An Approach to the Minimization of the Mumford-Shah Functional using \Gamma-convergence and Topological Asymptotic Expansion

TL;DR

This paper introduces a novel numerical method for minimizing the Mumford-Shah functional using al- gamma-convergence and topological asymptotic expansions, providing an alternative to classical approximations.

Contribution

It develops a new approach based on topological asymptotic analysis that converges to the Mumford-Shah functional and guides the placement of covering balls for edge detection.

Findings

Method converges to Mumford-Shah functional via al- gamma-limits.

Numerical examples demonstrate effectiveness and compare favorably with classical methods.

Topological asymptotic analysis informs optimal placement of edge-covering balls.

Abstract

In this paper, we present a method for the numerical minimization of the Mumford-Shah functional that is based on the idea of topological asymptotic expansions. The basic idea is to cover the expected edge set with balls of radius \epsilon > 0 and use the number of balls, multiplied with 2\epsilon, as an estimate for the length of the edge set. We introduce a functional based on this idea and prove that it converges in the sense of \Gamma-limits to the Mumford-Shah functional. Moreover, we show that ideas from topological asymptotic analysis can be used for determining where to position the balls covering the edge set. The results of the proposed method are presented by means of two numerical examples and compared with the results of the classical approximation due to Ambrosio and Tortorelli.