  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.
