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Boundaries in digital planes

Author

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  • Efim Khalimsky
  • Ralph Kopperman
  • Paul R. Meyer

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.

Suggested Citation

  • Efim Khalimsky & Ralph Kopperman & Paul R. Meyer, 1990. "Boundaries in digital planes," International Journal of Stochastic Analysis, Hindawi, vol. 3, pages 1-29, January.
  • Handle: RePEc:hin:jnijsa:454956
    DOI: 10.1155/S1048953390000041
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    Cited by:

    1. Sang-Eon Han, 2020. "The Most Refined Axiom for a Digital Covering Space and Its Utilities," Mathematics, MDPI, vol. 8(11), pages 1-21, October.

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