In this paper we propose an adaptation of the Eikonal equation on weighted graphs, using the framework of Partial difference Equations, and with the motivation of extending this equation's applications to any discrete data that can be represented by graphs. This adaptation leads to a finite difference equation defined on weighted graphs and a new efficient algorithm for multiple labels simultaneous propagation on graphs, based on such equation. We will show that such approach enables the resolution of many applications in image and high dimensional data processing using a unique framework. Keywords Eikonal equation • weighted graph • non-local image processing • active contour • PdE • fast marching • high dimensional data This work was supported under a doctoral grant of the Conseil Régional de Basse-Normandie and the Coeur et Cancer association.

In this paper, we introduce a new class of nonlocal p-Laplacian operators that interpolate between non-local Laplacian and infinity Laplacian. These operators are discrete analogous of the game p-laplacian operators on Euclidean spaces, and involve discrete morphological gradient on graphs. We study the Dirichlet problem associated with the new p-Laplacian equation and prove existence and uniqueness of it's solution. We also consider non-local diffusion on graphs involving these operators. Finally, we propose to use these operators as a unified framework for solution of many inverse problems in image processing and machine learning. Index Terms-p-Laplacian, PDEs-based morphology on graphs, image processing, machine learning, Tug-of-war games.