Second-Order Edge-Penalization in the Ambrosio--Tortorelli functional

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 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 numerically without any difficulties compared to the standard Ambrosio--Tortorelli functional. The computational results indicate several additional advantages. 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.