NOVEL BAYESIAN MULTISCALE METHODS FOR IMAGE DENOISING USING ALPHA ...(PhD Achim perilipsi)

UNIVERSITY OF PATRAS SCHOOL OF MEDICINE DEPARTMENT OF MEDICAL PHYSICS NATIONAL TECHNICAL UNIVERSITY OF ATHENS DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL TECHNICAL UNIVERSITY OF ATHENS DEPARTMENT OF MECHANICAL ENGINEERING NOVEL BAYESIAN MULTISCALE METHODS FOR IMAGE DENOISING USING ALPHA-STABLE DISTRIBUTIONS By Alin Achim SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY AT UNIVERSITY OF PATRAS PATRAS, GREECE JUNE 2003 Interdepartamental Program of Postgraduate Studies in BIOMEDICAL ENGINEERING c Copyright by Alin Achim, 2003 EPTAMELHS EXETASTIKH EPITROPH k A Mpezeriˆnos, Anaplhrwt s Kajhght s, Tm ma Iatrik s, Panepi mio Patr¸n k K Nik ta, Anaplhr¸tria Kajhg tria, Tm ma Hlektrolìgwn Mhqanik¸n kai Mhqanik¸n Upologi¸n, Ejnikì Metïbio Poluteqno k G Nikhforidhs, Kajhght s Iatrik s, Kajhght s, Tm ma Iatrik s, Panepi mio Patr¸n k N Pallhkarˆkhs, Kajhght s Iatrik s, Kajhght s, Tm ma Iatrik s, Panepi mio Patr¸n k G Panagiwtˆkhs, Anaplhrwt s Kajhght s, Tm ma Iatrik s, Panepi mio Patr¸n k A Stourths, Kajhght s, Tm ma Hlektrolìgwn Mhqanik¸n kai Mhqanik¸n Teqnologias Upologi¸n, Panepi mio Patr¸n k P T kalidhs, Anaplhrwt s Kajhght s, Tm ma Plhroforik s, Panepi - mio Kr ths TRIMELHS SUMBOULEUTIKH EPITROPH k A Mpezeriˆnos, Anaplhrwt s Kajhght s, Tm ma Iatrik s, Panepi mio Patr¸n, Prìedros k N Pallhkarˆkhs, Kajhght s Iatrik s, Kajhght s, Tm ma Iatrik s, Panepi mio Patr¸n, Mèlos k A Stourths, Kajhght s, Tm ma Hlektrolìgwn Mhqanik¸n kai Mhqanik¸n Teqnologias Upologi¸n, Panepi mio Patr¸n, Mèlos iii ΠΕΡΙΛΗΨΗ ∆Ι∆ΑΚΤΟΡΙΚΗΣ ∆ΙΑΤΡΙΒΗΣ Ο απώτερος σκοπός της έρευνας που παρουσιάζεται σε αυτή τη διδακτορική διατριβή είναι η διάθεση στην κοινότητα των κλινικών επιστηµόνων µεθόδων οι οποίες να παρέχουν την καλύτερη δυνατή πληροφορία για να γίνει µια σωστή ιατρική διάγνωση Οι εικόνες υπερήχων προσβάλλονται ενδογενώς από θόρυβο, ο οποίος οφείλεται στην διαδικασία δηµιουργίας των εικόνων µέσω ακτινοβολίας που χρησιµοποιεί σύµφωνες κυµατοµορφές Είναι σηµαντικό πριν τη διαδικασία ανάλυσης της εικόνας να γίνεται απάλειψη του θορύβου µε κατάλληλο τρόπο ώστε να διατηρείται η υφή της εικόνας, η οποία βοηθά στην διάκριση ενός ιστού από έναν άλλο Κύριος στόχος της διατριβής αυτής υπήρξε η ανάπτυξη νέων µεθόδων καταστολής του θορύβου σε ιατρικές εικόνες υπερήχων στο πεδίο του µετασχηµατισµού κυµατιδίων Αρχικά αποδείξαµε µέσω εκτενών πειραµάτων µοντελοποίησης, ότι τα δεδοµένα που προκύπτουν από τον διαχωρισµό των εικόνων υπερήχων σε υποπεριοχές συχνοτήτων περιγράφονται επακριβώς από µη-γκαουσιανές κατανοµές βαρέων ουρών, όπως είναι οι άλφα-ευσταθείς κατανοµές Μπεϋζιανούς εκτιµητές που αξιοποιούν αυτή τη στατιστική περιγραφή Κατόπιν, αναπτύξαµε Πιο συγκεκριµένα, χρησιµοποιήσαµε το άλφα-ευσταθές µοντέλο για να σχεδιάσουµε εκτιµητές ελάχιστου απόλυτου λάθος και µέγιστης εκ των υστέρων πιθανότητας για άλφα-ευσταθή σήµατα αναµεµειγµένα µε µηγκαουσιανό θόρυβο Οι επεξεργαστές αφαίρεσης θορύβου που προέκυψαν επενεργούν κατά µηγραµµικό τρόπο στα δεδοµένα και συσχετίζουν µε βέλτιστο τρόπο αυτή την µη-γραµµικότητα µε τον βαθµό κατά τον οποίο τα δεδοµένα είναι µη-γκαουσιανά Συγκρίναµε τις τεχνικές µας µε κλασσικά φίλτρα καθώς και σύγχρονες µεθόδους αυστηρού και µαλακού κατωφλίου εφαρµόζοντάς τες σε πραγµατικές ιατρικές εικόνες υπερήχων και ποσοτικοποιήσαµε την απόδοση που επιτεύχθηκε Τέλος, δείξαµε ότι οι προτεινόµενοι επεξεργαστές µπορούν να βρουν εφαρµογές και σε άλλες περιοχές ενδιαφέροντος και επιλέξαµε ως ενδεικτικό παράδειγµα την περίπτωση v Parintáilor mei, Ion ási Mariana ¸si surorii mele Laura vii viii Table of Contents Table of Contents ix Abstract xi Acknowledgements xiii Introduction 1.1 State of the Art 1.2 Contributions and Publications 1 Wavelets in Image Processing 2.1 Introduction to Wavelet Theory 2.1.1 Rationale for the Use of Wavelets in Signal Processing 2.1.2 Short-Time Fourier Transform vs Wavelet Transform 2.2 Dyadic Wavelet Transform 2.2.1 Multiresolution Analysis 2.2.2 Fast Discrete Wavelet Transform Algorithm in Two Dimensions 2.2.3 Daubechies’ Family of Regular Filters and Wavelets 2.3 Wavelet Shrinkage Principles 2.3.1 Hard and Soft Thresholding 2.3.2 Bayesian Wavelet Shrinkage 8 10 15 15 19 21 26 27 29 The 3.1 3.2 3.3 3.4 3.5 33 34 36 39 40 43 43 Alpha-Stable Family of Distributions Basic Properties of the Alpha-Stable Family The Class of Real SαS Distributions Bivariate Isotropic Stable Distributions Symmetric Alpha-Stable Processes Parameter Estimation for SαS Distributions 3.5.1 Maximum Likelihood Method ix 3.5.2 3.5.3 Method of Sample Quantiles Method of Sample Characteristic Function 44 45 Wavelet-based Ultrasound Image Denoising using an Alpha-Stable Prior Probability Model 4.1 Problem Formulation 4.2 Alpha-Stable Modeling of Ultrasound Wavelet Coefficients 4.3 A Bayesian Processor for Ultrasound Speckle Removal 4.4 Experimental Results 4.5 Summary 47 48 52 58 64 69 Ultrasound Image Denoising via Maximum a Posteriori Estimation of Wavelet Coefficients 5.1 SαS Parameters Estimation from Noisy Measurements 5.2 Design of a MAP Processor for Speckle Mitigation 5.3 Simulation Results 5.4 Discussions 73 74 76 78 80 Application to SAR Image Despeckling 6.1 Introduction 6.2 Modeling SAR Wavelet Coefficients with Alpha-Stable Distributions 6.3 Speckle Noise in SAR Images 6.4 Experimental Results 6.4.1 Synthetic Data Examples 6.4.2 Real SAR Imagery Examples 6.5 Discussions 83 84 86 93 95 95 99 103 Future Work Directions 105 Bibliography 107 x Abstract Before launching into ultrasound research, it is important to recall that the ultimate goal is to provide the clinician with the best possible information needed to make an accurate diagnosis Ultrasound images are inherently affected by speckle noise, which is due to image formation under coherent waves Thus, it appears to be sensible to reduce speckle artifacts before performing image analysis, provided that image texture that might distinguish one tissue from another is preserved The main goal of this thesis was the development of novel speckle suppression methods from medical ultrasound images in the multiscale wavelet domain We started by showing, through extensive modeling, that the subband decompositions of ultrasound images have significantly non-Gaussian statistics that are best described by families of heavy-tailed distributions such as the alpha-stable Then, we developed Bayesian estimators that exploit these statistics We used the alpha-stable model to design both the minimum absolute error (MAE) and the maximum a posteriori (MAP) estimators for alpha-stable signal mixed in Gaussian noise The resulting noise-removal processors perform non-linear operations on the data and we relate this non-linearity to the degree of non-gaussianity of the data We compared our techniques to classical speckle filters and current state-of-the-art soft and hard thresholding methods applied on actual ultrasound medical images and we quantified the achieved performance improvement Finally, we have shown that our proposed processors can find application in other areas of interest as well, and we have chosen as an illustrative example the case of synthetic aperture radar (SAR) images xi Acknowledgements First of all, I would like to thank Professor Anastasios Bezerianos, my main advisor, for his many suggestions and constant support during this research The door of his office was always open for me and he always found some time to hear my problems I am also grateful to Professor Nikolas Pallikarakis and Professor Athanasios Stouraitis for their participation in my advisory committee In particular, Professor Pallikarakis together with Professor Giorgos Kostopoulos have also helped me in a hard episode of my private live I will never forget their support Professor A Stouraitis expressed his interest in my work and provided me the reprints of some of his recent joint work with Y Karayiannis, which hopefully constitutes the ground for continued collaboration Besides the members in my advisory committee, there was a person without whom this work could not have been carried out in the way it has been done: Panos Tsakalides was the one who introduced me in the alpha-stable world and always helped me to keep the hope alive I’m sure that his advisory work put the bases for a long lasting collaboration and friendship Also, I should definitely mention here the great contribution that Dr Radu Negoescu from the Institute of Public Health and Professor F.M.G Tomescu from “Politehnica” University of Bucharest have had to my formation as an young researcher in Romania I would like to thank Dr C Frank Starmer and the IT Lab at the Medical University of South Carolina for providing most of the ultrasound images used in this thesis The image of the fetal chest used in Chapter has been provided by Acuson Corporation (Mountain Wiew, CA) Dr Daniel E Wahl from Sandia National Laboratories supplied me with part of the SAR imagery used in Chapter I am also grateful to Dr John P Nolan from American University who kindly provided his STABLE program in library form The State Scholarships Foundation (IKY) grant that was awarded to me for the period 1999–2003, was crucial to the successful completion of this project xiii 6.5 Discussions 6.5 103 Discussions We introduced a new statistical representation for the wavelet decomposition coefficients of SAR images, based on heavy-tailed alpha-stable models Consequently, we tested a MAP processor which relies on this representation and we found it to be more effective than traditional wavelet shrinkage methods both in terms of speckle reduction and signal detail preservation We evaluated the results on both synthetic data and real SAR images, all coded in 8-bit Naturally, our approach is more computationally expensive due to the fact that the prior distribution parameters need to be estimated at each decomposition scale of interest However, this is not a serious problem for off-line processing It should also be noted that in this work, the parameters of the SαS model are estimated globally within each decomposition scale For this reason, the shrinking functions shown in Figure 5.1 act the same for strong point target and for extended homogenous regions According to the results, the proposed filter achieves a global compromise between smoothing and edge preservation Chapter Future Work Directions Currently, we are addressing several issues related to the work we presented in this thesis One major issue is the choice of a statistical model for the speckle noise component of the wavelet coefficients that is more appropriate than the currently used Gaussian model It is to be tested whether the SαS family is a good model also for the noise component In this case, our problem will be formulated as Bayesian signal detection from measurements that are mixtures of SαS signal in SαS noise with different characteristic exponents, in general Statistical correlation between adjacent pixels is a result of diffraction effects in the transverse direction and intersymbol interference effects in the range direction [24] Speckle correlation was not considered in this work As we mentioned, this problem can be addressed by image subsampling at the expense of reduced spatial resolution A more sophisticated approach is to consider the speckle correlation structure into the MAP function A fully global Bayesian estimator based on alpha-stable statistics that takes into consideration both the inter- and intra-scale dependencies of the wavelet coefficient 106 Future Work Directions should be also developed This issue could eventually be addressed by first developing the theory of alpha-stable Markov random fields Following denoising, subsequent image analysis tasks should become easier to accomplish The alpha-stable model could be further used for developing image segmentation or texture classification/synthesis algorithms Finally, one issue that could be addressed and subsequently applied in 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Ultrasound Image Denoising using an Alpha- Stable Prior Probability Model 4.1 Problem Formulation 4.2 Alpha- Stable Modeling of Ultrasound Wavelet Coefficients 4.3 A Bayesian. .. state-of-the-art methods using simulated as well as real images The improvement is quantified using different quality measures The methods proposed in Chapters and can be easily adapted for the purpose of denoising. .. the wavelet transform, and (ii) a Bayesian denoising algorithm based on an alpha- stable prior for the signal First, the original image is logarithmically transformed to change multiplicative speckle