On the $p$-Laplacian and $\infty$-Laplacian on Graphs with Applications in Image and Data Processing

Abstract

International audienceIn this paper we introduce a new family of partial difference operators on graphs and study equations involving these operators. This family covers local variational $p$-Laplacian, $\infty$-Laplacian, nonlocal $p$-Laplacian and $\infty$-Laplacian, $p$-Laplacian with gradient terms, and gradient operators used in morphology based on the partial differential equation. We analyze a corresponding parabolic equation involving these operators which enables us to interpolate adaptively between $p$-Laplacian diffusion-based filtering and morphological filtering, i.e., erosion and dilation. Then, we consider the elliptic partial difference equation with its corresponding Dirichlet problem and we prove the existence and uniqueness of respective solutions. For $p=\infty$, we investigate the connection with Tug-of-War games. Finally, we demonstrate the adaptability of this new formulation for different tasks in image and point cloud processing, such as filtering, segmentation, clustering, and inpainting

A Lie symmetry analysis and explicit solutions of the two‐dimensional ∞‐Polylaplacian

“…In this section, we derive the determining equations of the generators of the symmetry group for both the ∞-Polylaplacian (6) and its reduction (8).…”

“…In what follows, we will see that this is not the case. In this way, we obtain the Lie algebra for the symmetry generators for both Equations (6) and (8) and thus, using the Lie series, derive the full groups of Lie point symmetries for both equations. At the end of this section, we discuss about the discrete symmetries of Equations (6) and (8).…”

“…In this section, we are concerned with the symmetry reductions and the construction of group invariant solutions of Equations (6) and (8). We construct solutions that are invariant under one-dimensional subgroups acting nontrivially on the independent variables.…”

“… and references therein. These equations are strongly nonlinear elliptic PDEs and appear in many important applications such as modes for traveling waves in suspension bridges, the modeling of granular matter, image processing, and game theory …”

Morphological PDEs on Graphs for Image Processing on Surfaces and Point Clouds

“…PDEs of the Laplacian or p-Laplacian type have been used successfully in many applications, such as image denoising, restoration, segmentation and inpainting; see [1][2][3][4] and the references therein. PDEs of the Hamilton-Jacobi type play a central role in continuous mathematical morphology.…”