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On the regularity of edges in image segmentation. (English) Zbl 0883.49004

The paper is concerned with the regularity of minimizers \((u,K)\) of the Mumford-Shah functional \[ J(u,K)=\int_{\Omega\setminus K}|\nabla u|^2+ \int_{\Omega\setminus K}(u-g)^2+ {\mathcal H}^1(K) \] (with \(\Omega\subset{\mathbb{R}}^2\) open and bounded, \(g\in L^\infty(\Omega)\)) where \(u\in C^1(\Omega\setminus K)\) and \(K\subset\Omega\) is relatively closed. The main result is very close to the original Mumford-Shah conjecture: the \(C^{1,1}\) regularity of any optimal set \(K\) out of a locally finite number of points. However, this is proved under the a priori assumption that \(K\) has finitely many connected components. This assumption is used to show that, at least locally, the ratio \[ {1\over\rho}\int_{B_\rho(x)}|\nabla u|^2 dx \] is nondecreasing in \(\rho\), and this leads through blow-up arguments to a classification of the singularities of \(K\). One of the main ingredients of the proof is also the introduction of a weak notion of local minimality (in which competitors essentially do not decrease the number of connected components of \(\Omega\setminus K\)) which is stable under blow-up limits. Without any topological assumption on \(K\) the author also proves that the singular set of \(K\) is \({\mathcal H}^1\)-negligible. A similar result has been obtained independently by G. David [SIAM J. Appl. Math. 56, No. 3, 783-888 (1996; Zbl 0870.49020)], and in any dimension by L. Ambrosio, N. Fusco and D. Pallara [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24, No. 1, 1-38, 39-62 (1997)].
Reviewer: L.Ambrosio (Pavia)

MSC:

49J10 Existence theories for free problems in two or more independent variables
68U10 Computing methodologies for image processing
49Q20 Variational problems in a geometric measure-theoretic setting
49N60 Regularity of solutions in optimal control

Citations:

Zbl 0870.49020
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References:

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