Abstract
Minimal partition problems consist in finding a partition of a domain into a given number of components in order to minimize a geometric criterion. In applicative fields such as image processing or continuum mechanics, it is standard to incorporate in this objective an interface energy that accounts for the lengths of the interfaces between components. The present work is focused on thetheoretical and numerical treatment of minimal partition problems with interface energies. The considered approach is based on a Gamma-convergence approximation and duality techniques.
Highlights
1.1 Problem descriptionConsider a partition of a bounded domain D of R2 into relatively closed subsets Ω1, . . . , ΩN called phases that may intersect only through their boundaries:D = ∪Nj=1Ωj, and Ωi ∩ Ωj = ∂Ωi ∩ ∂Ωj ∩ D for i = j.Denote the interface separating Ωi and Ωj by Γij: Γij = ∂Ωi ∩ ∂Ωj ∩ D for i = j, with the additional convention Γi,i = ∅
In particular we look at binary and multilabel minimal partition problems, including supervised or automatic image classification and deblurring
We show that the interface energy can be rewritten as a linear combination of perimeters and we are able to compute the coefficients
Summary
Je tiens à remercier Zakaria BELHACHMI et Édouard OUDET d’avoir accepté de rapporter sur ma thèse. Je voudrais aussi remercier Luc DINH THE et Jérôme FEHRENBACH d’avoir accepté de faire partie de mon jury de thèse. Un grand merci à Samuel AMSTUTZ pour ses conseils, recommandations et encouragements. Merci pour être toujours disponible et pour avoir une réponse à toutes mes questions. Je voudrais remercier tous les membres du LMA. Je remercie Anna et Chiara pour les bons moments qu’on a passés ensemble. Je veux remercier ma famille pour m’avoir encouragé à suivre ma passion pour les mathématiques. Merci Atika, Israa, Zahraa et Athraa pour être toujours près de moi. Tout ce que je fais c’est pour vous
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