On the approximation of free discontinuity problems. (English) Zbl 0776.49029

This paper compliments another one of the authors [Commun. Pure Appl. Math. 43, No. 8, 999-1036 (1990; Zbl 0722.49020)] both concerning the approximation (in the sense of \(\Gamma\)-convergence) of the Mumford-Shah type functional (or rather its lower semicontinuous envelope) by elliptic functionals which formally have simpler form.
The Mumford-Shah type functionals appear in a variational approach to the image segmentation problem and have as unknown also a hypersurface \(K\subset \mathbb{R}^ n\): \[ F(u,K)=\int_{\Omega\backslash K} (\alpha|\nabla u|^ 2 + \beta(u-g)^ 2)dx +{\mathcal H}^{n- 1}(K), \] where \(\Omega\) is an open and bounded set in \(\mathbb{R}^ n\), \(K\) is closed in \(\Omega\), \(u\in C^ 1(\Omega\backslash K)\), \(\alpha\), \(\beta\) are fixed positive constants and \(g\) is a given function in \(L^ \infty(\Omega)\), \({\mathcal H}^{n-1}\) is the \((n-1)\)-dimensional Hausdorff measure. Each of the two papers proposes a variational approximation of the lower semicontinuous envelope \(\overline F\) by two different families of functionals \(F_ h(u,s)\), \(G_ x(u,s)\), resp., \(s\) being now a functional variable, so that the minimizers “converge” (up to subsequences) to a minimizer of \(\overline F\). The so-called minimal partition problem is also considered.


49R50 Variational methods for eigenvalues of operators (MSC2000)
49Q12 Sensitivity analysis for optimization problems on manifolds
68U10 Computing methodologies for image processing


Zbl 0722.49020