Advertisement

A Model for Digital Topology

  • Gilles Bertrand
  • Michel Couprie
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1568)

Abstract

In the framework known as digital topology, two different adjacency relations are used for structuring the discrete space Z n . In this paper, we propose a model for digital topology based on the notion of order and discrete topology. We “validate” our model by considering the two fundamental notions of surface and simple point. At last, we give the different possible configurations that may appear in 2- and 3- dimensional surfaces in Z 4 which correspond to our model.

Keywords

discrete topology digital topology surface simple point 
Download to read the full conference paper text

References

  1. 1.
    P. Alexandroff, “Diskrete Räume”, Mat. Sbornik, 2, pp. 501–518, 1937.zbMATHGoogle Scholar
  2. 2.
    R. Ayala, E. Domínguez, A.R. Francés, A. Quintero, J. Rubio, “On surfaces in digital topology”, 5th Colloquium Discrete Geometry for Comp. Imagery, pp. 271–276, 1995.Google Scholar
  3. 3.
    R. Ayala, E. Domínguez, A.R. Francés, A. Quintero, “Determining the components of the complement of a digital (n-1)-manifold in Z n, Discrete Geometry for Computer Imagery, Vol. 1176, Lect. Notes in Comp. Science, Springer Verlag, pp. 163–176, 1996.Google Scholar
  4. 4.
    R. Ayala, E. Domínguez, A.R. Francés, A. Quintero, “Digital lighting functions”, Discrete Geometry for Computer Imagery, Vol. 1347, Lect. Notes in Comp. Science, Springer Verlag pp. 139–150, 1997.Google Scholar
  5. 5.
    G. Bertrand, “Simple points, topological numbers and geodesic neighborhoods in cubic grids”, Pattern Rec. Letters, Vol. 15, pp. 1003–1011, 1994.CrossRefGoogle Scholar
  6. 6.
    G. Bertrand, “New notions for discrete topology”, 8th Conf. on Discrete Geom. For Comp. Imag., Vol. 1568, Lect. Notes in Comp. Science, Springer Verlag, pp. 216–226, 1999.Google Scholar
  7. 7.
    A.V. Evako, R. Kopperman, Y.V. Mukhin, “Dimensional Properties of Graphs and Digital Spaces”, Jour. of Math. Imaging and Vision, 6, pp. 109–119, 1996.CrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Françon, “Discrete combinatorial surfaces”, CVGIP: Graphical Models and Image Processing, Vol. 57, pp. 20–26, 1995.CrossRefGoogle Scholar
  9. 9.
    J. Françon, “On recent trends in discrete geometry in computer science”, Conf. On Discrete Geometry for Comp. Imag., pp. 3–16, 1996.Google Scholar
  10. 10.
    E. Khalimsky, R. Kopperman, P.R. Meyer, “Computer Graphics and Connected Topologies on Finite Ordered Sets”, Topology and its Applications, 36, pp. 1–17, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    T.Y. Kong, A.W. Roscoe, “Continuous Analogs of Axiomatized Digital Surfaces”, Computer Vision, Graphics, and Image Processing, 29, pp. 60–86, 1985.CrossRefGoogle Scholar
  12. 12.
    T.Y. Kong and A. Rosenfeld, “Digital Topology: introduction and survey”, Comp. Vision, Graphics and Image Proc., 48, pp. 357–393, 1989.Google Scholar
  13. 13.
    T.Y. Kong, “Topology-Preserving Deletion of 1’s from 2-, 3-and 4-Dimensional Binary Images”, Conf. on Discrete Geometry for Comp. Imag., Lect. Notes in Comp. Science, Vol. 1347, Springer Verlag pp. 3–18, 1997.Google Scholar
  14. 14.
    T.Y. Kong, R. Kopperman, P.R. Meyer, “A Topological Approach to Digital Topology”, American Mathematical Monthly, 38, pp. 901–917, 1991.CrossRefMathSciNetGoogle Scholar
  15. 15.
    R. Kopperman, P.R. Meyer and R.G. Wilson, “A Jordan surface theorem for three-dimensional digital spaces”, Discrete Comput. Geom., 6, pp. 155–161, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    V.A. Kovalevsky, “Finite Topology as Applied to Image Analysis”, Computer Vision, Graphics, and Image Processing, 46, pp. 141–161, 1989.CrossRefGoogle Scholar
  17. 17.
    V.A. Kovalevsky, “Topological foundations of shape analysis”, in Shape in Pictures, NATO ASI Series, Series F, Vol. 126, pp. 21–36, 1994.Google Scholar
  18. 18.
    R. Malgouyres, “A definition of surfaces of Z 3”, Conf. on Discrete Geometry for Comp. Imag., pp. 23–34, 1993. See also Doctoral dissertation, Université d’Auvergne, France, 1994.Google Scholar
  19. 19.
    D.G. Morgenthaler and A. Rosenfeld, “Surfaces in three-dimensional images”, Information and Control, 51, 227–247, 1981.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    G.M. Reed and A. Rosenfeld, “Recognition of surfaces in three-dimensional digital images”, Information and Control, Vol. 53, pp. 108–120, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    G.M. Reed, “On the characterization of simple closed surfaces in three-dimensional digital images”, Computer Vision, Graphics and Image Proc., Vol. 25, pp. 226–235, 1984.Google Scholar
  22. 22.
    A. Rosenfeld, “Digital Topology”, Amer. Math. Monthly, 86, pp. 621–630, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    J. Serra, Image Analysis and Mathematical Morphology, Academic Press, 1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Gilles Bertrand
    • 1
  • Michel Couprie
    • 1
  1. 1.Laboratoire A2SINoisy-Le-Grand CedexFrance

Personalised recommendations

Cite paper