The Weighted Ambrosio - Tortorelli Approximation Scheme

TL;DR

This paper investigates a weighted Ambrosio-Tortorelli approximation scheme, proving its convergence to a Mumford-Shah functional that depends on a weight function in SBV space, extending the classical model.

Contribution

It introduces a weighted version of the Ambrosio-Tortorelli scheme and proves its -convergence to a Mumford-Shah functional with a weight  in SBV space, broadening the theoretical understanding.

Findings

The scheme -converges to a weighted Mumford-Shah functional.

The convergence depends on the weight  and its jump values.

The analysis extends classical Ambrosio-Tortorelli approximation results.

Abstract

The Ambrosio-Tortorelli approximation scheme with weighted underlying metric is investigated. It is shown that it {\Gamma}-converges to a Mumford-Shah image segmentation functional depending on the weight $\omega dx$, where $\omega\in SBV(\Omega)$, and on its value $\omega^-$.