Variational Approximation of a Second Order Free Discontinuity Problem in Computer Vision

Abstract

We consider a functional, proposed by Blake and Zisserman for computer vision problems, which depends on free discontinuities, free gradient discontinuities, and second order derivatives. We show how this functional can be approximated by elliptic functionals defined on Sobolev spaces. The approximation takes place in a variational sense, the De Giorgi $\Gamma$-convergence, and extends to this second order model an approximation of the Mumford--Shah functional obtained by Ambrosio and Tortorelli. For the purpose of illustration an algorithm based on the $\Gamma$-convergent approximation is applied to the problem of computing depth from stereo images and some numerical examples are presented.