Second-order edge-penalization in the Ambrosio-Tortorelli functional

TL;DR

This paper introduces two second-order variants of the Ambrosio-Tortorelli functional that better approximate the Mumford-Shah functional, offering smoother edge contours and improved convergence in computational algorithms.

Contribution

It proposes novel second-order penalization methods for the Ambrosio-Tortorelli functional, enhancing approximation quality and computational performance.

Findings

Smoother and clearer edge contours in minimizers.

Improved convergence of alternating minimization algorithms.

Equivalent elliptic approximation of Mumford-Shah functional.

Abstract

We propose and study two variants of the Ambrosio-Tortorelli functional where the first-order penalization of the edge variable $v$ is replaced by a second-order term depending on the Hessian or on the Laplacian of $v$, respectively. We show that both the variants as above provide an elliptic approximation of the Mumford-Shah functional in the sense of $\Gamma$-convergence. In particular the variant with the Laplacian penalization can be implemented without any difficulties compared to the standard Ambrosio-Tortorelli functional. The computational results indicate several advantages however. First of all, the diffuse approximation of the edge contours appears smoother and clearer for the minimizers of the second-order functional. Moreover, the convergence of alternating minimization algorithms seems improved for the new functional. We also illustrate the findings with several computational results.