0 distribution in Intensity format.. 75 5.8.2 G0 Multiplicative Model representation............... 76 5.8.3 G0 Compound representation..................... 76 5.8.4 Shanbhag and Sreehari, 1977 theorem................ 76 5.8.5 Stable distributions in Amplitude format.............. 78 5.8.6 Log-compression of stable distributions............... 78 5.9 The multiplicative model and the Generalized Central Limit Theorem. 78 5.10 Compound representation and the multiplicative model......... 78 5.11 Concluding remarks.............................. 79 6 Applications 81 6.1 B-scan image filtering.............................. 81 6.2 B-scan image segmentation.......................... 81 6.3 Heart ejection fraction estimation....................... 81 7 Conclusions and future work 85 7.1 Conclusions................................... 85 7.2 Future Work................................... 85 A Moments generating function for HG0 distribution 87 B MATLAB code used in this work 89 B.1 Simulating and plotting K-distributed data................. 89 B.1.1 PDF and CDF evaluation....................... 89 B.1.2 Plotting PDF and CDF......................... 89 B.2 Simulating and plotting Homodyned K-distributed data......... 90 B.2.1 Simulating Homodyned-K random numbers............ 90 B.2.2 Montecarlo experiments........................ 90 B.2.3 Polynomial fit for HG0 parameters estimation........... 91 B.2.4 Polynomial objective function for for HG0 parameters estimation 93 B.2.5 Parameters Estimation......................... 93 x B.2.6 Kullback-Leibler divergence..................... 94 C MATLAB Code for Stable Distributions 95 C.1 Stable Distribution PDF............................ 95 C.2 Stable Distribution CDF............................ 102 C.3 Stable Distributed Random Numbers.................... 106 C.4 Stable Distribution data fitting......................... 107 C.5 Stable Distribution CDF inversion...................... 118 D MATLAB Code for Applications 137 D.1 Speckle Filtering................................. 137 D.2 Image Segmentation.............................. 138 Bibliography 141 xi List of Figures 2.1 Rayleigh Distribution.............................. 13 2.2 Rice Distribution Motivation......................... 14 2.3 Rice