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Discrete approximation of the Mumford-Shah functional in dimension two. (English) Zbl 0943.49011
The Mumford-Shah functional, which appears in studying image segmentation problems, is considered on the space of special functions of bounded variation \((\text{SBV}(\Omega))\) introduced by De Giorgi and Ambrosio. In the study of this functional \[ \int_{\Omega}|\nabla u(x)|^2 dx + {\mathcal{H}}^1(S_u) + \int_{\Omega}|u(x) - g(x)|^2 dx \] the main difficulty is the presence of the linear term \( {\mathcal{H}}^1(S_u)\), where \({\mathcal{H}}^1\) denotes the one-dimensional Hausdorff measure, \(S_u\) is the set of essential discontinuity points of \(u \in \text{SBV}(\Omega)\) (\(\Omega\) being an open and bounded set in \({R}^2\)). Thus many papers have been devoted to the problem of approximating this functional by simpler ones defined on Sobolev spaces. In this paper, concerning only the two-dimensional case, the authors propose a new approximation of the Mumford-Shah functional based on adaptive finite elements. Namely, they prove that this functional can be approximated in the sense of \(\Gamma\)-convergence by a sequence of integral functionals defined on piecewise affine functions. The construction of an appropriate triangulation of \(\Omega\) needed in their proof of convergence is provided in the Appendix.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
68U10 Computing methodologies for image processing
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