Discrete stochastic approximations of the Mumford-Shah functional

TL;DR

This paper introduces a new discrete approximation method for the Mumford-Shah functional using stochastic lattices, enabling better modeling of anisotropic effects and homogenization in multiple dimensions.

Contribution

It presents a Γ-convergent discrete approximation on stochastic lattices that accounts for anisotropy and general finite differences, advancing the numerical analysis of Mumford-Shah models.

Findings

The approximation converges to the Mumford-Shah functional.

Statistically isotropic lattices can effectively model isotropic properties.

The method applies to vectorial cases in any dimension.

Abstract

We propose a $\Gamma$-convergent discrete approximation of the Mumford-Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford-Shah functional in any dimension.