Associating Cell Complexes to Four Dimensional Digital Objects

  • Ana Pacheco
  • Pedro Real
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6607)


The goal of this paper is to construct an algebraic-topological representation of a 80–adjacent doxel-based 4–dimensional digital object. Such that representation consists on associating a cell complex homologically equivalent to the digital object. To determine the pieces of this cell complex, algorithms based on weighted complete graphs and integral operators are shown. We work with integer coefficients, in order to compute the integer homology of the digital object.


digital object integer homology integral operator weighted complete graph 


  1. 1.
    Diestel, R.: Graph Theory. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  2. 2.
    Fristch, R., Piccinini, R.A.: Cellular structure in topology. Cambridge University Press, Cambridge (1990)Google Scholar
  3. 3.
    Gonzalez-Diaz, R., Jimenez, M.J., Medrano, B., Molina-Abril, H., Real, P.: Integral Operators for Computing Homology Generators at Any Dimension. In: Ruiz-Shulcloper, J., Kropatsch, W.G. (eds.) CIARP 2008. LNCS, vol. 5197, pp. 356–363. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Herman, G.T.: Discrete multidimensional Jordan surfaces. CVGIP: Graphical Models and Image Processing 54, 507–515 (1992)Google Scholar
  5. 5.
    Kenmochi, Y., Imiya, A., Ichikawa, A.: Boundary Extraction of Discrete Objects. Computer Vision and Image Understanding 71, 281–293 (1998)CrossRefGoogle Scholar
  6. 6.
    Kalvin, A.D., Cutting, C.B., Haddad, B., Noz, M.E.: Constructing topologically connected surfaces for the comprehensive analysis of 3D medical structures. In: Proc. of SPIE, vol. 1445, pp. 247–258 (1991)Google Scholar
  7. 7.
    Lorensen, W.E., Cline, H.E.: Marching cubes: A high-resolution 3D surface construction algorithm. Computer Graphics 21, 163–169 (1988)CrossRefGoogle Scholar
  8. 8.
    Pacheco, A., Real, P.: Getting topological information for a 80-adjacency doxel-based 4D volume through a polytopal cell complex. In: Bayro-Corrochano, E., Eklundh, J.-O. (eds.) CIARP 2009. LNCS, vol. 5856, pp. 279–286. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Pacheco, A., Real, P.: Polyhedrization, homology and orientation. Progress in Combinatorial Image Analysis, 151–164 (2009)Google Scholar
  10. 10.
    Udupa, J.K.: Multidimensional digital boundaries. CVGIP: Graphical Models and Image Processing 56, 311–323 (1994)Google Scholar
  11. 11.
    Voss, K.: Discrete Images, Objects, and Functions in Zn. Algorithms and Combinatorics (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ana Pacheco
    • 1
  • Pedro Real
    • 1
  1. 1.Dpto. Matematica Aplicada I, E.T.S.I. InformaticaUniversidad de SevillaSevillaSpain

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