Discrete stochastic approximations of the Mumford–Shah functional

Ruf, Matthias

doi:10.1016/j.anihpc.2018.10.004

Ruf, Matthias

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 887-937.

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Résumé
Abstract

Nous proposons une nouvelle discrétisation de la fonctionnelle de Mumford–Shah convergente au sens de la Γ-convergence. Les fonctionnelles discrètes agissent sur des fonctions définies sur des réseaux aléatoires stationnaires et dépendent de différences finies générales via un potentiel non-convexe. Dans ce cadre, la géométrie du réseau aléatoire influence grandement l'anisotropie de la fonctionnelle limite. Ainsi, en utilisant des réseaux aléatoires statistiquement isotropes, on démontre par des techniques d'homogénéisation le résultat d'approximation de la fonctionnelle vectorielle de Mumford–Shah en toutes dimensions.

We propose a new Γ-convergent discrete approximation of the Mumford–Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford–Shah functional in any dimension.

DOI : 10.1016/j.anihpc.2018.10.004

Classification : 49M25, 68U10, 49J55, 49J45
Mots-clés : Mumford–Shah functional, Discrete approximation, Γ-convergence, Stochastic homogenization

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     title = {Discrete stochastic approximations of the {Mumford{\textendash}Shah} functional},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {887--937},
     publisher = {Elsevier},
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     number = {4},
     year = {2019},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2018.10.004/}
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Ruf, Matthias. Discrete stochastic approximations of the Mumford–Shah functional. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 887-937. doi : 10.1016/j.anihpc.2018.10.004. http://www.numdam.org/articles/10.1016/j.anihpc.2018.10.004/

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