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Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. (English) Zbl 0830.49015
The paper deals with an approximation of the Mumford-Shah functional \[ E^0(u)= \lambda^2 \int_{\Omega\backslash K} |u'(t)|^2 dt+ \alpha\text{ card}(K)+ \int_{\Omega\backslash K} |u(t)- g(t)|^2 dt \] by a family of discrete functionals \(E^h(f)\) defined by \[ E^h(f)= h \sum_{\{k: (kh, kh+ h)\in \Omega\}} W_h \Biggl({|f_{k+ 1}- f_k|\over h}\Biggr)+ h \sum_{\{k: kh\in \Omega\}} |f_k- g^h_k|^2, \] where \(g^h\) are discrete approximations of \(g\) and \(W_h(x)= \min(\lambda^2 x^2, \alpha/h)\) are truncated quadratic potentials. The convergence result of the discrete approximations to the continuous result can be rigorously formulated in terms of \(\Gamma\)-convergence. A two-dimensional version of the approximation theorem is also presented; in this case, to obtain convergence it is necessary to replace the length term \({\mathcal H}^1(K)\) of the two-dimensional Mumford-Shah functional with a different, anisotropic term: \[ \int_K |\nu_1(x)|+ |\nu_2(x)|d{\mathcal H}^1(x), \] where \(\nu(x)= (\nu_1(x), \nu_2(x))\) is the approximate unit normal to \(K\).
Reviewer: L.Ambrosio (Pisa)

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
26A45 Functions of bounded variation, generalizations
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