Advertisement

Integral Operators for Computing Homology Generators at Any Dimension

  • Rocio Gonzalez-Diaz
  • Maria Jose Jimenez
  • Belen Medrano
  • Helena Molina-Abril
  • Pedro Real
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5197)

Abstract

Starting from an nD geometrical object, a cellular subdivision of such an object provides an algebraic counterpart from which homology information can be computed. In this paper, we develop a process to drastically reduce the amount of data that represent the original object, with the purpose of a subsequent homology computation. The technique applied is based on the construction of a sequence of elementary chain homotopies (integral operators) which algebraically connect the initial object with a simplified one with the same homological information than the former.

Keywords

integer homology generators chain homotopies 

References

  1. 1.
    Björner, A.: Topological Methods. Handbook on Combinatorics 2, 1819–1872 (1995)zbMATHGoogle Scholar
  2. 2.
    Dahmen, W., Micchelli, C.A.: On the Linear Independence of Multivariate b-Splines. Triangulation of Simploids. SIAM J. Numer. Anal. 19 (1982)Google Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dumas, J.G., Saunders, B.D., Villad, G.: On Efficient Sparse Integer Matrix SNF Computations. J. Symb. Comput. 32(1), 71–99 (2001)CrossRefGoogle Scholar
  5. 5.
    Gonzalez-Diaz, R., Real, P.: Computation of Cohomology Operations on Finite Simplicial Complexes. HHA 5(2), 83–93 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gonzalez-Diaz, R., Real, P.: On the Cohomology of 3D Digital Images. Discrete Appl. Math. 147(2-3), 245–263 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gonzalez-Diaz, R., Medrano, B., Real, P., Sanchez-Pelaez, J.: Reusing Integer Homology Information of Digital Images. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 199–210. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Gonzalez-Diaz, R., Jimenez, M.J., Medrano, B., Real, P.: Chain Homotopies for Object Topological Representations. Discrete Appl. Math. (to appear)Google Scholar
  9. 9.
    Gonzalez-Diaz, R., Iglesias-Ham, M., Ion, A., Kropatsch, W.G.: Irregular Graph Pyramids, Integral Operators and Representative Cocycles of Cohomology Generators. In: Workshop on Computational Topology in Image Context, Poitiers, France (2008)Google Scholar
  10. 10.
    Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. In: AMS, Providence, Rhode Island, vol. 157 (2004)Google Scholar
  11. 11.
    Massey, W.M.: A Basic Course in Algebraic Topology. Graduate Texts in Math., vol. 127. Springer, New York (1991)zbMATHGoogle Scholar
  12. 12.
    MacLane, S.: Homology. Classics in Mathematics. Springer, Berlin (1995)zbMATHGoogle Scholar
  13. 13.
    Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Canada (1993)zbMATHGoogle Scholar
  14. 14.
    Peltier, S., Alayrangues, S., Fuchs, L., Lachaud, J.: Computation of Homology Groups and Generators. In: Andrès, É., Damiand, G., Lienhardt, P. (eds.) DGCI 2005. LNCS, vol. 3429, pp. 195–205. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Peltier, S., Ion, A., Haxhimusa, Y., Kropatsch, W.G., Damiand, G.: Computing Homology Generators Using Irregular Graph Pyramids. In: Escolano, F., Vento, M. (eds.) GbRPR 2007. LNCS, vol. 4538, pp. 283–294. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Rocio Gonzalez-Diaz
    • 1
  • Maria Jose Jimenez
    • 1
  • Belen Medrano
    • 1
  • Helena Molina-Abril
    • 1
  • Pedro Real
    • 1
  1. 1.Applied Math DepartmentUniversity of SevilleSpain

Personalised recommendations