Flatness and finiteness in the Mumford-Shah problem
type: Item
instance of: scholarly article
title: Flatness and finiteness in the Mumford-Shah problem
zbMATH Open document ID: 0942.49030
DOI: 10.1016/S0021-7824(99)00019-7
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2000-08-17T00:00:00Z
review text: The paper is concerned with the regularity properties of optimal pairs \((u,K)\) for the Mumford-Shah image segmentation problem. By a blow-up method introduced by A. Bonnet, this problem is closely related to the classification of the global minimizers (i.e., minimizers with respect to compactly supported perturbations which, loosely speaking, do not decrease the number of connected components at infinity). \textit{A. Bonnet} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 13, No. 4, 485-528 (1996; Zbl 0883.49004)] was able to classify all globally minimizing pairs \((u,K)\) with \(K\) connected, thus proving the Mumford-Shah conjecture for sets \(K\) which locally have finitely many connected components. The classification in the general case is still an open problem. The main result of the paper is the replacement of Bonnet's connectedness assumption by a ``mean flatness'' property, stated in terms of the David and Semmes ``Big Pieces of Lipschitz Graphs'' property. The results are achieved using a Cauchy-type representation formula of \(u\) as a function of \(K\) which has an independent interest and more potential applications.
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 49J10
zbMATH DE Number: 1344439
zbMATH Keywords: image segmentation
zbMATH Keywords: Mumford-Shah problem
zbMATH Keywords: regularity of minimizers
author: Jean-Christophe Léger
reviewed by: Luigi Ambrosio
MaRDI profile type: Publication
cites work: Q4364771
cites work: Q4364772
cites work: Q3999762
cites work: On the regularity of edges in image segmentation
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full work available at URL: s0021-7824(99)00019-7
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description: scientific article; zbMATH DE number 1344439