Singular sets of minimizers for the Mumford-Shah functional
type: Item
instance of: scholarly article
title: Singular sets of minimizers for the Mumford-Shah functional
zbMATH Open document ID: 1086.49030
author: Guy David
published in: Progress in Mathematics
publication date: 2005-04-05T00:00:00Z
review text: This monograph is the Ferran Sunyer i Balaguer 2004 prize winner. The prize is awarded yearly by the Fundació Ferran Sunyer i Balaguer and the Institute d'Etudis Catalans for a mathematical monograph of expository nature. In the author's intentions, the book wants to provide a wide survey on the machineries recently developed in the framework of the study of the Mumford-Shah functional, and on the progresses in the subject, in the hope that this can lead to some achievements in the analysis of the Mumford-Shah conjecture, or even in some other contexts. Recall that the Mumford-Shah conjecture says that, in dimension 2, the singular set of a reduced minimizer of the Mumford-Shah functional is locally a \(C^1\)-curve, except at a finite number of points. The book follows an approach to the problem different from the bounded variation one, and tries to be as much as possible accessible and self-contained. It consists of nine chapters. The first one provides a general presentation of the Mumford-Shah functional and its origin in the framework of image segmentation. Then, the Mumford-Shah conjecture and some known results on it are described. After a review on simple facts about Sobolev spaces, the first regularity results for minimizers and quasiminimizers in dimension 2 are presented with proofs very close to the original ones. In particular, the Ahlfors regularity results, the Carleson measure estimates, the projection and concentration properties, and the local uniform rectifiability property are discussed. Then, the Dal Maso-Morel-Solimini concentration property is used in order to prove that limits of minimizers or of almost-minimizers are still minimizers or almost-minimizers. The almost everywhere \(C^1\)-regularity result for almost-minimizers in dimension 2 is provided in Chapter 5 by means of a decay estimate for the normalized energy coming from a variant of a Bonnet's monotonicity argument. The properties of global minimizers in the plane are described in Ch…
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 94A08
Mathematics Subject Classification ID: 68U10
Mathematics Subject Classification ID: 49-02
zbMATH DE Number: 2152355
zbMATH Keywords: Mumford-Shah functional
zbMATH Keywords: regularity of singular sets
zbMATH Keywords: image segmentation
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2152355