Finite difference approximation of the Mumford‐Shah functional

We study the pointwise convergence and the Γ of a family of nonlocal functionals defined in \input amstex \loadmsbm $L^1_{\roman {loc}}(\Bbb R^n)$ to a local functional ${\cal F}(u)$ 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. © 1998 John Wiley & Sons, Inc.