A Non-Local Mean Curvature Flow and its semi-implicit time-discrete approximation

TL;DR

This paper introduces a novel non-local mean curvature flow for curve denoising, addressing degeneracies with a viscosity solutions framework and a minimizing movements approach, and compares it to standard mean curvature flow.

Contribution

It develops a new non-local curvature flow model with a semi-implicit approximation, providing existence, uniqueness, and comparison with classical flows.

Findings

Existence and uniqueness of viscosity solutions for the non-local flow

Construction of an exact flow via minimizing movements

Comparison showing differences from standard mean curvature flow

Abstract

We address in this paper the study of a geometric evolution, corresponding to a curvature which is non-local and singular at the origin. The curvature represents the first variation of the energy recently proposed as a variant of the standard perimeter penalization for the denoising of nonsmooth curves. To deal with such degeneracies, we first give an abstract existence and uniqueness result for viscosity solutions of non-local degenerate Hamiltonians, satisfying suitable continuity assumption with respect to Kuratowsky convergence of the level sets. This abstract setting applies to an approximated flow. Then, by the method of minimizing movements, we also build an "exact" curvature flow, and we illustrate some examples, comparing the results with the standard mean curvature flow.