Nonlocal Perimeters and Curvature Flows on Graphs with Applications in Image Processing and High-Dimensional Data Classification

Abstract

In this paper, we revisit the notion of perimeter on graphs, introduced in El Chakik, Elmoataz, and Desquesnes [Signal Process., 105 (2014), pp. 449–463], and we extend it to so-called inner and outer perimeters. We will also extend the notion of total variation on graphs. Thanks to the co-area formula, we show that discrete total variations can be expressed through these perimeters. Then, we propose a novel class of curvature operators on graphs that unifies both local and nonlocal mean curvature on an Euclidean domain. This leads us to translate and adapt the notion of the mean curvature flow on graphs as well as the level set mean curvature, which can be seen as approximate schemes. Finally, we exemplify the usefulness of these methods in image processing, 3D point cloud processing, and high dimensional data classification.