Advertisement

Generating Second Order (Co)homological Information within AT-Model Context

  • Pedro Real
  • Helena Molina-Abril
  • Fernando Díaz del Río
  • Darian Onchis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11382)

Abstract

In this paper we design a new family of relations between (co)homology classes, working with coefficients in a field and starting from an AT-model (Algebraic Topological Model) AT(C) of a finite cell complex C These relations are induced by elementary relations of type “to be in the (co)boundary of” between cells. This high-order connectivity information is embedded into a graph-based representation model, called Second Order AT-Region-Incidence Graph (or AT-RIG) of C. This graph, having as nodes the different homology classes of C, is in turn, computed from two generalized abstract cell complexes, called primal and dual AT-segmentations of C. The respective cells of these two complexes are connected regions (set of cells) of the original cell complex C, which are specified by the integral operator of AT(C). In this work in progress, we successfully use this model (a) in experiments for discriminating topologically different 3D digital objects, having the same Euler characteristic and (b) in designing a parallel algorithm for computing potentially significant (co)homological information of 3D digital objects.

Keywords

Cell complex Algebraic-topological model Homology computation Primal and dual AT-segmentation AT-model region-incidence-graph nD digital object 

References

  1. 1.
    Alexandroff, P.S.: Combinatorial Topology. Dover, New York (1998)zbMATHGoogle Scholar
  2. 2.
    Ayala, R., Domínguez, E., Francés, A.R., Quintero, A.: Homotopy in digital spaces. In: Borgefors, G., Nyström, I., di Baja, G.S. (eds.) DGCI 2000. LNCS, vol. 1953, pp. 3–14. Springer, Heidelberg (2000).  https://doi.org/10.1007/3-540-44438-6_1CrossRefGoogle Scholar
  3. 3.
    Boykov, Y.Y., Jolly, M.P.: Interactive graph cuts for optimal boundary and region segmentation of objects in ND images. In: Proceedings of Eighth IEEE International Conference on Computer Vision, vol. 1, pp. 105–112 (2001)Google Scholar
  4. 4.
    Cadek, M., Krcal, M., Matousek, J., Vokrinek, L., Wagner, U.: Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension. SIAM J. Comput. 43(5), 1728–1780 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Carr, H.A., Weber, G.H., Sewell, C.M., Ahrens, J.P.: Parallel peak pruning for scalable SMP contour tree computation. In: IEEE 6th Symposium on Large Data Analysis and Visualization (LDAV), pp. 75–84 (2016)Google Scholar
  6. 6.
    Couprie, M., Bertrand, G.: Asymmetric parallel 3D thinning scheme and algorithms based on isthmuses. Pattern Recogn. Lett. 76, 22–31 (2016)CrossRefGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Díaz-del-Río, F., Real, P., Onchis, D.: A parallel homological spanning forest framework for 2D topological image analysis. Pattern Recogn. Lett. 83, 49–58 (2016)CrossRefGoogle Scholar
  9. 9.
    De Floriani, L., Mesmoudi, M.M., Morando, F., Puppo, E.: Decomposing non-manifold objects in arbitrary dimensions. Graph. Models 65(1), 2–22 (2003)CrossRefGoogle Scholar
  10. 10.
    Dumas, J.G., Saunders, B.D., Villard, G.: On efficient sparse integer matrix Smith normal form computations. J. Symbol. Comput. 32(1), 71–99 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Eilenberg, S., Mac Lane, S.: On the groups \(H (\pi, n)\), II: methods of computation. Ann. Math. 60, 49–139 (1954)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Forman, R.: Morse theory for cell complexes. Adv. Math. 134, 90–145 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hilaga, M., Shinagawa, Y., Kohmura, T., Kunii, T.L.: Topology matching for fully automatic similarity estimation of 3D shapes. In: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, pp. 203–212. ACM (2001)Google Scholar
  14. 14.
    De Floriani, L., Fugacci, U., Iuricich, F.: Homological shape analysis through discrete morse theory. In: Breuß, M., Bruckstein, A., Maragos, P., Wuhrer, S. (eds.) Perspectives in Shape Analysis. MV, pp. 187–209. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-24726-7_9CrossRefzbMATHGoogle Scholar
  15. 15.
    Dumas, J.G., Heckenbach, F., Saunders, D., Welker, V.: Computing simplicial homology based on efficient Smith normal form algorithms. In: Joswig, M., Takayama, N. (eds.) Algebra, Geometry and Software Systems, pp. 177–206. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-662-05148-1_10CrossRefGoogle Scholar
  16. 16.
    Fiorio, C.: A topologically consistent representation for image analysis: the frontiers topological graph. In: Miguet, S., Montanvert, A., Ubéda, S. (eds.) DGCI 1996. LNCS, vol. 1176, pp. 151–162. Springer, Heidelberg (1996).  https://doi.org/10.1007/3-540-62005-2_13CrossRefGoogle Scholar
  17. 17.
    González-Díaz, R., Real, P.: On the cohomology of 3D digital images. Discret. Appl. Math. 147(2), 245–263 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    González-Díaz, R., Jiménez, M.J., Medrano, B., Real, P.: Chain homotopies for object topological representations. Discret. Appl. Math. 157(3), 490–499 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gonzalez-Lorenzo, A., Bac, A., Mari, J.L., Real, P.: Allowing cycles in discrete Morse theory. Topol. Appl. 228, 1–35 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Günther, D., Reininghaus, J., Wagner, H., Hotz, I.: Efficient computation of 3D Morse-Smale complexes and persistent homology using discrete Morse theory. Vis. Comput. 28(10), 959–969 (2012)CrossRefGoogle Scholar
  21. 21.
    Haarmann, J., Murphy, M.P., Peters, C.S., Staecker, P.C.: Homotopy equivalence in finite digital images. J. Math. Imaging Vis. 53(3), 288–302 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Harker, S., Mischaikow, K., Mrozek, M., Nanda, V.: Discrete Morse theoretic algorithms for computing homology of complexes and maps. Found. Comput. Math. 14(1), 151–184 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hurewicz, W.: Homology and homotopy theory. In: Proceedings of the International Mathematical Congress, p. 344 (1950)Google Scholar
  24. 24.
    Klette, R.: Cell complexes through time. In: International Symposium on Optical Science and Technology, pp. 134–145. International Society for Optics and Photonics (2000)Google Scholar
  25. 25.
    Kong, T.Y., Rosenfeld, A.: Topological Algorithms for Digital Image Processing, vol. 19. Elsevier, Amsterdam (1996)CrossRefGoogle Scholar
  26. 26.
    Kovalevsky, V.: Algorithms in digital geometry based on cellular topology. In: Klette, R., Žunić, J. (eds.) IWCIA 2004. LNCS, vol. 3322, pp. 366–393. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-30503-3_27CrossRefGoogle Scholar
  27. 27.
    Lefschetz, S.: Algebraic Topology, American Mathematical Society Colloquium Publications, vol. 27. American Mathematical Society, New York (1942)Google Scholar
  28. 28.
    Lienhardt, P.: Topological models for boundary representation: a comparison with n-dimensional generalized maps. Comput. Aided Des. 23(1), 59–82 (1991)CrossRefGoogle Scholar
  29. 29.
    Menger, K.: Allgemeine Räume und Cartesische Räume, Teil I, Amsterdam, pp. 476–482 (1926)Google Scholar
  30. 30.
    Molina-Abril, H., Real, P., Nakamura, A., Klette, R.: Connectivity calculus of fractal polyhedrons. Pattern Recogn. 48(4), 1150–1160 (2015)CrossRefGoogle Scholar
  31. 31.
    Molina-Abril, H., Real, P.: Homological spanning forest framework for 2D image analysis. Ann. Math. Artif. Intell. 64, 1–25 (2012)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Molina-Abril, H., Real, P.: Homological optimality in Discrete Morse Theory through chain homotopies. Pattern Recogn. Lett. 11, 1501–1506 (2012)CrossRefGoogle Scholar
  33. 33.
    Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Boston (1984)zbMATHGoogle Scholar
  34. 34.
    Palmieri, J.H: Sage Module: Algebraic-Topological Model for a Cell Complex (2015). http://doc.sagemath.org/
  35. 35.
    Pilarczyk, P., Real, P.: Computation of cubical homology, cohomology and (co)homological operations via chain contractions. Adv. Comput. Math. 41(1), 253–275 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Pudney, C.: Distance-ordered homotopic thinning: a skeletonization algorithm for 3D digital images. Comput. Vis. Image Underst. 72(3), 404–413 (1998)CrossRefGoogle Scholar
  37. 37.
    Real, P., Molina-Abril, H., Gonzalez-Lorenzo, A., Bac, A., Mari, J.L.: Searching combinatorial optimality using graph-based homology information. Appl. Algebra Eng. Commun. Comput. 26(1–2), 103–120 (2015)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Real, P., Diaz-del-Rio, F., Onchis, D.: Toward parallel computation of dense homotopy skeletons for nD digital objects. In: Brimkov, V.E., Barneva, R.P. (eds.) IWCIA 2017. LNCS, vol. 10256, pp. 142–155. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-59108-7_12CrossRefGoogle Scholar
  39. 39.
    Romero, A., Rubio, J., Sergeraert, F.: Effective homology of filtered digital images. Pattern Recogn. Lett. 83, 23–31 (2016)CrossRefGoogle Scholar
  40. 40.
    Robins, V., Wood, P.J., Sheppard, A.P.: Theory and algorithms for constructing discrete Morse complexes from grayscale digital images. IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1646–1658 (2011)CrossRefGoogle Scholar
  41. 41.
    Saha, P.K., Borgefors, G., di Baja, G.S.: A survey on skeletonization algorithms and their applications. Pattern Recogn. Lett. 76, 3–12 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Pedro Real
    • 1
  • Helena Molina-Abril
    • 1
  • Fernando Díaz del Río
    • 1
  • Darian Onchis
    • 2
    • 3
  1. 1.H.T.S. Informatics’ EngineeringUniversity of SevilleSevilleSpain
  2. 2.Faculty of MathematicsUniversity of ViennaViennaAustria
  3. 3.Faculty of Mathematics and Computer ScienceWest University of TimisoaraTimişoaraRomania

Personalised recommendations