Towards Digital Cohomology

  • Rocio Gonzalez–Diaz
  • Pedro Real
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2886)


We propose a method for computing the Z 2–cohomology ring of a simplicial complex uniquely associated with a three–dimensional digital binary–valued picture I. Binary digital pictures are represented on the standard grid Z 3, in which all grid points have integer coordinates. Considering a particular 14–neighbourhood system on this grid, we construct a unique simplicial complex K(I) topologically representing (up to isomorphisms of pictures) the picture I. We then compute the cohomology ring on I via the simplicial complex K(I). The usefulness of a simplicial description of the digital Z 2–cohomology ring of binary digital pictures is tested by means of a small program visualizing the different steps of our method. Some examples concerning topological thinning, the visualization of representative generators of cohomology classes and the computation of the cup product on the cohomology of simple 3D digital pictures are showed.


Digital topology chain complexes cohomology ring 


  1. 1.
    Ayala, R., Domínguez, E., Francés, A.R., Quintero, A.: Homotopy in Digital Spaces. In: Nyström, I., Sanniti di Baja, G., Borgefors, G. (eds.) DGCI 2000. LNCS, vol. 1953, pp. 3–14. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  2. 2.
    Berrio, J.M., González–Díaz, R., Leal, F., López, M.M., Real, P.: Visualizing Cohomology Aspects of 3D Objects. In: Proc. of the 6th Asian Tech. Conf. in Math, pp. 459–468 (2001)Google Scholar
  3. 3.
    Björner, A.: Topological Methods. Handbook on Combinatorics 2, 1819–1872 (1995)Google Scholar
  4. 4.
    Delfinado, C.J.A., Edelsbrunner, H.: An Incremental Algorithm for Betti Numbers of Simplicial Complexes on the 3–Sphere. Comput. Aided Geom. Design 12, 771–784 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Forman, R.: Combinatorial Differential Topology and Geometry. New Perspective in Geom. Combinatorics. MSRI Public. 8, 177–206 (1999)MathSciNetGoogle Scholar
  6. 6.
    González–Díaz, R., Real, P.: Computation of Cohomology Operations on Finite Simplicial Complexes. Homology, Homotopy and Applications 5 (2), 83–93 (2003)zbMATHMathSciNetGoogle Scholar
  7. 7.
    González–Díaz, R., Real, P.: Geometric Objects and Cohomology Operations. In: Proc. of the 5th Workshop on Computer Algebra in Scientific Computing, pp. 121–130 (2002)Google Scholar
  8. 8.
    Kenmochi, Y., Imiya, A.: Discrete Polyhedrization of Lattice Point Set. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 150–162. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Khalimsky, E.D., Kopperman, R.D., Meyer, P.R.: Computer Graphics and Connected Topologies on Finite Ordered Sets. Topology and Appl. 36, 1–17 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kong, T.Y.: A digital Fundamental Group. Comput. Graphics 13, 159–166 (1989)CrossRefGoogle Scholar
  11. 11.
    Kong, T.Y., Roscoe, A.W., Rosenfeld, A.: Concepts of Digital Topology. Topology and its Applications 8, 219–262 (1992)CrossRefMathSciNetGoogle Scholar
  12. 12.
    MacLane, S.: Homology. Classic in Math. Springer, Heidelberg (1995)Google Scholar
  13. 13.
    Kovalevsky, V.A.: Discrete Topology and Contour Definition. Pattern Recognition Letter 2, 281–288 (1984)CrossRefGoogle Scholar
  14. 14.
    Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley Co., Reading (1984)zbMATHGoogle Scholar
  15. 15.
    Rosenfeld, A.: 3D Digital Topology. Inform. and Control 50, 119–127 (1981)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Rocio Gonzalez–Diaz
    • 1
  • Pedro Real
    • 1
  1. 1.Applied Math Dept.University of SevilleSpain

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