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An approach to the minimization of the Mumford–Shah functional using Γ-convergence and topological asymptotic expansion

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Grasmair, Markus ; Muszkieta, Monika ; Scherzer, Otmar:
An approach to the minimization of the Mumford–Shah functional using Γ-convergence and topological asymptotic expansion.
In: Interfaces and free boundaries : modelling, analysis and computation. Bd. 15 (2013) Heft 2. - S. 141-166.
ISSN 1463-9963 ; 1463-9971

Kurzfassung/Abstract

In this paper, we present a method for the numerical minimization of the Mumford–Shah functional that is based on the idea of topological asymptotic expansions. The basic idea is to cover the expected edge set with balls of radius ϵ>0 and use the number of balls, multiplied with 2ϵ, as an estimate for the length of the edge set. We introduce a functional based on this idea and prove that it converges in the sense of Γ-limits to the Mumford–Shah functional. Moreover, we show that ideas from topological asymptotic analysis can be used for determining where to position the balls covering the edge set. The results of the proposed method are presented by means of two numerical examples and compared with the results of the classical approximation due to Ambrosio and Tortorelli.

Weitere Angaben

Publikationsform:Artikel
Schlagwörter:Topological asymptotic expansion, Γ-convergence, Mumford-Shah functional, image segmentation
Institutionen der Universität:Mathematisch-Geographische Fakultät > Mathematik > Lehrstuhl für Mathematik - Wissenschaftliches Rechnen/Informatik
Peer-Review-Journal:Ja
Titel an der KU entstanden:Ja
Eingestellt am:02. Okt 2013 08:15
Letzte Änderung:09. Jun 2016 17:56
URL zu dieser Anzeige:http://edoc.ku-eichstaett.de/13566/