Boundaries in digital planes. (1990)

• Record Type: Journal Article
• Authors: Khalimsky, Efim; Kopperman, Ralph; Meyer, Paul R.
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 3:Number 1(1990)
• Page Start: 27
• Page End: 55
• Publication Date: 1990
• Subjects: Jordan curve -- linearly ordered topological space -- connected ordered topological space -- digital plane -- computer graphics -- finite topological spaces -- specialization order -- scene -- cartoon -- screen -- separator; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953390000041
• Languages: English
• ISSNs: 1048-9533

Abstract

Abstract : The importance of topological connectedness properties in processing digital pictures is well known. A natural way to begin a theory for this is to give a definition of connectedness for subsets of a digital plane which allows one to prove a Jordan curve theorem. The generally accepted approach to this has been a non-topological Jordan curve theorem which requires two different definitions, 4-connectedness, and 8-connectedness, one for the curve and the other for its complement. In [KKM] we introduced a purely topological context for a digital plane and proved a Jordan curve theorem. The present paper gives a topological proof of the non-topological Jordan curve theorem mentioned above and extends our previous work by considering some questions associated with image processing: How do more complicated curves separate the digital plane into connected sets? Conversely given a partition of the digital plane into connected sets, what are the boundaries like and how can we recover them? Our construction gives a unified answer to these questions. The crucial step in making our approach topological is to utilize a natural connected topology on a finite, totally ordered set; the topologies on the digital spaces are then just the associated product topologies. Furthermore, this permits us to define path, arc, and curve as certain continuous functions on such a parameter interval.