Algorithms for motion by mean curvature flow of networks of surfaces

Selim Esedoglu
Department of Mathematics, University of Michigan, Ann Arbor

I will review some of the most popular numerical methods for simulating the motion of interfaces, including networks of them, by mean curvature and related flows. Motion by mean curvature arises as gradient descent for the sum of areas of surfaces in the network. This energy comes up in a number of applications ranging from image processing, computer vision, and machine learning, to materials science. In image processing, it is used for denoising of images while preserving sharp boundaries of objects (edges). In computer vision, it appears in the Mumford-Shah functional -- one of the most important variational models for image segmentation. In machine learning, it can be used for graph partitioning, e.g. in the context of supervised classification and recognition tasks. Finally, in materials science, the evolution of boundaries of single crystal pieces (grains) that make up a polycrystalline material (e.g. most metals and ceramics) is described by mean curvature flow. In all these applications, numerous topological changes are to be expected, and any relevant numerical method should be prepared to handle them. I will touch on Monte Carlo Potts, level set, phase field, and threshold dynamics methods that have been applied to these problems, and try to give a sense of where things stand.