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

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 𝒢 h ( u, z ) with z ranging between 0 and 1 and related to the set K . The minimizing z h 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 𝒢 h to 𝒢 takes place in a variational sense, the De Giorgi F‐convergence.