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Séminaire du jeudi 19/02/2015

Date: 
Jeudi, février 19, 2015 - 14:00

Date:
Jeudi, Février 19, 2015 - 14:00 to 16:00
localisation: IMS

Orateur : Elodie Pozzi
Affiliation : IMB

Titre : Méthodes d’approximation, bases de reconstruction et exemple du signal IRM de diffusion

Résumé :

Dans cet exposé, nous présenterons une gamme de problèmes d’approximation (sous contraintes) liés aux questions de reconstruction de données physiques (potentiel électrique, magnétique, par exemple) à partir de mesures faites sur le bord d’un domaine. Les méthodes de reconstruction amènent la question du choix d’une base de représentation adequate pour le signal reconstruit. En ce sens, nous donnerons l’exemple de la reconstruction du signal IRM de diffusion en donnant une analyse des bases de reconstruction utilisées en imagerie cérébrale.

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Orateur : Laure Blanc Feraud

Affiliation : I3S, Sophia

Titre : The Continuous Exact L0 (CEL0) penalty: An alternative to L0-norm

Résumé :

Many signal processing areas are concerned with sparse solution recovery as compressed sensing, variable selection, source separation, learning… Strictly speaking, sparsity is modeled by considering “l0-norm” constraints. This leads to nonconvex optimization problems which are well known to be NP-hard. For tractability reasons, these problems are traditionally relaxed to convex optimization, using l1-norm. However using l1 rather than l0 norm can lead to a diminution of performances. In this talk we are interested in l0 regularized least squares problem, called l2-l0 problem.

We propose to replace the l0 penalty by a continuous nonconvex approximation, which we call “Continuous Exact l0 penalty (CEL0)”. This continuous approximation CEL0 is such that global minimizers of the objective function l2-CEL0 are proved to be the same as for the l2-l0 problem in a given sense. We also demonstrate that all local minimizers of this functional are local minimizers for l2-l0. However, the reciprocal is false and some local minimizers of the initial functional are not critical points of l2-CEL0. Then solving the l2-CEL0 problem provides better properties in the sense that the functional is continuous and has less local minimizers. Finally, recent nonsmooth nonconvex algorithms are used to address this relaxed problem within a macro algorithm ensuring the convergence to a critical point of the relaxed functional which is also a (local) optimum of the initial l2-l0 problem.