Non-local approximation of the Mumford-Shah functional
type: Item
instance of: scholarly article
title: Non-local approximation of the Mumford-Shah functional
zbMATH Open document ID: 0873.49009
author: G. Dal Maso
author: Andrea Braides
published in: Calculus of Variations and Partial Differential Equations
publication date: 1997-10-01T00:00:00Z
review text: The paper deals with the approximation of the Mumford-Shah functional \[ F(u):=\int_\Omega |\nabla u|^2 dx+\beta{\mathcal H}^{n-1}(\Omega\cap S_u) \] defined in the class SBV\((\Omega)\) of special functions with bounded variation (here \(S_u\) is the Lebesgue discontinuity set of \(u\)). It is easy to see that \(F\) can't be approximated (in the sense of \(\Gamma\)-convergence) by integral functionals defined on Sobolev spaces and depending only on one variable. The main result of the paper is the proof that an approximation can be achieved with non local functionals, whose lack of locality tends to \(0\). Specifically, the approximating functionals are given by \[ F_\varepsilon(u):={1\over\varepsilon}\int_\Omega f\biggl(\varepsilon \hbox{\vrule height3.5pt depth-2.8pt width6pt}\mkern-18mu\int\nolimits_{B_\varepsilon(x)\cap\Omega} |\nabla u|^2(y) dy\biggr) dx,\qquad\qquad \varepsilon\downarrow 0, \] where \(f(t)\) is any continuous increasing function such that \(f'_+(0)=1\) and \(f(\infty)=\beta/2\). Allowing \(f\) to depend on \(\varepsilon\), suitable convergence theorems of minimum points to minimum points for the Mumford-Shah image segmentation problem \[ \min\biggl\{F(u)+\alpha\int_\Omega (u-g)^2 dx:\;u\in \text{SBV}(\Omega)\biggr\} \] are estabilished.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 49M30
zbMATH DE Number: 1019672
zbMATH Keywords: Mumford-Shah functional
zbMATH Keywords: \(\Gamma\)-convergence
zbMATH Keywords: non local problems
zbMATH Keywords: SBV
zbMATH Keywords: special functions with bounded variation
reviewed by: Luigi Ambrosio
MaRDI profile type: Publication
full work available at URL: s005260050068
OpenAlex ID: W2087655772
DOI: 10.1007/S005260050068
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description: scientific article; zbMATH DE number 1019672