Digital topologies revisited: An approach based on the topological point-neighbourhood

  • Pavel Ptak
  • Helmut Kofler
  • Walter Kropatsch
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1347)


Adopting the point-neighbourhood definition of topology, which we think may in some cases help acquire a very good insight of digital topologies, we unify the proof technique of the results on 4-connectedness and on 8-connectedness in ℤ2. We also show that there is no topology compatible with 6-connectedness. We shortly comment on potential further use of this approach.

Key words

Image processing digital topology adjacency path-connectedness and topological connectedness in ℤ2 


  1. 1.
    E. Cech: Topological Spaces, Interscience, Wiley, New York, 1966.Google Scholar
  2. 2.
    J. M. Chassery: Connectivity and consecutivity in digital pictures, Computer Graphics and Image Processing 9, 294–300, 1979.Google Scholar
  3. 3.
    E. Khalimsky, R. Kopperman and P. R. Meyer: Computer graphics and connected topologies on finite ordered sets, Topology App1.36, 117, 1980.Google Scholar
  4. 4.
    V. A. Kovalevsky: Finite topology as applied to image analysis, Computer Vision, Graphics and Image Processing 46, 141–161, 1989.Google Scholar
  5. 5.
    W. G. Kropatsch: Equivalent Contraction Kernels and the Domain of Dual Irregular Pyramids, Technical Report PRIP-TR-042, TU-Vienna, 1996.Google Scholar
  6. 6.
    L. Latecki:Digitale und Allgemeine Topologie in der bildhaften Wissensreprdsentation, Ph.D.-Thesis, Hamburg, 1992.Google Scholar
  7. 7.
    L. Latecki: Topological connectedness and 8-connectedness in digital pictures, Computer Vision, Graphics and Image Processing: Image Understanding 57, 261–262, 1993.Google Scholar
  8. 8.
    A. Rosenfeld: Digital topology, Am. Math. Monthly 86, 621–630, 1979.Google Scholar
  9. 9.
    F. Wyse and D. Marcus et al.: Solution to Problem 5712, Am. Math. Monthly 77, 1119, 1970.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Pavel Ptak
    • 1
  • Helmut Kofler
    • 2
  • Walter Kropatsch
    • 2
  1. 1.Faculty of Electrical EngineeringCenter for Machine Perception CMP — Czech Technical UniversityPraha 2Czech Republic
  2. 2.Pattern Recognition and Image Proc. Group — PRIPVienna University of TechnologyViennaAustria

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