Nonlocal discrete ∞-Poisson and Hamilton Jacobi equations from stochastic game to generalized distances on images, meshes, and point clouds

Matthieu Toutain 1 Abderrahim Elmoataz 1 François Lozes 1 Amin Mansouri 2
1 Equipe Image - Laboratoire GREYC - UMR6072
GREYC - Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen
Abstract : In this paper we propose an adaptation of the ∞-Poisson equation on weighted graphs, and propose a finer expression of the ∞-Laplace operator with gradient terms on weighted graphs, by making the link with the biased version of the tug-of-war game. By using this formulation, we propose a hybrid ∞-Poisson Hamilton-Jacobi equation, and we show the link between this version of the ∞-Poisson equation and the adaptation of the eikonal equation on weighted graphs. Our motivation is to use this extension to compute distances on any discrete data that can be represented as a weighted graph. Through experiments and illustrations , we show that this formulation can be used in the resolution of many applications in image, 3D point clouds, and high dimensional data processing using a single framework.
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Journal of Mathematical Imaging and Vision, Springer Verlag, 2015, 13p. 〈10.1007/s10851-015-0592-x〉
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Matthieu Toutain, Abderrahim Elmoataz, François Lozes, Amin Mansouri. Nonlocal discrete ∞-Poisson and Hamilton Jacobi equations from stochastic game to generalized distances on images, meshes, and point clouds. Journal of Mathematical Imaging and Vision, Springer Verlag, 2015, 13p. 〈10.1007/s10851-015-0592-x〉. 〈hal-01188784〉

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