Non-Local Morphological PDEs and $p$-Laplacian Equation on Graphs With Applications in Image Processing and Machine Learning

Abstract

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.

Nonlinear Operators on Graphs via Stacks

“…Secondly, several papers use the MBO algorithm on a graph to approximate motion by mean curvature [17,40]. It is important to note that the curvature is defined by means of the isotropic total variation [17] instead of the one-Laplacian as it is the case in [10,11,12,13].…”

“…On the other hand, some regularisation techniques and nonlinear operators have been introduced on function evaluated on graph via directional derivative [10] [11] [12] or discrete version of the p-Laplacian [13]. We adopt a different viewpoint, our approach is inspired from the signal processing approach on graphs [14,15,16,17].…”

Morphological PDEs on graphs for filtering and inpainting of point clouds

“…Additionnaly, it was shown in [EDL12] that the non-local ∞−Laplacian (Eq. 8) depends on these NLD and NLE operators:…”

“…3) Interpolation and filtering algorithms: Whatever the problems (interpolation or filtering), works of [EDL12] have shown that both problems have a unique solution that can be obtained using the following iterative algorithm:…”

“…In the first section, we present PdEs on graphs [ELB08], [EDL12] and derive morphological operators that are used to filter or inpaint point clouds. Second section presents the construction of a patchbased weighted graph from a point cloud.…”