Finite difference approximation of the Mumford‐Shah functional

We study the pointwise convergence and the Γ of a family of nonlocal functionals defined in amstex L¹_ loc( Rⁿ) to a local functional that depends on the gradient of u and on the set of discontinuity points of u. We apply this result to approximate a minimum problem introduced by Mumford and Shah to study edge detection in computer vision theory.