Partial Difference Operators on Weighted Graphs for Image Processing on Surfaces and Point Clouds

Partial difference equations (PDEs) and variational methods for image processing on Euclidean domains spaces are very well established because they permit to solve a large range of real computer vision problems. With the recent advent of many 3D sensors, there is a growing interest in transposing and solving PDEs on surfaces and point clouds. In this paper, we propose a simple method to solve such PDEs using the framework of PDEs on graphs. This latter approach enables us to transcribe, for surfaces and point clouds, many models and algorithms designed for image processing. To illustrate our proposal, three problems are considered: 1) \(p\) -Laplacian restoration and inpainting; 2) PDEs mathematical morphology; and 3) active contours segmentation.