Approximation of functional depending on jumps by elliptic functional via t‐convergence

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Abstract
We show how it is possible to approximate the Mumford‐Shah (see [29]) image segmentation functional magnified image by elliptic functionals defined on Sobolev spaces. The heuristic idea is to consider functionals 𝒢h(u, z) with z ranging between 0 and 1 and related to the set K. The minimizing zh are near to 1 in a neighborhood of the set K, and far from the neighborhood they are very small. The neighborhood shrinks as h → + ∞. For a similar approach to the problem compare Kulkarni; see [25]. The approximation of 𝒢h to 𝒢 takes place in a variational sense, the De Giorgi F‐convergence.

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