Searching combinatorial optimality using graph-based homology information

  • Pedro RealEmail author
  • Helena Molina-Abril
  • Aldo Gonzalez-Lorenzo
  • Alexandra Bac
  • Jean-Luc Mari
Original Paper


This paper analyses the topological information of a digital object \(O\) under a combined combinatorial-algebraic point of view. Working with a topology-preserving cellularization \(K(O)\) of the object, algebraic and combinatorial tools are jointly used. The combinatorial entities used here are vector fields, \(V\)-paths and directed graphs. In the algebraic side, chain complexes with extra \(2\)-nilpotent operators are considered. By mixing these two perspectives we are able to explore the problems of combinatorial and homological optimality. Combinatorial optimality is understood here as the problem for constructing a discrete gradient vector field (DGVF) in the sense of Discrete Morse Theory, such that it has the least possible number of critical cells. Fixing \({\mathbb {Z}}/2{\mathbb {Z}}\) as field of coefficients, by homological ‘optimality’ we mean the problem of constructing a \(2\)-nilpotent codifferential map \(\phi :C_*(K(O)) \rightarrow C_{*+1}(K(O))\) for finite linear combinations of cells in \(K(O)\), called homology integral operator. The homology groups associated to the chain complex \((C(K(O)), \phi )\) are isomorphic to those of \((C(K(O)), \partial )\), being \(\partial \) the canonical boundary or differential operator of the cell complex \(K(O)\). Relations between these two problems are tackled here by using a type of discrete graphs associated to a homology integral operator, called Homological Spanning Forests (HSF for short). Informally, an HSF for a cell complex can be seen as a kind of combinatorial compressed representation of a homology integral operator. As main result, we refine the heuristic for computing DGVFs based on the iterative Morse complex reduction technique of [1], reducing the search space for an optimal DGVF to an HSF associated to a homology integral operator.


Discrete gradient vector field Optimal discrete gradient vector field Chain homotopy Homological spanning forest Homological information 


  1. 1.
    Molina-Abril, H., Real, P.: Homological optimality in discrete morse theory through chain homotopies. Pattern Recogn. Lett. 33(11), 1501–1506 (2012)CrossRefGoogle Scholar
  2. 2.
    Whitehead, J.H.C.: Combinatorial homotopy. I. Bull. Am. Math. Soc. 55, 213–245 (1949)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Mari, J., Real, P.: Simplicialization of digital volumes in 26-adjacency: application to topologicalanalysis. Pattern Recognit. Image Anal 19(2), 231–238 (2009)CrossRefGoogle Scholar
  4. 4.
    Molina-Abril, H., Real, P.: Cell AT-models for digital volumes. GbR 2009, LNCS. 5534, 314–323 (2009)Google Scholar
  5. 5.
    Munkres, J.: Elements of Algebraic Topology. Addison Wesley, Menlo Park (1984)zbMATHGoogle Scholar
  6. 6.
    Delfinado, C., Edelsbrunner, H.: An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere. Comput. Aided Geom. Des. 12, 771–784 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Forman, R.: Topology and physics for Raoul Bott. In: Yau, S.T. (ed.) A Discrete Morse Theory for Cell Complexes. International Press, Cambridge (1995)Google Scholar
  8. 8.
    Sergeraert, F.: The computability problem in algebraic topology. Adv. Math. 104, 1–29 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    González-Diaz, R., Medrano, B., Real, P., Sanchez-Pelaez, J.: Algebraic topological analysis of time-sequence of digital images. Lect. Notes Comput. Sci. 3718, 208–219 (2005)CrossRefGoogle Scholar
  10. 10.
    Pilarczyk, P., Real, P.: Computation of cubical homology, (co)homology and (co)homological operations via chain contractions. Adv. Comput. Math. (2014). doi: 10.1007/s10444-014-9356-1
  11. 11.
    Joswig, M., Pfetsch, M.E.: Computing optimal morse matchings. SIAM J. Discrete Math. 20(1), 11–25 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Molina-Abril, H., Real, P.: Homological spanning forest framework for 2d image analysis. Ann. Math. Artif. Intell. 64(4), 385–409 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Molina-Abril, H., Real, P., Nakamura, A., Klette, R.: Connectivity calculus of fractal polyhedrons. Pattern Recognit. 48(4), 1146–1156 (2014)Google Scholar
  14. 14.
    Molina-Abril, H., Real, P.: Advanced homological information on 3d digital volumes. In: SSPR 2008, LNCS 5342, pp. 361–371 (2008)Google Scholar
  15. 15.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2001)Google Scholar
  16. 16.
    Forman, R.: Combinatorial vector fields and dynamical systems. Math. Z. 228(4), 629–681 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Malgouyres, R., Francés, A.: Determining whether a simplicial 3-complex collapses to a 1-complex is np-complete. In: Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 4992. Springer, Berlin, pp. 177–188 (2008)Google Scholar
  18. 18.
    Lewiner, T., Lopes, H., Tavares, G.: Optimal discrete Morse functions for 2-manifolds. Comput. Geom. 26(3), 221–233 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Pedro Real
    • 1
    Email author
  • Helena Molina-Abril
    • 1
  • Aldo Gonzalez-Lorenzo
    • 2
  • Alexandra Bac
    • 2
  • Jean-Luc Mari
    • 2
  1. 1.Institute of Mathematics of University of Seville (IMUS)SevilleSpain
  2. 2.CNRS, LSIS UMR 7296Aix-Marseille UniversitéMarseilleFrance

Personalised recommendations