Computation of Homology Groups and Generators

  • Samuel Peltier
  • Sylvie Alayrangues
  • Laurent Fuchs
  • Jacques-Olivier Lachaud
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3429)

Abstract

Topological invariants are extremely useful in many applications related to digital imaging and geometric modelling, and homology is a classical one. We present an algorithm that computes the whole homology of an object of arbitrary dimension: Betti numbers, torsion coefficients and generators. Results on classical shapes in algebraic topology are presented and discussed.

Keywords

Simplicial Complex Chain Complex Homology Group Betti Number Incidence Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [ACZ04]
    Allili, M., Corriveau, D., Ziou, D.: Morse homology desriptor for shape characterization. In: Proc. ICPR 2004 (2004)Google Scholar
  2. [ADFQ03]
    Ayala, R., Dominguez, E., Francès, A.R., Quintero, A.: Homotopy in digital spaces. DAMATH: Discrete Appl. Math. and Combin. Oper. Research and Comput. Science 125 (2003)Google Scholar
  3. [Ago76]
    Agoston, M.K.: Algebraic Topology, a first course. Marcel Dekker, New York (1976)MATHGoogle Scholar
  4. [Box99]
    Boxer, L.: A classical construction for the digital fundamental group. J. Math. Imaging Vision 10, 51–62 (1999)MATHCrossRefMathSciNetGoogle Scholar
  5. [Cai61]
    Cairns, S.S.: Introductory Topology. Ronald Press Company (1961)Google Scholar
  6. [Cur71]
    Curtis, E.: Simplicial homotopy theory. Adv. Math. 6, 107–209 (1971)MATHCrossRefMathSciNetGoogle Scholar
  7. [DE95]
    Delfinado, C.J.A., Edelsbrunner, H.: An incremental algorithm for betti numbers of simplicial complexes on the 3-sphere. Comput. Aided Geom. Design 12(7), 771–784 (1995)MATHCrossRefMathSciNetGoogle Scholar
  8. [DG03]
    Desbarats, P., Gueorguieva, S.: Topological mainframe for numerical representations of objects. In: Kumar, V., Gavrilova, M.L., Tan, C.J.K., L’Ecuyer, P. (eds.) ICCSA 2003. LNCS, vol. 2668, pp. 498–507. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. [DSV01]
    Dumas, J.G., Saunders, B.D., Villard, G.: On efficient sparse integer matrix smith normal form computations. Journal of Symbolic Computation (2001)Google Scholar
  10. [GDR03]
    Gonzalez–Diaz, R., Real, P.: Towards digital cohomology. In: Sanniti di Baja, G., Svensson, S., Nyström, I. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 92–101. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. [Gie95]
    Giesbrecht, M.: Nearly optimal-algorithms for canonical matrix-forms. SIAM J. COMPUT. 24(5), 948–969 (1995)MATHCrossRefMathSciNetGoogle Scholar
  12. [Hat02]
    Hatcher, A.: Algebraic Topology. Cambridge University Press (2002), disponible sur, http://www.math.cornell.edu/~hatcher/AT/ATpage.html
  13. [KMS98]
    Kaczynski, T., Mrozek, M., Slusarek, M.: Homology computation by reduction of chain complexes. Computers & Math. Appl. 34(4), 59–70 (1998)CrossRefMathSciNetGoogle Scholar
  14. [Kon89]
    Kong, T.Y.: A digital fundamental group. Computers and Graphics 13(2), 159–166 (1989)CrossRefGoogle Scholar
  15. [KR89]
    Kong, T.Y., Rosenfeld, A.: Digital Topology: Introduction and Survey. Comput. Vision, Graphics, and Image Processing 48(3), 357–393 (1989)CrossRefGoogle Scholar
  16. [LE93]
    Lienhardt, P., Elter, H.: Different combinatorial models based on the map concept for the representation of different types of cellular complexes. In: Proceedings of IFIP TC 5/WG II Work. Conf. on Geom. Modeling in Comp.Graphics, Springer, Heidelberg (1993)Google Scholar
  17. [LL95]
    Lang, V., Lienhardt, P.: Geometric modeling with simplicial sets. In: Proc. of Pacific Graphics 1995, Seoul, Korea (1995)Google Scholar
  18. [Mal01]
    Malgouyres, R.: Computing the fundamental group in digital spaces. IJPRAI 15(7), 1075–1088 (2001)Google Scholar
  19. [May67]
    May, J.P.: Simplicial Objects in Algebraic Topology. Van Nostrand (1967)Google Scholar
  20. [Mun84]
    Munkres, J.R.: Elements of algebraic topology. Perseus Books (1984)Google Scholar
  21. [Sto96]
    Storjohann, A.: Near optimal algorithms for computing Smith normal forms of integer matrices. In: Lakshman, Y.N. (ed.) Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, pp. 267–274. ACM Press, New York (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Samuel Peltier
    • 1
  • Sylvie Alayrangues
    • 2
  • Laurent Fuchs
    • 1
  • Jacques-Olivier Lachaud
    • 2
  1. 1.SIC (FRE 2731 CNRS)Université de PoitiersFuturoscope ChasseneuilFrance
  2. 2.LaBRIUniversité Bordeaux 1TalenceFrance

Personalised recommendations