Border Operator for Generalized Maps

  • Sylvie Alayrangues
  • Samuel Peltier
  • Guillaume Damiand
  • Pascal Lienhardt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5810)


In this paper, we define a border operator for generalized maps, a data structure for representing cellular quasi-manifolds. The interest of this work lies in the optimization of homology computation, by using a model with less cells than models in which cells are regular ones as tetrahedra and cubes. For instance, generalized maps have been used for representing segmented images. We first define a face operator to retrieve the faces of any cell, then deduce the border operator and prove that it satisfies the required property : border of border is void. At last, we study the links between the cellular homology defined from our border operator and the classical simplicial homology.


Homology Group Edge Incident Klein Bottle Face Operator Compact Cell 
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  1. 1.
    Alayrangues, S., Daragon, X., Lachaud, J.-O., Lienhardt, P.: Equivalence between closed connected n-g-maps without multi-incidence and n-surfaces. J. Math. Imaging Vis. 32(1), 1–22 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andres, E., Breton, R., Lienhardt, P.: Spamod: design of a spatial modeler. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 90–107. Springer, Heidelberg (2002)Google Scholar
  3. 3.
    Bertrand, G.: New notions for discrete topology. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 218–228. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  4. 4.
    Brisson, E.: Representing geometric structures in d dimensions: Topology and order. Discrete & Computational Geometry 9, 387–426 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    Dumas, J.-G., Heckenbach, F., Saunders, B.D.: Computing simplicial homology based on efficient smith normal form algorithms. In: Algebra, Geometry, and Software Systems, pp. 177–206 (2003)Google Scholar
  7. 7.
    Dupas, A., Damiand, G.: First results for 3D image segmentation with topological map. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds.) DGCI 2008. LNCS, vol. 4992, pp. 507–518. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Elter, H., Lienhardt, P.: Cellular complexes as structured semi-simplicial sets. International Journal of Shape Modeling 1(2), 191–217 (1995)CrossRefzbMATHGoogle Scholar
  9. 9.
    Giesbrecht, M.: Probabilistic computation of the smith normal form of a sparse integer matrix. In: Cohen, H. (ed.) ANTS 1996. LNCS, vol. 1122, pp. 173–186. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  10. 10.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  11. 11.
    Hu, S.T.: On the realizability of homotopy groups and their operations. Pacific J. of Math. 1(583-602) (1951)Google Scholar
  12. 12.
    Lienhardt, P.: Topological models for boundary representation: a comparison with n-dimensional generalized maps. Comput. Aided Design 23(1), 59–82 (1991)CrossRefzbMATHGoogle Scholar
  13. 13.
    Lienhardt, P.: N-dimensional generalized combinatorial maps and cellular quasi-manifolds. Int. J. on Comput. Geom. & App. 4(3), 275–324 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    May, J.P.: Simplicial Objects in Algebraic Topology. Van Nostrand (1967)Google Scholar
  15. 15.
    Munkres, J.R.: Elements of algebraic topology. Perseus Books, Cambridge (1984)zbMATHGoogle Scholar
  16. 16.
    Peltier, S., Alayrangues, S., Fuchs, L., Lachaud, J.-O.: Computation of homology groups and generators. Comput. & Graph. 30, 62–69 (2006)CrossRefzbMATHGoogle Scholar
  17. 17.
    Peltier, S., Fuchs, L., Lienhardt, P.: Simploidals sets: Definitions, operations and comparison with simplicial sets. Discrete App. Math. 157, 542–557 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rossignac, J., O’Connor, M.: A dimension-independant model for pointsets with internal structures and incomplete boundaries. Geometric modeling for Product Engineering, 145–180 (1989)Google Scholar
  19. 19.
    Sergeraert, F.: Constructive algebraic topology. SIGSAM Bulletin 33(3), 13–13 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Sylvie Alayrangues
    • 1
  • Samuel Peltier
    • 1
  • Guillaume Damiand
    • 2
  • Pascal Lienhardt
    • 1
  1. 1.XLIM-SICUniversité de Poitiers, CNRS, Bât. SP2MI, Téléport 2, Bvd Marie et Pierre CurieFuturoscope Chasseneuil CedexFrance
  2. 2.CNRS, LIRIS, UMR5205Université de LyonFrance

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