Numerical minimization of the Mumford--Shah functional
type: Item
instance of: scholarly article
title: Numerical minimization of the Mumford--Shah functional
zbMATH Open document ID: 1017.49017
published in: Calcolo
publication date: 2003-04-22T00:00:00Z
review text: In this paper the authors are interested in the numerical implementation of the Mumford-Shah functional, a functional used in the calculus of variations to approach the segmentation problems; the approach is taken by using the finite elements method (the minima of the discretized functional \(\Gamma\)-converge in the strong \(L^2\) topology to the minima of a functional, similar to the Mumford-Shah functional, but taking into account also the geometry of the triangulation used in the finite elements method). To find the minimum of the discrete functional, the authors use a quasi-Newton algorithm (taking into account that the Hessian of the discrete functional is not in general symmetric). The authors also give a one-dimensional model problem, pointing out the fact that there are local mimima, both for the discrete as well as for the continuous functional. To this aim, the authors propose two algorithms, both based on a multiscaled technique, which give the convergence to the global minimizer.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49M25
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 94A08
zbMATH DE Number: 1898327
zbMATH Keywords: Mumford-Shah functional
zbMATH Keywords: finite elements
zbMATH Keywords: quasi-Newton algorithm
zbMATH Keywords: gamma-convergence
zbMATH Keywords: segmentation
author: Matteo Negri
author: Maurizio Paolini
reviewed by: Michele jun. Miranda
MaRDI profile type: Publication
full work available at URL: s100920170004
OpenAlex ID: W2158144798
DOI: 10.1007/S100920170004
description: scientific article; zbMATH DE number 1898327