Discrete & Computational Geometry

, Volume 54, Issue 1, pp 42–77 | Cite as

Homology of Cellular Structures Allowing Multi-incidence

  • Sylvie Alayrangues
  • Guillaume Damiand
  • Pascal Lienhardt
  • Samuel Peltier


This paper focuses on homology computation over ‘cellular’ structures that allow multi-incidence between cells. We deal here with combinatorial maps, more precisely chains of maps and subclasses such as maps and generalized maps. Homology computation on such structures is usually achieved by computing simplicial homology on a simplicial analog. But such an approach is computationally expensive because it requires computing this simplicial analog and performing the homology computation on a structure containing many more cells (simplices) than the initial one. Our work aims at providing a way to compute homologies directly on a cellular structure. This is done through the computation of incidence numbers. Roughly speaking, if two cells are incident, then their incidence number characterizes how they are attached. Having these numbers naturally leads to the definition of a boundary operator, which induces a homology. Hence, we propose a boundary operator for chains of maps and provide optimization for maps and generalized maps. It is proved that, under specific conditions, the homology of a combinatorial map as defined in the paper is equivalent to the homology of its simplicial analogue.


Homology computation Boundary operator Combinatorial maps 

Mathematics Subject Classification




The authors wish to thank the reviewers for their many useful remarks and suggestions and in particular Pol Vanhaecke and Francis Sergeraert for many informative discussions.


  1. 1.
    Agoston, M.K.: Algebraic Topology: A First Course. Pure and Applied Mathematics. Marcel Dekker Ed., New York (1976)Google Scholar
  2. 2.
    Alayrangues, S., Damiand, G., Lienhardt, P., Peltier, S.: A boundary operator for computing the homology of cellular structures. Research Report 2012–1, XLIM-Sic Laboratory, University of Poitiers, France. (2011). Accessed 12 Mar 2014
  3. 3.
    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)Google Scholar
  4. 4.
    Alayrangues, S., Lienhardt, P., Peltier, S.: Conversion between chains of maps and chains of surfaces; application to the computation of incidence graphs homology. Research Report, Université de Poitiers. (2015). Accessed 12 Mar 2014
  5. 5.
    Alayrangues, S., Peltier, S., Damiand, G., Lienhardt, P.: Border operator for generalized maps. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 5810, pp. 300–312. Springer, Berlin/Heidelberg (2009)Google Scholar
  6. 6.
    Basak, T.: Combinatorial cell complexes and poincaré duality. Geom. Dedicata 147, 357–387 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Baumgart, B.: A polyhedron representation for computer vision. Proc. AFIPS Natl. Conf. 44, 589–596 (1975)Google Scholar
  8. 8.
    Bellet, T., Poudret, M., Arnould, A., Fuchs, L., Le Gall, P.: Designing a topological modeler kernel: a rule-based approach. In: Shape Modeling International (SMI’10), Aix-en-Provence, France, 2010Google Scholar
  9. 9.
    Bertrand, Y., Damiand, G., Fiorio, C.: Topological encoding of 3D segmented images. In: Borgefors, G., et al. (eds.) Proceedings of 9th Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 1953, pp. 311–324. Springer, Berlin (2000)Google Scholar
  10. 10.
    Brandel, S., Schneider, S., Perrin, M., Guiard, N., Rainaud, J.F., Lienhardt, P., Bertrand, Y.: Automatic building of structured geological models. J. Comput. Inf. Sci. Eng. 5(2), 138–148 (2005)CrossRefGoogle Scholar
  11. 11.
    Braquelaire, A., Damiand, G., Domenger, J.-P., Vidil, F.: Comparison and convergence of two topological models for 3D image segmentation. Proceedings of 4th IAPR-TC15 Workshop on Graph-Based Representations in Pattern Recognition. Lecture Notes in Computer Science, vol. 2726, pp. 59–70. Springer, New York (2003)Google Scholar
  12. 12.
    Braquelaire, J.-P., Guitton, P.: A model for image structuration. In: Proceedings of the Computer Graphics International’88, Genève, Switzerland, May 1988Google Scholar
  13. 13.
    Brisson, E.: Representing geometric structures in \(d\) dimensions: topology and order. Discrete Comput. Geom. 9(1), 387–426 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Cardoze, D., Miller, G., Phillips, T.: Representing topological structures using cell-chains. In: Kim, M.-S., Shimada, K. (eds.) Geometric Modeling and Processing—GMP 2006. Lecture Notes in Computer Science, vol. 4077, pp. 248–266. Springer, Berlin/Heidelberg (2006)Google Scholar
  15. 15.
    Cavalcanti, P.R., Carvalho, P.C.P., Martha, L.F.: Non-manifold modeling: an approach based on spatial subdivisions. Comput. Aided Des. 29(3), 299–320 (1997)CrossRefGoogle Scholar
  16. 16.
    Choi, Y., Gursoz, E.L., Prinz, F.B.: Vertex-based representation of non-manifolds boundaries. In: Turner, J., Wozny, M., Preiss, K. (eds.) Geometric Modeling for Product Engineering, pp. 107–130. North-Holland, Amsterdam (1990)Google Scholar
  17. 17.
    Colin de Verdière, E., Lazarus, F.: Optimal pants decompositions and shortest homotopic cycles on an orientable surface. J. ACM 54, 18 (2007)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Damiand, G., Bertrand, Y., Fiorio, C.: Topological model for two-dimensional image representation: definition and optimal extraction algorithm. Comput. Vis. Image Underst. 93(2), 111–154 (2004)CrossRefGoogle Scholar
  19. 19.
    Damiand, G., Lienhardt, P.: Combinatorial Maps: Efficient Data Structures for Computer Graphics and Image Processing. A K Peters/CRC Press, Boca Raton (2014)CrossRefGoogle Scholar
  20. 20.
    Damiand, G., Peltier, S., Fuchs, L.: Computing homology generators for volumes using minimal generalized maps. In: Brimkov, V.E., et al. (eds.) International Workshop on Combinatorial Image Analysis. Lecture Notes in Computer Science, vol. 4958, pp. 63–74. Springer, Berlin (2008)Google Scholar
  21. 21.
    Delfinado, C.J.A., Edelsbrunner, H.: An incremental algorithm for betti numbers of simplicial complexes on the 3-sphere. Comput. Aided Geom. Des. 12(7), 771–784 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Dlotko, P., Kaczynski, Tomas, Mrozek, M., Wanner, T.: Coreduction homology algorithm for regular CW-complexes. Discrete Comput. Geom. 46, 361–388 (2010)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Dobkin, D., Laszlo, M.: Primitives for the manipulation of three-dimensional subdivisions. Algorithmica 5(4), 3–32 (1989)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Dumas, J.G., Saunders, B.D., Villard, G.: On efficient sparse integer matrix Smith normal form computations. J. Symb. Comput. 32, 71–99 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Dupas, A., Damiand, G.: First results for 3D image segmentation with topological map. In: Coeurjolly, D., et al. (eds.) Proceedings of 14th International Conference on Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 4992, pp. 507–518. Springer, Berlin (2008)Google Scholar
  26. 26.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Edmonds, J.: A combinatorial representation for polyhedral surfaces. Not. Am. Math. Soc. 7, A646 (1960)Google Scholar
  28. 28.
    Elter, H., Lienhardt, P.: Cellular complexes as structured semi-simplicial sets. Int. J. Shape Model. 1(2), 191–217 (1994)CrossRefzbMATHGoogle Scholar
  29. 29.
    Fritsch, R., Piccinini, R.A.: Cellular Structures in Topology. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  30. 30.
    Giesbrecht, M.: Probabilistic computation of the Smith normal form of a sparse integer matrix. In: Cohen, H. (ed.) Proceedings of the Second International Symposium on Algorithmic Number Theory. Lecture Notes in Computer Science, vol. 1122, pp. 173–186. Springer, Berlin (1996)Google Scholar
  31. 31.
    González-Díaz, R., Jiménez, M.J., Medrano, B., Real, P.: Chain homotopies for object topological representations. Discrete Appl. Math. 157(3), 490–499 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Guibas, L., Stolfi, G.: Primitives for the manipulation of general subdivisions and the computation of voronoi diagrams. Trans. Graph. 4(2), 74–123 (1985)CrossRefzbMATHGoogle Scholar
  33. 33.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  34. 34.
    Hu, S.T.: On the realizability of homotopy groups and their operations. Pac. J. Math. 1, 583–602 (1951)CrossRefzbMATHGoogle Scholar
  35. 35.
    Jacque, A.: Constellations et graphes topologiques. Colloq. Math. Soc. Janos Bolyai 2, 657–672 (1970)Google Scholar
  36. 36.
    Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. Springer, New York (2004)CrossRefzbMATHGoogle Scholar
  37. 37.
    Kaczynski, T., Mrozek, M., Slusarek, M.: Homology computation by reduction of chain complexes. Comput. Math. Appl. 34(4), 59–70 (1998)CrossRefMathSciNetGoogle Scholar
  38. 38.
    Kannan, R., Bachem, A.: Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput. 8(4), 499–507 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Lang V., Lienhardt P.: Simplicial sets and triangular patches. In: Proceedings of CGI’96, Pohang, Korea (1996)Google Scholar
  40. 40.
    Lee, C.N., Poston, T., Rosenfeld, A.: Holes and genus of 2D and 3D digital images. CVGIP: Graph. Model. Image Process. 55(1), 20–47 (1993)Google Scholar
  41. 41.
    Lee, S.H., Lee, K.: Partial entity structure: a fast and compact non-manifold boundary representation based on partial topological entities. In: 6th ACM Symposium on Solid Modeling and Applications, Ann Arbor, USA (2001)Google Scholar
  42. 42.
    Lienhardt, P.: Topological models for boundary representation: a comparison with \(n\)-dimensional generalized maps. Comput. Aided Des. 23(1), 59–82 (1991)CrossRefzbMATHGoogle Scholar
  43. 43.
    Lienhardt, P.: N-dimensional generalized combinatorial maps and cellular quasi-manifolds. Int. J. Comput. Geom. Appl. 4(3), 275–324 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Lienhardt, P., Skapin, X., Bergey, A.: Cartesian product of simplicial and cellular structures. Int. J. Comput. Geom. Appl. 14(3), 115–159 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Mac Lane, S.: Homology, Grundlehren Series, Springer 1963. Fourth printing, Classics in Mathematics, Springer (1995)Google Scholar
  46. 46.
    Massey, W.S.: A Basic Course in Algebraic Topology. Graduate Texts in Mathematics. Springer, New York (1991)Google Scholar
  47. 47.
    May, J.P.: Simplicial Objects in Algebraic Topology. Van Nostrand, Princeton (1967)Google Scholar
  48. 48.
    Meine, H., Köthe, U.: The geomap: a unified representation for topology and geometry. In: Brun, L., Vento, M. (eds.) Proceedings of the IAPR Graph-Based Representations in Pattern Recognition. Lecture Notes in Computer Science, vol. 3434, pp. 132–141. Springer, Berlin (2005)Google Scholar
  49. 49.
    Munkres, J.R.: Elements of Algebraic Topology. Addison Wesley, Cambridge (1984)zbMATHGoogle Scholar
  50. 50.
    Niethammer, M., Stein, A.N., Kalies, W.D., Pilarczyk, P., Mischaikow, K., Tannenbaum, A.: Analysis of blood vessel topology by cubical homology. In: IEEE Proceedings of the International Conference on Image Processing, vol. 2, pp. 969–972 (2002)Google Scholar
  51. 51.
    Peltier, S., Alayrangues, S., Fuchs, L., Lachaud, J.-O.: Computation of homology groups and generators. Comput. Graph. 30, 62–69 (2006)CrossRefGoogle Scholar
  52. 52.
    Peltier, S., Fuchs, L., Lienhardt, P.: Simploidals sets: definitions, operations and comparison with simplicial sets. Discret. Appl. Math. 157, 542–557 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    Serre, J.-P.: Homologie singuliere des espaces fibres. Ann. Math. 54(3), 425–505 (1951)CrossRefzbMATHGoogle Scholar
  54. 54.
    Spehner, J.-C.: Merging in maps and pavings. Theor. Comput. Sci. 86, 205–232 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    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, New York (1996)CrossRefGoogle Scholar
  56. 56.
    Terraz, O., Guimberteau, G., Mérillou, S., Plemenos, D., Ghazanfarpour, D.: 3Gmap L-systems: an application to the modelling of wood. Vis. Comput. 25(2), 165–180 (2009)CrossRefGoogle Scholar
  57. 57.
    Tutte, W.: Graph Theory. Encyclopaedia of Mathematics and Its Applications. Addison-Wesley, Menlo Park (1984)Google Scholar
  58. 58.
    Untereiner, L., Cazier, D., Bechmann, D.: n-Dimensional multiresolution representation of subdivision meshes with arbitrary topology. Graph. Model. 75(5), 231–246 (2013)CrossRefGoogle Scholar
  59. 59.
    Vidil, F., Damiand, G.: Moka. (2003). Accessed 12 Mar 2014
  60. 60.
    Vince, A.: Combinatorial maps. J. Comb. Theory Ser. B 34, 1–21 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  61. 61.
    Weiler, K.: The radial-edge data structure: a topological representation for non-manifold geometry boundary modeling. In: Proceedings of IFIP WG 5.2 Working Conference, Rensselaerville, USA (1986)Google Scholar
  62. 62.
    Zomorodian, A., Carlsson, G.: Localized homology. Comput. Geom. 41(3), 126–148 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Sylvie Alayrangues
    • 1
  • Guillaume Damiand
    • 2
  • Pascal Lienhardt
    • 1
  • Samuel Peltier
    • 1
  1. 1.Université de Poitiers, Laboratoire XLIM, Département SIC, CNRS 7252, Bâtiment SP2MI - Téléport 2Futuroscope-Chasseneuil CedexFrance
  2. 2.Université de Lyon, CNRS, LIRIS, UMR5205VilleurbanneFrance

Personalised recommendations