Digital n-Pseudomanifold and n-Weakmanifold in a Binary (n + 1)-Digital Image

  • Mohammed Khachan
  • Patrick Chenin
  • Hafsa Deddi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1953)

Abstract

We introduce the notion of digital n-pseudomanifold and digital n-weakmanifold in (n+1)-digital image, in the context of (2n, 3 n -1)- adjacency, and prove the digital version of the Jordan-Brouwer separation theorem for those classes. To accomplish this objective, we construct a polyhedral representation of the n-digital image, based on cubical complex decomposition. This enables us to translate some results from polyhedral topology into the digital space. Our main result extends the class of “thin” objects that are defined locally and verifies the Jordan-Brouwer separation theorem.

Keywords

digital topology combinatorial topology discrete spaces combinatorial manifolds 

References

  1. 1.
    Agoston, M. K.: Algebraic topology. Marcel Dekker, New York, 1976. 38, 39MATHGoogle Scholar
  2. 2.
    Aleksandrov, P. S.: Combinatorial topology. Vol 2. Rochester, New York, 1957.Google Scholar
  3. 3.
    Aleksandrov, P. S.: Combinatorial topology. Vol 3. Rochester, New York, 1957. 39Google Scholar
  4. 4.
    Ayala, R., E. Dominguez, A. R. Francés, A. Quintero: Determining the components of the complement of a digital (n-1)-manifold in ℤn. 6th International Workshop on Discrete Geometry for Computer Imagery, DGCI’96, Lectures Notes in Computer Science.1176 (1996), 163–176. 38Google Scholar
  5. 5.
    Ayala, R., E. Dominguez, A. R. Francés, A. Quintero: A Digital Lighting Function for strong 26-Surfaces. Discrete Geometry for Computer Imagery, DGCI’96, Lectures Notes in Computer Science, Springer Verlag, 1568 (1999), 91–103. 38CrossRefGoogle Scholar
  6. 6.
    Bertrand, G., R. Malgouyres: Local Property of Strong Surfaces. SPIE Vision Geometry, Vol 3168, (1997), 318–327. 38Google Scholar
  7. 7.
    Couprie, M., G. Bertrand: Simplicity surfaces: a new definition of surfaces in ℤ3 SPIE Vision Geometry, Vol 3454, (1998), 40–51. 38Google Scholar
  8. 8.
    Hudson, J. F. P.: Piecewise linear topology. W.A Benjamin, New York, 1969. 38, 39Google Scholar
  9. 9.
    Kenmochi, Y., A. Imiya, N. Ezquerra: Polyhedra generation from lattice points. 6th International Workshop on Discrete Geometry for Computer Imagery, DGCI’96,Lectures Notes in Computer Science.1176 (1996), 127–138.Google Scholar
  10. 10.
    Khachan, M.,P. Chenin, H. Deddi: Polyhedral representation and adjacency graph in n-digital image, CVIU, Computer Vision and Image Understanding (accepted, to appear). 41Google Scholar
  11. 11.
    Kong, T. Y., A.W. Roscoe: Continuous analogs of axiomatized digital surfaces. CVGIP 29 (Computer Vision, Graphics, and Image Processing) (1985), 60–86. 37Google Scholar
  12. 12.
    Kong, T. Y., A. W. Roscoe: A Theory of Binary Digital Pictures. CVGIP 32 (1985), 221–243.Google Scholar
  13. 13.
    Kong, T. Y., A. Rosenfeld: Survey Digital Topology: Introduction and Survey. CVGIP 48 (1989), 357–393. 40Google Scholar
  14. 14.
    Kong, T. Y., A. W. Roscoe, A. Rosenfeld: Concepts of digital topology. Topology and its Application 46 (1992), 219–262. 38, 40MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Malgouyres, R.: About surfaces in ℤ3. Fifth International Workshop on Discrete Geometry for Computer Imagery, DGCI’95, pp.243–248, 1995. 38Google Scholar
  16. 16.
    Morgenthaler, D. G.,A. Rosenfeld: Surfaces in three-dimensional digital images. Information and Control 51 (1981), 227–247. 37MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Rosenfeld, A., T. Y. Kong, A.Y Wu: Digital surfaces. CVGIP 53, 1991, 305–312 40MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Mohammed Khachan
    • 1
  • Patrick Chenin
    • 2
  • Hafsa Deddi
    • 3
  1. 1.Department of Computer Science (IRCOM-SIC)University of PoitiersFuturoscope cedexFrance
  2. 2.LMC-IMAGUniversity Joseph FourierGrenoble cedex 9France
  3. 3.LACIMUniversity of Quebec at MontrealMontrealCanada

Personalised recommendations