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Approximation of functionals depending on jumps by elliptic functionals via \(\Gamma\)-convergence. (English) Zbl 0722.49020
Summary: We show how it is possible to approximate the Mumford-Shah image segmentation functional [D. Mumford and J. Shah, “Boundary detection by minimizing functionals”, in: Proc. IEEE Computer Soc. Conf. Comput. Vision Pattern Recognition, San Francisco/CA 1985, 22-26 (Piscataway/NJ 1985)] \[ {\mathcal G}(u,K)=\int_{\Omega \setminus K}[| \nabla u|^ 2+\beta (u-g)^ 2]dx\quad +\quad \alpha {\mathcal H}^{n- 1}(K), \]
\[ u\in W^{1,2}(\Omega \setminus K),\quad K\subset \Omega \quad closed\text{ in } \Omega \] by elliptic functionals defined on Sobolev spaces. The heuristic idea is to consider functionals \({\mathcal G}_ h(u,z)\) with z ranging between 0 and 1 and related to the set K. The minimizing \(z_ h\) are near to 1 in a neighborhood of the set K, and far from the neighborhood they are very small. The neighborhood shrinks as \(h\to +\infty.\) For a similar approach to the problem see S. Mitter, T. Richardson and S. R. Kulkarni [in: Signal processing, Part I: Signal processing theory, Proc. Lect., Minneapolis/MN (USA) 1988, IMA Vol. Math. Appl. 22, 189-210 (1990; Zbl 0701.49003)]. The approximation of \({\mathcal G}_ h\) to \({\mathcal G}\) takes place in a variational sense, the De Giorgi \(\Gamma\)-convergence.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
49J27 Existence theories for problems in abstract spaces
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