{"id":690,"date":"2014-03-14T15:51:25","date_gmt":"2014-03-14T15:51:25","guid":{"rendered":"http:\/\/image.math.u-bordeaux.fr\/?p=690&lang=fr"},"modified":"2016-01-08T12:36:18","modified_gmt":"2016-01-08T12:36:18","slug":"study-of-sparse-methods-for-tomography","status":"publish","type":"post","link":"https:\/\/image.math.u-bordeaux.fr\/?p=690","title":{"rendered":"Study of sparse methods for tomography"},"content":{"rendered":"

<\/p>\n

As part of tomography with a limited number of angles, the resulting inverse problem is ill-conditioned. The algorithms “classic”, analytic and algebraic produce many artifacts. The optimization methods solve this problem by using a regularization. In my work, I guess the parsimonious or sparse images in a given representation. A sparse image is an image that has few non-zero components. To promote this parsimony, I will rectify the problem with a standard \"\ell^1\" and solve one of the two equivalent problems below.<\/p>\n

<\/a><\/p>\n

(3) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

<\/a><\/p>\n

(4) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

My job is to assess the theoretical limits of these problems while being the closest to reality. Recent studies [3] have highlighted sufficient conditions ensuring the reconstruction of the image by minimizing \"\ell^1\" and provide an error of this reconstruction [1]. These conditions require the existence of a particular vector called dual certificate <\/em>. Depending on the application, it may be possible to calculate a candidate who will be a conditional dual certificate <\/em> [2,4].<\/p>\n

This work primarily cover the construction of such candidates. I propose in my work, a greedy approach to calculate a candidate who is a dual certificate <\/em> for almost all images “rebuildable” and minimizing the theoretical reconstruction error.<\/p>\n

<\/p>\n

[1] Charles Dossal and Remi Tesson. Consistency of l1 recovery from noisy deterministic measurements. Applied and Computational Harmonic Analysis, 2013.<\/p>\n

[2] Jean-Jacques Fuchs. Recovery of exact sparse representations in the presence of bounded noise. Information Theory, IEEE Transactions on, 51(10) :3601\u20133608, 2005.<\/p>\n

[3] Markus Grasmair, Otmar Scherzer, and Markus Haltmeier. Necessary and su\ufb03cient conditions for linear convergence of l1-regularization. Commun. Pure and Appl. Math., 64(2) :161\u2013182, 2011.<\/p>\n

[4] Samuel Vaiter, Gabriel Peyr\u00e9, Charles Dossal, and Jalal Fadili. Robust sparse analysis regularization. 2011.<\/p>","protected":false},"excerpt":{"rendered":"

As part of tomography with a limited number of angles, the resulting inverse problem is ill-conditioned. The algorithms “classic”, analytic and algebraic produce many artifacts. The optimization methods solve this problem by using a regularization. In my work, I guess the parsimonious or sparse images in a given representation. A sparse image is an image […]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[24,16],"tags":[],"class_list":["post-690","post","type-post","status-publish","format-standard","hentry","category-optimization","category-traitement-dimages"],"_links":{"self":[{"href":"https:\/\/image.math.u-bordeaux.fr\/index.php?rest_route=\/wp\/v2\/posts\/690","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/image.math.u-bordeaux.fr\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/image.math.u-bordeaux.fr\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/image.math.u-bordeaux.fr\/index.php?rest_route=\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/image.math.u-bordeaux.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=690"}],"version-history":[{"count":6,"href":"https:\/\/image.math.u-bordeaux.fr\/index.php?rest_route=\/wp\/v2\/posts\/690\/revisions"}],"predecessor-version":[{"id":774,"href":"https:\/\/image.math.u-bordeaux.fr\/index.php?rest_route=\/wp\/v2\/posts\/690\/revisions\/774"}],"wp:attachment":[{"href":"https:\/\/image.math.u-bordeaux.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=690"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/image.math.u-bordeaux.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=690"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/image.math.u-bordeaux.fr\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=690"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}