Partial differential equations and mathematical morphology
type: Item
instance of: scholarly article
title: Partial differential equations and mathematical morphology
zbMATH Open document ID: 0920.35040
DOI: 10.1016/S0021-7824(01)80003-9
author: Frédéric Cao
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1999-09-01T00:00:00Z
review text: The paper is devoted to the behaviour of a class of iterated filters arising in image processing. In particular, the author considers the so-called morphological filters (i.e. monotone and commuting with translations and with strictly increasing contrast changes) defined by \[ IS_su(x):=\inf_{B\in{\mathbb B}}\sup_{y\in B}u(x+s^{1/N}y), \qquad SI_su(x):=\sup_{B\in{\mathbb B}}\inf_{y\in B}u(x+s^{1/N}y), \] where \(N\) is the space dimension, and \({\mathbb B}\) is the so-called set of structuring elements of the filter. Under suitable assumptions on \({\mathbb B}\), one of the main results of the paper is the convergence of the iterates of the alternate operator \(T_h=SI_s IS_s\) (with \(h=s^{2/(N+1)}\)), namely \(u(x,t)=T_h^{[t/h]}(x)\) uniformly converge on compact subsets of \({\mathbb R}^N\times (0,\infty)\) to the solution, in the viscosity sense, of the fully nonlinear PDE \[ u_t=c| Du| \Biggl(\Pi_{i=1}^{N-1}\lambda_i\Biggr)^{1/(N+1)} H(\lambda_1,\ldots,\lambda_{N-1}) \] where \(\lambda_1,\ldots,\lambda_{N-1}\) are the principal curvatures of the level set of \(\{y: u(t,y)=u(t,x)\}\) containing \(x\) and \(H(t_1,\ldots,t_{N-1})=1\) of all \(t_i\)'s are positive, \(H(t_1,\ldots,t_{N-1})=-1\) of all \(t_i\)'s are negative and \(H(t_1,\ldots,t_{N-1})=0\) in all the other cases. The author also proves convergence of the iterated median filters to the classical mean curvature motion.
Mathematics Subject Classification ID: 35G20
Mathematics Subject Classification ID: 68U10
zbMATH DE Number: 1248318
zbMATH Keywords: morphological operators
zbMATH Keywords: scale space
zbMATH Keywords: structuring elements
zbMATH Keywords: viscosity solutions
zbMATH Keywords: mean curvature motion
reviewed by: Luigi Ambrosio
MaRDI profile type: Publication
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full work available at URL: s0021-7824(01)80003-9
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description: scientific article; zbMATH DE number 1248318