Advertisement

Homological Shape Analysis Through Discrete Morse Theory

  • Leila De FlorianiEmail author
  • Ulderico Fugacci
  • Federico Iuricich
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

Homology and persistent homology are fundamental tools for shape analysis and understanding that recently gathered a lot of interest, in particular for analyzing multidimensional data. In this context, discrete Morse theory, a combinatorial counterpart of smooth Morse theory, provides an excellent basis for reducing computational complexity in homology detection. A discrete Morse complex, computed over a given complex discretizing a shape, drastically reduces the number of cells of the latter while maintaining the same homology. Here, we consider the problem of shape analysis through discrete Morse theory, and we review and analyze algorithms for computing homology and persistent homology based on such theory.

Keywords

Simplicial Complex Regular Grid Cell Complex Critical Cell Forman Gradient 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work has been partially supported by the US National Science Foundation under grant number IIS-1116747 and by the University of Genova through PRA 2013. The authors wish to thank Davide Bolognini, Emanuela De Negri and Maria Evelina Rossi for their helpful comments and suggestions.

References

  1. 1.
    Agoston, M.K.: Computer Graphics and Geometric Modeling: Mathematics. Springer, London (2005)zbMATHGoogle Scholar
  2. 2.
    Alexandroff, P., Hopf, H.: Topologie i, vol. 1035. Springer, Berlin (1935)zbMATHCrossRefGoogle Scholar
  3. 3.
    Artin, M.: Algebra. Prentice Hall, Englewood Cliffs (1991)zbMATHGoogle Scholar
  4. 4.
    Bendich, P., Edelsbrunner, H., Kerber, M.: Computing robustness and persistence for images. IEEE Trans. Vis. Comput. Graph. 16 (6), 1251–1260 (2010)CrossRefGoogle Scholar
  5. 5.
    Benedetti, B., Lutz, F.H.: Random discrete Morse theory and a new library of triangulations. Exp. Math. 23 (1), 66–94 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Boissonnat, J.D., Dey, T.K., Maria, C.: The compressed annotation matrix: an efficient data structure for computing persistent cohomology. In: Algorithms–ESA 2013, Sophia Antipolis, pp. 695–706. Springer (2013)Google Scholar
  7. 7.
    Boltcheva, D., Canino, D., Merino Aceituno, S., Léon, J.C., De Floriani, L., Hétroy, F.: An iterative algorithm for homology computation on simplicial shapes. Comput. Aided Des. 43 (11), 1457–1467 (2011)CrossRefGoogle Scholar
  8. 8.
    Bremer, P.T., Hamann, B., Edelsbrunner, H., Pascucci, V.: A topological hierarchy for functions on triangulated surfaces. IEEE Trans. Vis. Comput. Graph. 10 (4), 385–396 (2004)CrossRefGoogle Scholar
  9. 9.
    Canino, D., De Floriani, L., Weiss, K.: IA*: an adjacency-based representation for non-manifold simplicial shapes in arbitrary dimensions. Comput. Graph. 35 (3), 747–753 (2011)CrossRefGoogle Scholar
  10. 10.
    Carlsson, G., Ishkhanov, T., De Silva, V., Zomorodian, A.J.: On the local behavior of spaces of natural images. Int. J. Comput. Vis. 76 (1), 1–12 (2008)CrossRefGoogle Scholar
  11. 11.
    Cazals, F., Chazal, F., Lewiner, T.: Molecular shape analysis based upon the Morse-Smale complex and the Connolly function. In: Proceedings of 9th Annual Symposium on Computational Geometry, pp. 351–360. ACM Press, New York (2003)Google Scholar
  12. 12.
    Cerri, A., Ferri, M., Giorgi, D.: Retrieval of trademark images by means of size functions. Graph. Models 68 (5), 451–471 (2006)CrossRefGoogle Scholar
  13. 13.
    Chung, M.K., Bubenik, P., Kim, P.T.: Persistence diagrams of cortical surface data. In: Information Processing in Medical Imaging, pp. 386–397. Springer, Berlin/New York (2009)Google Scholar
  14. 14.
    Čomić, L., De Floriani, L., Iuricich, F.: Simplification operators on a dimension-independent graph-based representation of Morse complexes. In: Hendriks, C.L.L., Borgefors, G., Strand R. (eds.) ISMM. Lecture Notes in Computer Science, vol. 7883, pp. 13–24. Springer, Berlin/New York (2013)Google Scholar
  15. 15.
    Čomić, L., De Floriani, L., Iuricich, F., Fugacci, U.: Topological modifications and hierarchical representation of cell complexes in arbitrary dimensions. Comput. Vis. Image Underst. 121, 2–12 (2014)CrossRefGoogle Scholar
  16. 16.
    Connolly, M.L.: Measurement of protein surface shape by solid angles. J. Mol. Graph. 4 (1), 3–6 (1986)MathSciNetCrossRefGoogle Scholar
  17. 17.
    De Floriani, L., Hui, A.: Data structures for simplicial complexes: an analysis and a comparison. In: Desbrun, M., Pottmann, H. (eds.) Proceedings of 3rd Eurographics Symposium on Geometry Processing. ACM International Conference on Proceeding Series, vol. 255, pp. 119–128. Eurographics Association, Aire-la-Ville (2005)Google Scholar
  18. 18.
    Dequeant, M.L., Ahnert, S., Edelsbrunner, H., Fink, T.M., Glynn, E.F., Hattem, G., Kudlicki, A., Mileyko, Y., Morton, J., Mushegian, A.R., et al.: Comparison of pattern detection methods in microarray time series of the segmentation clock. PLoS One 3 (8), e2856 (2008)CrossRefGoogle Scholar
  19. 19.
    Dey, T.K., Fan, F., Wang, Y.: Computing topological persistence for simplicial maps. arXiv preprint arXiv:1208.5018 (2012)Google Scholar
  20. 20.
    Dey, T.K., Hirani, A.N., Krishnamoorthy, B., Smith, G.: Edge contractions and simplicial homology. arXiv preprint arXiv:1304.0664 (2013)Google Scholar
  21. 21.
    Dłotko, P., Kaczynski, T., Mrozek, M., Wanner, T.: Coreduction homology algorithm for regular cw-complexes. Discret. Comput. Geom. 46 (2), 361–388 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Dłotko, P., Wagner, H.: Simplification of complexes of persistent homology computations. Homol. Homotopy Appl. 16 (1), 49–63 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Berlin (1987)zbMATHCrossRefGoogle Scholar
  24. 24.
    Edelsbrunner, H., Harer, J.: Persistent homology-a survey. Contemp. Math. 453, 257–282 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.J.: Topological persistence and simplification. Discret. Comput. Geom. 28 (4), 511–533 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Fellegara, R., Iuricich, F., De Floriani, L., Weiss, K.: Efficient computation and simplification of discrete Morse decompositions on triangulated terrains. In: 22th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems, ACM-GIS 2014, Dallas, 4–7 Nov 2014 (2014)Google Scholar
  27. 27.
    Forman, R.: Combinatorial vector fields and dynamical systems. Mathematische Zeitschrift 228, 629–681 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Forman, R.: Morse theory for cell complexes. Adv. Math. 134, 90–145 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Fugacci, U., Iuricich, F., De Floriani, L.: Efficient computation of simplicial homology through acyclic matching. In: Proceedings of 5th International Workshop on Computational Topology in Image Context (CTIC 2014), Timisoara (2014)Google Scholar
  30. 30.
    Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. 45 (1), 61–75 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    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
  32. 32.
    Gyulassy, A., Bremer, P.T., Pascucci, V.: Computing Morse-Smale complexes with accurate geometry. IEEE Trans. Vis. Comput. Graph. 18 (12), 2014–2022 (2012). doi:10.1109/TVCG.2012.209CrossRefGoogle Scholar
  33. 33.
    Gyulassy, A., Bremer, P.T., Hamann, B., Pascucci, V.: A practical approach to Morse-Smale complex computation: scalability and generality. IEEE Trans. Vis. Comput. Graph. 14 (6), 1619–1626 (2008)CrossRefGoogle Scholar
  34. 34.
    Gyulassy, A., Bremer, P.T., Hamann, B., Pascucci, V.: Practical considerations in Morse-Smale complex computation. In: Pascucci, V., Tricoche, X., Hagen, H., Tierny, J. (eds.) Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications, Mathematics and Visualization, pp. 67–78. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  35. 35.
    Gyulassy, A., Kotava, N., Kim, M., Hansen, C., Hagen, H., Pascucci, V.: Direct feature visualization using Morse-Smale complexes. IEEE Trans. Vis. Comput. Graph. 18 (9), 1549–1562 (2012)CrossRefGoogle Scholar
  36. 36.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Harker, S., Mischaikow, K., Mrozek, M., Nanda, V., Wagner, H., Juda, M., Dłotko, P.: The efficiency of a homology algorithm based on discrete Morse theory and coreductions. In: Proceedings of 3rd International Workshop on Computational Topology in Image Context (CTIC 2010), Cádiz. Image A, vol. 1, pp. 41–47 (2010)Google Scholar
  38. 38.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge/New York (2002)zbMATHGoogle Scholar
  39. 39.
    King, H., Knudson, K., Mramor, N.: Generating discrete Morse functions from point data. Exp. Math. 14 (4), 435–444 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Lewiner, T., Lopes, H., Tavares, G.: Optimal discrete Morse functions for 2-manifolds. Comput. Geom. 26 (3), 221–233 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Lewis, R.H., Zomorodian, A.J.: Multicore homology via Mayer Vietoris. arXiv preprint arXiv:1407.2275 (2014)Google Scholar
  42. 42.
    Lundell, A.T., Weingram, S.: The topology of CW complexes. Van Nostrand Reinhold Company, New York (1969)zbMATHCrossRefGoogle Scholar
  43. 43.
    Martin, S., Thompson, A., Coutsias, E.A., Watson, J.P.: Topology of cyclo-octane energy landscape. J. Chem. Phys. 132 (23), 234115 (2010). doi:10.1063/1.3445267CrossRefGoogle Scholar
  44. 44.
    Milnor, J.: Morse Theory. Princeton University Press, Princeton (1963)zbMATHCrossRefGoogle Scholar
  45. 45.
    Mischaikow, K., Nanda, V.: Morse theory for filtrations and efficient computation of persistent homology. Discret. Comput. Geom. 50 (2), 330–353 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Mrozek, M., Batko, B.: Coreduction homology algorithm. Discret. Comput. Geom. 41 (1), 96–118 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Mrozek, M., Wanner, T.: Coreduction homology algorithm for inclusions and persistent homology. Comput. Math. Appl. 60 (10), 2812–2833 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Munkres, J.: Elements of Algebraic Topology. Advanced Book Classics. Perseus Books, New York (1984)zbMATHGoogle Scholar
  49. 49.
    Nanda, V.: The Perseus software project for rapid computation of persistent homology. http://www.math.rutgers.edu/~vidit/perseus/index.html
  50. 50.
    Rieck, B., Leitte, H.: Structural analysis of multivariate point clouds using simplicial chains. Comput. Graph. Forum 33 (8), 28–37 (2014). doi:10.1111/cgf.12398CrossRefGoogle Scholar
  51. 51.
    Rieck, B., Mara, H., Leitte, H.: Multivariate data analysis using persistence-based filtering and topological signatures. IEEE Trans. Vis. Comput. Graph. 18 (12), 2382–2391 (2012). doi:10.1109/TVCG.2012.248CrossRefGoogle Scholar
  52. 52.
    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
  53. 53.
    Rosenfeld, A., Kak, A.C.: Digital Picture Processing. Academic Press, London (1982)zbMATHGoogle Scholar
  54. 54.
    Shivashankar, N., Maadasamy, S., Natarajan, V.: Parallel computation of 2D Morse-Smale complexes. IEEE Trans. Vis. Comput. Graph. 18 (10), 1757–1770 (2012)CrossRefGoogle Scholar
  55. 55.
    Shivashankar, N., Natarajan, V.: Parallel computation of 3D Morse-Smale complexes. Comput. Graph. Forum 31 (3), 965–974 (2012)CrossRefGoogle Scholar
  56. 56.
    de Silva, V., Ghrist, R.: Coverage in sensor networks via persistent homology. Algebr. Geom. Topol. 7, 339–358 (2007). doi:10.2140/agt.2007.7.339MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Wang, Y., Agarwal, P.K., Brown, P.H.E., Rudolph, J.: Coarse and reliable geometric alignment for protein docking. In: Proceedings of Pacific Symposium on Biocomputing, Hawaii, vol. 10, pp. 65–75 (2005)Google Scholar
  58. 58.
    Weiss, K., De Floriani, L., Fellegara, R., Velloso, M.: The PR-star octree: a spatio-topological data structure for tetrahedral meshes. In: GIS, Chicago, pp. 92–101 (2011)Google Scholar
  59. 59.
    Weiss, K., Iuricich, F., Fellegara, R., De Floriani, L.: A primal/dual representation for discrete Morse complexes on tetrahedral meshes. Comput. Graph. Forum 32 (3), 361–370 (2013)CrossRefGoogle Scholar
  60. 60.
    Van de Weygaert, R., Vegter, G., Edelsbrunner, H., Jones, B.J., Pranav, P., Park, C., Hellwing, W.A., Eldering, B., Kruithof, N., Bos, E., et al.: Alpha, Betti and the megaparsec universe: on the topology of the cosmic web. In: Transactions on Computational Science XIV, pp. 60–101. Springer, Berlin/New York (2011). http://arxiv.org/abs/1306.3640
  61. 61.
    Zomorodian, A.J.: Topology for Computing, vol. 16. Cambridge University Press, Cambridge/New York (2005)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Leila De Floriani
    • 1
    Email author
  • Ulderico Fugacci
    • 2
  • Federico Iuricich
    • 3
  1. 1.Department of Computer Science, Bioengineering, Robotics, and Systems EngineeringUniversity of GenovaGenovaItaly
  2. 2.Department of Computer Science and UMIACSUniversity of MarylandMDUSA
  3. 3.Department of Geographical Sciences and UMIACSUniversity of MarylandMDUSA

Personalised recommendations