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 [19], 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.

Tug of War games and PDEs on graphs with applications in image and high dimensional data processing

“…Consequently, designing new methods for processing and analyzing data on graphs has been the object of many works. We mention among others [1][2][3][4][5][6][7] where the authors propose the adaptation of many continuous PDEs and variational models such as the total variation flow, mean curvature flow and Hamilton-Jacobi equations to the framework of graphs. Naturally, this made discretizing and solving PDEs on graphs and networks gain attention and interest due to numerous applications in imaging, computer vision and machine learning, where data are given in the form of graphs or functions defined on graphs (see Fig.…”

“…A large class of PDEs on graphs recovered from ( 8)-( 9), which are obtained by taking particular parameter values in the TOW games ( 5)- (6).…”