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The Mumford-Shah conjecture in image processing. (English) Zbl 0879.49002

Séminaire Bourbaki. Volume 1995/96. Exposés 805–819. Paris: Société Mathématique de France, Astérisque. 241, 221-242, Exp. No. 813 (1997).
The paper is concerned with the Mumford-Shah functional \[ J(u,K)=\int_{\Omega\setminus K}\nabla u{}^2+ \int_{\Omega\setminus K}(u-g)^2+ {\mathcal H}^{n-1}(K) \] (with \(\Omega\subset\mathbb{R}^n\) open and bounded, \(g\in L^\infty(\Omega)\)) where \(u\in C^1(\Omega\setminus K)\) and \(K\subset\Omega\) is relatively closed. Mumford and Shah proposed, in connection with a variational approach to image segmentation problems, the minimization of this kind of functionals. The state of the art of the current research in this subject is reviewed. In particular, the paper is concerned with the properties of optimal segmentations, i.e., sets \(K\) such that, choosing \(u\) to be the solution of \(\Delta u=(u-g)\) in \(\Omega\setminus K\) with homogeneous Neumann conditions, the pair \((u,K)\) minimizes \(J\). Among these properties we mention density lower bounds, concentration and projection properties, (uniform) rectifiability, almost everywhere regularity.
For the entire collection see [Zbl 0866.00026].
Reviewer: L.Ambrosio (Pavia)

MSC:

49J10 Existence theories for free problems in two or more independent variables
68U10 Computing methodologies for image processing
35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49Q05 Minimal surfaces and optimization
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