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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.

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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