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Axiomatic Digital Topology

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Abstract

The paper presents a new set of axioms of digital topology, which are easily understandable for application developers. They define a class of locally finite (LF) topological spaces. An important property of LF spaces satisfying the axioms is that the neighborhood relation is antisymmetric and transitive. Therefore any connected and non-trivial LF space is isomorphic to an abstract cell complex. The paper demonstrates that in an n-dimensional digital space only those of the (a, b)-adjacencies commonly used in computer imagery have analogs among the LF spaces, in which a and b are different and one of the adjacencies is the “maximal” one, corresponding to 3n− 1 neighbors. Even these (a, b)-adjacencies have important limitations and drawbacks. The most important one is that they are applicable only to binary images. The way of easily using LF spaces in computer imagery on standard orthogonal grids containing only pixels or voxels and no cells of lower dimensions is suggested.

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Authors and Affiliations

  1. Department of Computer Science, University of Applied Sciences Berlin, Luxemburger Str. 10, 13353, Berlin, Germany

    Vladimir Kovalevsky

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  1. Vladimir Kovalevsky
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Correspondence to Vladimir Kovalevsky.

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Vladimir Kovalevsky received his diploma in physics from the Kharkov University (Ukraine) in 1950, the first doctor degree in technical science from the Central Institute of Metrology (Leningrad) in 1957, the second doctor degree in computer science from the Institute of Cybernetics (Kiev) in 1968. Since 1961 he has been the head of Department of Pattern Recognition at that Institute. Lives since 1980 in Germany. Currently he is professor of computer science at the University of Applied sciences Berlin. His research interest include digital geometry, digital topology, computer vision, image processing and pattern recognition.

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Kovalevsky, V. Axiomatic Digital Topology. J Math Imaging Vis 26, 41–58 (2006). https://doi.org/10.1007/s10851-006-7453-6

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  • DOI: https://doi.org/10.1007/s10851-006-7453-6

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