UNIVERSITY OF PATRAS INTERDEPARTMENTAL POSTGRADUATE PROGRAM IN MEDICAL PHYSICS. PhD Thesis

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1 UNIVERSITY OF PATRAS SCHOOL OF HEALTH SCIENCES FACULTY OF MEDICINE SCHOOL OF NATURAL SCIENCES DEPARTMENT OF PHYSICS INTERDEPARTMENTAL POSTGRADUATE PROGRAM IN MEDICAL PHYSICS PhD Thesis Image Processing and Analysis Methods in Thyroid Ultrasound Imaging Tsantis Stavros Patras, 007, Hellas We thank the Operational Program for Educational and Vocational Training II (EPEAEK II) for funding this PhD thesis.

2 ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΤΜΗΜΑ ΙΑΤΡΙΚΗΣ ΤΜΗΜΑ ΦΥΣΙΚΗΣ ΔΙΑΤΜΗΜΑΤΙΚΟ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ ΙΑΤΡΙΚΗ ΦΥΣΙΚΗ Διδακτορική Διατριβή Μέθοδοι Επεξεργασίας και Ανάλυσης Υπερηχογραφικής Εικόνας του Θυρεοειδούς Αδένα Σταύρος Τσαντής Πάτρα, 007 Ευχαριστούμε το Επιχειρησιακό Πρόγραμμα Εκπαίδευσης και Αρχικής Επαγγελματικής Κατάρτισης (ΕΠΕΑΕΚ ΙΙ) για την χρηματοδότηση της διδακτορικής διατριβής

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4 SUPERVISING COMMITTEE 1. George Nikiforidis, Professor, Department of Medical Physics, University of Patras (Supervisor), Greece. Dionisis Cavouras, Professor, Department of Medical Instruments Technology, Technological Institute of Athens, Greece 3. Vassilis Anastasopoulos, Professor, Department of Physics, University of Patras, Greece EXAMINING COMMITTEE 1. George Nikiforidis, Professor, Department of Medical Physics, University of Patras, Greece.. Vassilis Anastasopoulos, Professor, Department of Physics, University of Patras, Greece. 3. George Panayiotakis, Professor, Department of Medical Physics, University of Patras, Greece. 4. Anastasios Bezerianos, Professor, Department of Physics, University of Patras, Greece. 5. Dimitrios Siamplis, Professor, School of Medicine, University of Patras, Greece. 6. Kostas Berberidis, Associate Professor, Department of Computer Engineering and Informatics, University of Patras, Greece. 7. George Oikonomou, Assistance Professor, Department of Physics, University of Patras, Greece.

5 Η έμπνευση διαψεύδει την αντίληψη Σοφοκλής ( π.χ) Inspiration denies perception Sofoklis ( b.c)

6 ACKNOWLEDGEMENTS I wish to express my gratitude to my supervisor Professor G. Nikiforidis for the assignment of this proect and for his suggestions and guidance throughout this thesis. I am also grateful to Professor D. Cavouras for his faith and confidence towards me and for his contribution in the fulfillment of this thesis. I would also like to thank him for his valuable guidelines in writing scientific articles. I would like to thank Dr N. Dimitropoulos for his selfness support throughout this thesis, his proposition regarding the topic of this thesis and his valuable medical guidelines regarding thyroid imaging. I would also like to thank N. Arikidis mainly for his friendship and for the long conversations regarding wavelet theory, Dr I. Kalantzis and Dr N. Piliouras for their important guidance in pattern recognition theory and algorithms. Finally, I wish to express my gratitude to my parents for their constant support and encouragement during the years of that work.

7 TABLE OF CONTENTS ΕΛΛΗΝΙΚΗ ΠΕΡΙΛΗΨΗ-SUMMARY IN GREEK CHAPTER 1 Introduction Power of Ultrasound 1 1. Need for Image Processing and Analysis Methods in Thyroid Ultrasound Images Aims and Novelties of Thesis Wavelet Based Image Processing 1.3. Image Analysis Publications Dissertation Layout 7 CHAPTER Thyroid Gland 9.1 Introduction 9. Thyroid Disorders 10.3 Management of Solitary Nodules 11.4 Grading 13 CHAPTER 3 Physics & Instrumentation of Ultrasound Nature of Ultrasound Propagation in Tissue Pulse Echo Imaging Instrumentation Quality control of the ultrasound system Data Acquisition and Storage 18 CHAPTER 4 The Wavelet Transform 1 Summary Wavelet Theory 1 4. Continuous Wavelet transform 4.3 Redudant Dyadic Wavelet Transform (1-D) Redudant Dyadic Wavelet Transform (-D) Multiscale edge representation 3 CHAPTER 5 Singularity Detection 35 Summary Singularity and mathematical description Wavelet transform and singularity Singularity Detection (1-D) Singularity Detection (-D) 41

8 CHAPTER 6 Pattern Recognition 43 Summary Pattern Recognition Theory Obect Isolation Feature Generation Textural Features First order statistical features Second order statistical features Co-Occurrence matrix features Run-Length matrix features Shape and Geometrical features Local maxima features Data Normalization Classification task Minimum distance classifiers Bayesian classifier Neural networks classifiers The Support Vector Machines Classifier 57 CHAPTER 7 Wavelet-based speckle suppression in ultrasound images 61 Summary Review of the Literature Materials and methods Overview and Implementation of the Algorithm Speckle Model Inter-scale Wavelet Analysis Dyadic Wavelet Transform Gradient Vector Modulus Maxima Lipschitz Regularity Detection of singularities Experimental Results and Evaluation Tissue mimicking Phantom Validation US image Case Study Observer evaluation study Discussion and Conclusions 85 CHAPTER 8 - Thyroid Nodule Boundary Detection in ultrasound images 87 Summary Review of the Literature Materials and Methods Overview and implementation of the algorithm US data Acquisition Edge Detection Procedure 94

9 Multiscale Edge Representation Coarse to Fine Analysis Multi-scale Structure Model Maxima Linking Structure Identification Nodule s Boundary Extraction Constrained Hough Transform Accumulator Local Maxima Detection Results Discussion and Conclusion 106 CHAPTER 9 Development of a Support Vector Machine Based Image Analysis System for Assessing the Thyroid Nodule Malignancy Risk on Ultrasound 109 Summary Review of the Literature Materials and Methods US image data acquisition Data pre-processing Classification Support vector machine classifier Multilayer perceptron (MLP) classifier Quadratic least squares minimum distance classifier Quadratic Bayesian classifier Support Vector Machines Wavelet Kernels Wavelet Kernels implementation System performance evaluation Results and discussion SVM Classification Outcome MLP Classification Outcome GLSMD & QB Classification Outcome SVM with Wavelet Kernels Classification Outcome 1 CHAPTER 10 Pattern Recognition Methods Employing Morphological and Wavelet Local Maxima Features towards Evaluation of Thyroid Nodules Malignancy Risk in Ultrasonography 17 Summary Materials and Methods Patients Feature extraction Feature selection and classification Results and discussion SVM & PNN model Evaluation without the presence of Speckle SVM & PNN model Evaluation with the presence of Speckle Conclusion 139

10 CHAPTER 11 Conclusion and Future Work Conclusion Feature Work 14 REFERENCES 143 APPENDIX I List of Figures 157 APPENDIX II List of Tables 161 APPENDIX III Abbreviations 163 APPENDIX III Index of Terms 165

11 ΜΕΘΟΔΟΙ ΕΠΕΞΕΡΓΑΣΙΑΣ ΚΑΙ ΑΝΑΛΥΣΗΣ ΥΠΕΡΗΧΟΓΡΑΦΙΚΗΣ ΕΙΚΟΝΑΣ ΤΟΥ ΘΥΡΕΟΕΙΔΟΥΣ ΑΔΕΝΑ Εισαγωγή Η καθιέρωση της υπερηχογραφίας ως ένα πολύτιμο εργαλείο στην πλειονότητα των ιατρικών εφαρμογών παγκοσμίως, συνδέεται άμεσα με την ραγδαία τεχνολογική εξέλιξη των συστημάτων απεικόνισης που υιοθετούνται στην ιατρική και τη βιολογία. Ο σχεδιασμός και η υλοποίηση ολοένα και πιο σύγχρονων συστημάτων απεικόνισης έδωσε την δυνατότητα στην υπερηχογραφία να διεισδύσει σε ιατρικές ειδικότητες, όπως η Ορθοπεδική, η Εντατική θεραπεία, η Διαβητολογία κλπ., στις οποίες λίγα έτη πριν, η εφαρμογή της ήταν απαγορευμένη. Το γεγονός αυτό μετέβαλε ριζικά την φύση της προληπτικής ιατρικής στην σύγχρονη εποχή. Στην πραγματικότητα, η υπερηχογραφική εξέταση αναγνωρίζεται πλέον ως μια θεμελιώδης τεχνική στην πρόληψη, τη διάγνωση και τη θεραπεία ενός συνεχώς διευρυνόμενου φάσματος ασθενειών. Η υπερηχογραφία επέτυχε να επιβάλει την παρουσία της σε κάθε ιατρό. Από μια μικρή ιδιωτική κλινική μέσω μιας φορητής μονάδος, ως ένα γενικό νοσοκομείο μέσω ενός ακριβού συστήματος απεικόνισης τεσσάρων διαστάσεων, ο υπέρηχος αποδεικνύει την αποτελεσματικότητα και την ακρίβειά του σε καθημερινή βάση. Η ψηφιακή απεικόνιση του θυρεοειδούς αδένα, τόσο μέσω της κλασσικής δισδιάστατης B-Mode εικόνας όσο και μέσω της έγχρωμης απεικόνισης Doppler, καθιστά την υπερηχογραφία ένα αξιόπιστο και εύχρηστο μέσο για την κλινική αξιολόγηση του. Η υψηλή διακριτική ικανότητα των σύγχρονων αυτών συστημάτων παρέχει στον Ιατρό την δυνατότητα να εντοπίσει την ύπαρξη όζων είτε συμπαγών είτε κολλοειδών στον θυρεοειδή αδένα ακόμα και με διαστάσεις πολύ μικρές (1mm) [14-8]. Επίσης, η υπερηχογραφική εικόνα επιτρέπει την λήψη βιοψίας (Fine Needle Aspiration - FNAΒ) σε πραγματικό χρόνο για την περαιτέρω αξιολόγηση του όζου [9-33]. H διεθνής αρθογραφία αμφισβήτησε την προηγούμενη δεκαετία την υπερηχογραφική εικόνα, ως ένα αξιόπιστο μέσο για τον διαχωρισμό ενός κακοήθους από έναν καλοήθη όζο. Στην I

12 κλινική διάγνωση ωστόσο, πληθώρα υπερηχογραφικών χαρακτηριστικών όπως η ηχογένεια, το περίγραμμα, η παρουσία αποτιτανώσεων και η διόγκωση με τον χρόνο έχουν καθιερωθεί τα τελευταία χρόνια ως ενδείξεις πιθανής κακοήθειας [ ]. Αυτό το γεγονός, σε συνδυασμό με την έλλειψη πρόσφατων ποσοτικών μελετών στην αξιολόγηση της φύσης των όζων, καθιστά απαραίτητη την έρευνα για τον σχεδιασμό και υλοποίηση αλγορίθμων επεξεργασίας και αναγνώρισης προτύπων στην υπερηχογραφική εικόνα. Οι αλγόριθμοι αυτοί έχουν ως σκοπό την αύξηση της ακρίβειας ταξινόμησης των όζων και την παροχή βοήθειας στους ιατρούς κατά την προ-εγχειρητική αντιμετώπιση των ασθενών. Η παρούσα διατριβή πραγματεύεται τον σχεδιασμό, την ανάπτυξη και υλοποίηση νέων μεθόδων: 1. Επεξεργασίας εικόνας βασισμένες στον μετασχηματισμό μικροκυματιδίων (Wavelet Transform) με σκοπό την αφαίρεση θορύβου και την τμηματοποίηση των όζων.. Ανάλυσης εικόνας με σκοπό την αυτόματη ταξινόμηση των όζων σε όζους υψηλού και χαμηλού κινδύνου. 1. Μέθοδοι Επεξεργασίας Εικόνας με βάση τον Wavelet 1.1 Εισαγωγή Transform (WT) Παρά τα μεγάλα πλεονεκτήματα της υπερηχογραφίας, στις εικόνες εμφανίζεται μια κοκκώδης υφή η οποία αποτελεί έναν σημαντικό παράγοντα υποβάθμισης της ποιότητας εικόνας. Όταν μια δέσμη υπερήχων προσπίπτει σε μια ανομοιογενή επιφάνεια ή σε σωματίδια με μέγεθος ή αποστάσεις μεταξύ τους μικρότερες από το όριο της χωρικής διακριτικής ικανότητας του συστήματος, παρουσιάζονται φαινόμενα συμβολής (αφαιρετικής και ενισχυτικής) με αποτέλεσμα τόσο την παραμόρφωση των ανατομικών δομών όσο και την διαφοροποίηση στην ένταση των ανακλώμενων από αυτές ηχητικών κυμάτων (διαφοροποίηση των τόνων του γκρι). Οι έντονες αυτές διακυμάνσεις στην ένταση των ανακλώμενων κυμάτων μέσα σε μια ομοιόμορφη ανατομική περιοχή συνθέτουν το speckle [90,111]. Η υπερηχογραφική εικόνα με την παρουσία του speckle πολλές φορές δεν αντιστοιχεί στην πραγματική δομή των εικονιζόμενων ιστών. Το speckle υποβαθμίζει τις μικρές λεπτομέρειες και τον σαφή καθορισμό των οριογραμμών της περιοχής ενδιαφέροντος. Επιπλέον, αποτελεί έναν περιοριστικό παράγοντα για τον σχεδιασμό αλγορίθμων τμηματοποίησης και αναγνώρισης προτύπων. II

13 Εκτός της παρουσίας του speckle, διάφορες ιδιότητες των υπερηχητικών κυμάτων μπορούν να προκαλέσουν την ποιοτική υποβάθμιση και ενδεχομένως ψευδείς ενδείξεις στις εικόνες υπερηχογραφίας. Η αντήχηση, η σκίαση, η διάθλαση και οι πλευρικοί λοβοί υποβαθμίζουν την διακριτική ικανότητα της απεικόνισης και κατά συνέπεια την συνολική ποιότητά της [47,49]. Τα προαναφερθέντα προβλήματα που προκύπτουν από τη σύνθετη φύση της υπερηχογραφικής απεικόνισης, καθιστούν την ακριβή ανίχνευση των ορίων μιας συγκεκριμένης ανατομικής δομής αρκετά δύσκολη ακόμα και για τους ακτινολόγους με μεγάλη πείρα. Ένας ακριβής υπολογισμός του περιγράμματος ενός όζου του θυρεοειδούς αποτελεί καθοριστικό παράγοντα τόσο στον εντοπισμό του μεγέθους και της θέση του όζου όσο και στην ταξινόμηση των όζων με βάση διάφορα μορφολογικά χαρακτηριστικά. Επιπλέον, μπορεί βοηθήσει στην ακριβή τοποθέτηση της βελόνας βιοψίας σε πραγματικό χρόνο κατά την διαδικασία της FNAB [9-33]. Οι περισσότεροι σύγχρονοι αλγόριθμοι επεξεργασίας εικόνας δεν είναι αποτελεσματικοί όταν εφαρμόζονται απευθείας στις τιμές έντασης των τόνων του γκρι της υπερηχογραφικής εικόνας. Αυτές οι τιμές έντασης είναι ιδιαίτερα πλεονάζουσες, ενώ το ποσοστό σημαντικών πληροφοριών μέσα στην εικόνα μπορεί να είναι σχετικά μικρό. Η απεικόνιση των τόνων του γκρι υπό μια διαφορετική γωνία μπορεί να αναδείξει διάφορα σημαντικά χαρακτηριστικά γνωρίσματα που δεν είναι ευδιάκριτα στην αρχική εικόνα. Ο μετασχηματισμός με βάση τα μικροκυματίδια (wavelet transform) της αρχικής υπερηχογραφικής εικόνας σε μια απεικόνιση χαρακτηριστικών αποκαλύπτει τα χρήσιμα γνωρίσματα μιας εικόνας χωρίς την απώλεια ουσιαστικών πληροφοριών, μειώνει τον πλεονασμό των διάφορων τόνων του γκρι και αποβάλλει οποιεσδήποτε μη χρήσιμες πληροφορίες [55]. Όταν σε μια εικόνα οι σημαντικές δομές έχουν διαφορετικά μεγέθη, οι κλίμακες πρέπει να ποικίλουν. Οι αιχμές σε διάφορες κλίμακες αντιστοιχούν σε διαφορετικές φυσικές οντότητες. Τα μεγάλα αντικείμενα απεικονίζονται με ακρίβεια στις μεγάλες κλίμακες ενώ οι μικρές δομές εντοπίζονται ευκολότερα στις μικρές κλίμακες. Ο μετασχηματισμός μικρο-κυματιδίων σε πολλαπλές κλίμακες παρέχει την δυνατότητα μελέτης και ανάλυσης διαφόρων αιχμών μεταβάλλοντας το μέγεθος του οπτικού παραθύρου εξέτασης [57,58]. Οι Mallat και Zong [6] έχουν αποδείξει ότι η απεικόνιση των αιχμών σε διαφορετικές κλίμακες παρέχει μια πλήρη και σταθερή αντιπροσώπευση της αρχικής εικόνας, γεγονός το οποίο σημαίνει ότι το σύνολο των πληροφοριών της εικόνας υπάρχει σε αυτές τις αιχμές. Η διαπίστωση αυτή υποδηλώνει ότι οι διάφορες μέθοδοι επεξεργασίας εικόνας μπορούν να χρησιμοποιηθούν απευθείας στην απεικόνιση των αιχμών αντί στην αρχική εικόνα. Οι διάφορες αιχμές ή τα περιγράμματα (ομάδες αιχμών με παρόμοιες ιδιότητες) αντιστοιχούν στις έντονες αντιθέσεις των τόνων του γκρι και μπορούν να αντιπροσωπευθούν από τα τοπικά III

14 μέγιστα της μετατροπής μικρο-κυματιδίων. Η αναπαράσταση τοπικών μεγίστων είναι μια αναδιοργάνωση των πληροφοριών της εικόνας, η οποία παρέχει με μεγαλύτερη ανάλυση την απεικόνιση διαφόρων δομών. Μια ξεχωριστή ιδιότητα του μετασχηματισμού μικροκυματιδίων (WT) είναι η ικανότητα του να χαρακτηρίζει την τοπική κανονικότητα διαφόρων χαρακτηριστικών της εικόνας όπως ασυνέχειες ή έντονες αιχμές. Η τοπική κανονικότητα υπολογίζεται μέσω των εκθετών Lipschitz [61]. Η απεικόνιση των αιχμών σε διάφορες κλίμακες παρέχει τη δυνατότητα της ανίχνευσης των διαφόρων ιδιομορφιών (singularities σημεία στον χώρο όπου λαμβάνουν χώρα απότομες μεταβολές) που βρίσκονται σε μια εικόνα και του υπολογισμού της κανονικότητας τους. Σε αυτήν την διατριβή πραγματοποιήθηκε μια έρευνα σχετικά με την συμπεριφορά των ιδιομορφιών (singularities) σε μια υπερηχογραφική εικόνα του θυρεοειδούς αδένα υπολογίζοντας την Lipschitz κανονικότητα τους. Τα τοπικά μέγιστα που δημιουργούνται από τις ιδιομορφίες (singularities) που αντιστοιχούν σε θόρυβο έχουν διαφορετική συμπεριφορά από αυτά που επηρεάζονται κυρίως από τις ωφέλιμα χαρακτηριστικά της εικόνας. Η έρευνα κατέληξε στην διαπίστωση ότι η κατανομή του speckle το μοντέλο θορύβου το οποίο υιοθετήθηκε σε αυτήν την διατριβή θεωρεί το speckle σαν έναν αθροιστικό παράγοντα, άμεσα εξαρτώμενο από το αρχικό σήμα είναι σχεδόν παντού σημειακή (singular) ή ασυνεχής με μη θετικούς εκθέτες Lipschitz. Αντιθέτως, οι χρήσιμες ιδιομορφίες (singularities) που προέρχονται από την φυσιολογική κατανομή των τόνων του γκρι στην υπερηχογραφική εικόνα αποτελούν έντονες αιχμές με θετικούς εκθέτες Lipschitz. 1. Αφαίρεση Θορύβου στις Υπερηχογραφικές Εικόνες 1..1 Εισαγωγή Σαν εφαρμογή της προαναφερθείσας διαπίστωσης, ένας αλγόριθμος αναπτύχθηκε η οποίος αφαιρεί το speckle από τις υπερηχογραφικές εικόνες υπερήχου με την ανάλυση της βαθμιαίας εξέλιξης των τοπικών μεγίστων στις διάφορες κλίμακες. Αυτή η αναζήτηση πραγματοποιήθηκε με τον εντοπισμό των αιχμών στις μεγάλες κλίμακες και την προοδευτική ανίχνευση τους προς την μικρότερη δυνατή κλίμακα ώστε μελετηθεί η συμπεριφορά τους. Στην διεθνή βιβλιογραφία έχει εφαρμοσθεί πληθώρα μεθόδων για την αφαίρεση του speckle σε εικόνες υπερηχογραφίες διαφόρων ανατομικών δομών. Οι πρώτες μέθοδοι χρησιμοποίησαν διάφορες παραμέτρους από το ιστόγραμμα της εικόνας για την αφαίρεση του speckle [9,93,94]. Στην δεκαετία του 1990 η αύξηση της υπολογιστικής ισχύος των Η/Υ προκάλεσε την ταυτόχρονη αύξηση της πολυπλοκότητας των φίλτρων για την καταστολή του speckle. Οι μέθοδοι IV

15 εφαρμόζονταν κατά βάση στο πεδίο του χρόνου [95-100]. Επίσης, στο δεύτερο μισό της δεκαετίας του 90 καινούργιες μέθοδοι προτάθηκαν με βάση τον μετασχηματισμό μικροκυματιδίων [ ]. 1.. Μέθοδοι Η τεχνική αφαίρεσης θορύβου περιλαμβάνει τα ακόλουθα βήματα: 1. Δυαδικός Μετασχηματισμός μικροκυματιδίων (Dyadic Wavelet Transform). Αναπαράσταση Τοπικών Μεγίστων (Modulus maxima representation) 3. Ομαδοποίηση των τοπικών μεγίστων από την μεγαλύτερη στην μικρότερη κλίμακα του μετασχηματισμού (Coarse to fine grouping of local maxima) 4. Υπολογισμός κανονικότητας Lipschitz (Lipschitz regularity calculation) 5. Αντίστροφος Δυαδικός Μετασχηματισμός μικροκυματιδίων (Inverse Dyadic Wavelet Transform) 1. Δυαδικός Μετασχηματισμός μικροκυματιδίων: Η ανάλυση μικροκυματιδίων που παρουσιάζεται σε αυτήν την διατριβή χρησιμοποιεί τον Δυαδικό Μετασχηματισμό μικροκυματιδίων για το χαρακτηρισμό των σημάτων από τις αιχμές σε διάφορες κλίμακες (Multiscale Edge Representation) [6]. Η ανάλυση μικροκυματιδίων σε διάφορες κλίμακες της αρχικής εικόνας εφαρμόστηκε με μια τράπεζα φίλτρων η οποία αποκαλείται ως `algorithme a atrous (αλγόριθμος με τρύπες) [63].. Αναπαράσταση Τοπικών Μεγίστων: O δυσδιάστατος μετασχηματισμός μικροκυματιδίων μιας εικόνας μπορεί να θεωρηθεί σαν ένα άνυσμα κλίσης το οποίο μπορεί να αναπαρασταθεί από το μέτρο και τη γωνία του. Ένα σημείο θεωρείται ως τοπικό μέγιστο εάν το πλάτος του ανύσματος κλίσης είναι μεγαλύτερο έναντι των γειτονικών του κατά μήκος της κατεύθυνσης που δίνεται από τη γωνία του [6,66]. 3. Ομαδοποίηση των τοπικών μεγίστων από την μεγάλη στην μικρή κλίμακα του μετασχηματισμού: Στην παρούσα μελέτη η πληροφορία η οποία βρίσκεται στις διαφορετικές ζώνες συχνότητας (κλίμακες) αναλύεται ώστε να υπολογιστεί η τοπική κανονικότητα των χαρακτηριστικών της εικόνας. Αυτές οι πληροφορίες στις διάφορες κλίμακες υπολογίζονται μέσω μια διαδικασίας ανίχνευσης τοπικών μεγίστων με κοινές ιδιότητες τα οποία διαδίδονται από την τελευταία κλίμακα μέχρι την πρώτη (Back-propagation Tracking). Η ομαδοποίηση των τοπικών μεγίστων σε όλες τις διαθέσιμες κλίμακες βασίζεται στην αρχή ότι: εάν μια αιχμή υπάρχει σε ένα μεγαλύτερο επίπεδο, μπορεί επίσης να εντοπιστεί σε ένα μικρότερο επίπεδο V

16 1. Δύο τοπικά μέγιστα από δύο διαδοχικές κλίμακες ομαδοποιούνται εάν κατέχουν κοντινές θέσεις και γωνίες στο επίπεδο της εικόνας [68,69]. 4. Υπολογισμός κανονικότητας Lipschitz: Η διακύμανση του πλάτους των τοπικών μεγίστων σε διαφορετικές κλίμακες συσχετίζεται με την τοπική κανονικότητα των αντίστοιχων δομών που αντιπροσωπεύουν τα τοπικά μέγιστα. Η κλίση της καμπύλης των τοπικών μεγίστων σε λογαριθμική κλίμακα αποτελεί την κανονικότητα αυτών των μεγίστων και αξιολογείται μέσω των εκθετών Lipschitz "α". Όταν το πλάτος των τοπικών μεγίστων σε μια ομάδα μειώνεται όσο μειώνεται και η κλίμακα η κανονικότητα Lipschitz είναι θετική (θετικοί εκθέτες Lipschitz). Όταν το πλάτος των τοπικών μεγίστων σε μια ομάδα αυξάνεται όσο μειώνεται η κλίμακα η κανονικότητα Lipschitz είναι αρνητική (αρνητικοί εκθέτες Lipschitz). Το speckle αντιστοιχεί σε αρνητικούς εκθέτες Lipschitz ενώ οι διάφορες δομές της υπερηχογραφικής εικόνας αντιστοιχούν σε θετικούς εκθέτες Lipschitz [68-71]. 5. Αντίστροφος Δυαδικός Μετασχηματισμός μικροκυματιδίων: Η διάκριση των τοπικών μεγίστων που αντιστοιχούν σε speckle ή σε κανονικές δομές της υπερηχογραφικής εικόνας βασίζεται στον πρόσημο των εκθετών Lipschitz. Οι συντελεστές του μετασχηματισμού που αντιστοιχούν στα μέγιστα με τους αρνητικούς εκθέτες Lipschitz μηδενίζονται. Οι υπόλοιποι συντελεστές χρησιμοποιούνται στον αντίστροφο μετασχηματισμό για την δημιουργία της εικόνας χωρίς speckle Αποτελέσματα Η αποτελεσματικότητα της προτεινόμενης μεθόδου με βάση τον μετασχηματισμό μικροκυματιδίων αξιολογήθηκε μέσω ενός ομοιώματος ανθρώπινων ιστών και μιας υπερηχογραφικής εικόνας του θυρεοειδούς αδένα. Η προτεινόμενη μέθοδος (Inter-scale wavelet speckle suppression) συγκρίθηκε με τρεις αντιπροσωπευτικές μεθόδους αφαίρεσης θορύβου σε εικόνες υπερηχογραφίας: (α) Adaptive speckle suppression filter (ASSF) [97], (β) Soft Thresholding [104] και (γ) Hard thresholding [104]. Τα αποτελέσματα της αξιολόγησης έδειξαν ότι η προτεινόμενη μέθοδος ήταν ανώτερη έναντι των υπολοίπων με βάση τους παρακάτω δείκτες: speckle index και signal-to-mean-square-error ratio για την απόδοση του αλγορίθμου στην μείωση του speckle και την παράμετρο β για την ταυτόχρονη διατήρηση των αιχμών και του περιγράμματος των διαφόρων δομών (Πίνακας 1.1). VI

17 Πίνακας 1.1 Δείκτες ποιότητας υπερηχογραφικής εικόνας από τις τέσσερις μεθόδους αφαίρεσης speckle σε εικόνα υπερηχογραφίας/ομοίωμα ανθρώπινου ιστού Μέθοδος SI S/mse (db) β (db) (Ποσοστιαία βελτίωση) ASSF 14% / 1% / / Soft Thresholding 19% / 18% / / Hard Thresholding 16% / 15% / / Inter-scale wavelet speckle suppression 3% / 1% / / Επίσης υλοποιήθηκε μια μελέτη αξιολόγησης της μεθόδου από δύο παρατηρητές περιλαμβάνοντας 63 εικόνες υπερηχογραφίας θυρεοειδή 63 ασθενών μέσω ενός ερωτηματολογίου σχετικά με την απόδοση του προτεινόμενου αλγορίθμου. Το ερωτηματολόγιο περιλάμβανε επτά ερωτήσεις με διάφορες οπτικές παρατηρήσεις σχετικά με την προτεινόμενη αποτελεσματικότητα του αλγορίθμου. Οι απαντήσεις των παρατηρητών Ιατρών ήταν απόλυτα συνυφασμένες με την ποσοτική αξιολόγηση της μεθόδου. Πρέπει να σημειωθεί ότι και οι δύο Ιατροί στο σύνολο των εικόνων παρατήρησαν ότι η τεχνική αφαίρεσης του speckle είχε μια ιδιαιτέρως ικανοποιητική επίδραση στην ποιότητα της υπερηχογραφικής εικόνας αυξάνοντας την διακριτική ικανότητα και την αντίθεση των εικονιζόμενων δομών. Επίσης είναι χαρακτηριστικό ότι παρά την αφαίρεση του speckle στο σύνολο της εικόνας, οι διάφορες αιχμές που υπήρχαν στις εικόνες δεν επηρεάστηκαν καθόλου από την εφαρμογή του αλγορίθμου Συμπεράσματα Σαν τελικό συμπέρασμα μπορούμε να πούμε ότι υλοποιήθηκε μια αποτελεσματική μέθοδος αφαίρεσης του speckle σε εικόνες υπερηχογραφίας. Αυτή η μέθοδος βασίστηκε στον μετασχηματισμό μικροκυματιδίων, στην οποία ο πρωταρχικός στόχος ήταν να απομονωθούν οι αιχμές που υπάρχουν στις διάφορες κλίμακες και να ελεγχθεί η κανονικότητα τους. Η επιτυχής καταστολή του θορύβου από τον προτεινόμενο αλγόριθμο μπορεί να υιοθετηθεί ως ένα βήμα προ-επεξεργασίας για την τμηματοποίηση των όζων και σαν ένα βοηθητικό εργαλείο στη βελτίωση της συνολικής διαγνωστικής διαδικασίας. VII

18 1.3 Τμηματοποίηση των όζων του θυρεοειδούς αδένα Εισαγωγή Οι ιδιότητες των αιχμών στις διάφορες κλίμακες με βάση της κανονικότητα τους χρησιμοποιήθηκαν επίσης ως δεδομένα σε ένα υβριδικό μοντέλο για την αυτόματη τμηματοποίηση των όζων του θυρεοειδούς αδένα Πολυάριθμες αυτόματες ή ημι-αυτόματες μέθοδοι τμηματοποίησης έχουν παρουσιαστεί στην διεθνή αρθρογραφία σε εικόνες υπερηχογραφίας του προστάτη, των νεφρών, της καρδιακής ανατομίας, των ωοθηκών, του εμβρυϊκού κρανίου και των εικόνων μαστού. Όλοι αυτοί οι αλγόριθμοι μπορούν να ταξινομηθούν χονδρικά σε πέντε κατηγορίες ανάλογα με τη στρατηγική που επιλέγεται για την τμηματοποίηση της εκάστοτε περιοχής ενδιαφέροντος (ROI). Οι μέθοδοι αυτές τμηματοποίησης βασίζονται: (α) στην ανίχνευση αιχμών [118,13], (β) στην ανάλυση χαρακτηριστικών υφής [14 130], (γ) σε διάφορα μοντέλα (deformable and active models) [131,141], (δ) σε συνδυασμό των προαναφερθέντων [14 145], (ε) στον μετασχηματισμό μικροκυματιδίων σε διάφορες κλίμακες [ ] Μέθοδοι Η προτεινόμενη μέθοδος ενσωματώνει στο υβριδικό μοντέλο, τον μετασχηματισμό μικροκυματιδίων για την ανίχνευση αιχμών (Edge Detection), με τελικό σκοπό την ταυτόχρονη στερεοσκοπική ανίχνευση της συγκεκριμένης ανατομικής δομής σε όλες τις κλίμακες. Ο τελικός χάρτης περιγράμματος, που προέρχεται από το μοντέλο, χρησιμεύει σαν είσοδος στον μετασχηματισμό Hough με σκοπό την τελική τμηματοποίηση των όζων. Τα βήματα της μεθόδου είναι τα εξής: 1. Ανίχνευση αιχμών στο σύνολο των κλιμάκων του μετασχηματισμού (Edge detection procedure). Ανίχνευση δομών σε όλες τις κλίμακες του μετασχηματισμού (Multi-scale structure model) 3. Εξαγωγή του περιγράμματος του όζου (Nodule s boundary extraction) 1. Ανίχνευση αιχμών στο σύνολο των κλιμάκων του μετασχηματισμού. Η διαδικασία ανίχνευσης αιχμών βασίζεται στον μετασχηματισμό μικροκυματιδίων σε διάφορες δυαδικές κλίμακες. Ο τελικός χάρτης των αιχμών υπολογίζεται αναλύοντας τα τοπικά μέγιστα από την μεγαλύτερη στην μικρότερη κλίμακα του μετασχηματισμού. Τα τοπικά μέγιστα με θετικούς εκθέτες ` lipschitz " εισάγονται στο μοντέλο ανίχνευσης δομών που ακολουθεί. Τα τοπικά VIII

19 μέγιστα με τους αρνητικούς εκθέτες lipschitz ταξινομούνται ως speckle και αφαιρούνται από την συνέχεια της επεξεργασίας.. Ανίχνευση δομών στο σύνολο των κλιμάκων του μετασχηματισμού. Το επόμενο βήμα αυτής της μεθόδου είναι η ανίχνευση δομών σε όλες τις κλίμακες του μετασχηματισμού, η οποία θα συνέδεε μια ανατομική δομή στην υπερηχογραφική εικόνα με μια στερεοσκοπική αναπαράσταση της δομής αυτής με βάση τα τοπικά μέγιστα του μετασχηματισμού. Οι βασικές συνιστώσες του προτεινόμενου μοντέλου είναι: (α) τα τοπικά μέγιστα από το 1 ο βήμα, (β) οι αλυσίδες τοπικών μεγίστων που αποτελούν ομάδες τοπικών μεγίστων με παρόμοιες ιδιότητες στην ίδια κλίμακα, (γ) οι ανατομικές δομές που απαρτίζουν ένα σύνολο συνδεδεμένων αλυσίδων τοπικών μεγίστων, (δ) η σχέση που υπάρχει στο σύνολο των κλιμάκων για τον καθορισμό των κριτηρίων που υιοθετούνται από τον αλγόριθμο ώστε οι ομάδες αυτές να ενοποιηθούν για τον σχηματισμός της δομής (ε) και ένας αριθμητικός τελεστής ο οποίος αντιστοιχεί σε ποια δομή ανήκουν οι διάφορες ομάδες αλυσίδων τοπικών μεγίστων. Ο συνδυασμός αυτών των συνιστωσών υλοποιεί το μοντέλο για την όσο το δυνατόν αντιπροσωπευτικότερη παρουσίαση του περιγράμματος του όζου. 3. Εξαγωγή του περιγράμματος του όζου. Για την τελική τμηματοποίηση του όζου, η έξοδος του μοντέλου εισάγεται στον μετασχηματισμό Hough ο οποίος έχει την δυνατότητα της μερική αναγνώριση κοίλων αντικειμένων σε ένα περιβάλλον με μεγάλο θόρυβο από παρακείμενες δομές. Ο μετασχηματισμός Hough έχει την δυνατότητα να ανιχνεύει δομές ακόμα και αν το περίγραμμα αυτών δεν είναι συνεχές Αποτελέσματα Για την αξιολόγηση της απόδοσης της προτεινόμενης μεθόδου τμηματοποίησης, μια συγκριτική μελέτη υλοποιήθηκε, περιλαμβάνοντας 40 εικόνες υπερηχογραφίας του θυρεοειδή αδένα από 40 ασθενείς μεταξύ 40 και 65 ετών. Όλες οι εικόνες επιλέχτηκαν τυχαία από μια μεγάλη βάση δεδομένων. Τα αποτελέσματα τμηματοποίησης της μεθόδου συγκρίθηκαν με βάση με την χειροκίνητη σκιαγράφηση των όζων (θεωρούμενη ως ground truth) που προήλθαν από δύο έμπειρους παρατηρητές (OB1 και OB) και με βάση κάποια μορφολογικά χαρακτηριστικά όπως το εμβαδόν, η σφαιρικότητα (roundness), η κοιλότητα (concavity) και η μέση απόλυτη απόσταση μεταξύ των περιγραμμάτων της μεθόδου και των δύο παρατηρητών (Mean Absolute Distance). IX

20 Στη συγκριτική μελέτη, η ακρίβεια τμηματοποίησης των όζων του θυρεοειδούς προσέγγισε ποσοστά ταύτισης των περιγραμμάτων με τους δύο παρατηρητές της τάξης των 90,14 και 89,33%. Το ποσοστό ταύτισης της μεθόδου με τα αποτελέσματα των δύο παρατηρητών είναι εφάμιλλο με το ποσοστό ταύτισης μεταξύ των δύο παρατηρητών (91,83%) (Πίνακας.1). Πίνακας 1. Μέση ποσοστιαία συμφωνία μεταξύ της αυτόματης μεθόδου (AU) και την χειροκίνητης μεθόδου τμηματοποίησης από τους δύο παρατηρητές (OB 1, OB ) σε σχέση με το εμβαδόν, την στρογγυλότητα, Κοιλότητα και MAD %. Εμβαδόν Στρογγυλότητα Κοιλότητα MAD % AU- OB 1 AU- OB AU- OB 1 AU- OB AU- OB 1 AU- OB AU- OB 1 AU- OB Μέση Ποσοστιαία Συμφωνία 88,83 87,58 91,77 91,1 89,1 89,08 90,77 89,53 Η inter-observer μελέτη κατέδειξε την υψηλή συμφωνία συντελεστή kappa περίπου 0,83. Αν και ο αλγόριθμος είναι αυτόματος, τα αποτελέσματα αξιολόγησης μπορούν να θεωρηθούν αρκετά ενθαρρυντικά πάντα σε σχέση με την εν γένει χαμηλή ποιότητα γένει της υπερηχογραφικής εικόνας. Ο προτεινόμενος αλγόριθμος μπορεί να αποτελέσει ένα πρωτεύων εργαλείο σε συστήματα αυτόματης διάγνωσης για την ταξινόμηση των όζων αλλά και ως ένα δευτερεύων εργαλείο κατά την ίδια την υπερηχογραφική εξέταση Συμπεράσματα Σαν συμπέρασμα, μια νέα τεχνική τμηματοποίηση όζων του θυρεοειδούς αδένα προτείνεται με αρκετά καλά αποτελέσματα. Ο προτεινόμενος αλγόριθμος είναι σε θέση να ανιχνεύσει τους όζους ανεξαρτήτως ηχογένειας και πιθανής ύπαρξης ασυνέχειας στο περίγραμμα του όζου. Επίσης κατορθώνει και εξάγει την περιοχή ενδιαφέροντος ανεξαρτήτου του ποσοστού δομικού θορύβου από παρακείμενες δομές. Η χρησιμοποίηση της υβριδικής μεθόδου μπορεί να βοηθήσει στην κατηγοριοποίηση των όζων από τον παθολόγο και να ενισχύσει την ακρίβεια της διαδικασίας λήψης βιοψίας (FNAB). Επιπλέον μπορεί να χρησιμοποιηθεί ως εκπαιδευτικό εργαλείο για τους άπειρους ακτινολόγους. X

21 . Αυτόματη διάγνωση των όζων του θυρεοειδούς αδένα.1 Εισαγωγή Οι όζοι αναπτύσσονται από τα θυλακιώδη κύτταρα του θυρεοειδούς και απαντώνται στους θυρεοειδείς αδένες φυσιολογικού μεγέθους και στους διογκωμένους θυρεοειδείς αδένες (βρογχοκήλες). Το 95% των όζων του θυρεοειδούς αδένα είναι καλοήθεις. Μόνο ένα μικρό ποσοστό των όζων ταξινομείται σαν κακοήθεις. Υπάρχουν διάφορες μορφές του καρκίνου του θυρεοειδούς: Θηλώδες καρκίνωμα (Papillary carcinoma 75%), Θυλακιώδες ή λεμφοζιδιακό καρκίνωμα (follicular carcinoma 15%, Σε αυτήν την περίπτωση η χρήση της έγχρωμης απεικόνισης Doppler βοηθά στην διάγνωση), Μυελλοειδές καρκίνωμα (Medullary carcinoma 7%), Αναπλαστικό καρκίνωμα (Anaplastic carcinoma 3%). Στους νέους ανθρώπους αν η διάγνωση είναι έγκαιρη, το ποσοστό ίασης του Θηλώδους και Θηλακιώδους καρκινώματος αγγίζει το 95%. Στις άλλες δύο περιπτώσεις η ακρίβεια πρόγνωσης είναι μικρή. Ωστόσο, η ακριβής εκτίμηση και ταξινόμηση των κακοήθων όζων, εξαιτίας της φύσεως της υπερηχογραφικής εικόνας, παραμένει μια δύσκολη απόφαση [1-1]. Η μέτρηση του μεγέθους, της θέσης αλλά και άλλων χαρακτηριστικών (περιγράμματος και εσωτερικής μορφολογίας) του κάθε όζου μέσα από την εικόνα υπερηχογραφίας βοηθά στην σωστή αξιολόγηση του από τον ιατρό. Άλλες παράμετροι της εικόνας, όπως η παρουσία ηχοδιαυγαστικής ζώνης (άλω) που περιβάλει τους χαμηλού κινδύνου όζους και η παρουσία αποτιτανώσεων σε υψηλού κινδύνου όζους βοηθούν στην προσέγγιση της διάγνωσης. Οι συμπαγείς όζοι του θυρεοειδούς αδένα, σε ένα μεγάλο ποσοστό, δεν είναι δυνατόν να ταξινομηθούν αυστηρά σε καλοήθεις ή κακοήθεις με τις διάφορες απεικονιστικές μεθόδους που εφαρμόζονται σήμερα (Υπερηχογραφική εικόνα, γ-κάμερα), με αποτέλεσμα οι ασθενείς να υποβάλλονται σε βιοψία για κυτταρολογική εξέταση από διάφορες περιοχές του όζου [1]. Η δημοφιλέστερη τεχνική λήψης κυτταρολογικού υλικού (βιοψία) από το θυρεοειδή είναι η Fine Needle Aspiration λόγω της υψηλής ακρίβειας διάγνωσης, ασφάλειας και μικρού κόστους εξέτασης. Το κυτταρολογικό υλικό που λαμβάνεται μέσω της FNA τοποθετείται σε γυάλινα πλακάκια μικροσκοπίου και κατόπιν εξετάζεται από τον παθολογοανατόμο μέσω συστημάτων μικροσκοπίας. Η εξέταση στο πλακάκι μικροσκοπίας αποτελεί το τελευταίο στάδιο της διαγνωστικής αλυσίδας είτε για την αναγνώριση τόσο των καλοηθών από τους κακοήθεις όζους είτε του βαθμού επικινδυνότητας των όζων. Η διεθνής βιβλιογραφία έχει αναδείξει την κυτταρολογική εξέταση ως τη σημαντικότερη για τον εντοπισμό και την ταξινόμηση των όζων του θυρεοειδούς με ακρίβεια αναγνώρισης που ξεπερνάει το 85%. Αποκλείοντας τις σπάνιες περιπτώσεις του XI

22 χαρακτηριστικού νεοπλάσματος του θυρεοειδή η κυτταρολογική εκτίμηση διαχωρίζει του όζους σε δύο σημαντικές κατηγορίες: (α) επιθηλιακή υπερπλασία που μπορεί να χαρακτηριστεί ως υψηλού κινδύνου με πιθανή εμφάνιση κακοήθειας έχοντας ως αποτέλεσμα τις επαναλαμβανόμενες εξετάσεις υπερήχου και κυτταρολογίας του ασθενή, και (β) στις καλοήθεις περιοχές (κολλοειδείς όζοι), όπου οι υπερηχογραφικές εξετάσεις μπορούν να πραγματοποιηθούν ανά μεγάλα χρονικά διαστήματα. Η ανάγκη για την όσο τον δυνατόν πιο αντικειμενική εκτίμηση και ανάδειξη των υπερηχογραφικών χαρακτηριστικών που εμφανίζει η κάθε μια κατηγορία οδήγησε στην ανάγκη σχεδιασμού και υλοποίησης ενός αυτόματου συστήματος ταξινόμησης των όζων στις δύο αυτές κατηγορίες. Υπάρχουν δημοσιευμένες μελέτες σχετικά με την υλοποίηση αυτόματων προγραμμάτων διάγνωσης του πιθανότητας εμφάνισης κακοήθειας όζων του θυρεοειδούς αδένα. Αυτές οι μελέτες βασίζονται κυρίως σε χαρακτηριστικά από το ιστόγραμμα της υπερηχογραφικής εικόνας [164,165], διάφορα χαρακτηριστικά υφής [166] και στην εφαρμογή ανάλυσης διαχωρισμού (discriminant analysis) [ ]. Σε εκείνες τις μελέτες, τα ποσοστά διαχωρισμού μεταξύ των καλοηθών και των κακοήθων όζων ήταν 83,9% [166] και 85% [164]. Σε αυτήν μελέτη περιλαμβάνονται ασθενείς με όζους του θυρεοειδούς αδένα που εξετάσθηκαν σε ένα υπερηχογραφικό σύστημα και στη συνέχεια υποβλήθηκαν σε διαγνωστική βιοψία με τη μέθοδο FNA στην ιδιωτική κλινική EUROMEDICA. Τα βασικά βήματα της μελέτης είναι: 1. Συλλογή εικόνων υπερηχογραφίας από ασθενείς με όζους του θυρεοειδούς και δημιουργία ψηφιακής βάσης δεδομένων για την καταγραφή ιστορικού ασθενούς και κλινικών δεδομένων.. Σχεδιασμός-προτυποποίηση παραμέτρων λήψης εικόνας. Πραγματοποιήθηκε έλεγχος ποιότητας του υπερηχογραφικού συστήματος από την εταιρία κατασκευής. 3. Ανάλυση υπερηχογραφικών εικόνων για την δημιουργία χαρακτηριστικών (υφής, περιγράμματος και τοπικών μεγίστων) του όζου με τη βοήθεια στατιστικών παραμέτρων (στατιστικές παράμετροι πρώτης, δεύτερης [170] και ανώτερης τάξης [73,74] κ.α. Επιλογής χαρακτηριστικών με μεθόδους μονομεταβλητής και πολυμεταβλητής στατιστικής ανάλυσης [70, 83, 170]. 4. Ταξινόμηση των όζων του θυρεοειδούς σε δύο βασικές (υψηλού κινδύνου χαμηλού κινδύνου) με την υλοποίηση κλασσικών ταξινομητών, Μηχανών Υποστήριξης Διανυσμάτων (SVM) και Πιθανοκρατικών Νευρωνικών Δικτύων (PNN). XII

23 . Υπολογιστικό Σύστημα Αναγνώρισης Προτύπων βασισμένο σε Χαρακτηριστικά Υφής..1 Βάση Δεδομένων Υπερηχογραφικές εικόνες από 10 διαφορετικούς ασθενείς, οι οποίοι έχουν ηλικία από 30 έως 75 ετών και ανήκουν και στα δύο φύλα συλλέχθηκαν στο χρονικό διάστημα από τον Οκτώβριο του 003 μέχρι τον Σεπτέμβριο του 004 από τον ιατρικό διαγνωστικό κέντρο EUROMEDICA, Κατεχάκη 4, Αθήνα, Ελλάδα. Το Υπερηχογραφικό σύστημα απεικόνισης το οποίο χρησιμοποιήθηκε για την απόκτηση των εικόνων είναι ο υπερηχοτομογράφος HDI 3000 ATL- PHILΙPS (PHILIPS, USA). Ο υπεύθυνος ιατρός που πραγματοποιεί τις εξετάσεις είναι ο Δρ. Νίκος Δημητρόπουλος. Οι εικόνες που αποκτήθηκαν είναι δισδιάστατες με 56 τόνους του γκρι, ενώ η χρησιμοποιούμενη ηλεκτρονική κεφαλή είναι γραμμικής διάταξης (Linear Array) και κεντρικής συχνότητας συντονισμού 7 MHz με δυνατότητα ανίχνευσης ενός εύρους συχνοτήτων γύρω από την ονομαστική (5-9 ΜΗz - Broadband). Για την βέλτιστη απεικόνιση του όζου λαμβάνονται διάφορες εικόνες από τον ίδιο όζο σε εγκάρσιο, οβελιαίο και πρόσθιο επίπεδο. Για την ψηφιοποίηση του σήματος Video του υπερηχογράφου χρησιμοποιήσαμε την κάρτα Video, Miro PCTV (Pinnacle Systems), η οποία είναι εγκατεστημένη σε έναν Η/Υ. Διάφορα χαρακτηριστικά υφής υπολογίστηκαν αυτόματα από την περιοχή ενδιαφέροντος κάθε όζου. Τα χαρακτηριστικά αυτά συσχετίζονται με τη δομή των τόνων του γκρι αποτελούν χρήσιμη πληροφορία για την πιθανότητα εμφάνισης κακοήθειας. Τέσσερα χαρακτηριστικά υπολογίστηκαν από το ιστόγραμμα, 6 από τη μήτρα co-occurrence [ 73 ] και 10 από τη μήτρα run-length [ 74 ]... Αξιολόγηση Επίδοσης Συστήματος Ταξινόμησης Η επίδοση των αλγόριθμων ταξινόμησης εξετάστηκε με την μέθοδο leave-one-out. Σύμφωνα με την leave-one-out ο ταξινομητής σχεδιάζεται με όλα τα διανύσματα του σχεδιαστικού σετ πλην ενός. Το τελευταίο θεωρείται αγνώστου κατηγορίας και χρησιμοποιείται σαν είσοδος στον ταξινομητή για χαρακτηρισμό. Όλη η διαδικασία επαναλαμβάνεται για όλα τα διανύσματα του εκπαιδευτικού σετ σε συνδυασμούς των δύο, τριών, τεσσάρων σε καθολική έρευνα. Κλασσικοί ταξινομητές: Υλοποιήθηκαν κλασσικοί ταξινομητές όπως ο ελάχιστης απόστασης (minimum distance - MD), ελαχίστων τετραγώνων ελάχιστης απόστασης (least square minimum distance LSMD), Bayesian, και τεχνητών νευρωνικών δικτύων (artificial neural networks - MLP),[7,83,84,170] των οποίων η ακρίβεια ταξινόμησης χρησιμοποιήθηκε ως αναφορά για την αξιολόγηση της απόδοσης των σύγχρονων ταξινομητών που περιγράφονται στην συνέχεια. XIII

24 Σύγχρονοι ταξινομητές: Ταξινομητής Μηχανών Διανυσμάτων Στήριξης (Support Vector Machines-SVMs) [87,88]. Η βασική ιδέα για την εφαρμογή των SVMs για την αντιμετώπιση προβλημάτων ταξινόμησης συνίσταται σε δύο βήματα: 1. Μετασχηματισμός των ανυσμάτων εκπαίδευσης (που στην γενική περίπτωση δεν είναι γραμμικώς διαχωρίσιμα) σε ένα μεγαλύτερων διαστάσεων χώρο (feature space) μέσω μιας κατάλληλης συνάρτησης (συνάρτηση πυρήνα), ώστε να καταστούν γραμμικώς διαχωρίσιμα.. Υπολογισμός του βέλτιστου διαχωριστικού ορίου (margin) ανάμεσα στις δύο κατηγορίες με τέτοιο τρόπο ώστε να ικανοποιούνται οι ακόλουθες συνθήκες: α) το εύρος του διαχωριστικού ορίου να είναι μέγιστο, και β) το πλήθος των σημείων που βρίσκονται εντός του ορίου αυτού να είναι ελάχιστο. Στην περίπτωση δύο κατηγοριών, η συνάρτηση διάκρισης (discriminant function) ενός ταξινομητή βασισμένο στα SVMs είναι: g(x) = sign N i= 1 α y K( x, x i i i ) + b όπου x το προς ταξινόμηση διάνυσμα, Ν το πλήθος των ανυσμάτων εκπαίδευσης, xi το i-οστό άνυσμα εκπαίδευσης, yi {-1,+1} αναλόγως την κατηγορία, αi, b συντελεστές βαρύτητας και Κ(x,xi) η συνάρτηση πυρήνα. Οι συναρτήσεις πυρήνα πρέπει να ικανοποιούν συγκεκριμένες συνθήκες (συνθήκες Mercer [177]) και, ενδεικτικά, οι πλέον διαδεδομένες στη διεθνή βιβλιογραφία είναι: Η πολυωνυμική συνάρτηση: όπου d ο βαθμός του πολυωνύμου, Η Gaussian Radial Basis συνάρτηση: i T ( x ) ) d K( x, x ) = x + 1 i όπου σ η τυπική απόκλιση. K x, x ) = exp i x xi ( σ..3 Αποτελέσματα Για τον SVM-ταξινομητή με τον πολυωνυμική συνάρτηση πυρήνα 3 ου βαθμού, ο καλύτερος συνδυασμός χαρακτηριστικών ήταν ο μέση τιμή (mean value) των τόνων του γκρι από ιστόγραμμα και η sum variance από την μήτρα co-occurrence επιτυγχάνοντας ποσοστό επιτυχίας 96.7%. Ο καλύτερος συνδυασμός χαρακτηριστικών για όλες τις wavelet συναρτήσεις πυρήνες ήταν παρόμοιος με αυτόν του πολυωνυμικού πυρήνα βαθμού 3 rd (μέση τιμή και sum variance). XIV

25 Για τον ταξινομητή MLP η υψηλότερη ακρίβεια ταξινόμησης επιτεύχθηκε από τον συνδυασμό της μέσης τιμής και του χαρακτηριστικού Run Length Non Uniformity από τη μήτρα run-length επιτυγχάνοντας ποσοστό επιτυχίας 95.0%. Για τους ταξινομητές (QLSMD, QB) ο καλύτερος συνδυασμός χαρακτηριστικών ήταν η μέση τιμή από το ιστόγραμμα, η sum variance από την μήτρα co-occurrence και η Run Length Non Uniformity μήτρα run-length επιτυγχάνοντας ποσοστό επιτυχίας 9.5% και οι δύο (Πίνακας 3.1). Πίνακας.1 Ποσοστά ταξινόμησης διαφόρων ταξινομητών για τις μεθόδους leave-one-out και re-substitution. Ακρίβεια Ταξινόμησης LOO + (%) Resub.* (%) N SV ** SVM με πολυωνυμική συνάρτηση πυρήνα 1 ου βαθμού SVM με πολυωνυμική συνάρτηση πυρήνα ου βαθμού SVM με πολυωνυμική συνάρτηση πυρήνα 3 ου βαθμού SVM με πολυωνυμική συνάρτηση πυρήνα 4 ου βαθμού SVM με RBF συνάρτηση πυρήνα SVM με Daubechies Wavelet συνάρτηση πυρήνα SVM με Coiflet Wavelet συνάρτηση πυρήνα SVM με Symmlet Wavelet συνάρτηση πυρήνα MLP QLSMD Bayesian Leave-one-out Μέθοδος * Re-substitution Μέθοδος ** Αριθμός των διανυσμάτων στήριξης κατά την re-substitution μέθοδο..4 Συμπεράσματα: Ο βέλτιστος συνδυασμός χαρακτηριστικών γνωρισμάτων του υπολογιστικού συστήματος ανίχνευσης προτύπων (μέση τιμή και sum variance) αντιστοιχεί με τις διάφορες παραμέτρους υφής (ηχογένεια, παρουσία αποτιτανώσεων) που αποτελούν κλινικές ενδείξεις κακοήθειας του όζου [17,18,5,7]. Αναλυτικά, η μέση τιμή των τόνων του γκρι συσχετίζεται άμεσα με την ηχογένεια του όζου σε σχέση με τον περιβάλλοντα ιστό, ενώ το δεύτερο χαρακτηριστικό (sum variance) εκφράζει τις χρήσιμες χωρικές πληροφορίες και αντιστοιχεί με την παρουσία δομών όπως διάφορες αποτιτανώσεις στο εσωτερικό του όζου. Διάφορα χαρακτηριστικά που πηγάζουν από το ιστόγραμμα και την μήτρα co-occurrence έχουν επίσης υποδειχθεί στις προηγούμενες ποσοτικές μελέτες [164] με απώτερο σκοπό την πρόγνωση κακοήθειας των όζων, σημειώνοντας ακρίβειας ταξινόμησης της τάξης του 85%. Εφάμιλλες ακρίβειες (83,9%) επιτεύχθηκαν από μια άλλη μελέτη υιοθετώντας τη συνάρτηση διαχωρισμού (discriminant function) των όζων [166]. Η XV

26 υψηλότερη ακρίβεια που επιτεύχθηκε στην παρούσα μελέτη οφειλόταν πιθανότατα στην καλύτερη διακριτική ικανότητα του υπερηχογραφικού συστήματος καθώς και στην μη γραμμική φύση του ιδιαίτερα περίπλοκου αλγορίθμου SVM που υιοθετήθηκε. Τελικά, ένα ικανό σύστημα ταξινόμησης σχεδιάστηκε, βασισμένο στον αλγόριθμο SVM, για την αξιολόγηση του ποσοστού κινδύνου κακοήθειας σε όζους του θυρεοειδή αδένα σε εικόνες υπερηχογραφίας. Αυτό το σύστημα θα μπορούσε να είναι ένα χρήσιμο διαγνωστικό εργαλείο σαν μια δεύτερη άποψη στον Ιατρό και μπορεί να αποβεί σημαντικό για την ορθή διαχείριση των σθενών με σκοπό την αποφυγή άσκοπων βιοψιών..3 Υπολογιστικό Σύστημα Αναγνώρισης Προτύπων βασισμένο σε Χαρακτηριστικά Περιγράμματος και Τοπικών Μεγίστων.3.1 Βάση Δεδομένων Υπερηχογραφικές εικόνες από 86 διαφορετικούς ασθενείς, οι οποίοι ανήκουν και στα δύο φύλα συλλέχθηκαν στο χρονικό διάστημα από τον Φεβρουάριο του 005 μέχρι τον Αύγουστο του 006 από τον ιατρικό διαγνωστικό κέντρο EUROMEDICA, Κατεχάκη 4, Αθήνα, Ελλάδα. Το σύστημα υπερηχογραφίας, απόκτησης εικόνων και ο Ιατρός παραμένουν οι ίδιοι με την προηγούμενη έρευνα. 1 χαρακτηριστικά περιγράμματος υπολογίστηκαν αυτόματα από την περιοχή ενδιαφέροντος κάθε όζου. Τα χαρακτηριστικά αυτά σχετίζονται με τη μορφολογία του περιγράμματος των όζων και αποτελούν κλινικές ενδείξεις για την πιθανότητα ύπαρξης ή όχι κακοήθειας σε αυτούς. Επίσης υπολογίστηκαν 8 χαρακτηριστικά βασισμένα στα τοπικά μέγιστα από τον μετασχηματισμό μικροκυματιδίων. Θα πρέπει να αναφερθεί ότι τα χαρακτηριστικά αυτά χωριστήκαν σε δύο κατηγορίες.. Στην μία, τα χαρακτηριστικά προέρχονται από τα τοπικά μέγιστα που δεν αντιστοιχούν σε speckle, ενώ στην άλλη προέρχονται από το σύνολο των τοπικών μεγίστων πριν την ανάλυση της κανονικότητας τους. Ο σκοπός είναι να διερευνηθεί η επίδραση που έχει το speckle στην σωστή αξιολόγηση της εικόνας..3. Αξιολόγηση Επίδοσης Συστήματος Ταξινόμησης Η αξιολόγηση των αλγόριθμων ταξινόμησης πραγματοποιήθηκε με την ROC ανάλυση υλοποιώντας την μέθοδο leave-one-out. Όλη η διαδικασία επαναλαμβάνεται για όλα τα διανύσματα του εκπαιδευτικού σετ σε συνδυασμούς των δύο, τριών, τεσσάρων σε εκτεταμένη έρευνα. Ο ταξινομητής με την μεγαλύτερη τιμή της περιοχής κάτω από την καμπύλη ROC XVI

27 (AUC) για έναν συγκεκριμένο συνδυασμό χαρακτηριστικών θεωρείται ότι επέτυχε τον μεγαλύτερη ποσοστό επιτυχίας Σύγχρονοι ταξινομητές: Εκτός του Ταξινομητή Μηχανών Στήριξης Διανυσμάτων (Support Vector Machines-SVMs) στην παρούσα έρευνα χρησιμοποιήθηκαν και τα Πιθανοκρατία Νευρωνικά δίκτυα. Πιθανοκρατία Νευρωνικά δίκτυα (Probabilistic Neural Networks-PΝΝ): Τα PNN μοντελοποιούν τον Bayesian ταξινομητή, ο οποίος ελαχιστοποιεί το ενδεχόμενο σφάλμα ταξινόμησης ενός προτύπου σε λάθος κατηγορία. Ενώ ο Bayesian ταξινομητής θεωρεί κανονική την κατανομή της πιθανότητας, τα PNN εκτιμούν την συνάρτηση πυκνότητας πιθανότητας μέσω μη παραμετρικών μεθόδων (Parzen). Τα PNN είναι feed-forward δίκτυα δομημένα σε τέσσερα επίπεδα. α) επίπεδο εισόδου: σε αυτό το επίπεδο υπάρχουν τόσα στοιχεία, όσα και τα πρότυπα εκπαίδευσης β) πρότυπο επίπεδο: αυτό το επίπεδο περιέχει κ στοιχεία, ένα για κάθε δείγμα εκπαίδευσης. Σε κάθε κόμβο του επιπέδου υπολογίζεται η απόσταση του διανύσματος εισόδου από τα πρότυπα εκπαίδευσης. γ) επίπεδο άθροισης: έχει τόσα στοιχεία όσες και οι προς διαχωρισμό τάξεις και απλά προσθέτει τις εξόδους του προηγούμενου επιπέδου δ) επίπεδο εξόδου: τα στοιχεία εξόδου κρίνουν το αποτέλεσμα της ταξινόμησης Το κυριότερο πλεονέκτημα των PNN έγκειται στο γεγονός ότι η εκπαίδευση είναι εύκολη και γρήγορη και σε σχέση με άλλα είδη νευρωνικών δικτύων και ότι δεν απαιτεί ανάδραση. Η συνάρτηση διαχωρισμού ενός PNN για κατηγορίες δίνεται από την κάτωθι εξίσωση [84,85,86]: g N x xi 1 σ ( x ) = p p e / (π ) σ N i= 1 όπου x είναι το άνυσμα προς κατηγοριοποίηση, x i αίναι το i άνυσμα εκπαίδευσης, N ο αριθμός των ανυσμάτων στην κατηγορία, σ είναι μια παράμετρος εξομάλυνσης, και p είναι ο αριθμός των χαρακτηριστικών. Η ταξινόμηση του εκάστοτε ανύσματος πραγματοποιείται στην κατηγορία με την μεγαλύτερη τιμή της συνάρτησης διαχωρισμού Αξιολόγηση των δύο ταξινομητών χωρίς την παρουσία του speckle Όσον αφορά τον ταξινομητή SVM, η υψηλότερη ακρίβεια ταξινόμησης με τον ελάχιστο αριθμό χαρακτηριστικών (AUC 0,96) επιτεύχθηκε με τον συνδυασμό των εξής χαρακτηριστικών: Την XVII

28 ομαλότητα του περιγράμματος (Smoothness) την συμμετρία του όζου (Symmetry) και την τυπική απόκλιση των τοπικών μεγίστων, χρησιμοποιώντας την πολυωνυμική συνάρτηση πυρήνα ου βαθμού. Ο ταξινομητής PNN επέτυχε την υψηλότερη ακρίβεια ταξινόμησης (AUC 0.91) με τον εξής συνδυασμό χαρακτηριστικών "fractal dimension, την παρουσία κοιλοτήτων (Concavity) και την τυπική απόκλιση των τοπικών μεγίστων " (Πίνακας.). Πίνακας. Αποτελέσματα ανάλυσης ROC για τους ταξινομητές SVM και PNN για τους αντίστοιχους καλύτερους συνδυασμούς χαρακτηριστικών Model SVM με πολυωνυμική συνάρτηση πυρήνα 1 ου βαθμού SVM με πολυωνυμική συνάρτηση πυρήνα ου βαθμού SVM με πολυωνυμική συνάρτηση πυρήνα 3 ου βαθμού SVM με πολυωνυμική συνάρτηση πυρήνα 4 ου βαθμού SVM με RBF συνάρτηση πυρήνα AUC (Lower Upper 95.0% Confidence Limit) 0.88 ( ) 0.96 ( ) 0.9 ( ) 0.89 ( ) 0.91 ( ) PNN 0.91 ( ) Sensitivity (SN) Specificity (SP) Likelihood Ratio SN/(1-SP) Number of Support Vectors Αξιολόγηση των δύο ταξινομητών με την παρουσία του speckle Ο ταξινομητής SVM επέτυχε την υψηλότερη ακρίβεια ταξινόμησης χρησιμοποιώντας τον πολυωνυμικό πυρήνα 3 ου βαθμού (AUC 0.88) με τη μέθοδο leave-one-out χρησιμοποιώντας το συνδυασμό χαρακτηριστικών "τυπική απόκλισης των ακτινών του περιγράμματος και της συμμετρίας του όζου (Symmetry)". Ο ταξινομητής PNN σημείωσε την υψηλότερη ακρίβεια ταξινόμησης (AUC 0.86) υιοθετώντας τον εξής: συνδυασμό χαρακτηριστικών γνωρισμάτων: Η εντροπία των ακτινών του περιγράμματος και η παρουσία των κοιλοτήτων (concavity) (Πίνακας.3). XVIII

29 Πίνακας.3 Αποτελέσματα ανάλυσης ROC για τους ταξινομητές SVM και PNN για τους αντίστοιχους καλύτερους συνδυασμούς χαρακτηριστικών Model SVM με πολυωνυμική συνάρτηση πυρήνα 1 ου βαθμού SVM με πολυωνυμική συνάρτηση πυρήνα ου βαθμού SVM με πολυωνυμική συνάρτηση πυρήνα 3 ου βαθμού SVM με πολυωνυμική συνάρτηση πυρήνα 4 ου βαθμού SVM με RBF συνάρτηση πυρήνα SVM με πολυωνυμική συνάρτηση πυρήνα 1 ου βαθμού.3.3 Συμπεράσματα AUC (Lower Upper 95.0% Confidence Limit) 0,83 (0,63 0,93) 0,86 (0,68 0,94) 0,88 (0,68 0,97) 0,78 (0,5 0,86) 0,79 (0,66 0,91) 0,86 (0,74 0,90) Sensitivity (SN) Specificity (SP) Likelihood Ratio SN/(1-SP) Number of Support Vectors Σχετικά με την έρευνα χωρίς την παρουσία του speckle, και οι δύο ταξινομητές SVM και PNN αξιοποίησαν την περίπλοκη σχέση μεταξύ της μορφής των όζων και των χαρακτηριστικών που προέρχονται από τα τοπικά μέγιστα προς μια ακριβή διαφοροποίηση μεταξύ των όζων χαμηλού και υψηλού κινδύνου. Η παρουσία μικρών κοίλων περιοχών (concativity) και η αυξανόμενη ανομοιομορφία (ομαλότητα) στο περίγραμμα των όζων από κοινού με την αξιοπρόσεχτη διαφορά μεταξύ του πλάτους και του μήκους της (συμμετρία) και την αυξανόμενη μεταβλητότητα των αποτιτανώσεων (MCs) έχουν αποδειχθεί ως σημαντικά και λεπτομερή χαρακτηριστικά που οδηγούν προς την υποψία καρκίνου. Αντίθετα, η έλλειψη κοίλων σημείων, η ύπαρξη μιας ομαλής οριογραμμής, ένα σχεδόν στρογγυλό σχήμα των όζων μαζί με την παρουσία αποτιτανώσεων χωρίς διαφορά στον τόνο του γκρι αποτελούν χαρακτηριστικά τα οποία τείνουν να αθωώσουν τον όζο. Σχετικά με την έρευνα με την παρουσία του speckle, Ο αντίκτυπος που είχε στη διαδικασία επιλογής των χαρακτηριστικών προς το βέλτιστο συνδυασμό με το υψηλότερο ποσοστό ακρίβειας ήταν μεγαλύτερος από τον αναμενόμενο στην αρχή αυτής της μελέτης. Η συνολική ακρίβεια και των δύο ταξινομητών παραμένει σχετικά υψηλή (AUC SVM 0.88, AUC PNN 0.86), εντούτοις η χρησιμοποίηση των χαρακτηριστικών που προέρχονται μόνο από τη μορφολογική ομάδα είναι ένα ζήτημα που απαιτεί την περαιτέρω έρευνα. Η μείωση του αριθμού XIX

30 χαρακτηριστικών γνωρισμάτων χωρίς την σημαντική μείωση της συνολικής ακρίβειας μπορεί να θεωρηθεί ως σημαντικό προτέρημα στην πολυπλοκότητα αλγορίθμων όπως οι SVM και PNN. Παρόλα αυτά, δεν μπορεί να αντισταθμίσει τη διαγραφή των σημαντικών πληροφοριών (παρουσία αποτιτανώσεων ή όχι, συνάθροιση αυτών ή όχι, κ.λ.π...) που παρέχουν τα χαρακτηριστικά που προέρχονται από τα τοπικά μέγιστα. Σαν τελικό συμπέρασμα μπορεί να ειπωθεί ότι, το συνεχώς αυξανόμενο ποσοστό πληροφοριών, που προέρχεται από την εικόνα υπερήχου, έχει μετατρέψει την απόφαση για το κατά πόσον θα πρέπει να υποβληθεί σε βιοψία ένας ασθενής, σε μια μάλλον σύνθετη διαδικασία. Αυτό το γεγονός καθιστά τους αλγορίθμους αναγνώρισης προτύπων ως ουσιαστικό βοηθητικό εργαλείο, προκειμένου να παραμετροποιηθούν και να ποσοτικοποιηθούν όλες οι διαθέσιμες πληροφορίες. XX

31 CHAPTER 1 Introduction 1.1 Power of Ultrasound The establishment of Ultrasonography (US) as a leading tool in the maority of medical applications worldwide, is directly associated with the evolution of imaging technology employed in medicine and biology. The design and implementation of novel and -state of the art- ultrasound systems, allowed US to infiltrate into medical applications such as Orthopedics, I.C.U, Diabetology etc, in which few years ago the performing of US examinations was prohibited. The latter modified the use of prognostic medicine nowadays in a radical way. In fact, ultrasonography is recognized as the fundamental technique in prevention, diagnosis and therapy of a constantly broadened spectrum of diseases. US imposes itself on the hands of every physician anywhere he practices medicine. From a small private clinic via a portable unit to a general hospital through an expensive four-dimensional system, ultrasound proves its efficiency and accuracy in a daily basis. 1. Need for Image Processing and Analysis Methods in Thyroid Ultrasound Images Despite its non-invasive nature, low cost and easy-to-use real time application, US imaging suffers from the presence of a granular pattern termed as speckle. It is the result of various constructive and destructive interference phenomena, which occur when the distances between the tissue scatterers are smaller than the axial resolution limit of the system. It causes deformities of anatomic structures as well as random fluctuations in the image s intensity profile. If an image is corrupted with speckle there are no regions of approximately constant intensity profile even if the reflecting tissue is entirely uniform. In addition, several US properties can lead to misleading effects in the ultrasound image. Reverberation, shadowing, refraction, side and grating lobes deteriorate the resolution of the US image, thus degrade its overall quality. The aforementioned problems arising by the complex nature of US imaging constitute speckle suppression and accurate boundary detection as important steps towards US image quality and diagnostic procedure enhancement in medical ultrasound imaging. The sonographic evaluation of malignancy risk in thyroid nodules represents a typical example of the way ultrasonography is accomplished to gain the confidence of medical community throughout the past years. The medical interest regarding biopsy s necessity based on sonographic - 1 -

32 Introduction criteria is extremely high. New and more detailed features that have derived from US examinations of thyroid nodules are investigated to decide whether to proceed or not into Fine Needle Aspiration Biopsy (FNAB). Features, such as the nodule s echo-structure and echogenicity (solid or colloid, hyper-hypo or iso-echogenic), its shape differentiation (round, eggshape, wide or tall), its boundary irregularity degree (from normal to highly irregular borderline), its calcifications pattern (massive, snow-storm etc), are employed towards an improved prognosis. The increasingly amount of information provided by high resolution US systems constitutes the clinical decision procedure rather difficult, therefore the quantification of sonographic findings and the implementation of computer-based algorithms could be of assistance as a second opinion tool. 1.3 Aims, Contributions and Novelties of Thesis The aim of the present thesis was the design and implementation of new image processing and analysis methods in ultrasound thyroid images. The research procedure comprised two main concepts towards optimization of thyroid ultrasonography. The design and implementation of: 1. Wavelet-based image processing methods towards speckle suppression and thyroid nodule segmentation. Image analysis methods in order to evaluate the thyroid nodules malignancy risk factor Wavelet Based Image Processing Most contemporary vision algorithms cannot efficiently perform on image intensity values that are directly derived from the initial gray-level representation. These intensity values are highly redundant, while the amount of important information within the image may be small. The depiction of ultrasound intensity values under a different angle of view can reveal several significant features that are not easily distinguishable in the original image. The wavelet-based transformation from the initial ultrasound image representation into a feature representation explicitly reveals the useful image features without the loss of essential image information, reduces the redundancy of the image data and eliminates any irrelevant information [55]. When an image contains meaningful structures of various sizes, the scale parameter should vary. Edges at different scales correspond to different physical entities. Large obects are well represented in large scale whereas small structures are localized in small scales. The multiresolution wavelet analysis provides information content of images by viewing any sharp variations (edges) at different scales by investigating the neighbors of these edges with the neighboring size varying [57,58]. - -

33 Chapter 1 Since edges are considered as efficient descriptors of images, the multi-resolution formalism has the ability to detect and record them towards edge detection and segmentation purposes. Wavelet theory offers a mathematical framework for the multiscale processing and relates the behavior of edges across scales to local image properties. Mallat and Zong [6] have proved that a multiscale edge representation can provide a complete and stable representation of a signal, which in turn means that the whole signal information is carried by those multiscale edges. The latter denotes that image processing methods can be preferably utilized on the edge representation than directly onto the intensity value representation. Isolated edges or contours (group of edges with similar properties) correspond to sharp contrasts and can be detected from the local maxima of the wavelet transform. The multiscale local maxima representation is a reorganization of image information that provides higher level description of structures. A remarkable property of the wavelet transform is its ability to characterize the local regularity of image features such as discontinuities and sharp cusps. In mathematics, this local regularity is often measured with Lipschitz exponents [61]. The multiscale edge representation provides the ability to detect all the singularities (small points in space of sudden localized changes, which often indicate the most important features) of an image and to measure their Lipschitz regularity. In this thesis, an investigation has been made to the local behavior of image singularities in terms of Lipschitz regularity in thyroid ultrasound images. The wavelet transform modulus maxima created by noise singularities have a different behavior than those that are mainly affected by image singularities. The following realization has been made; the additive signal dependent noise random field, obtained by the speckle model implemented in this thesis, is a distribution which is almost everywhere singular or discontinuous, with non-positive Lipschitz exponents. On the contrary, the worth singularities derived from non irregular texture are sharp cusps that have positive Lipschitz exponents. As an application, an algorithm has been developed that removes speckle noise from ultrasound images by analyzing the evolution of wavelet transform modulus maxima across scales. This inter-scale search has been implemented by zooming into edges, beginning at low resolution (large scales) and adaptively increasing the resolution (small scales) to acquire the necessary details. In the resulting de-speckled ultrasound image, contrast enhancement of various structures in regard with the surrounding environment without the creation of blurring has been observed. In addition, according two independent observers the disclosure of structures that are not easily distinguishable by the human eye has also been reported. The two-microlocalization properties [69] of these edges that provide characterization of singularities have also been utilized in the subsequent segmentation algorithm. The multiscale - 3 -

34 Introduction information acquired by the aforementioned study has been integrated towards the nodule boundary detection. A multi-scale hybrid model has been introduced that employed wavelet local maxima, after the regularity estimation, towards obect identification in order to extract the thyroid nodule s boundary. The proposed model transfers the multiscale local maxima representation into a multiscale obect representation. Each obect that occupies a physical region has been detected by means of local maxima adacency in all available scales. The multiscale structure representation associates an anatomical obect in the image with a volume in the multiscale edge transform. This structure representation serves as input to a constrained Hough transform for nodule detection. The segmentation method offered an additional tool in the shapebased thyroid nodule categorization from the physician and accuracy enhancement during fine needle aspiration procedure Image Analysis Finally, an investigation of various pattern recognition methods for automatic thyroid nodule discrimination in terms of high and low risk of malignancy has been made. Various pattern recognition algorithms such as Support Vectors Machines (SVMs), the Probabilistic Neural Network (PNN), the classical quadratic least squares minimum distance (QLSMD), the quadratic Bayesian (QB) and the multilayer perceptron (MLP) classifiers, have been implemented throughout this thesis. This research comprised two independent studies that employed initially textural features and subsequently morphological and wavelet based features derived from the segmentation procedure. The texture-based classification scheme implemented in this study has managed to quantify several textural parameters visually evaluated by physicians in assessing the thyroid nodule s risk factor and succeeded high classification rates. These parameters mainly involved echogenicity in regard with the surrounding environment, presence of calcifications within the nodule and increased vascularity. An additional study has also been made that aimed at the employment of quantified morphological and wavelet-based features, in order to evaluate the malignancy risk factor in ultrasound thyroid nodules. In this research, a novel approach has been made that utilized the image singularities in order to evaluate the effect of speckle in the classification procedure. In a parallel study (with and without speckle), the pattern recognition algorithms employed various wavelet features so as to evaluate the discrimination importance of speckle. As a conclusion, speckle noise, even if in the original US image its effect cannot be easily evaluated; in the wavelet feature level its presence had a negative influence

35 Chapter 1 The quantification of various sonographic observations (such as echogenicity, the boundary irregularity degree, the non-circular boundary and the presence of micro-calcifications) led to a more obective evaluation towards biopsy necessity and could be of assistance in the decision making procedure. 1.4 Publications The research work of this thesis has resulted or contributed in publications and presentations in international ournals and conferences Publications in peer reviewed international ournals Journal Published Papers: 1. S. Tsantis, D. Cavouras, I. Kalatzis, N. Piliouras, N. Dimitropoulos, and G. Nikiforidis: Development of a support vector machine-based image analysis system for assessing the thyroid nodule malignancy risk on ultrasound, Ultrasound in Medicine and Biology, Vol. 31, No. 11, pp , 005. S. Tsantis, N. Dimitropoulos, D. Cavouras and G. Nikiforidis: A Hybrid Multi-Scale Model for Thyroid Nodule boundary detection on Ultrasound Images, Computer Methods and Programs in Biomedicine, Volume 84, Issues -3, Pages 86-98, S. Tsantis, N. Dimitropoulos, M. Ioannidou, D. Cavouras and G. Nikiforidis: Inter-Scale Wavelet Analysis for Speckle Reduction in Thyroid Ultrasound Images, Computerized Medical Imaging and Graphics, Volume 31, Issue 3, Pages , 007 Journal Submitted Papers: 4. S. Tsantis, N. Dimitropoulos, D. Cavouras, and G. Nikiforidis: Pattern Recognition Methods Employing Morphological and Wavelet Local Maxima Features towards Evaluation of Thyroid Nodules Malignancy Risk in Ultrasonography, Submitted in Ultrasound in Medicine and Biology, April Publications in International Conference Proceedings 1. S. Tsantis, N.Dimitropoulos, D. Cavouras and G. Nikiforidis: Morphological Features towards Ultrasound Thyroid Nodules Malignancy Evaluation, nd IC-EpsMsO, Athens, 4-7 July, 007. S. Tsantis, N.Dimitropoulos, D. Cavouras and G. Nikiforidis: 1 st Order vs. nd Order Derivatives towards Wavelet-Based Speckle Suppression in Ultrasound Images, nd IC- EpsMsO, Athens, 4-7 July,

36 Introduction 3. S. Tsantis, D. Gklotsos, I. Kalantzis, N. Piliouras, P. Spyridonos, N.Dimitropoulos, G. Nikiforidis and D. Cavouras: Computer Assisted Diagnosis of Thyroid Nodules Malignancy Risk.. European Congress of Radiology, S. Tsantis, I.Kalantzis, N Piliouras, D. Cabouras, N Dimitropoulos, G. Nikiforidis: Computeraided characterization of thyroid nodules by image analysis methods, Proceedings in International Conference of Computational Methods in Sciences and Engineering 003 (ICCMSE 003), pp 639:64, September S. Tsantis, D. Cabouras, N Dimitropoulos, G. Nikiforidis: Denoising sonographic images of thyroid nodules via singularity detection employing the wavelet transform modulus maxima, Proceedings in International Conference of Computational Methods in Sciences and Engineering 003 (ICCMSE 003), pp 643:646, September S. Tsantis, N.Piliouras, N.Dimitropoulos, D. Cavouras and G.Nikiforidis: Evaluation of Support Vector Machines Wavelet kernels for the automatic categorization of thyroid nodules, 4 th European Symposium on Biomedical Engineering, Patra, 5 th - 7 th June Stavros Tsantis, Dimitris Glotsos, Giannis Kalatzis, Nikos Dimitropoulos, George Nikiforidis, Dionisis Cavouras: Automatic contour delineation of thyroid nodules in ultrasound images employing the wavelet transform modulus-maxima chains, 1 st International Conference From Scientific Computing to Computational Engineering, IC-SCCE, Athens, 8-10 September, Stavros Tsantis, Dimitris Glotsos, Panagiota Spyridonos, Giannis Kalatzis, Nikos Dimitropoulos, George Nikiforidis, Dionisis Cavouras: Improving Diagnostic Accuracy in the classification of thyroid cancer by combining quantitative information extracted from both ultrasound and cytological images, 1 st International Conference From Scientific Computing to Computational Engineering, IC-SCCE, Athens, 8-10 September, Contributions in Publications in International Conference Proceedings 1. Glotsos D., Spyridonos P., Tsantis S., Kalatzis I., Dimitropoulos N., Nikiforidis G., Cavouras D: Unsupervised Segmentation of Fine Needle Aspiration Nuclei Images of Thyroid Cancer using a Support Vector Machine Clustering Methodology, 1 st International Conference From Scientific Computing to Computational Engineering, IC-SCCE, Athens, 8-10 September,

37 Chapter 1. D. Glotsos, S. Tsantis, J. Kybic 1, I. Kalatzis, P. Ravazoula, N. Dimitropoulos, G. Nikiforidis, D. Cavouras, Pattern recognition based segmentation versus wavelet maxima chain edge representation for nuclei detection in microscopy images of thyroid nodules, 3rd European Medical and Biological Engineering Conference, Prague, Czech Republic, 0-5 November, Glotsos D., Spyridonos P., Ravazoula I., Kalatzis I., Tsantis S., Nikiforidis G., Cavouras D: Evaluating the Generalization Performance of a Support Vector Machine based Classification Methodology in Brain Tumor Astrocytomas Grading, 1 st International Conference From Scientific Computing to Computational Engineering, IC-SCCE, Athens, 8-10 September, Contribution in Dimitris Glotsos and Jan Kybic: Development of a wavelet-assisted edgedetection algorithm for boundary detection of fine needle aspiration images of thyroid nodules. Research Report CTU-CMP , Center for Machine Perception, K13133 FEE Czech Technical University, Prague, Czech Republic, March Dissertation Layout In Chapter, a background on thyroid physiology and anatomy is provided as well as the grading categories of solitary thyroid nodules. In Chapter 3, a theoretical background on physics and instrumentation of ultrasound in general is presented. Moreover, the quality control procedure and the data acquisition system employed in this thesis are also given. Chapter 4 provides an overview of classic wavelet theory and wavelet transforms. Emphasis is given to redundant dyadic wavelet transform since it is the basis of the wavelet-based techniques used in this thesis. In Chapter 5 the regularity theory along with its correlation to wavelet transform modulus maxima is depicted. Chapter 6 describes the fundamentals of pattern recognition theory and various classification algorithms with a thorough study in feature selection and generation methods. In Chapter 7, a survey of various speckle suppression methods in ultrasonography is presented first. Then, a new wavelet-based method for speckle reduction in thyroid ultrasound imaging is explained in detail. Chapter 8 contains at first an extensive review of various segmentation algorithms in ultrasound imaging. Consequently a novel hybrid model is presented towards boundary extraction of thyroid nodules in ultrasound. In Chapter 9, an SVM model is designed and implemented in order to assess thyroid nodule malignancy risk factor that employed several textural characteristics of the sonographic image. Chapter 10 encloses a pattern recognition study based on two well known classification algorithms (SVMs and PNN) that - 7 -

38 Introduction employed morphological in conunction with various wavelet local maxima features directly derived from the segmentation procedure. In Chapter 11, a general conclusion and some future perspectives of the present thesis are provided. In Appendixes I, II, III, IV the List of Figures, List of Tables, Abbreviations and Index of this manuscript are listed respectively. 1.6 Research Funding The present research was funded by the Operational Program for Educational and Vocational Training II (EPEAEK II)

39 CHAPTER Thyroid Gland.1 Introduction The thyroid gland is a brownish-red and highly vascular organ, located in the front of the lower neck and attached between the lower part of the larynx and the upper part of the trachea. The gland varies from an H to a U shape formed by two elongated lateral lobes. Both lobes are about 4 cm long and 1- cm wide and are linked together by a median isthmus [1] (Figure.1). Figure.1 The thyroid gland. The thyroid gland produces, stores and secrets thyroid hormones, which are peptides containing iodine. The two most important hormones are tetraiodothyronine (thyroxine or T4) and triiodothyronine (T3). They are essential for humans and have many effects on body metabolism, growth, and development. The thyroid gland s function is influenced by hormones produced by two organs: 1. The pituitary gland, located at the base of the brain which produces thyroid stimulating hormone (TSH) and,. The hypothalamus, a small part of the brain above the pituitary that produces thyrotropin releasing hormone (TRH) (Figure.). Low levels of thyroid hormones in the blood are detected by the hypothalamus and the pituitary. TRH is released, stimulating the pituitary to release TSH. Increased levels of TSH, in turn, stimulate the thyroid to produce more thyroid hormone, thereby returning the level of thyroid hormone in the blood back to normal. The three glands and the hormones produce the "Hypothalamic - Pituitary - Thyroid axis". Once thyroid hormone levels are restored, TSH secretion stabilizes at a high level. [-4]

40 Thyroid Gland Figure. Schematic representation of Hypothalamic - Pituitary - Thyroid axis. Thyroid Disorders The enlargement of the thyroid gland is called goitre. Goitre does not always indicate a disease, since thyroid enlargement can also be caused by physiological conditions such as puberty and pregnancy [5,6]. The main causes of thyroid disease are: 1. Excessive thyroid hormone production or hyperthyroidism.. Decreased thyroid hormone production or hypothyroidism. 3. The state of normal thyroid function is called euthyroidism. All thyroid disorders are much more common in women than in men. Other disorders termed as "Autoimmune" of the thyroid gland are also common. These are caused by abnormal proteins, (called antibodies), and the white blood cells which act together to stimulate or damage the thyroid gland. Graves' disease (hyperthyroidism) and Hashimoto's thyroiditis, are diseases of this type [7-1]. Graves' Disease: Graves' disease (thyrotoxicosis) is due to a unique antibody called "thyroid stimulating antibody" which stimulates the thyroid cells to grow larger and to produce excessive amounts of thyroid hormones. In this disease, the goitre is due not to TSH but to this antibody. Hashimoto's Thyroiditis: In Hashimoto's thyroiditis, the goitre is caused by an accumulation of white blood cells and fluid (inflammation) in the thyroid gland. This leads to destruction of the thyroid cells and, eventually, thyroid failure (hypothyroidism). As the gland is destroyed, thyroid hormone production decreases; as a result, TSH increases, making the goitre even larger. Hyperthyroidism is treated mostly by medical means, but occasionally it may require the surgical removal of the thyroid gland. Sometimes, thyroid enlargement is restricted to one part of the gland. The most common cause of this is a cyst or nodule, which may be benign or malignant. Occasionally there are many nodules. This, so called "multinodular goitre", is probably caused by mutations of follicular cells. Thyroid nodules are not expression of a single disease but constitute the

41 Chapter clinical indication of a wide range of different diseases. An initial differentiation within thyroid nodules subects to quantitative criteria. A single nodule is called solitary nodule whereas the presence of multiple nodules is often called as multinodular goitre. Although as many as 50% of the population will have a nodule somewhere in their thyroid, the vast maority of these are benign. Occasionally, solitary thyroid nodules can take on characteristics of malignancy and require either a needle biopsy or surgical excision..3 Management of solitary nodules The parameters that must be considered into a clinical decision, regarding solitary nodules, include the history of the lesion, age, sex, and family history of the patient, physical characteristics of the gland, local symptoms, and laboratory evaluation. The age of the patient is an important consideration since the ratio of malignant to benign nodules is higher in youth and lower in older age. Male sex also carries a similar importance. The basics steps towards an efficient management of solitary nodules include [13-16]: Clinical examination Thyroid function tests: TSH, antibodies Ultrasound Τhyroid scan Cytology of fine needle aspirate (FNA) Clinical examination: The physician inspects the neck and feels the thyroid gland (palpation). The size and consistency of the thyroid gland, how painful it is, and the extent to which it may have moved out of position from surrounding structures are also assessed by the physician. Thyroid function tests: Through a blood test the thyroid gland s functionality can be evaluated. Measurements of T3, T4 and hormones that control thyroid gland activity (the TSH test) are taken and compared with the norm. Ultrasound: High-resolution ultrasonography (US) can be used to determine the size and presence of nonpalpable nodules as small as 1 mm within the thyroid tissue (Figure.3). Furthermore, any solid or cystic components within a thyroid nodule can be detected with high precision [17-7]. In this examination technique, different types of body tissue conduct and reflect sound in different ways. The reflecting echo is recorded and displayed as an image from the part of the body from which it resonated. Ultrasound has now established itself as a standard means of examination in thyroid gland morphology. In cases where a nodule presents some suspicious US characteristics, a fine needle aspirate biopsy (FNAB) is performed. In a review of published studies, the use of conventional thyroid ultrasonography did not allow accurate prediction between malignant and benign cases of solitary thyroid nodules. Its main

42 Thyroid Gland indications are accurate measurement of size and as a guide for FNAB. However, certain US features such as irregular borders of the nodule, lack of a "halo", echogenicity, evidence of calcium flakes, marginal nodules in a cyst, increased blood flow, and growth on consecutive ultrasounds, are suggestive signs of malignancy. Figure.3 Ultrasonographic examination in the transverse plane of the thyroid containing a solid nodule in the right lobe and a homogeneous appearance on the left lobe. Thyroid scan: By using thyroid gland scintigraphy, a morphological and functional image of the thyroid gland is produced simultaneously. This means that the way different areas of the thyroid gland are depicted relates to how they are functioning (normally, hyperactively or hypoactively). Such examinations were obligatory before, with the use of radio-iodine for treating thyroid gland disorders [8,9]. Thyroid scan can differentiate a solitary nodule as cold (Hypo-functioning) nodule or hot (Hyper-functioning) nodule (Figure.4). (a) (b) Figure.4 Scintiscans of thyroid. (a)the scan on the left is normal. (b) A typical scan of a "cold" thyroid nodule failing to accumulate iodide isotope is shown on the right. The thyroid scan can also provide evidence for a diagnosis in a multinodular goiter, in Hashimoto s thyroiditis, and rarely in thyroid cancer when functioning cervical metastases are seen. Malignant tumors usually fail to accumulate iodide to a degree equal to that of the normal gland

43 Chapter FNAB: Fine needle aspiration biopsy has become the diagnostic tool of choice for the initial evaluation of solitary thyroid nodule because of its accuracy, safety, and cost effectiveness. In most but not all cases, FNAB is the only non-surgical method which can differentiate malignant and benign nodules [30,31]. Fewer patients have undergone thyroidectomy for benign disease as a result of FNAB, with resultant decreased health care costs. Although needle biopsy can be performed easily, consistently obtaining adequate tissue and processing the specimens to achieve accurate cytopathological interpretation, requires expertise and experience. The needle is placed into the nodule several times and cells are aspirated into a syringe. The cells are placed on a microscope slide, stained, and examined by a pathologist. Often a small percentage of FNA s are termed as Nondiagnostic, which indicate that there are an insufficient number of thyroid cells in the aspirate and no diagnosis is possible. A nondiagnostic aspirate should be repeated..4 Grading Various alternative classifications of thyroid nodules have been proposed. A summarized classification approach based on cytology findings is illustrated in Figure.5 [3,33]. Thyroid Nodules Benign Nodules Malignant Nodules Simple Cyst Inflammatory Focal Hemorrhage Indeterminate Papillary Carcinoma Medullary Carcinoma Lymphoma Anaplastic Carcinoma Colloid Cyst Follicular Cells (hyperplasia) non-functioning follicular adenoma Figure.5 Classification of thyroid solitary nodules Follicular carcinoma Bening nodules: The great maority of solitary thyroid nodules are benign (>90%). Common types of the benign thyroid nodules are simple thyroid cysts, inflammatory cysts and cysts with focal haemorrhage [34]

44 Thyroid Gland Adenomatous Hyperplasia: An interesting fact regarding bening nodules, is that a certain category (epithelial hyperplastic nodules) can, under particular circumstances, be transformed into malignant ones. The thyroid cells on these aspirates are neither clearly benign nor malignant [45,46]. Twenty five percent of such suspicious lesions are found to be malignant when these patients undergo thyroid surgery. These are usually follicular cell cancers (Figure.6). Therefore, surgery is recommended for the treatment of thyroid nodules from which a suspicious aspiration has been obtained. (a) (b) Figure.6 (a) Thyroid nodule with epithelial hyperplasia, (b) Colloid nodule. Malignant nodules: Most thyroid cancers are very curable. In fact, the most common types of thyroid cancer (papillary and follicular) are the most curable. In younger patients, both papillary and follicular cancers can be expected to have better than 97% cure rate if treated appropriately. Both papillary and follicular cancers are typically treated with complete removal of the lobe of the thyroid which harbors the cancer [35-38]. Medullary cancer of the thyroid is significantly less common, but has a worse prognosis. Medullary cancers tend to spread to large numbers of lymph nodes very early on therefore requiring a much more aggressive operation than does the more localized cancers such as papillary and follicular. The least common type of thyroid cancer is anaplastic which has a very poor prognosis [39-40].Most primary thyroid lymphomas occur in middle-aged or elderly patients with a ratio of women-men ranging from :1 to 8:1. Patients present with a relatively rapid thyroid enlargement accompanied by hoarseness, dysphagia and/or dyspnea in approximately 5% of cases and cord paralysis in about 17% [41-43]. Anaplastic thyroid cancer tends to be found after it has spread and is not cured in most cases. Often an operation cannot remove the entire tumor [44]. One of the obectives of this thesis is to explore all available ultrasonic characteristics by means of image analysis methods in order to predict the malignancy risk factor between high risk nodules (adenomatus or epithelial hyperplasia) and low risk nodules (simple or colloid cysts)

45 CHAPTER 3 Physics & Instrumentation of ultrasound 3.1 Nature of Ultrasound A sound or ultrasound (US) wave consists of a mechanical disturbance of a medium (gas, liquid or solid) which passes through the medium at a fixed speed. The rate at which particles in the medium vibrate is the frequency of the sound and is measured in hertz (cycles/second). In medical ultrasound, the disturbance which is characterized by the local pressure change of the particles of the medium from the resting positions originates at a piezoelectric transducer in a probe placed on the skin surface. The transducer (operating as a transmitter) transforms electrical signals to mechanical movement. The same transducer can transform the reflecting mechanical vibrations into electrical signals (operating as a receiver). The ultrasound frequencies used in contemporary US systems range from 1 to 0 MHz [47,48]. 3. Propagation in Tissue Ultrasound is altered by the tissue through which it passes. At the boundaries between different tissue types, the US beam can be partially reflected, refracted, scattered by small tissue structures or subected to energy loss by absorption [49,50]. Reflection: When ultrasound is incident on a smooth boundary (interface) between two media some ultrasound is transmitted through the interface and some reflected. If the interface is perpendicular to the direction of propagation the intensity of the reflected ultrasound beam is proportional to the acoustic impedances of the two media. Refraction: The transmitted beam at an interface between media having different speeds of ultrasound deviates from the path of incident beam, provided the angle of incident is nonzero. The beam deviation is dependent on the difference of ultrasound speed (not impedances) and the refracted beam bends away from the perpendicular if the speed in the second medium is higher than in the first and vice versa. Scattering: If ultrasound is incident on a rough surface or on particles with size small or comparable with the beam s wavelength then the ultrasound is scattered in all directions. If the scattering particles are small compared with the wavelength then the scattered power is proportional to the fourth power of the ultrasound frequency. Absorption: Ultrasound power is also subected to absorption in which the energy of the ultrasound is converted into heat. The loss due to absorption increases with frequency

46 Physics & Instrumentation of Ultrasound 3.3 Pulse Echo Imaging At a boundary between two tissues a proportion of ultrasound passes on and the rest is reflected. The degree of reflection depends on the acoustic impedances of the two tissues which are depended on density and compressibility. A large difference in acoustic impedance (i.e. soft tissue bone or soft tissue air interfaces) leads to a high degree of reflection. At the boundary between two different types of soft tissue (i.e. muscle fat) the degree of reflection is small [51]. In ultrasonic imaging the transducer is periodically driven by an electrical pulse leading to the transmission of an ultrasound pulse which is received back after reflection or scattering at tissue interfaces. The time of arrival of the echo from a given interface depends on its depth and the US system employs the time of echo s arrival after the transmission as an indication of the depth of the interface. Since the amplitude of an echo is determined by the structure and physical decomposition of the reflector or scatterer, it is used to determine the brightness of the echo in a display. B Mode: In this imaging technique the ultrasound beam is scanned through the tissue. The echo signals received at each beam position are displayed as spots on the monitor screen in which the brightness indicate the echo amplitude (grayscale display). The positions of the spots are determined by the orientation of the beam and by the time of arrival of the echoes. M Mode: The movement of echo-generating tissues can be displayed as a function of time by means of the M-mode display. Doppler Mode: Movement of reflectors or scatterers also changes the frequency of the received signal. This change from the transmitted signal frequency is known as the Doppler Effect and the magnitude of the change (Doppler shift) is proportional to the reflector or scatterer velocity. By measuring the Doppler shift, the cyclical variation of blood velocity can be monitored. The integration of real time Doppler instruments produces a color-flow image which is superimposed in the grayscale display. 3.4 Instrumentation All US examinations throughout this thesis were performed on an HDI-3000(Figure 3.1) ATL digital ultrasound system Philips Ultrasound P.O. Box 3003 Bothel, WA , USA with a wide band (5-1 MHz) linear probe (L7-4) using various scanning methods such as longitudinal and transversal cross sections of the thyroid gland. The system is located in Medical Imaging Department, EUROMEDICA Medical Center, Mesogeion Avenue, Athens, Greece

47 Chapter 3 Figure 3.1 HDI-3000-ATL digital ultrasound system The wide-band linear ultrasound transducer has the capacity to resonate at multiple frequencies which in turn gives the system the ability to acquire a wide range of frequencies (5-9 MHz), contrary to conventional transducers that detects only the nominal frequency (7 MHz), producing US images of high quality (Figure 3.). Figure 3. US image of the thyroid gland with a cystic nodule. 3.5 Quality Control of the Ultrasound System Before the acquisition and storage of US image in the computer a thorough quality control is performed on the system. The quality control procedure utilized a tissue mimicking phantom RMI 403 LE, GAMMEX RMI P.O. Box 6037 Middleton, WI USA with the same attenuation and speed of sound as human soft tissue with uniform scatter distribution

48 Physics & Instrumentation of Ultrasound that yields a smooth image texture. The image quality indicators employed in the procedure are presented below: Depth of Penetration: The point at which usable tissue information disappears or maximum depth of penetration is reached can be defined simply as how far one can see into the phantom. Equipment sensitivity and noise determines the deepest echo signal which can be detected and clearly displayed. Image Uniformity: Ultrasound systems can produce various image artefacts and nonuniformities which in some cases mask variations in tissue texture. Common non-uniformities are horizontal bands in the image caused by inadequate handling of transitions between focal zones or vertical bands indicating inactive or damaged transducer elements. Axial Resolution: Axial resolution describes the scanner s ability to detect and clearly display closely spaced obects that lie on the beam s axis. Using pin targets of decreased vertical spacing, the system s axial resolution is determined by locating the two resolvable pins with the smallest separation. Distance Accuracy: Vertical and horizontal distance measurement errors can easily go unnoticed on clinical images. Distance accuracy as a quality indicator is determined by comparing the measured distance between selected pin targets in the phantom with the known distance. Lateral Resolution: Lateral resolution is described as the distinction of small adacent structures perpendicular to the beam s maor axis. The lateral resolution is measured indirectly by measuring the width of pin targets at depths corresponding to near, mid, and far field ranges of the transducer. Dead Zone: The dead or ring down zone is the portion of the image directly under the transducer where image detail is missing or distorted. The depth of an instrument s dead zone is determined by identifying the shallowest pin target that can be clearly visualized. All steps carried out during the quality control procedure were upon the guidelines determined both from the ultrasound system and Phantom manufacturer user manuals and the performance of the system was accordingly to the standard specifications. 3.6 Data Acquisition and Storage An image processing system for acquisition and storage of US images consists of the US system, a personal computer and an interface between the system and the computer (Figure 3.3). The interface converts analog information into digital data which the computer can process. This takes place in a special piece of hardware, the frame grabber, which also stores the image. Usually the frame grabber package contains a library of often-used routines which can be linked to the user s program

49 Chapter 3 The frame grabber utilized in the present study is the Miro PCTV (Pinnacle Systems Inc. 80 N.Bernardo Avenue Mountain View, CA 94043) with the BT 848 chipset integrated. The video output of the ultrasound system (HDI 3000) is connected to the frame grabber of the image processing computer (Microsoft PC, PII at 600 MHz with 64 MB of RAM). Figure 3.3 Image processing system for acquisition and storage of US images. The BT848 chipset integrates an NTSC/PAL/SECAM composite, an S-Video decoder, a scaler, a DMA controller, and a PCI Bus master on a single device. It can place video data directly into host memory for video capture applications and into a target video display frame buffer for video overlay applications. BT848 is designed to efficiently utilize the available 13 MB/s PCI bus. The video stream consumes bus bandwidth with average data rates set to 44 MB/s for full size 768x576 PAL RGB3. Consecutive video frames can be written into the Video image buffer (continuous capture mode). The external trigger (Freeze command) signal temporarily stops this process thus freezing the video. When the external trigger is disabled the video buffer is then again written to, by every consecutive frame until the software through the radiologist issues another FREEZE command. The selected US Images are captured at frame rate (40 ms, 5 Hz - PAL standard video signal) and converted into JPEG format images with a resolution of 768x

50 Physics & Instrumentation of Ultrasound (full PAL resolution) pixels. The software of acquisition and storage of US images is the Icon-Print 000, it is written in Visual C ++ and uses technology VFW (Video for Windows)

51 CHAPTER 4 The Wavelet Transform Summary This chapter reviews the theory behind wavelets and wavelet transforms. At first a small review of the continuous wavelet transform is given, followed by an extensive study of the dyadic wavelet transform both in one dimensional (1-D) signals and two dimensional (-D) images. The -D redundant dyadic wavelet transform is implemented and utilized throughout this thesis. Special reference is made also to the spline wavelets employed in this thesis. Moreover, a summary is given regarding the advantages of the redundant dyadic wavelet transform. The multi-scale edge representation is also analyzed and depicted at the last part of this chapter. 4.1 Wavelet Theory Wavelets are an extension of windowed Fourier analysis by Gabor [5], in which through a fixed window a large number of oscillations are used for detecting high frequencies, whereas a small number is used to detect low frequencies. However, in the first case the window is blind to smooth events and in the second case the window probably will miss a brief change. Instead of a fixed window and a variable number of oscillations Morlet and Grossman [53] employed a mother wavelet which is stretched or compressed to change the size of the window, thus providing a decomposition of the signal at different scales (frequency bands). The dilation of the function called mother wavelet produces a family of functions. The wavelet transform of a signal is a sequence of signals obtained by the convolution of the signal with the wavelet family. The wavelets size variation due to dilation permits them to automatically adapt to the different components of the signal. A small window (high frequency band) detects rapid high-frequency components and a large window (low frequency band) traces slow low-frequency components. The wavelet transform is required to satisfy a so called admissibility condition so that it can form a complete and numerically stable representation. The wavelet transform gives a representation that has good localization in both frequency and space [54-56]. The localization in frequency implies a correspondence between a scale of the wavelet transform and a frequency band. The overall study across all available frequency bands is called multiresolution analysis [57-59]. The wavelet transform is divided in two main categories: the continuous wavelet transform (CWT) in which all values of the - 1 -

52 The Wavelet Transform parameters are employed and the discrete wavelet transform (DWT) in which only a discrete set of parameters are considered. 4. Continuous Wavelet Transform The continuous wavelet transform is shift invariant thus suitable for feature extraction and image analysis methods [6]. The CWT decomposes a signal by means of dilated and translated wavelets. Let a wavelet ψ ( x) L ( R) is a function of zero average: ψ ( x) dx = It is normalized ψ = 1 and centered in the neighborhood of x = 0. The function ψ(x) is used to create a wavelet family by dilating ψ with s: 1 x ψ s ( x) = ψ 4. s s All the functions in the wavelet family have the same shape as the wavelet. The continuous wavelet transform of a signal L ( R) by: f is a family of functions [ W f x) ] + s ( and defined s R Ws f ), If ψˆ ( ω) is the Fourier transform of ψ(x), then: + ( x) = ψ s f ( x s R 4.3 Wˆ s f ( ω) = ψˆ ( sω) f ( ω) 4.4 In order for the transform to be invertible, the wavelet ψ(x) must satisfy the admissibility condition [53]: C ψ = + dω ψˆ ( ω) < ω The function f(x) can be reconstructed from its wavelet transform [53]: W 1 : f ( x) = + 0 ds ψ ˆ s Ws f ( x) 4.6 s The admissibility condition also ensures that the wavelet transform is an isometry: - -

53 Chapter 4 ds s = + f ( x) W f ( x) 4.7 s 0 Equation 4.7 implies that the continuous wavelet transform is a complete and numerical stable representation. 4.3 Redundant Dyadic Wavelet Transform (1-D) In the translation-invariant dyadic wavelet transform the scale parameter s is discretized dyadically ([ ) to simplify the numerical calculations, while the spatial parameter is ] Z continuous [60,61]. Let a wavelet ψ ( x) L ( R) is a wavelet whose average is zero. The wavelet family by dilating ψ with s is: 1 x ψ ( x) = ψ ( ) 4.8 The dyadic wavelet transform of a function f(x) L, at a given scale obtained by the convolution of f(x) with the wavelet family W f ( x and at the position x x) = f ψ ( ) 4.9 We refer to the dyadic wavelet transform as the sequence of functions: Wf = W f ( x)) ( Z, 4.10 where W is the dyadic wavelet transform operator. In order to study the completeness and stability of the DWT we denote the Fourier transform of f ( ) as: W x ˆ W f ( ω) fˆ( ω) ψˆ ( ω = ) 4.11 Given that there are two strictly positive constants A & B such that: ω R, A ψˆ ( ω) B, 4.1 Z it is ensured that the whole frequency axis is covered by dilations of ψˆ ( ω) by ( ) Z so that f ˆ ( ω ) and consequently f (x) can be recovered from its dyadic wavelet transform. The reconstructing wavelet χ(x) is any function whose Fourier transform satisfies: - 3 -

54 The Wavelet Transform + = ψ ˆ ( ω) ˆ( χ ω) = If equation (4.1) is valid, an infinite number of functions ˆ χ( x) exist that satisfy equation (4.13). The inverse dyadic wavelet transform that recovers f (x) is given by the summation: + ( x) = W f ( χ = f χ ) In practice, we can compute a Wavelet Transform only over finitely many scales. This is because the observed data is limited between a non-zero small (fine) scale and a finite large (coarse) scale. According to Mallat [57], one can normalize the observable finest scale to 1( 0 ) and the coarsest scale to J where J is dependent on the sample size of the data. In order to model this scale limitation, a real function φ(x) is introduced, whose Fourier transform is an aggregation of ψˆ ( ω) and ˆ χ( ω) at scales larger than 1: + = 1 ˆ( φ ω) = ψˆ ( ω) ˆ( χ ω) 4.15 The reconstructive wavelet χ(ω) is such a function that ψˆ ( ω) ˆ( χ ω) is a positive, real and even function. The equation (4.13) implies that the integral of φ(x) is equal to 1 and hence that it is a smoothing function. Let S be the smoothing operator defined by: S 1 x x) = f φ ( ), φ ( x) = φ( ) 4.16 f ( x If the scale is larger, the more details of f(x) are removed by S. For any scale J >1 equation (4.15) yields: J ˆ J φ ( ω) ˆ( φ ω) = ψˆ ( ω) ˆ( χ ω) 4.17 = 1 From this equation it is derived that the higher frequencies of S f ( ), which have disappeared in f ( ) x 1 x S J can be recovered from the dyadic wavelet transform {[ W f } ] 1 J between the scale and J. In numerical applications, the input signal is measured at a finite resolution and thus the wavelet transform cannot be computed at any arbitrary scale. The original signal can be - 4 -

55 Chapter 4 considered as a discrete sequence D = [ ] of finite energy. If two constants C1>0 and d n n Z C>0 exist, such that ˆ φ ( ω) satisfies: + n= ω R, C1 φ( ω) + nπ C 4.18 From Equation 4.18 the periodic signal D can be considered as the sampling of a smoothed version of f ( x) L ( R) at the finest scale 1: n Z, S1 f ( n) = 4.19 d n The input signal can thus be rewritten as D = [ S 1 f ( n)]. Mallat and Zhong [6] have proposed the redundant discrete wavelet transform (RDWT), utilizing a particular class of wavelets, to compute a uniform sampling of the wavelet transform of f(x) at any scale larger than 1. S d f = S f ( n + w) Let us denote [ J ] n Z and W d n Z f = W f ( n + w)] [ n Z where w is a sampling shift that depends on ψ(x). For any coarse scale J the sequence of discrete signals: d d { S J f [ W f }, 4.0 ] 1 J is called the discrete dyadic wavelet transform of D = [ S 1 f ( n)]. The coefficient signal [ W d f ] provide the details of the input signal at scales 1 J and the coarse signal n Z S d J f provides the approximation of the input signal at the coarse scale J. The filter bank algorithm for computing 1-D RDWT is presented in Figure 4.1. The left size shows the decomposition into wavelet coefficients and the right the reconstruction from wavelet coefficients. G(ω) W 1 f(x) Κ(ω) f(x)=s 1 f(x) G(ω) W f(x) Κ(ω) + f(x) Η(ω) W 3 f(x) + Η*(ω) S f(x) G(4ω) Κ(4ω) Η(ω) + Η*(ω) S 3 f(x) Η(4ω) Η*(4ω) Figure 4.1 One-dimensional three level redundant discrete dyadic wavelet transform

56 The Wavelet Transform The algorithm does not involve sub-sampling and is similar to the algorithme ά trous (algorithm with holes) [63], which also does not involve sub-sampling. Filters H(ω), G(ω) and K(ω), are π periodic and satisfy the perfect reconstruction condition: H ( ω) + G( ω) K( ω) = At dyadic scale, the discrete filters H, G, K, are obtained by inserting -1 zeros between each of the coefficients of the corresponding filters at scale 1. The scaling (smoothing) function φ(x) defined in equation (4.15) can be derived from H(ω) using the equation: ˆ( φ ω) = iwω + ρω e Η( ) 4. ρ = 1 where the sampling shift parameter w is adusted so that φ(x) is symmetrical with respect to 0. Equation (4.) implies that ˆ φ(ω) = e iwω H ( ω) ˆ( φ ω) 4.3 A wavelet ψ(x) is defined, whose Fourier transform ψˆ ( ω) is given from the equation: ψˆ (ω) = e iwω G( ω) ˆ( φ ω) 4.4 The reconstruction wavelet χ(x) is derived from the equation: ˆ χ(ω) = e iwω Κ( ω) ˆ( φ ω) 4.5 A class of filters that satisfy equation (4.1) has been provided by Mallat and Zhong [6]. H(ω) was chosen to obtain a wavelet ψ(x) which is anti-symmetrical, as regular as possible and has a compact support. The wavelet ψ(x) is also equal to the first order derivative (gradient) of a smoothing function θ(x): dθ ( x) ψ ( x) = 4.6 dx Filters H(ω), G(ω) and K(ω) are given by: iω / ν + 1 H ( ω) = e (cos( ω / )) 4.7 iω / G ( ω) = 4e sin( ω / )

57 Chapter 4 1 H ( ω) K( ω) = 4.9 G( ω) All filters have compact support and are either symmetrical or anti-symmetrical. From equations (4. & 4.4) the corresponding scaling and wavelet functions can be derived as: sin( ω / ) ˆ( φ ω) = ω / n sin( / 4) ˆ ω ψ ( ω) = iω ω / 4 n The Fourier transform of the smoothing function θ(x) is therefore: sin( ω / 4) ˆ( θ ω) = ω / 4 n+ 4.3 We have chosen n+1=3. In order to have a wavelet anti-symmetrical with respect to 0 and φ(x) symmetrical with respect to 0, the shifting constant w is equal to ½. From equations (4.31 & 4.3) it can be proven that ψ(x) is a quadratic spline wavelet with compact support, while θ(x) is a Gaussian-like cubic spline whose integral is equal to 1. These functions are depicted in Figure 4.. (a) (b) Figure 4. (a) A cubic spline function and (b) a wavelet that is a quadratic spline of compact support. Mallat [64] have extended this class of filters and derived wavelet functions ψ(x) that are equal to the second order derivative (Laplacian) of a smoothing function. dθ ( x) ψ ( x) = 4.33 dx - 7 -

58 The Wavelet Transform 4.4 Redundant Dyadic Wavelet Transform (-D) x y 1 x y Let ψ ( x, y) = ψ (, ) and ψ (, ) ( x, y) = ψ. The dyadic wavelet transform of a -D function f ( x, y) L ( R ) has two components defined by: Wf 1 { W f ( x, y), W f ( x, y } Z = ) where the Fourier transforms of W f ( x, ) and f ( x, ) are given respectively by: y W y Wˆ Wˆ 1 f ( ω, ω ) = x f ( ω, ω ) = x y y 1 fˆ( ω, ω ) ψˆ ( ω, ω ) x fˆ( ω, ω ) ψˆ ( ω, ω ) x y y x x y y 4.35 and ˆ 1( ψ ω, ω ), ˆ ( ψ ω, ω ) are the Fourier transforms of the partial wavelet x y x y 1 functions ψ ( x, y), ψ ( x, y) respectively. The function f(x,y) can be reconstructed from its dyadic wavelet function with: f ( x, y) = + = 1 1 ( W ( x, y) + W ( x, y) ) χ 4.36 χ 1 where the partial reconstruction wavelets χ ( x, ) and χ ( x, ) satisfy the equation: + = y y 1 1 ( ˆ ( ω, ω ) ˆ χ ( ω, ω ) + ψˆ ( ω, ω ) ˆ χ ( ω, ω )) ψ = x y x y The scaling function is an aggregation of ψˆ ( ω) and ˆ χ( ω) at scale greater than 1: x y x y ˆ( φ ω, ω ) x y = + = ( ψˆ ( ω, ω ) ˆ χ ( ω, ω ) + ψˆ ( ω, ω ) ˆ χ ( ω, ω )) x y x y x y x y = The approximation of f(x) at scale function φ ( ) : x is defined as a convolution with a dilated scaling S f ( y x, y) = f φ ( x, ) 4.39 In practice, the input image is measured in finite resolution and thus the wavelet transform cannot be computed at any arbitrary fine scale. Similarly to the 1-D case, if the discrete signal has a finite number N x Ν pixels, it is symmetrically extended to N x Ν pixels. The discrete - 8 -

59 Chapter 4 periodic image D can then be considered as the sampling of a smoothed version of a function f(x,y) at the finest scale 1: n, m Z, S 1 f ( n, m) = d n, m 4.40 The -D RDWT of Mallat and Zhong [6], computes the uniform sampling of the wavelet transform of f(x,y) at any larger scale than 1, using a particular class of wavelets. For any coarse scale, the RDWT is defined as a sequence of discrete coefficients: where d 1, d, d { S f W f W f }, J, 1 J W W S 1, d, d d f f f = 1 ( W f ( n + w, m + w) ) ( W f ( n + w, m + w) ), ( S f ( n + w, m + w) ) = = 4.4 and w is a sampling shift that depends on the choice of wavelets. The coefficient images 1, W d f and, W d f, provide the details of the input image at scales 1 J and the coarse image S f provides the approximation of the input image at the coarse scale J. The filter bank algorithm for computing the -D RDWT is depicted in Figure 4.3. W 1 1 f G(ω x ) Κ(ω x )L(ω y ) G(ω y ) W 1 f Κ(ω y )L(ω x ) f(x,y)=s 1 f G(ω x ) W 1 f Κ(ω x )L(ω y ) + f(x,y) W f G(ω y ) Κ(ω y )L(ω x ) W 31 f Η(ω x )Η(ω y ) G(4ω x ) Κ(4ω x )L(4ω y ) + Η*(ω x )Η*(ω y ) W 3 f S f G(4ω y ) Κ(4ω y )L(4ω x ) + Η*(ω Η*(ω) x )Η*(ω y ) Η(ω Η(ω) x )Η(ω y ) S 3 f Η(4ω x )Η(4ω y ) Η*(4ω x )Η*(4ω y ) Figure 4.3 Two dimensional three level redundant dyadic wavelet transform

60 The Wavelet Transform The left side depicts the decomposition into wavelet coefficients, and the right the reconstruction from wavelet coefficients. Filters H(ω), G(ω), K(ω) and L(ω) are π periodic and satisfy the perfect reconstruction condition: H ( ω) + G( ω) K( ω) = 1 1+ H ( ω) L( ω) = 4.43 At dyadic scale, the discrete filters H, G, K, L and are obtained by inserting -1 zeros between each of the coefficients of the corresponding filters at scale 1. The -D wavelets are given by expressions analogous to that of the 1-D case. The class of spline functions described in the previous section is used. The wavelets are partial derivatives of -D smoothing functions: ψ ( x, y) = 1 1 θ ( x, y) ( x, y) and ψ ( x, y) = 4.44 x x θ where the functions θ 1 ( x, y) and θ ( x, y) are numerically closed to a single smoothing function θ ( x, y). The redundant wavelet representation presented in this section has several advantages with respect to orthogonal wavelet representation. The sub-band images are shift invariant [65], do not present aliasing and have the same number of pixels as the original, thus the representation is highly redundant. Moreover, smooth symmetrical or anti-symmetrical wavelet functions can be used, allowing the alleviation of any boundary effects via mirror extension of the signal. Due to these advantages it has been extensively used for denoising, segmenting and pattern recognition applications. An example of the RDWT employed in the Circle image is depicted in Figure

61 Chapter 4 S J S 1 f f W 1 f W f 1 1 W 1 f W f W 1 f W f 3 3 W 1 f W f 4 4 Figure 4.4 Redundant dyadic wavelet transform of the Circle image

62 The Wavelet Transform Multiscale Edge Representation (MER) Gradient Vector: Let ), ( y x θ be a symmetrical smoothing function approximating the Gaussian. As already explained in details in the precious section the RDWT of a function ) ( ), ( R L y x f is the set of functions )), ( ),, ( ( 1 y x f W y x f W, which are respectively the partial derivative along the horizontal and vertical orientation of the convolution of ), ( y x f by the smoothing function ), ( y x θ, dilated along a dyadic sequence Ζ ) ( and is given by [6]: ( )( ) ( )( ) = = y x f y y x f x f y x f W y x f W,, ), ( ), ( 1 1 θ θ ψ ψ 4.45 where ), ( 1 y x ψ and ), ( y x ψ are the analyzing wavelets and the dyadic scale. Equation 4.45 indicates that the above set of functions can be viewed as the two components of the gradient vector of ), ( y x f smoothed by ), ( y x θ at each scale : ), )( * ( ), ( ), ( 1 y x f y x f W y x f W θ = 4.46 The modulus angle representation of the gradient vector is given by: 1 ), ( ), ( ), ( y x f W y x f W y x f M + = 4.47 and ), ( ), ( arctan ), ( 1 y x f W y x f W y x f A = 4.48 Modulus Maxima: The sharper variation points of ), ( * y f χ θ at a scale correspond to edges, are obtained from the local maxima of ), ( y x f M along the gradient direction given by ), ( y x f A. The gradient direction values ), ( y x f A were constricted to the following values [66]: 4 7, 3, 4 5,, 4 3,, 4 0, π π π π π π π At each scale of the Dyadic Wavelet Transform, the point (x,y) where the modulus of the

63 Chapter 4 gradient vector M f ( x, ) is maximum compared with its neighbor s locally positioned in y the direction specified by f ( x, ), is called modulus maxima (Figure 4.5). Each time A y such a point is detected, the position of the resultant local maxima is recorded as well together with the values of the modulus M f ( x, ) and angle A f ( x, ) at the corresponding locations [67]. y y M f ( x, ) A f ( x, ) Local Maxima y y Figure 4.5 The gradient magnitude, the gradient directions and the local maxima of the Circle image

64 The Wavelet Transform In order to construct curves in the image plane individual wavelet modulus maxima are connected in a certain scale if they are neighbors and the vector that oins these two points is perpendicular to the angle direction at these points

65 CHAPTER 5 Singularity Detection Summary This chapter discusses at first the theory of singularities along with their classifications according to Lipschitz criteria. In the second part of the chapter a thorough review is given regarding the correlation between singularity detection and wavelet transform modulus maxima accompanied with suitable depictions. 5.1 Singularity and Mathematical Description Singularities are points of sharp variations, which often indicate the most important features of the estimated functions. A specific singularity called change-point, which describes sudden localized change, is of important interest in statistics and has been studied over years. Wavelet analysis is an ideal tool to study localized changes such as discontinuities and sharp cusps in a noisy function. The magnitudes and positions of singularities can be observed from the empirical wavelet coefficients. The modulus maxima of discrete wavelet transform are directly related to the Lipschitz regularity, a mathematical measurement of singularity [68]. The local regularity of a signal is measured with the Lipschitz criteria. Lipschitz exponents (also called Holder exponents) provide uniform regularity measurements over time intervals, but also at any point ν. If f has a singularity at ν, then the Lipschitz exponent at ν characterizes this singular behavior. Pointwise Lipschitz Regularity: A function f is pointwise Lipschitz a 0 at v, if there exist K>0 and a Taylor polynomial p v of degree m = a such that: a t R, f ( t) p ( t) K t ν 5.1 ν A function f is uniformly Lipschitz α over [a,b] if it satisfies condition (5.1) for all v [ a, b], with a constant K that is independent of ν. If f is uniformly Lipschitz α>m in the neighborhood of ν, then one can verify that f is necessarily m times continuously differentiable in this neighborhood. If 0 a < 1then p ( t) = f ( ν ) and the condition (5.1) becomes ν a t R, f ( t) f ( ν ) K t ν

66 Singularity Detection A function that is bounded but discontinuous at ν is Lipschitz 0 at ν. If the Lipschitz regularity is α<1 at ν, then f is not differentiable at ν and α characterizes the singularity type. 5. Wavelet Transform and Singularity A remarkable property of the wavelet transform is its ability to characterize the local regularity of functions. If the wavelet has n vanishing moments then we show that the wavelet transform can be interpreted as a multiscale differential operator of order n. This yields a first relation between the differentiability of f and its wavelet transform decay at fine scales. The following proposition proves that a wavelet with n vanishing moments can be written as the n th order derivative of a function θ [69]. The resulting wavelet transform is a multiscale differential operator. We suppose that ψ has a fast decay which means that for any decay exponent m N there is a C m such that: Cm t R,ψ ( t) 5.3 m 1+ t Theorem 1: A wavelet ψ with a fast decay has n vanishing moments if and only if there is a function θ with a fast decay such that: ψ ( t) = ( 1) n n d θ ( t) n dt 5.4 As a consequence the following equation is obtained: n n d Wf ( u, s) = s ( f θ s )( u), n du t with: θ s( t) = θ ( ). s s Moreover, ψ has no more vanishing moments if and only if + θ ( t) dt 0. The decay of the wavelet transform amplitude across scales is related to the uniform and pointwise Lipschitz regularity of the signal. Measuring this asymptotic decay is equivalent to zooming into signal structures with a scale that goes to zero. Mallat [68,70] also proved that the uniform Lipschitz regularity of f on an interval is related to the amplitude of its wavelet transform at fine scales. Theorem : If that: f L R is uniform Lipschitz α n over [ a, b] then there exists A>0 such ( u, s) [ a, b] R, a+ 1/ Wf ( u, s) As

67 Chapter 5 Conversely, if Wf ( u, s) satisfies condition (5.6) and if α<n is not an integer then f is uniformly Lipschitz α on [ a + ε, b ε ] for any ε>0. Condition (5.6) signifies that the wavelet transform Wf ( u, s) decays like the scale s goes to 0. a+1/ s over intervals where f is uniformly Lipschitz α, when Theorem 3: A necessary and sufficient condition on the wavelet transform for estimating the Lipschitz regularity of f at a point v is [69]: ( u, s) R R +, Wf ( u, s) As a+ 1/ 1+ u ν s a 5.7 Conversely, if α<n is not an integer and there exists a constant A, and α <α such that: ( u, s) R R +, Wf ( u, s) As a+ 1/ 1+ u ν s a' 5.8 then f is Lipschitz α at ν. To interpret more easily the necessary and the sufficient conditions (5.7 & 5.8), it is supposed that ψ has compact support equal to [-C, C]. The cone of influence of ν in the scale-space 1 t u plane is the set of points (u,s) such that ν is included in the support of ψ u, s( t) = ψ ( ). s s t u Since the support of ψ ( ) is equal to [u-cs, u+cs], the cone of influence of ν is defined s by: u ν Cs 5.9 Theorems and 3 prove that the local Lipschitz regularity of f at ν depends on the decay at fine scales of Wf ( u, s) in the neighborhood of ν. The decay of Wf ( u, s) can be controlled from its local maxima values. We use the term modulus maximum to describe any point ( o o u, s ) such that Wf u, s ) is locally maximum at u=u o. This implies that: ( o Wf ( uo, s u o ) = This local maximum should be a strict local maximum in either the right or the left neighborhood of u o, to avoid having any local maxima when Wf u, s ) is constant. We call ( o maxima line any connected curve s(u) in the scale-space plane (u,s) along which all point are

68 Singularity Detection local maxima. In Figure (5.1) high amplitude wavelet coefficients are in the cone of influence of each singularity. (a) (b) (c) (d) (e) Figure 5.1 (a),(b),(c),(d) Wavelet transform of f(t) calculated with quadratic spline wavelet ψ=-θ where θ is the cubic spline smoothing function approximating the Gaussian. The red stars are the local maxima of the wavelet coefficients along each scale. The scale increases from top to bottom. (e) Maxima line in the scale-space plane inside the cone of influence. 5.3 Singularity Detection (1-D) The singularities are detected by finding the abscissa where the wavelet modulus maxima converge at fine scales. If the wavelet has only one vanishing moment, wavelet modulus maxima are the maxima of the first order derivative of f smoothed byθ s. Hwang and Mallat [71] proved that if Wf ( u, s) has no modulus maxima at fine scales then f is locally regular

69 Chapter 5 Theorem 4: Suppose that ψ is C n with a compact support, and ψ=(-1) n θ (n) with + θ ( t) dt 0. Let f L 1 [ a, b]. If there exists s o >0 such that Wf ( u, s) has no local maximum for u [ a, b] and s<s o then f is uniformly Lipschitz n on [ a + ε, b ε ] for any ε>0. This Theorem implies that f can be singular (not Lipschitz 1) at a point ν only if there is a sequence of wavelet maxima points ( u, ) that converges towards ν at fine scales: n s n n N lim un =ν and lim s n = n + n + These modulus maxima may or may not be along the same maxima line. The result guarantees that all singularities are detected by following the wavelet transform modulus maxima at fine scales. Figure (5.) gives an example where all singularities are located by following the maxima lines. For a ψ=(-1) n θ (n) where θ is a Gaussian the modulus maxima of Wf ( u, s) of any f L R belong to connected curves that are never interrupted when the scale decreases. The decay of Wf ( u, s) in the neighborhood of ν is controlled by the decay of modulus maxima in the cone u ν Cs [70,71]. According to Theorem 1 a function f is uniformly Lipschitz α in the neighborhood of ν if and only if there exists A>0 such that each modulus maximum (u,s) in the cone u ν Cs satisfies: a+ 1/ Wf ( u, s) As 5.1 which is equivalent to: 1 log Wf ( u, s) log A + ( a + )log( s) 5.13 Therefore, the Lipschitz regularity at ν is the maximum slope of log Wf ( u, s) as a function of logs along the maxima lines converging to ν (Figure 5.3)

70 Singularity Detection Figure 5. Wavelet transform of f(t) calculated with quadratic spline wavelet ψ=-θ where θ is the cubic spline smoothing function approximating the Gaussian. The red stars are the local maxima of the wavelet coefficients along each scale. The scale increases from top to bottom

71 Chapter 5 Figure 5.3 The full line gives the decay of log Wf ( u, s) from Figure (5.) as a function of logs along the maxima line that converges to the abscissa t=11. The dashed line gives log Wf ( u, s) along the maxima line that converges at t=168. The last graph (Figure 5.3) depicts the maxima lines in the scale-space plane towards zero s for singularity detection. 5.4 Singularity Detection (-D) Theorems and 3 constitute an efficient proof that the wavelet transform is particularly well adapted to estimate the local regularity of functions. The decay of the two-dimensional wavelet transform depends on the regularity of f. We restrict the analysis to Lipschitz exponents 0 a < 1. A function f is said to be Lipschitz α at (x o,y o ) if there exists K>0 such that for all ( x, y) R f ( x, y) a / f ( xo, yo) K( x xo + y yo ) 5.14 If there exists K>0 such that condition (5.14) is satisfied for any ( o o x, y ) Ω then f is uniformly Lipschitz α over Ω. Τhe Lipschitz regularity of a function f(x,y) is related to the asymptotic decay at fine scales of wavelet transform along horizontal and vertical directions W 1 f(u,v, ) and W f(u,v, ) in the corresponding neighborhood. This decay is controlled by it s local maximum value Mf(u,v, ). Like in Theorem one can prove that f is uniformly Lipschitz α inside a bounded domain of R is and only if there exists A>0 such that for all (u,v) inside this domain and all scales : ( a+ 1) Mf ( u, v, ) A

72 Singularity Detection Suppose that the image has an isolated edge curve along which f has Lipschitz regularity α. The value of Mf ( u, v, ) in a two dimensional neighborhood of the edge curve can be bounded by the wavelet modulus values along the edge curve. The Lipschitz regularity of the edge is measured with condition (5.15) by estimating the decay exponent of the modulus amplitude across scales

73 CHAPTER 6 Pattern Recognition Summary This chapter discusses pattern recognition theory along with the implementation of various classifiers employed in this thesis. In the first part of the chapter a detailed explanation is given regarding the feature generation employed as input in the classification system. In the second part the classification algorithms implemented in this thesis are presented. 6.1 Pattern Recognition Theory Pattern recognition classifies obects into a number of categories or classes. This classification procedure is a two-folded process which at first generates a description of the obect (i.e., the pattern) and then classifies it based on that description (i.e., the recognition). The obect description involves feature generation techniques in order to produce certain attributes, whereas the classification task associates a predefined label with the obect based on those attributes. The main goal of each pattern recognition system is to determine the most accurate label for each obect analyzed. The pattern recognition procedure is accomplished with a training phase that configures the algorithms used in both the description and classification tasks based on a number of obects whose labels are known as the training set. During the training phase, a training set is analyzed to determine the attributes used to label the obects with the highest possible accuracy. Following the training phase, the classification takes place to an unlabeled obect based on the attributes of that obect. High coincidence between the known labels and those assigned by the pattern recognition system denotes high classification accuracy. The methodology for description and classification with known attributes is called supervised learning. In cases where the training set is not available the procedure employed is termed as unsupervised learning. A common step in the pattern recognition procedure that usually precedes the hierarchy presented before is the isolation of the obect to be recognized from the surrounding environment. This step is prerequisite in order for the feature extraction to take place

74 Pattern Recognition 6. Obect Isolation Thyroid nodules play a key role in US thyroid imaging diagnostic procedure. Thus it is very important to extract them from the noisy environment for further processing. This isolation procedure is termed as segmentation and a detailed review regarding the segmentation approach employed in this thesis is presented in chapter Feature Generation The feature generation stage is the process of computing features from an image or from a region within this image to be used in the classification task. The generated features must encode this kind of information in order to enhance the classification accuracy. In US thyroid nodule image analysis, the computed features should exhibit high separability attributes between high and low risk cases. In the current proect three categories of features were employed to assess the malignancy risk factor of thyroid nodules: (a) Textural features, (b) Shape and Geometrical features and (c) Wavelet Local Maxima features. 6.4 Textural Features The textural information extracted from the thyroid nodule can be employed as criteria in assessing the risk factor of malignancy and can be of value in patient management, i.e. whether to recommend or not surgical operation. Textural features are divided in two main categories: First and second order statistical features First Order Statistical Features The 1 st order statistical features determine the distribution of grey level values within the thyroid nodule [7]. The most important features are: 1. Mean value(m) m = i N g( i, ) 6.1 where g(i,) is the grey level value in the position (i,) and N the number of pixels.. Standard Deviation (std) std = i ( g( i, ) m) N 6. The standard deviation represents the variation of grey level value in comparison with the mean value m

75 Chapter 6 3 ( g( i, ) m) 3. Skewness (sk) 1 i sk = 3 N std 6.3 The skewness describes the degree of histogram asymmetry around the mean. 4 ( g( i, ) m) 4. Kurtosis (k) 1 i k = 4 N std 6.4 Kurtosis describes the sharpness of the grey level histogram Second Order Statistical Features Features resulting from the nd order statistics provide information regarding the spatial relationship between various grey level values within thyroid nodule. These textural features were derived from the co-occurrence and run-length matrices [73,74] Co-Occurrence Matrix Features In the co-occurrence matrices, grey level pixels are considered in pairs with a relative distance (d) and orientation φ among them [73]. The orientation φ is quantized in four directions (0 0, 45 0, 90 0, ). An example of co-occurrence matrix computation is depicted in Figure 6.1. Grey tones (a) Grey tones (0,0) (0,1) (0,) (0,3) 1 (1,0) (1,1) (1,) (1,3) (,1) (,) (,3) (,4) 3 (3,1) (3,) (3,3) (3,4) (b) Co-Occurrence φ=0 0 Co-Occurrence Co-Occurrence Co-Occurrence φ=45 0 φ=90 0 φ= (c) (d) (e) (f) Figure 6.1 (a) Image array with four grey levels. (b) General form of any grey-tone cooccurrence matrix. (c)-(f) Computation of all four co-occurrence matrices with distance d=

76 Pattern Recognition Let an image array I(m,n) with four grey levels (N g =4) ranging from 0 to 3. Figure 6.1(b) depicts the general form of any grey tone co-occurrence matrix. For example, the element in the (0,0) position of a distance d=1 is the total number of times two grey tones of value 0 and 0 occurred along the four quantized directions adacent to each other. Figures 6.1(c) - 6.1(f) demonstrate all possible grey tones combinations with distance set to 1 along all four directions. The textural features that can be extracted from the co-occurrence matrices are presented below: N g 1 N g 1 1. Angular Second Moment (ASM) ASM = ( p( i, )) 6.5 where N g is the number of grey levels in the image, i,=1,,ng, and p(i,) is the co-occurrence matrix. The ASM feature describes the degree of homogeneity within the thyroid nodule and takes small values in regions with no variability. N g 1 Ng Ng. Contrast (CON) CON = n 1 1 ( p( i, )), i = n 6.6 n= 0 i= 0 = 0 Con feature describe the amount of local variations present within the nodule and takes high values in regions great variability. The factor n enhances any possible existence of local variations. p( i, ) 3. Inverse Different Moment (IDM) IDM = 1+ ( i ) 6.7 i= 0 = 0 N g 1 N g 1 i= 0 = 0 IDM feature takes high values for low-contrast images due to the inverse (i-) dependence. N g 1 N g 1 4. Entropy(ENT) = p( i, )log( p( i, ) ) ENT 6.8 ENT feature describes the degree of randomness and takes low values for smooth images. i= 0 = 0 5. Correlation(COR) COR = Ng 1 Ng 1 i= 0 = 0 ( i) p( i, ) m x m y σ σ x y 6.9 where m x,m y,σ x,σ y are the mean values and standard deviations of p x and p y (equations ) respectively. COR feature describes the spatial dependencies of the grey tones within the thyroid nodule. N rows px ( i) = p( i, ) 6.10 = i

77 Chapter 6 Ncolumns p y ( ) = p( i, ) 6.11 = 1 Other features derived from the co-occurrence matrices are: N g 1 N g 1 6. Sum of Squares (SSQ) SSQ = (1 m) p( i, ) 6.1 i= 0 = 0 7. Sum Average (SAV) SAV = ip x + y ( i) 6.13 = i where p x+y is N g 1N g 1 x+ y ( k) = p( i, ), i + = k, k = p,3,...,n g 6.14 i= i = 1 8. Sum Entropy (SENT) SENT = p x+ y i)log( p x+ y ( i) ) N g = i N g ( Sum Variance (SVAR) SVAR = = Ng ( i SENT ) p x + y ( i) 6.16 i 10. Difference Variance (DVAR) DVAR = = i N g ( i SAV ) p x y ( i) Different Entropy (DENT) = p x y ( i)log( p x y ( i) ) where p x-y is Ng 1 i= 0 DENT 6.18 x y Ng Ng p ( k) = p( i, ), i = k, k =,3,..., N 1 i= i = 1 g Information Measure of Correlation (ICM1) ICM1 HXY HXY1 = 6.0 max{ HX, HY} 13. Information Measure of Correlation (ICM) ICM HXY 1/ = (1 exp[.0( HXY )]) 6.1 where Ng 1 Ng 1 HXY = p( i, )log( p( i, ) ) 6. i= 0 =

78 Pattern Recognition Ng 1 N g 1 i= 0 = 0 ( p ( i) p ( ) ) HXY 1 = p( i, )log x y 6.3 HXY = N g 1N g 1 i= 0 = 0 p ( i) p x y ( )log ( p ( i) p ( ) ) x y Run-Length Matrix Features The run length matrix encodes textural information based on the number each grey level appears in the image by itself [74]. Let an image array I(m,n) with four grey levels (N g =4) ranging from 0 to 3( Figure 6.(a) ). For each direction (0 0, 45 0, 90 0, ) the corresponding run length matrix is computed (Figure 6.(b) - Figure 6.(e)) (a) Run Length Grey Level (b) Run Length Grey Level Run Length Grey Level (c) Run Length Grey Level (d) (e) Figure 6. (a) Image array with four grey levels. (b)-(e) Computation of all four run length matrices for texture analysis. Each matrix element specifies the number of times that the picture contains a run of length (0 3) in the given direction. The first element of the first row of the matrix is the number of times grey level 0 appears by itself, the second element is the number of times it appears in pairs and so on. The textural features that can be extracted from the run length matrices are presented below:

79 Chapter 6 1. Short Run Emphasis(SRE) SRE = N N g r i= 1 = 1 N N g r i= 1 = 1 r( i, ) r( i, ) 6.5 where r(i,) is the run length matrix, Ng is the number of gray values in the image, Nr is the largest possible run, i=1,,ng, =1,,Nr. SRE tends to emphasize short runs due to the division with. It takes large values for nodules with high variability.. Long Run Emphasis(LRE) LRE = N N g r i= 1 = 1 N N g r i= 1 = 1 r( i, ) r( i, ) 6.6 LRE tends to emphasize long runs. It takes large values for nodules with low variability. 3. Grey Level Non Uniformity (GLNU) GLNU = N g r i= 1 = 1 N N N g r i= 1 = 1 r( i, ) r( i, ) 6.7 GLNU is proportional with large run length values that are uniformly distributed. It takes large values for nodules with high variability. 4. Run Length Non Uniformity (RLNU) RLNU = N r g i= 1 = 1 N N N g r i= 1 = 1 r( i, ) r( i, ) 6.8 RLNU encodes long runs that are non-uniformly distributed. It takes small values for nodules with high variability. 5. Run Percentage (RP) N g Nr r( i, ) i= = RP = 1 1 P 6.9 where P is the total possible number of runs in the nucleus image. This feature takes its lowest value in nodules with low variability

80 Pattern Recognition Shape and Geometrical Features Besides textural sonographic criteria of the thyroid nodule, various shape and geometrical features such as irregular margins and circular boundaries are employed in the decision making procedure. In the present thesis several geometrical and shape based features were computed in order to quantify all the observations made by the physicians throughout the thyroid nodule literature [7, 75, 76]. These features were: 1. Average Radius Ravg N i= = 1 ( x( i) Xo) + ( y( i) Yo) ) Ravg is computed by averaging the Euclidean distance from the nodule s centroid (Xo,Yo) to each of the boundary points (x,y) (Figure 6.3). N 6.30 Figure 6.3 Line segments used to compute radius.. Radius Standard Deviation RST N i= 1 = ( R( i) Ravg) N 6.31 RST encodes information regarding the irregularity of the nodule s borderline. It takes high values in cases where the boundary is not circular. 3. Perimeter P = N 6.3 P is measured by summing the number or pixels on the border of the nodule. 4. Area Area is computed by counting the number of pixels on the interior of the nodule s boundary. 5. Radial Entropy RE = p k log( p k ) k =

81 Chapter 6 where pk is the probability that the radius distance will be between d(i) and d(i) d(i). The parameter pk was computed by the radial histogram. The amplitude range between the minimum and maximum values of the radius distance measure was divided into 100 bins and the number of times the radius distance plot passed through each bin was summed. Afterwards the sums were divided by the total number of samples. P 6. Circularity or Roundness Circ = 6.34 Area Circ is minimized for a circle and is proportional with the nodule s shape irregularity. 7. Smoothness SM = po int s Ri Ri P 1 + Ri where the Ri, R i-1 and R i+1 are depicted in Figure 6.4. SM is computed by measuring the difference between the length of a radius and the mean length of the two radiuses surrounding it. It takes small values for nodules with regular borders. Figure 6.4 Line segments used to compute Smoothness. 8. Concavity Concavity 1 N 1 M N i= 1 = M i= 1 ( Y ( Y Centroid Centroid Y Y CV AB i i ) ) + ( X + ( X Centroid Centroid X X CV AB i i ) ) 6.36 Concavity is computed by measuring the size of any indentations in the thyroid nodule. In fact is the average of the convex hull (CV) distances from the center (Centroid) of the nodule and the distances from the actual nodule boundary (AB) (Figure 6.5). Apparently it takes minimum values for circular or elliptical nodules. 9. Concave Points

82 Pattern Recognition Concave points are the number of points in the actual nodule s boundary that lies in the concave region (Figure 6.5). The greater the number of concave points the more irregular the nodule s border. Figure 6.5 Convex hull is used to compute concavity and concave points. 10. Average Convex Hull Radius RCHavg N i= = 1 ( CHx( i) Xo) + ( CHy( i) Yo) ) N 6.37 RCHavg is computed by averaging the Euclidean distance from the nodule s centroid (Xo,Yo) to each of the convex hull boundary points (CH x,ch y ). 11. Symmetry SYM Σ left right i i i = 6.38 Σ left + right i i i SYM is computed by measuring the relative difference in length between pairs of line segments perpendicular to the maor axis of the nodule. The maor axis is determined by finding the diameter of the nodule and the line segments were drawn at regular intervals (Figure 6.6). It encodes geometrical information regarding shape variability of the nodule [77,78]. Figure 6.6 Line segments used to compute Symmetry. The lengths of perpendicular segments on the right of the maor axis are compared to those on the left

83 Chapter 6 1. Fractal Dimension Fractal dimension of the nodule s boundary is approximated using the box-counting method [7,79,80].The perimeter of the nodule is measured using decreasingly smaller rulers to construct a box that contains the nodule. As the ruler size decrease, increasing the precision of the measurement, the observed perimeter increases. Plotting these values on a log scale and measuring the upward slope gives the approximation of the fractal dimension (Figure 6.7). Fractal dimension is a measure of how complicated is the boundary of the nodule. Figure 6.7 Fractal dimension estimation. N is the number of covering boxes and s is the number of rules or the size (perimeter) of each box Local Maxima Features A comprehensive study regarding the local maxima features employed in the present thesis is presented in Chapter Data Normalization In most cases the features values have different dynamic ranges. In order to overcome this problem all features values are normalized so that they lie within similar dynamic ranges. The normalization technique used in this thesis is made via the mean and variance of the feature values [7,81]. x x ik ik xˆ ik = 6.39 σ k

84 Pattern Recognition where xik and xˆ ik are the k th feature values before and after the normalization. xik and mean value and standard deviation of the k th feature. 6.6 Classification Task σ k are the Given a specific classifier, classification performance is tested employing the leave-one-out method and for all possible combinations ( s, 3 s, etc) of the computed features during the feature generation stage. The aim is to determine the optimum combination of features that achieves the highest classification accuracy with the minimum number of features. According the leave-one-out method, the classifier is designed by all but one the training sets of feature vectors. The left-out-feature vector is treated us unknown class and it is classified by the system. The whole procedure is repeated until all feature vectors have been tested. Results are then presented in a two way truth table or confusion matrix [8,83]. The classifiers designed throughout this thesis are presented below Minimum Distance Classifiers Minimum distance classifier: In the minimum distance classifier the pattern classes in the feature space cluster around their respective means. The decision boundary which separates the two patterns is the perpendicular bisector of the line oining both means. Each feature vector is classified whether its positions is on the left or the right of the bisector [83]. The discriminant function of the minimum distance classifier using Euclidean metrics is presented in the following equation: T 1 T d ( x) = x m ( m m ) 6.40 where x is the input feature vector, and m the mean value of class. Least square minimum distance (LSMD) classifier: The LSMD classifier maps via a non-linear transformation the input data set into a decision space where each class is clustered around a preselected point [83]. The classification of a given test point is based on its minimum distance from each pre-selected point. For the LSMD, the discriminant function is given by: g i (x) = d =1 α α i(d+1) 6.41 x i where d is the number of features, α i are weight elements and x are the input vector feature elements

85 Chapter Bayesian Classifier The Bayes decision theory develops a probabilistic approach to pattern recognition, based on the statistical nature of the generated features. The Bayes discriminant function [7,83] for class i and for pattern vector x is given by: g i (x) = lnp i 1 ln C i 1 [(x m i ) T C -1 i (x m i )] 6.4 where P i is the probability of occurrence of class i, m i is the mean feature vector of class i, and C i is the covariance matrix of class i Neural Networks Classifiers Artificial Neural Networks are basic input and output models, with the neurones organised into layers. Simple perceptrons consist of a layer of input neurones, coupled with a layer of output neurones, and a single layer of weights between them. The learning process consists of finding the correct values for the weights between the input and output layer. The principle weakness of simple perceptron was that it could only solve problems that were linearly separable. To obtain a bilinear solution more layers of weights are added to the simple perceptron model obtaining the multilayer perceptron network [83,84]. Multilayer Perceptron (MLP) Classifier: In MLP classifier [83,84] (Figure 6.8), each node of a hidden layer or output layer and the output y() of node is related to its input by: N y = 1 1+ e 6.43 ( ) S ( ) where S( ) = i = y( i) w( i, ) and w(i,) are connections weights between the previous node i and 1 the current node ; y(i)w(i,) is the weighted output of the previous node i, which is used as input to node ; N is number of inputs to node ; and S() is the sum of all weighted inputs y(i)w(i,) of the previous layer to node

86 Pattern Recognition Figure 6.8 Schematic diagram of the multilayer perceptron neural network employed, with two input features, two classes, two hidden layers and four nodes in each hidden layer. The connection weights w(i,) between different layer nodes of the MLP are calculated iteratively until they stabilize, by the following equation: w(i,) n+1 = w(i,) n + αd()y() + z(w(i,) n w(i,) n-1 ) 6.44 where (n+1), n, (n-1) correspond to next, present, and previous respectively, α, z are constants, d() is the error between the desired t() and actual y(), and is given by: d() = (t() y())y()(1 y()) 6.45 and for a hidden layer node by: d() = y()(1 y()) d ( k) w(, k) 6.46 k where k is associated with all layers nodes to the right of the current node. Probabilistic Neural Network (PNN) classifier: The PNN is implemented by a feed-forward and one-pass structure (see Figure 6.9) and encapsulate the Bayes s decision rule together with the use of Parzen estimators of data s probability distribution function. The discriminant function of a PNN for class is given by the following equation, as described at [84,85,86]: g T ( x xi ) ( x x ) N 1 σ ( ) = p p e / (π ) σ N i= 1 x 6.47 i

87 Chapter 6 where x is the test pattern vector to be classified, x i is the i-th training pattern vector of the -th class, N is the number of patterns in class, σ is a smoothing parameter, and p is the number of features employed in the feature vector. The PNN architecture comprises 4 layers (Figure 6.9). The input layer that has a node for each feature of input data. The pattern layer in which, one pattern node corresponds to each training pattern. The summation layer, which receives the outputs from pattern nodes associated with a given class and the output layer which has as many nodes as the input classes. The test pattern x is classified to the class with the larger discriminant function value. Figure 6.9 Schematic diagram of the probabilistic neural network employed, with two input features and two classes Support Vector Machines Classifier A classifier based on support vector machines (SVM) [84,87,88] is a general classifier that it can be applied to linearly as well to non-linearly separable data, with or without overlap between the classes. In the most general case of overlapped and non-linearly separable data, the problem is (a) to transform the training patterns from the input space to a feature space with higher dimensionality (x R d a Φ(x) R h ) where the classes become linearly separable, and (b) find two parallel hyperplanes with maximum distance between them and at the same time with minimum number of training points in the area between them (also called the margin). The separating hyperplanes in the transformed feature space are defined by the following equation: w Φ(x) + b = ±

88 Pattern Recognition where +1 is referred to class 1, 1 is referred to class, x if the pattern vector, w is the normal vector to the hyperplanes, and b the bias or threshold which describes the distance of the decision hyperplane from the origin (that is equal to b/ w ). The discriminant function is given by: g(x) =sign(w Φ(x) + b) 6.49 The parameters w and b are calculated as follows: Let N training pattern vectors x i R d, i=1 N (where d is the number of features) belonging to two classes identified by the label y i { 1, +1}. The conditions for the hyperplanes may take the following mathematical formulation: (i) minimize the number of training pattern vectors that lie between the two hyperplanes, so: y i (w Φ(x i) + b) + ξ i where ξ i 0, i=1 N are real non-negative slack-variables. (ii) the distance between the two hyperplanes (which is equal to / w ) must be maximized, so 1 w must be minimized. The above conditions lead to minimizing 1 w + Cξ i subect to (6.49), where C is a positive constant that reflects a trade-off between the classification errors and the size of the margin. Introducing Lagrangian multipliers α i, β i, i=1 N, the Lagrangian is given by: L P = 1 N i= 1 N w + C ξ α ( y ( w Φ( x ) + b) + ξ 1) β ξ 6.51 i i= 1 L The problem is now to maximize L P subect to P L = 0, P = 0 b w α i, β i 0). These constraints give respectively: i i i i N i= 1 i i L P and = 0 ξ i (with N i= 1 α = i y i and N w = α y i iφ(xi ) 6.53 i= 1 C = α i + β i 6.54 The equation (A.7), in combining with α i, β i 0, results that 0 α i C. Substituting equations (6.51, 6.5, 6.53) in relation with (6.50), we take the dual variables Lagrangian L D :

89 Chapter 6 L D = N i= 1 N 1 α i α iα yi y Φ( xi ) Φ( x ) 6.55 i, = 1 By use of a kernel function, that it can replace the inner product Φ(x i ) Φ(x ) in the higher dimensional feature space, the dual Lagrangian L D can take the form of (6.55): L D = N i= 1 N 1 α i α iα yi y k( xi, x ) 6.56 i, = 1 A function can be used as a kernel function if it satisfies the following Mercer s condition: Any symmetric function k(x,y) in the input space is equivalent to an inner product in the feature k, for any function g(x) for which g ( x) dx < space, if ( x, y) g( x) g( y) dxdy 0 Using equations (6.48), (6.5), and (6.55), it may be seen that ξ i and β i have vanished, so the discriminant function of the SVM classifier may be written as: g(x) = sign N S i= 1 α i yik( xi, x) + b 6.57 where N s is the number of pattern vectors (also called the support vectors) with non-zero α i s. Combining the equations (6.47), (6.5), and (6.55), the threshold b may be found as: 1 N S b = y N S = 1 i= 1 N α i yik( xi, x ) 6.58 and the coefficients α i are obtained by solving the dual problem, which is maximization of L D (equation (6.55)) subect to equation (6.51), with 0 α i C. Functions that are commonly used as kernels are: 1. The linear kernel k(x i,x ) = x. i x The polynomial kernel k(x i,x ) = (x. i x +θ ) d 6.60 where d is the degree of the polynomial and θ an offset parameter, 3. The Gaussian radial basis kernel k(x i,x ) = where σ is the standard deviation. exp T ( xi xj ) ( xi xj ) σ The sigmoidal kernel k(x i,x ) = tanh(κ(x i. x ) + θ ) 6.6 where κ the gain and θ the offset

90 Pattern Recognition 5. The inverse multiquadric kernel where c a non-negative real number. k(x,y) = ((x i x ) T (x i x ) + c 1 ) The wavelet kernel K( x, y) = ψ, k ( x) ψ, k ( y) 6.64 where ψ is a translated wavelet of resolution., k, k

91 CHAPTER 7 Wavelet-Based Speckle Suppression in Ultrasound Images Summary In this chapter a wavelet-based method for speckle suppression in ultrasound images of the thyroid gland is introduced. The chapter is organized as follows: At first an extensive review of the literature regarding noise reduction in US images is presented. Afterwards, the speckle model adopted by this study is presented followed by the proposed strategy based on the inter-scale wavelet analysis. In the results section the speckle removal efficiency and edge preserving are compared to that of current speckle suppressing methods. Moreover 63 US images of the thyroid gland are subected to review by experienced observers via questionnaire for qualitative evaluation of the proposed despeckling process. In the last section an extensive discussion regarding the proposed algorithm is given. 7.1 Review of the Literature A variety of speckle reduction techniques have been implemented in the past two decades. Part of them reduce speckle by acquiring the radio frequency (RF) pulse echo signals from the US devices directly after log compression and time gain compensation (TGC) and before scan conversion [89,90]. However, access to raw RF data is somewhat complex and sometimes impossible, especially in modern US scanners, which in turn renders the application of such methods difficult for research purposes. In the early years of computer image processing speckle removal in US images was achieved via simple averaging, median filtering and Wiener filtering [91]. Simple averaging not only failed to eliminate speckle but introduced blurring and edge loss in areas where anatomic boundaries prevail. Median filtering enhances edges and speckle indiscriminately, while Wiener filter manages to remove considerable amounts of speckle but also tends to oversmooth the boundaries of important image features. Various adaptive filters based on local statistics, such as mean and variance, have been implemented for noise reduction not solely in medical imaging but for image denoising in general [9,93,94]. As the computing technology boosted during the 90s, along with the processors speed and power, new and more complicated filters were introduced. They were employed mainly in time domain such as the adaptive-weighted median filter by Loupas et al. [95], the segmentation based L-filtering by Kofidis et al. [96], the adaptive speckle suppression by Karaman et al. [97], the aggressive region growing method by Chen et al. [98], the symmetrical speckle reduction filter by Huang et al. [99] and the diffusion stick model by Hiao et al. [100]

92 Wavelet-based speckle suppression in ultrasound images Loupas introduced a new class of non-linear adaptive filters employing some local statistics, such as the ratio of σ / m where σ, m are the local variance and mean inside a moving window with pre-specified dimensions. Through these local statistical parameters the adaptive filter was turned into a general low pass filter in homogenous areas and into an enhanced median filter into areas with small structures or boundaries. Karaman s method depended on the same statistical principles employed by Loupas regarding the moving window and its local statistics. The filter was transformed into a mean filter or a median depending on an estimated homogeneity criterion. Kofidis segmented the US image to various stationary regions employing a combination of the Learning Vector Quintizer (LVQ) and the maximum likelihood estimator of the original noiseless signal (L mean). Subsequently, these subimages are filtered by a set of L-filters. L-filter design process is based on order statistics (i.e. autocorrelation matrix) derived from the previous stage. Chen presented a new region growing filtering method based on a trade off between a trimmed mean filter and a median filter. In order to overcome some of the limitations of the above methods he added an adaptive homogeneity criterion. Through this criterion a homogenous area is differentiated from a heterogeneous area, thus altering the filter applied to that area to mean or median respectively. Huang divided his filtering strategy based on the slope facet model in two stages: firstly he introduced two criteria in the region growing process to approximate the largest despeckling window within an 11x11 matrix. The first is the widely used variance to mean criterion and the second is the gradient criterion. In the second stage after the maor removal of speckle he used only the gradient criterion for the final noise elimination. In both stages the filter acts generally as common mean filter. Xiao exhibits an interesting oriented filter with 4 asymmetrical diffusion sticks inside a symmetric moving matrix. Through a variation function applied in every stick, the algorithm smoothes the sticks with high homogeneity and penalizes smoothing within heterogeneous regions. The smoothing function comprised the weighted sum of averages along each stick. For optimization of the results the whole filtering process is done iteratively. The empirical choice of some parameters such as window size, weight calculation, homogeneity criterion or various thresholds employed of the above mentioned methods degraded their generalization ability thus made them US machine and anatomical region depended. In the past decade a new approach in US images denoising emerged based on the wavelet transform. Some of the wavelet based proposed methods for US image despeckling, are the homomorphic wavelet shrinkage by Zong et al. [101], the multiscale nonlinear processing method by Hao et al. [10], and the Bayesian wavelet method by Achim et al. [103]. Zong applied Mallat s Dyadic Wavelet transform [6] on a US image, which is logarithmically transformed due to the multiplicative nature of speckle. In the resulted decompositions he applied a combination of soft and hard thresholding, introduced by Donoho [104], at fines and - 6 -

93 Chapter 7 middle scales respectively in order to eliminate the presence of speckle. Besides wavelet coefficients shrinkage, Zong achieved boundary enhancement by means of an adaptive gain operator and some predefined thresholds. Hao presented a combination of Loupas method and wavelet transform shrinkage. Initially, he divided the image via the adaptive-weighted median filter in two parts that approximate signal and noise. These two parts are decomposed through the wavelet transform and a modification of Donoho s soft thresholding is used to remove speckle. The final denoised image is the sum of the two reconstructed image parts, into which the original image was split in the first stage. Both methods are mainly adopting the denoising thresholding procedure presented by Donoho. In this method thresholds are calculated empirically and in an ad hoc manner without taking into account the special statistical properties of speckle. Achim in his attempt to overcome the limitations arising from empirical thresholding of wavelet coefficients employed a Bayesian approach for signal extraction and speckle suppression. The log-transformed US image was decomposed in different frequency scales via the wavelet transform. In each scale a Bayesian estimator is used, based on symmetric alpha stable distribution of the wavelet decomposition, to differentiate the signal coefficients from the noise coefficients. Besides US images, speckle dominates Synthetic Aperture Radar (SAR) images as well, introducing difficulties on their correct interpretation. Various attempts are made in the wavelet domain in order to efficiently reduce the resulting granular pattern. Sveinsson et al [105] and Pantaleoni et al [106], via orthogonal wavelet transform and the Daubechies wavelet family, applied soft thresholding and the enhanced Lee filter respectively on the wavelet coefficients to reduce the presence of speckle. The aforementioned wavelet-based approaches use the logarithmic transform to convert the multiplicative model of speckle into additive model with signal independent noise before performing the speckle reduction method. After that, an exponential transform is applied to convert the denoised image to its original format. The fact that the mean of the log transformed speckle noise is not zero, whereas additive white Gaussian noise (AWGN) is considered with zero mean from the above methods, led to the need of a correction step regarding the mean bias in the processing stages, to avoid distortion in the de-speckled image [107]. Several recent studies avoid the log transform and directly apply the wavelet transform onto the SAR images. Foucher et al [108] in order to discriminate reflectivity coefficients from speckle coefficients implemented a Bayesian analysis based on the Pearson system for probability density function (pdf) approximation of the wavelet coefficients. Argenti et al [109] applied a minimum mean-square error (MMSE) filtering in the un-decimated wavelet domain to suppress speckle noise coefficients and Dai et al [110] combined a Bayesian shrinkage factor and a ratio edge detector applied in the wavelet coefficients for speckle

94 Wavelet-based speckle suppression in ultrasound images reduction with edge preservation in homogenous areas. An efficient discrimination between speckle noise and reflected signal in US or SAR images either in time or wavelet domain is still under discussion in the scientific society. As already mentioned an accurate despeckling algorithm is very important in the decision making process especially in US images of the thyroid gland. Often, thyroid nodules, which play the most important role in estimating the malignancy risk factor, are of low contrast in a noisy background. Likewise the presence of various structures inside the nodules comprises a critical factor for a proper interpretation of the US image. The dominating speckle noise in all US images can lead to misleading analysis thus obstructing the physician s diagnosis. 7. Materials and Methods 7..1 Overview and Implementation of the Algorithm A wavelet based method is introduced in this thesis for efficient speckle suppression in sonographic images of the thyroid gland while important edges and boundaries are preserved. The proposed wavelet approach avoids both log and exponential transform, considering the fully developed speckle as additive signal-dependent noise with zero mean. The proposed method throughout the wavelet transform has the capacity to combine the information at different frequency bands and accurately measure the local regularity of image features. The inter-scale information is acquired by means of a coarse to fine connectivity of the wavelet transform modulus maxima (WTMM). Two structures, represented by the modulus maxima, in two consecutive scales belong to the same anatomic area if the pair position & angle pixel of the maximum wavelet coefficient value in the upper scale is also approximately present in the lower scale. The decay across scales of wavelet transform maxima is related to local regularity of these structures and is assessed by the Lipschitz exponent a [70]. The purpose of the present study is to employ the knowledge given from the evolution of the wavelet transform maxima across scales to discriminate image singularities from speckle singularities. All the proposed method s steps (i.e Redundant wavelet transform, multiscale edge representation, coarse to fine analysis together with the wavelet coefficient and maxima display) were all implemented in Matlab 6.5. The methods to which the proposed algorithm is compared with were also implemented and integrated with the same software packet. The computer used for processing has an AMD Athlon XP+ processor running at 1.8 GHz and 51 of RAM. The ultrasound system used for this study was the HDI-3000 ATL digital ultrasound system with a broadband linear array with 7 MHz central frequency. The digitization of the output Video signal of the ultrasound system was made via the video card Miro PCTV(Pinnacle Systems), which is installed in a PC. US images are stored in JPEG format and their size is 768 x 576 x 8. The primary steps of the proposed method are illustrated in Figure

95 Chapter 7 Multiscale Edge Representation Function Regularity Estimation 'Atrous' DWT Coarse to Fine Interscale Analysis Speckle Image Gradient Vector Computation IDWT De-Speckled Image Singularity Detection Modulus Maxima Figure 7.1 Block Diagram of the proposed wavelet based algorithm for speckle suppression. 7.. Speckle Model Despite the profound advantages of ultrasonography, US images carry a granular pattern, so called speckle, which constitutes a maor image quality degradation factor. Speckle pattern is created when an ultrasonic wave with uniform intensity is incident either on a rough surface or on tissue particles that are spaced at less than the axial resolving distance of the US system. In that case, the reflection beam profile will not have a uniform intensity. Instead it will be composed of many regions with strong and weak intensities. This complex intensity profile arises because sound is reflected in many different directions from the rough surface or from the small scatterers, thus leading US waves that have travelled different scan lines to interfere constructively and destructively towards the ultrasonic transducer. The intensity fluctuations within a uniform anatomic area, caused by the above phenomenon, constitute speckle [111]. The resulting degraded by speckle US image does not correspond to the actual tissue microstructure. In fact, speckle noise deteriorates image quality, fine details and edge definition. Speckle also tends to mask the presence of low-contrast lesions, therefore reducing the physician s ability for accurate interpretation. Moreover, it constitutes a limiting factor in the performance of quantitative procedures such as segmentation and pattern recognition algorithms. Hence, effective speckle suppression is considered of value for improving US image quality and possibly the diagnostic potential of medical ultrasound imaging, except in rare cases of abdominal and breast US images where the presence of speckle may assist in assessing liver cirrhosis or breast cancer [11,113]. The speckle model employed in the despeckling strategies in time domain [95,97,98,99, 100,10] considers the envelope detected RF signal having a Rayleigh distribution, thus speckle can be considered as multiplicative noise. However, due to US device s undergoing signal processing stages, the finally formatted US image speckle is no longer multiplicative and can be thought as Gaussian additive noise independent of the noise-free signal. Most

96 Wavelet-based speckle suppression in ultrasound images wavelet based methods [101,103,105,106,70], adapted for additive Gaussian white noise, applied a logarithmic transform in the speckle image and approximated speckle as additive noise. The proposed method adopted Foucher et al [108], Argenti et al [109] and Dai et al [110] approaches, by omitting the log-transform to avoid the mean bias correction problem and decomposing the multiplicative speckle model into an additive signal dependent noise model. The multiplicative speckle model at pixel position [x,y] is expressed in the following form: I( x, y) = f ( x, y) r( x, y) 7.1 where f(x,y) is an unknown -d function, such as the original image to be recovered without noise. I(x,y) is the corrupted with noise formatted US image, and r(x,y) a random variable that represents speckle. We consider speckle as fully developed (large number of small scatterers in each resolution cell) whose magnitude follows the Rayleigh pdf [114]: p r Its mean and variance are [114]: πr πr ( r) = exp, 0 4 r 7. 4 E ( r) = 1, var( r ) = π We convert the multiplicative model into an additive model: I ( x, y) = f ( x, y) + f ( x, y)[ r( x, y) 1] = f ( x, y) + N ( x, y) 7.4 Where [ r ( x, y) 1] is a random variable with zero mean and varianceσ N σ ( x, y) represents an additive signal dependent noise term, which is proportional to the signal to be estimated. σ

97 Chapter Inter-Scale Wavelet Analysis Dyadic Wavelet Transform Employing wavelet theory, the correlation of the inter-scale edge information may be applied to characterize different types of edges. Wavelet analysis was performed by means of the Dyadic Wavelet Transform (DWT) introduced by Mallat and Zong [6] for characterization of signals from multiscale edges. DWT is based on a wavelet function ψ(x) with compact support, which is the first order derivative of cubic spline function. The wavelet decomposition across scales of the original image was implemented with a filter bank algorithm, so called algorithme a atrous (algorithm with holes). The proposed transform is in fact a fast bi-orthogonal discrete wavelet transform, in which the size of the decomposed sub-band images is the same as that of the original image thus making the transform highly redundant. Let θ ( x, y) be a symmetrical smoothing function approximating the Gaussian. The twodimensional Dyadic Wavelet Transform of a function f ( x, y) L ( R ) is the set of 1 functions ( W f ( x, y), W f ( x, )), which are respectively the partial derivative along the y horizontal and vertical orientation of the convolution of f ( x, y) with the smoothing function θ ( x, y) dilated along a dyadic sequence ( ) Ζ. The DWT is given by: W W 1 f ( x, y) = f ( x, y) f 1 ψ ψ = x y ( f θ )( x, y) ( )( ) f θ x, y 7.5 where ψ 1 ( x, y) and ψ ( x, y) are the analyzing wavelets and the dyadic scale. We performed the dyadic wavelet transform using Mallat s filters. These filters are suitable for fast implementation of discrete algorithms and they offer exact reconstruction. At a dyadic scale the dilation of the discrete filters is obtained by inserting ( -1) zeros (holes) between each of the coefficients of the corresponding filters. In Figure 7. the wavelet decomposition with the atrous algorithm in 3 dyadic scales of a US image of the thyroid gland is presented

98 Wavelet-based speckle suppression in ultrasound images Figure 7. At the top is the original US image. The two columns show respectively the 1 horizontal and vertical wavelet transform { W f x, y), W f ( x, y } ( 1 3 ) 1 3 along three dyadic scales. The scale increases from top to bottom. The redundant wavelet transform presented is in fact shift-invariant and it is widely used for pattern recognition, feature extraction and edge detection purposes. The wavelet coefficients

99 Chapter 7 comprise the intensity profile of an image s local variations for a given scale. They can be considered as a classification map in which any kind of change (abrupt or smooth) exist in an image can be localized on a particular scale. The latter conclusion indicates the importance of an accurate selection of the dyadic scale in which the image will be decomposed. The choice of that scale is in fact a trade off between the suppression of wavelet coefficients characterizing image s irregularities and the blurring effect caused by the dilation of the smoothing function. In small scales the wavelet coefficients mostly characterize high frequency events mainly caused by noise. In bigger scales low frequency events are detected such as smooth image variations Gradient Vector Equation 5 indicates that the above set of functions can be viewed as the two components of the gradient vector of f ( x, y) smoothed by θ ( x, y) at each scale : 1 W f ( x, y) = ( f * θ )( x, y) W f ( x, y) 7.6 The modulus angle representation of the gradient vector is given by: M 1 f ( x, y) W f ( x, y) + W f ( x, y) = 7.7 and 1 W f ( x, y) A f ( x, y) = arctan 7.8 W f ( x, y) Modulus Maxima The sharper variation points of f * θ ( χ, y) at a scale correspond to edges, are obtained from the local maxima of M f ( x, ) along the gradient direction given by A f ( x, ). y y At each scale of the Dyadic Wavelet Transform, the point (x,y) where the modulus of the gradient vector M f ( x, ) is maximum compared with its neighbor s locally positioned in y the direction specified by A f ( x, ), is called modulus maxima (Figure 7.3). y

100 Wavelet-based speckle suppression in ultrasound images Figure 7.3 At the top is the original US image. The first column displays the modulus images f ( x, ). High intensity values correspond to black pixels whereas low intensity M y values to white pixels for optimized visual interpretation of the results. At the second column the angle images A f ( x, ) are shown. The angle value turns from 0 (white) to π (black) y along the circle contour. At the last column, the image points where M f ( x, ) has local y

101 Chapter 7 maxima in the direction indicated by A f ( x, ) are presented (black pixels). Each time such y a point is detected, the position of the resultant local maxima is recorded as well together with the values of the modulus M f ( x, ) and angle A f ( x, ) at the corresponding locations. y y Lipschitz Regularity The aim of the present study is to efficiently characterize the image s singularities, via an inter-scale wavelet analysis, in order to discriminate speckle-noise from signal. The classification of singularities depends upon their local regularity. This regularity is quantified by Lipschitz exponents [68]. A function f(x,y) is said to be Lipschitz α, 0 α 1, at (x 0,y 0 ) if there exists K>0 such that for all points ( x, y ) R : 0, y0 ) ( x x0 + y y0 ) f ( x, y ) f ( x K 7.9 If there exists a constant K>0 such that equation (9) is satisfied for any ( x 0, y0 ) Ω, then f is uniformly Lipschitz a over Ω. The larger the a, the more regular is the function. Τhe Lipschitz regularity of a function f(x,y) is related to the asymptotic decay from coarse to fine scales of it s wavelet transform along horizontal and vertical directions W 1 f(u,v,s) and W f(u,v,s) in the corresponding neighborhood. This decay is controlled by the wavelet transform local maximum value Mf(u,v,s) [70]. A function f (x, y) is uniformly Lipschitz α, 0 α 1 inside a bounded domain of R if and only if there exists a constant A >0 such that for all (u,v) inside this domain and for any dyadic scale s relation 10 holds: a / a+ 1 Mf ( u, v, s) As 7.10 By measuring from (10) the Lipschitz exponent a through the computation of the decay slope of log Mf(u,v,s) we derive an estimate of the Lipschitz regularity along the edge Detection of Singularities All singularities of f(x,y) can be located by following the wavelet transform modulus maxima up to the finer scale. The terms coarse and fine are relative. Conventionally, coarse scales are referred to bigger dyadic scales ( 3, 4 ), whereas fine scales are referred to smaller dyadic scales ( 1, ). The main obective of that inter-scale analysis is to isolate different structures exist in the image, beginning at a coarse scale and adaptively decrease the scale to gather the necessary details. If an edge appears in a coarser level, it should also appear in finer level - 1. The latter can be rephrased that any wavelet transform modulus maximum at a coarse scale belongs to a connected inter-scale chain that is never interrupted when the scale decreases [70], which in turn means that any structure represented by the corresponding maxima is

102 Wavelet-based speckle suppression in ultrasound images located in a coarse scale can also be found in a finer scale with an approximate position and angle value. Mallat [6,68] implements the maxima chaining procedure starting from a scale and considers that it propagates to coarser scale +1 having similar position and angle values. In this study the forming of maxima chains in the scale-space domain is made with a backpropagation approach starting the inter-scale connectivity from the coarser scale ( 3 ) computed and complete it at the finer scale ( 1 ) available. With this approach the computational complexity of the implemented algorithm is reduced even more since we employ the inter-scale exhaustive search with the smaller possible number of local maxima (as the scale increases the number of local maxima decreases). Before we apply this back-propagation tracking of modulus maxima in wavelet space, the maority of some false maxima at the coarser scale ( 3 ) that either are not suppressed by the smoothing function or created by numerical errors in regions where the wavelet transform is close to zero are removed through a simple 70 th percentile thresholding (all maxima values below the 70% of the maximum modulus value are discarded) [115]. The chaining of modulus maxima, after the thresholding procedure, across scales employs a two-folded interscale investigation based on the parameter pair: position angle. Two modulus maxima at two successive scales [(X k,y k,m k,a k ), (X k,y k,m k,a k ) -1 ] are chained if they have a close position in the image plane(x k,y k ) and similar angle value(a k ). If a single coarse local maximum computed to back-propagate in more than one finer local maxima, only the one having the largest maximum (M k ) is considered to belong to the maxima chain. The maxima matching between different scales was not an easy task. The position angle investigation was implemented at each scale within a different neighbourhood taking into account the different size of the decomposition Mallat s filter (different number of zeros holes between the coefficients). The coarse information is traced within large neighbourhoods whereas the fine information in small neighbourhoods. The adaptively decreasing investigation window from coarse to fine scales avoids matching errors created either from small maxima groups that might not constitute an exact match or large maxima groups that may produce inaccurate inter-scale linking if the two successive structures are locally distorted. A square window of width K at a coarse resolution corresponds to square windows with approximate size of K/ and K/4 at finer resolutions -1 and - (Figure 7.4) - 7 -

103 Chapter 7 Figure 7.4 Inter-scale back-propagation maxima connectivity in wavelet space. At each of the maxima chains, acquired from back-propagation tracking, the decay of the modulus maxima amplitude across scales is calculated in order to discriminate speckle singularities from image singularities. In maxima chains where the amplitude of the wavelet transform modulus maxima decreases when the scale decreases the Lipschitz regularity is positive (positive Lipschitz exponents). On the contrary, when the maxima amplitude increases when the scale decreases the Lipschitz regularity is negative (negative Lipschitz exponents). The Lipschitz regularity was calculated between those scales in which the amplitude of the decay slope was the greater. In our case was between the 3 & scales. The different decay behaviour of the modulus maxima is the main criterion in an accurate discrimination of image and noise singularities. Image singularities belong to regular curves with positive Lipschitz regularity that varies smoothly along these curves. Speckle singularities give rise to negative Lipschitz regularity and considered as irregular variations of the positions, angle and modulus values of the maxima. The propagating maxima chains with positive Lipschitz regularity can be considered as an edge map that corresponds to important image structures. The despeckling procedure implemented in this article removes all wavelet coefficients at all scales that correspond to those maxima whose amplitude increase when the scale decreases or do not belong to a backpropagating maxima chain. The maxima recognized by the algorithm as speckle and as edges for all scales are presented in Figure

104 Wavelet-based speckle suppression in ultrasound images Figure 7.5 At the first column are the wavelet transform modulus maxima (non-propagating maxima and propagating maxima with negative Lipschitz exponents) classified as speckle. In the second are the propagating maxima with positive Lipschitz exponents classified as important edges

105 The remaining coefficients at all scales including the coarse image for completeness are utilized in the inverse Dyadic Wavelet Transform to obtain the speckle suppressed US image. The ability to isolate all important structures at all scales (figure 7.5 right column) via the proposed wavelet inter-scale analysis gave us the opportunity to perform the despeckling strategy (maxima removal) at all computed scales, even at the finer one ( 1 ), contrary to Mallat [9] which avoids to incorporate that scale to his denoising procedure due to signal domination from noise. In our method as we can see in figure 7.5 left column although at the finer scale ( 1 ) available the human eye cannot discriminate the contours of the anatomical structures, after the back-propagation tracking and singularity detection, in the same scale these contours with approximate positions and angles became prominent. 7.3 Experimental Results and Evaluation The effectiveness of the introduced wavelet based de-speckling approach was tested using a tissue mimicking digital phantom and a US image of the thyroid gland. An observer evaluation study was also undertaken involving 63 US thyroid images of 63 patients via a questionnaire regarding the performance of the proposed algorithm. The proposed inter-scale wavelet analysis method was compared with three representative denoising methods: (a) Karaman s adaptive speckle suppression filter ASSF [97] (b) Donoho s soft thresholding and (c) Donoho s hard thresholding [104]. Wavelet shrinkage was implemented with Daubechies 8 mother wavelet in three decomposition scales, produced by the wavelet toolbox in matlab. The quantification of the speckle suppression performance of all methods (in both phantom and thyroid US image) was carried out by means of the speckle index (SI: mean to standard deviation), on a homogenous area with uniformly distributed echoes, and the signal-to-meansquare-error ratio (S/mse), introduced by Cagnon [116], on the same homogenous area (Local region of interest) and on the entire image (Total). S/mse is defined as: S mse = 10 log 10 K I i i= 1 K i= 1 I i I i 7.11 Where, I are the intensity values of the speckle image, I are the intensity values of the despeckled image and K is the image size. The S/mse index can be considered as an index of signal-to-noise within an image. High S/mse index values refer to efficient speckle suppression while low to inadequate performance. The S/mse index is expressed in db. The evaluation of the edge preservation capacity, both locally (area where boundaries prevail) and totally (entire image), of all methods was made by means of the parameter β, which has been introduced by Hao [10] as shown in relation (7.1): 75

106 Wavelet-based speckle suppression in ultrasound images Γ ΔI ΔI, ΔI ΔI β = 7.1 Γ ΔI ΔI, ΔI ΔI Γ ΔI ΔI, ΔI ΔI Where: ΔI, ΔI are speckle and de-speckled images respectively, filtered by a 3x3 pixel standard approximation of the Laplacian operator, and Γ is given by: Γ K ( I, 1 I ) = i= 1 I 1i I i 7.13 In case of optimum edge preservation, β approximates to 1. The closer the β is to 1 the better are the edge preservation properties of each algorithm Tissue Mimicking Phantom Validation The phantom used in this study was the 403 LE model manufactured by GAMMEX. It comprises of three groups of three anechoic cystic targets (approximating thyroid nodules) with, 4 and 6 mm diameter positioned at 3, 8 and 14 cm respectively. The attenuation coefficient for the tissue mimicking materials is 0.5 db/cm/mhz whereas for the anechoic cysts is 0.05 db/cm/mhz. Regarding the phantom image, the value of SI prior to despeckling was 7.18 and the values of SI, S/mse and β after despeckling are shown in Table 7.1. Table 7.1 Image quality measures obtained by four denoising methods tested on a digital ultrasound phantom image Method SI (Percentage Improvement) S/mse (Local/ Total) (db) β (Local / Total) (db) ASSF 6.98 (1%) / / Soft Thresholding 7.33 (18%) / / Hard Thresholding 7.11 (15%) / / Wavelet inter-scale analysis Denoising 7.3. US Image Case Study 7.50 (1%) / / All despeckling methods were also applied to an US image of the thyroid gland and their results are demonstrated in Figure

107 Chapter 7 Figure 7.6 (a) US image of the thyroid gland. (b) ASSF method, (c) wavelet shrinkage with soft thresholding, (d) wavelet shrinkage with hard thresholding, (e) wavelet inter-scale analysis denoising. Both parameters (S/mse & β) are calculated locally and totally. The selected regions are 77

108 Wavelet-based speckle suppression in ultrasound images presented in Figure 7.7. The values of S/mse and β for all methods applied to the US image are given in Table 7.. The Si value prior to despeckling was Figure 7.7 Locally selected area containing the thyroid nodule Box A for β calculation. Locally selected area corresponding to homogenous tissue Box B for S/mse calculation. Table 7. Image quality measures obtained by four denoising methods tested on an ultrasound image of the thyroid gland Method SI (Percentage Improvement) S/mse (Local /Total) (db) β (Local /Total) (db) ASSF 3.98 (14%) / /0.359 Soft Thresholding 4.15 (19%) / / Hard Thresholding 4.05 (16%) / / Wavelet inter-scale analysis Denoising 4.30 (3%) / / A more detailed description regarding despeckling and edge preservation may also be obtained by the profile signals depicted in Figure 7.9 derived from figure 7.6. The corresponding scan line is presented in figure 7.8. Figure 7.8 The scan line including the borders (A&B) of the thyroid nodule. 78

109 Chapter 7 Figure 7.9 Scan profiles of US thyroid image. High intensity line corresponds to the denoised scan line whereas low intensity line to original image. (a) ASSF method, (b) Soft thresholding, (c) hard thresholding, (d) wavelet inter-scale analysis denoising The performance of the proposed method on various US images is depicted in figures 7.10, 7.11,

110 Wavelet-based speckle suppression in ultrasound images (a) Figure 7.10 (a) Original US image, (b) De-speckled US image (b) (a) Figure 7.11 (a) Original US image, (b) De-speckled US image (b) (a) Figure 7.1 (a) Original US image, (b) De-speckled US image (b) 80

111 Chapter Observer Evaluation Study Sixty three US images of the thyroid gland were included in a questionnaire (Figure 7.13) that comprised seven queries concerning various visual observations regarding the proposed algorithm s effectiveness. Figure 7.13 Microsoft access interface employing the questionnaire for US images denoising evaluation The image dataset was acquired following the same parameter s protocol in the time interval from October 003 to September 004. The questionnaire was implemented in Microsoft Access. The seven queries were: 1) Removal of speckle granular pattern. ) Improvement of micro-calcification detection inside the thyroid nodule. 3) Creation during denoising of artefacts and ghost images. 4) Preservation of nodules boundaries, resolvable details and anatomical structures. 5) Contrast enhancement between nodule and surrounding environment. 6) Revealing of small structures invisible in the original image. 7) Improvement of diagnostic evaluation procedure. The study was performed by two experienced qualified radiologists specialized in ultrasonography. The reviewing of all cases was done independently on a high resolution monitor. The ranking for each query ranged from 0 to 100 with a 5-point step corresponding to fail (0), poor, good, very good and excellent performance (100), except query number 3 where low score refers to high effectiveness (Tables 7.3 and 7.4). The performance of the 81