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

In 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, -Laplacian, nonlocal p-Laplacian and -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=, 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.