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Local Geometric Transformations in Image Analysis

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Local Geometric Transformations in Image Analysis Local Geometric Transformations in Image Analysis THÈSE NO 7014 (2016) PRÉSENTÉE LE 17 JUIN 2016 À LA FACULTÉ DES SCIENCES ET TECHNIQUES DE L'INGÉNIEUR LABORATOIRE D'IMAGERIE BIOMÉDICALE PROGRAMME DOCTORAL EN GÉNIE ÉLECTRIQUE ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES PAR Zsuzsanna PÜSPÖKI acceptée sur proposition du jury: Prof. P. Vandergheynst, président du jury Prof. M. Unser, directeur de thèse Dr J.-C. Olivo-Marin, rapporteur Dr U. Köthe, rapporteur Prof. D. Van de Ville, rapporteur Suisse 2016 ◦ Thèse N 7014 (juin 2016) Thèse présentée à la faculté des sciences et techniques de l’ingénieur pour l’obtention du grade de docteur ès sciences et acceptée sur proposition du jury Prof. Pierre Vandergheynst, président Prof. Michaël Unser, directeur de thèse Prof. Dimitri Van De Ville, rapporteur Dr. Jean-Christophe Olivo-Marin, rapporteur Dr. Ullrich Köthe, rapporteur École polytechnique fédérale de Lausanne—2016 Cover design by Annette Unser Printing and binding by Repro-EPFL Typeset with LATEX Copyright © 2016 by Zsuzsanna Püspöki Available at http://bigwww.epfl.ch/ Abstract The characterization of images by geometric features facilitates the precise analysis of the structures found in biological micrographs such as cells, proteins, or tissues. In this thesis, we study image representations that are adapted to local geometric transfor- mations such as rotation, translation, and scaling, with a special emphasis on wavelet representations. In the first part of the thesis, our main interest is in the analysis of directional pat- terns and the estimation of their location and orientation. We explore steerable rep- resentations that correspond to the notion of rotation. Contrarily to classical pattern matching techniques, they have no need for an a priori discretization of the angle and for matching the filter to the image at each discretized direction. Instead, it is sufficient to apply the filtering only once. Then, the rotated filter for any arbitrary angle can be determined by a systematic and linear transformation of the initial filter. We derive the Cramér-Rao bounds for steerable filters. They allow us to select the best harmonics for the design of steerable detectors and to identify their optimal ra- dial profile. We propose several ways to construct optimal representations and to build powerful and effective detector schemes; in particular, junctions of coinciding branches with local orientations. The basic idea of local transformability and the general principles that we utilize to design steerable wavelets can be applied to other geometric transformations. Ac- cordingly, in the second part, we extend our framework to other transformation groups, with a particular interest in scaling. To construct representations in tune with a notion of local scale, we identify the possible solutions for scalable functions and give specific criteria for their applicability to wavelet schemes. Finally, we propose discrete wavelet frames that approximate a continuous wavelet transform. Based on these results, we present a novel wavelet-based image-analysis software that provides a fast and auto- i ii matic detection of circular patterns, combined with a precise estimation of their size. Keywords Circular harmonic wavelets, Cramér-Rao bounds, detection of junctions, directional patterns, estimation of orientation, Fourier multipliers, F-test, image analysis, image representations, local geometric transformations, local symmetries, scalability, steer- ability, steerable filters, steerable wavelets, tight wavelet frames, transformability Résumé La compréhension des caractéristiques géométriques d’une image facilite l’analyse des structures des données microbiologiques, tels que les cellules, les protéines ou les tis- sues. Dans cette thèse, nous étudions les représentations d’images, en particulier les représentations en ondelettes, qui sont adaptées aux transformations géométriques lo- cales comme les rotations, les translations et les dilatations. Dans une première partie, nous portons notre attention sur l’analyse des motifs orientés contenus dans une image et sur l’estimation de leur localisation et orientation. Nous explorons des représentations dites orientables qui correspondent à la notion de rotation. Par opposition aux techniques classiques de filtrage par motif, ces représen- tations ne nécessitent ni la discrétisation de l’angle a priori, ni l’adéquation du filtre à l’image pour chaque direction discrétisée, et il est au contraire suffisant d’appliquer le filtre une seule fois. Ainsi, le filtre auquel on applique une rotation d’un angle arbitraire peut être déterminé par une transformation systématique et linéaire à partir du filtre initial. Nous établissons les bornes de Cramér-Rao pour les filtres orientables, ce qui nous permet de sélectionner les meilleures harmoniques pour la conception de détecteurs orientables et d’identifier le profil radial optimal. Nous proposons différents moyens de construire une représentation optimale et de construire des méthodes performantes et opérantes pour la détection de motifs, en particulier pour la détection de jonctions dans les images. L’idée de base des transformations locales et les principes généraux que nous uti- lisons pour la conception d’ondelettes orientables peuvent être appliqués á d’autres transformations géometriques. Ainsi, nous étendons ces principes à d’autres groupes de transformation dans la seconde partie de cette thèse, avec un intérêt particulier pour les dilatations. Afin de construire des représentations en accord avec la notion de chan- iii iv gement d’échelle local, nous identifions les solutions possibles de fonctions “dilatables" et donnons des critères spécifiques pour leur application à des schémas d’ondelettes. Enfin, nous proposons des ondelettes discrètes redondantes approximant une tranfor- mée en ondelette continue. À partir de ces résultats, nous présentons un nouveau soft- ware pour l’analyse d’image permettant une détection automatique et rapide de motifs circulaires, ainsi qu’une estimation précise de leur taille. Mots clefs Analyse d’image, bornes de Cramér-Rao, détection de jonctions, dilatations, estimation d’orientation, filtres orientables, F-test, motifs orientés, multiplicateurs de Fourier, on- delettes harmoniques circulaires, ondelettes orientables, ondelettes redondantes, re- présentation d’images, symétries locales, transformations géométriques locales. Clouds of all different sizes. They come and they go, while the sky remains the same sky always. The clouds are mere guests in the sky that pass away and vanish, leaving behind the sky. The sky both exists and doesn’t exist. It has substance and at the same time doesn’t. And we merely accept that vast expanse and drink it in. HARUKI MURAKAMI Dedicated to my Grandmother, my Parents, and to Dr. Erika Taróczy and Dr. Zsolt Nagy Acknowledgement First of all, I would like to express my sincere gratitude to my advisor Prof. Michaël Unser, for the continuous support of my Ph.D studies and related research, and for his patience, motivation, and immense knowledge. Besides my advisor, I would like to thank the rest of my thesis committee: Prof. Pierre Vandergheynst, Prof. Dimitri Van De Ville, Dr. Jean-Christophe Olivo-Marin, and Dr. Ullrich Köthe for their encouragement, insightful comments, and thought- provoking questions. I am grateful to Manuelle Mary and Nadia Macor for helping me out with various administrative matters inside and outside EPFL. I have been very fortunate to be a PhD student in the Biomedical Imaging Group (BIG). I take this opportunity to thank its present and past members: Prof. Arash Amini, Anaïs Badoual, Ayush Bhandari, Dr. Emrah Bostan, Dr. Aurélien Bourquard, Prof. Kunal N. Chaudhury, Dr. Nicolas Chenouard, Dr. Ning Chu, Prof. Adrien Depeursinge, Laurène Donati, Julien Fageot, Dr. Denis Fortun, Emmanuel Froustey, Harshit Gupta, Dr. Ulugbek Kamilov, Dr. Hagai Kirshner, Dr. Stamatis Lefkimmiatis, Ramtin Madani, Dr. Michael McCann, Junhong Min, Dr. Ha Nguyen, Dr. Masih Nilchian, Pedram Pad, Dr. Daniel Sage, Daniel Schmitter, Dr. Tomáš Škovránek, Dr. Ferréol Soulez, Dr. Martin Storath Kurt, Dr. Pouya Dehghani Tafti, Dr. Philippe Thévenaz, Virginie Uhlmann, Prof. Dimitri Van De Ville, Dr. Cédric Vonesch, and Dr. John Paul Ward. Over the last couple of years, I had plenty of collaborations, discussions and brain- storming with several members of the group. Especially, I would like to thank my co- authors, John Paul, Daniel Sage, Arash, Cédric, Virginie, Adrien and Martin for all the work we did together. Also, I would like to express my gratitude to Philippe for his de- tailed feedback and comments on my writing. ix x I spent four wonderful years in the lab sharing an office with Julien and Masih. Thank you both, it was great fun to be with you. I would like to thank Prof. Fred Hamprecht and Dr. Ullrich Köthe for hosting me at the Heidelberg Collaboratory for Image Processing over a month. Also, a big thanks to Dr. Ferran Diego and Dr. Melih Kandemir for sharing their office with me. I am grateful to all my friends in Lausanne and Zürich, who made Switzerland my second home. In particular, a very special thanks to Daniel for teaching me how to board (and not to break my neck), John Paul for transforming me into a runner, Julien for introducing me to “graph theory”, Martin for making me addicted to “grillade de saumon”, Pouya for accompanying me in our US trip, Ulugbek for the Saturday morning shopping sessions, and Virginie for co-organizing several evenings. Also, I would like to thank all my Hungarian friends from the region, especially Dr. Ágnes Antal, Brigitta Kazi, Gergely Orbán and Dr. Tamás András Tóth for their support, inspiration and friendship. I really appreciate all the good moments that we shared together. I would like to say special thanks to my friends from Budapest, who are geograph- ically remote, but still part of the most important moments of my life. In particular, I would like to thank Nóra Kamarás, Dr. Nóra Fanna Kovács, Lili Nagy, and Eszter Tagányi for all their love and care. I am grateful to have several good friends all around the world; and I am very glad that we could maintain our friendship despite the long distance.
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