{"id":44,"date":"2008-10-10T03:37:31","date_gmt":"2008-10-10T01:37:31","guid":{"rendered":"http:\/\/www.freehackers.org\/thomas\/?page_id=44"},"modified":"2026-02-12T20:14:02","modified_gmt":"2026-02-12T18:14:02","slug":"phd","status":"publish","type":"page","link":"https:\/\/freehackers.org\/thomas\/research\/phd\/","title":{"rendered":"Ph.D."},"content":{"rendered":"\n<p>This page is about the work done during my Ph.D. at the <a href=\"http:\/\/www.upmc.fr\" target=\"_blank\">university Pierre &amp; Marie Curie (Paris 6)<\/a> \/&nbsp; <a href=\"http:\/\/www.ann.jussieu.fr\/\" target=\"_blank\">laboratoire J.-L. Lions<\/a> between 2004 and 2008. My advisor was the Professor <a href=\"http:\/\/www.ann.jussieu.fr\/~plc\/\" target=\"_blank\">P. L. Combettes<\/a>. I gave my defense talk on june, 10th, 2008.<\/p>\n\n\n\n<p>The jury was composed of<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Patrick L. Combettes (Advisor \/ Directeur)<\/li>\n\n\n\n<li>Christine De Mol (Rapporteur)<\/li>\n\n\n\n<li>Fr\u00e9d\u00e9ric Hecht<\/li>\n\n\n\n<li>Alfred Hero<\/li>\n\n\n\n<li>G\u00e9rard Kerkyacharian (President \/ Pr\u00e9sident)<\/li>\n\n\n\n<li>Pierre Mar\u00e9chal (Rapporteur)<\/li>\n\n\n\n<li>Ali Mohammad-Djafari<\/li>\n<\/ul>\n\n\n\n<p>The full text of the manuscript can be downloaded from here :<a href=\"https:\/\/freehackers.org\/media\/orzel\/TheseTC.tgz\"><img decoding=\"async\" title=\"pdf_icon\" width=\"80\" class=\"alignnone size-medium wp-image-58\" src=\"http:\/\/www.freehackers.org\/thomas\/wp-content\/uploads\/2008\/10\/pdf_icon.jpg\" alt=\"\" srcset=\"https:\/\/freehackers.org\/thomas\/wp-content\/uploads\/2008\/10\/pdf_icon.jpg 192w, https:\/\/freehackers.org\/thomas\/wp-content\/uploads\/2008\/10\/pdf_icon-150x150.jpg 150w\" sizes=\"(max-width: 192px) 100vw, 192px\" \/><\/a><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">English summary:<\/h2>\n\n\n\n<p class=\"has-text-align-center\"><strong>Generalized convex projection algorithms and applications to<br>\nmedical imaging<\/strong><\/p>\n\n\n\n<p>This thesis focuses on the use of generalized projection operators in convex optimization algorithms and on their applications to medical imaging. We describe various extensions of the notion of a projection onto a closed convex set in a Hilbert space. These include subgradient projectors, proximity operators, and Bregman projectors. We propose a new generalized projection algorithm for projecting onto a countable intersection of closed convex sets and prove its convergence. This contribution unifies several existing results on the convergence of projection methods in Hilbert spaces. We then study original applications of projection methods in medical imaging. We propose a new strategy to incorporate Poisson noise in continuous tomography, as well as a new approach to discrete tomography via convex programming and total variation. We also discuss the connections between total variation, compressive sensing, and tomographic reconstruction. Finally, we present what seem to be the first numerical results on the use of Bregman distances in projection algorithms. The software tools that have been developed to carry out this work have been designed so as to be made available to the scientific community.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">English Keywords:<\/h2>\n\n\n\n<p>Bregman distance, compressive sensing, convex programming, discrete tomography, image restoration, image reconstruction, nonexpansive operator, projection algorithm, radiotherapy, statistical estimation, tomography, total variation.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">French summary:<\/h2>\n\n\n\n<p class=\"has-text-align-center\"><strong>Algorithmes de projections convexes g\u00e9n\u00e9ralis\u00e9es et applications<br>\nen imagerie m\u00e9dicale<\/strong><\/p>\n\n\n\n<p>Cette th\u00e8se porte sur l&#8217;utilisation d&#8217;op\u00e9rateurs de projection g\u00e9n\u00e9ralis\u00e9s dans les algorithmes d&#8217;optimisation convexe et sur leur application en imagerie m\u00e9dicale. Nous d\u00e9crivons diverses g\u00e9n\u00e9ralisations de la notion de projection sur un convexe ferm\u00e9 dans un Hilbert. Celles-ci incluent les projections sous-diff\u00e9rentielles, les op\u00e9rateurs proximaux et les projecteurs de Bregman. Nous proposons un nouvel algorithme de projections g\u00e9n\u00e9ralis\u00e9es pour projeter sur une intersection d\u00e9nombrable de convexes et d\u00e9montrons sa convergence. Cette contribution unifie plusieurs r\u00e9sultats existants sur la convergence des m\u00e9thodes de projection dans les Hilbert. Nous \u00e9tudions ensuite des applications originales des m\u00e9thodes de projection en imagerie m\u00e9dicale, en proposant une nouvelle strat\u00e9gie pour la prise en compte du bruit poissonnien en tomographie continue, ainsi qu&#8217;une nouvelle approche en tomographie discr\u00e8te par la programmation convexe et l&#8217;utilisation de la variation totale. Nous formulons \u00e9galement quelques r\u00e9flexions sur les liens entre variation totale, acquisition comprim\u00e9e, et reconstruction tomographique. Nous pr\u00e9sentons enfin ce qui semble \u00eatre les premiers r\u00e9sultats num\u00e9riques sur l&#8217;utilisation des distances de Bregman dans les algorithmes de projection. Les outils informatiques d\u00e9velopp\u00e9s dans le cadre de cette th\u00e8se ont \u00e9t\u00e9 con\u00e7us de mani\u00e8re \u00e0 \u00eatre mis \u00e0 disposition de la communaut\u00e9 scientifique.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">French keywords:<\/h2>\n\n\n\n<p>acquisition comprim\u00e9e, algorithme de projection, contraction, distance de Bregman, estimation statistique, programmation convexe, radioth\u00e9rapie, restauration d&#8217;image, reconstruction d&#8217;image, tomographie, tomographie discr\u00e8te, variation totale.<\/p>\n<br class=\"fixfloat\" \/>","protected":false},"excerpt":{"rendered":"<p>This page is about the work done during my Ph.D. at the university Pierre &amp; Marie Curie (Paris 6) \/&nbsp; laboratoire J.-L. Lions between 2004 and 2008. My advisor was the Professor P. L. Combettes. I gave my defense talk on june, 10th, 2008. The jury was composed of The full text of the manuscript [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":20,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":["post-44","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/freehackers.org\/thomas\/wp-json\/wp\/v2\/pages\/44","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/freehackers.org\/thomas\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/freehackers.org\/thomas\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/freehackers.org\/thomas\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/freehackers.org\/thomas\/wp-json\/wp\/v2\/comments?post=44"}],"version-history":[{"count":12,"href":"https:\/\/freehackers.org\/thomas\/wp-json\/wp\/v2\/pages\/44\/revisions"}],"predecessor-version":[{"id":801,"href":"https:\/\/freehackers.org\/thomas\/wp-json\/wp\/v2\/pages\/44\/revisions\/801"}],"up":[{"embeddable":true,"href":"https:\/\/freehackers.org\/thomas\/wp-json\/wp\/v2\/pages\/20"}],"wp:attachment":[{"href":"https:\/\/freehackers.org\/thomas\/wp-json\/wp\/v2\/media?parent=44"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}