A tutorial on well-composedness
type: Item
instance of: scholarly article
title: A tutorial on well-composedness
zbMATH Open document ID: 1435.68337
author: Nicolas Boutry
author: Thierry Géraud
author: Laurent Najman
published in: Journal of Mathematical Imaging and Vision
publication date: 2018-10-19T00:00:00Z
review text: The well-known ``paradox'' of digital topology, coming from the early works of Rosenfeld, is that for binary images on a square/cubic grid (in $\mathbb{Z}^n$), with connectivity given by graph adjacency, the preservation of standard Euclidean topological properties requires that one chooses opposite adjacencies on foreground and background: axial adjacency (linked to the $L_1$-norm) for one, and diagonal adjacency (linked to the $L_\infty$-norm) for the other. This presents several problems: a binary digital image will have two topologies corresponding to the two choices of adjacency for foreground, and the idea does not extend easily to non-binary images, whose values can for instance form an anti-chain of labels, or a chain of grey-levels. To overcome that problem, Latecki and co-workers introduced the notion of \textit{well-composed} images, those for which the topology does not depend on the choice of adjacencies. This paper gives a survey of this concept, covering its various definitions and its possible applications. First, four definitions of well-composedness are presented and analysed: \begin{itemize} \item DWC: digital well-composedness, characterised by the absence of ``critical configurations'' (for instance in 2D, a $2\times2$-square with one diagonal with one value, and the other diagonal with another value). \item AWC: well-composedness in the Alexandrov sense, where the image is embedded in a cellular complex with Alexandrov topology, then the boundary of each connected component will be a discrete $(n-1)$-surface. \item EWC: well-composedness by connectivity equivalence: all choices of adjacencies will lead to the same connected components for all image values. \item CWC: well-composedness in the continuous sense: replacing each pixel/voxel by the corresponding Euclidean closed unit square/cube, one obtains an object whose boundary is an $(n-1)$-manifold. \end{itemize} These four definitions are related as follows: \begin{itemize} \item DWC implies …
reviewed by: Christian Ronse
Mathematics Subject Classification ID: 68U03
Mathematics Subject Classification ID: 05C40
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 68U10
zbMATH DE Number: 6958494
zbMATH Keywords: well-composedness
zbMATH Keywords: digital topology
zbMATH Keywords: mathematical morphology
zbMATH Keywords: critical configurations
zbMATH Keywords: discrete surfaces
zbMATH Keywords: manifolds
describes a project that uses: VIGRA
MaRDI profile type: Publication
full work available at URL: s10851-017-0769-6
OpenAlex ID: W2761766963
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A tutorial on well-composedness (DOI)
DOI: 10.1007/S10851-017-0769-6