SIAM J. Multiscale Model. Simul.
Inserted: 20 apr 2015
Last Updated: 10 nov 2015
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 numerically 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.
Keywords: $\Gamma$-convergence, Free-discontinuity problems, Ambrosio-Tortorelli approximation, variational image segmentation