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Discrete approximation of a free discontinuity problem. (English) Zbl 0806.49002
Authors’ summary: We approximate by discrete \(\Gamma\)-convergence a functional proposed by Mumford-Shah for a variational approach to image segmentation. Such a functional is first relaxed with a sequence of nonconvex functionals, which in turn, are discretized by piecewise linear finite elements. Under a suitable relation between the relaxation parameter \(\varepsilon\) and the mesh size \(h\), the convergence of the discrete functionals and the compactness of any sequence of discrete minimizers are proved. The proof relies on the techniques of \(\Gamma\)- convergence and on the properties of the Lagrange interpolation and Clement operators.
Reviewer: N.Medhin (Atlanta)

MSC:
49J10 Existence theories for free problems in two or more independent variables
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