Many problems in computer vision,  in fracture mechanics or in mathematical physics are modelized by variational problems  in which the unknown function  admits discontinuities. Part of the energy is a volume density, the other part being concentrated along the (free) discontinuity zone.

 In particular in image segmentation problemtinuity zone.

 In particular in image segmentation problems  one has to minimize the Mumford and Shah functional. Since the first existence result obtained  by De Giorgi, Carriero and Leaci, a lot of work has been directed in view of numerical approaches or in view of defining a parabolic evolution model. These questions are highly difficult because of the lack of convexity and regularity .  In fact we need to define a smooth approximation of the Mumford and Shah functional.

For a survey on the approximation of free-discontinuity problems, we refer to Braides.

In this paper we propose a family of functionals  which are regular approximations of  functionals which  generalize of the Mumford and Shah functional. The spirit of the method originates from a previous work concerning capillary equilibrium of droplets,  where the main idea was to consider degenerated potentials at infinity.

The main limitation of this paper is that we encompass only the one-dimensional case. However it seems that, when the potential is increasing,  the convergence  can be deduced in higher dimension by using the so called  slicing method. The other case  is more intricate and remains completely open although we think that the convergence result still holds true.