## 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.

### MSC:

 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