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.
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