A digital analogue of the Jordan curve theorem
type: Item
instance of: scholarly article
title: A digital analogue of the Jordan curve theorem
zbMATH Open document ID: 1062.54001
published in: Discrete Applied Mathematics
publication date: 2004-08-06T00:00:00Z
review text: The usual approach to digital topology of the plane pioneered by \textit{A. Rosenfeld} [see for example Am. Math. Mon. 86, 621--630 (1979; Zbl 0432.68061)] is to use a 4-adjacency relation for the background and an 8-adjacency relation for the image in the rectangular grid \({\mathbb Z}^ 2\). The need to use two distinct types of adjacency arises in order to be able to prove an analogue of the Jordan curve theorem for digital images. Later a purely topological theory for 2-dimensional digital spaces was proposed by \textit{E. Khalimsky, R. Kopperman} and \textit{P. R. Meyer} [Topology Appl. 36, 1--17 (1990; Zbl 0709.54017)] and extended to dimension 3 by \textit{R. Kopperman, P. R. Meyer} and the reviewer in [Discrete Comput. Geom. 6, 155--161 (1991; Zbl 0738.68086)]. The purpose of this paper is to show that similar results can be obtained using \(n\)-ary relations on \({\mathbb Z}^2\) and the (non-idempotent) closure operators associated with them. Specifically after defining a certain \(n\)-ary relation \(v_n\), it is shown that if \(J\) is a Jordan curve in \(({\mathbb Z}^2,v_n)\), then \(J\) separates \(({\mathbb Z}^2,v_n)\) into precisely two components. Whether this theory has any advantages over more traditional approaches to image processing remains to be seen.
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 68U10
zbMATH DE Number: 2083762
zbMATH Keywords: digital topology
zbMATH Keywords: Khalimsky topology
zbMATH Keywords: Khalimsky plane
zbMATH Keywords: \(n\)-ary relation
zbMATH Keywords: Jordan curve
author: Josef Šlapal
reviewed by: Richard G. Wilson
MaRDI profile type: Publication
full work available at URL: j.dam.2002.11.003
OpenAlex ID: W2081819469
cites work: Q5554111
cites work: Computer graphics and connected topologies on finite ordered sets
cites work: Boundaries in digital planes
cites work: A Topological Approach to Digital Topology
cites work: A theory of binary digital pictures
cites work: A Jordan surface theorem for three-dimensional digital spaces
cites work: Connectivity in Digital Pictures
cites work: A converse to the Jordan curve theorem for digital curves
cites work: Digital Topology
cites work: Q4277391
cites work: A GALOIS CORRESPONDENCE BETWEEN CLOSURE SPACES AND RELATIONAL SYSTEMS
cites work: Closure operations for digital topology.
cites work: Semi-metrics, closure spaces and digital topology
DOI: 10.1016/J.DAM.2002.11.003
description: scientific article; zbMATH DE number 2083762