On Median Filters for Motion by Mean Curvature

TL;DR

This paper explores median filters as a natural level set discretization of motion by mean curvature, establishing their connection to threshold dynamics and introducing energy-based stability and multiphase extensions.

Contribution

It provides a variational formulation and Lyapunov function for median filters, enabling energy-based stability and multiphase generalizations without frequent redistancing.

Findings

Median filters evolve level sets by threshold dynamics.

Energy-based analysis shows unconditional stability.

Multiphase median filters handle complex networks with varied surface tensions.

Abstract

The median filter scheme is an elegant, monotone discretization of the level set formulation of motion by mean curvature. It turns out to evolve every level set of the initial condition precisely by another class of methods known as threshold dynamics. Median filters are, in other words, the natural level set versions of threshold dynamics algorithms. Exploiting this connection, we revisit median filters in light of recent progress on the threshold dynamics method. In particular, we give a variational formulation of, and exhibit a Lyapunov function for, median filters, resulting in energy based unconditional stability properties. The connection also yields analogues of median filters in the multiphase setting of mean curvature flow of networks. These new multiphase level set methods do not require frequent redistancing, and can accommodate a wide range of surface tensions.