Removal Operations in nD Generalized Maps for Efficient Homology Computation

  • Guillaume Damiand
  • Rocio Gonzalez-Diaz
  • Samuel Peltier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7309)


In this paper, we present an efficient way for computing homology generators of nD generalized maps. The algorithm proceeds in two steps: (1) cell removals reduces the number of cells while preserving homology; (2) homology generator computation is performed on the reduced object by reducing incidence matrices into their Smith-Agoston normal form. In this paper, we provide a definition of cells that can be removed while preserving homology. Some results on 2D and 3D homology generators computation are presented.


nD Generalized Maps Cellular Homology Homology Generators Removal Operations 


  1. 1.
  2. 2.
  3. 3.
    Agoston, M.K.: Algebraic Topology, a first course. In: Dekker, M. (ed.) Pure and Applied Mathematics (1976)Google Scholar
  4. 4.
    Alayrangues, S., Damiand, G., Lienhardt, P., Peltier, S.: A boundary operator for computing the homology of cellular structures. Discrete & Computational Geometry (under submission)Google Scholar
  5. 5.
    Alayrangues, S., Peltier, S., Damiand, G., Lienhardt, P.: Border Operator for Generalized Maps. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 300–312. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Damiand, G., Dexet-Guiard, M., Lienhardt, P., Andres, E.: Removal and contraction operations to define combinatorial pyramids: Application to the design of a spatial modeler. Image and Vision Computing 23(2), 259–269 (2005)CrossRefGoogle Scholar
  7. 7.
    Damiand, G., Lienhardt, P.: Removal and Contraction for n-Dimensional Generalized Maps. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 408–419. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Dumas, J.-G., Heckenbach, F., Saunders, B.D., Welker, V.: Computing simplicial homology based on efficient smith normal form algorithms. In: Algebra, Geometry, and Software Systems, pp. 177–206 (2003)Google Scholar
  9. 9.
    Gonzalez-Diaz, R., Ion, A., Iglesias-Ham, M., Kropatsch, W.G.: Invariant representative cocycles of cohomology generators using irregular graph pyramids. Computer Vision and Image Understanding 115(7), 1011–1022 (2011)CrossRefGoogle Scholar
  10. 10.
    Kaczynski, T., Mrozek, M., Slusarek, M.: Homology computation by reduction of chain complexes. Computers & Math. Appl. 34(4), 59–70 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lienhardt, P.: Topological models for boundary representation: a comparison with n-dimensional generalized maps. CAD 23(1), 59–82 (1991)zbMATHGoogle Scholar
  12. 12.
    Lienhardt, P.: N-dimensional generalized combinatorial maps and cellular quasi-manifolds. Computational Geometry & Applications 4(3), 275–324 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    MacLane, S.: Homology. Classic in Mathematics. Springer (1995)Google Scholar
  14. 14.
    Peltier, S., Ion, A., Kropatsch, W.g., Damiand, G., Haxhimusa, Y.: Directly computing the generators of image homology using graph pyramids. Image and Vision Computing 27(7), 846–853 (2009)CrossRefGoogle Scholar
  15. 15.
    Storjohann, A.: Near optimal algorithms for computing smith normal forms of integer matrices. In: Lakshman, Y.N. (ed.) Proceedings of the 1996 Int. Symp. on Symbolic and Algebraic Computation, pp. 267–274. ACM (1996)Google Scholar
  16. 16.
    Vidil, F., Damiand, G.: Moka (2003),

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Guillaume Damiand
    • 1
  • Rocio Gonzalez-Diaz
    • 2
  • Samuel Peltier
    • 3
  1. 1.CNRS, LIRIS, UMR5205Université de LyonFrance
  2. 2.Dpto. de Matemática Aplicada IUniversidad de SevillaSpain
  3. 3.CNRS, XLIM-SIC, UMR6172Université de PoitiersFrance

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