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Journal of Mathematical Imaging and Vision

, Volume 61, Issue 2, pp 174–192 | Cite as

Acyclic Partial Matchings for Multidimensional Persistence: Algorithm and Combinatorial Interpretation

  • Madjid AlliliEmail author
  • Tomasz Kaczynski
  • Claudia Landi
  • Filippo Masoni
Article
  • 56 Downloads

Abstract

Given a simplicial complex and a vector-valued function on its vertices, we present an algorithmic construction of an acyclic partial matching on the cells of the complex compatible with the given function. This implies the construction can be used to build a reduced filtered complex with the same multidimensional persistent homology as of the original one filtered by the sublevel sets of the function. The correctness of the algorithm is proved, and its complexity is analyzed. A combinatorial interpretation of our algorithm based on the concept of a multidimensional discrete Morse function is introduced for the first time in this paper. Numerical experiments show a substantial rate of reduction in the number of cells achieved by the algorithm.

Keywords

Multidimensional persistent homology Discrete Morse theory Acyclic partial matchings Matching algorithm Reduced complex 

Notes

References

  1. 1.
    Allili, M., Kaczynski, T., Landi, C.: Reducing complexes in multidimensional persistent homology theory. J. Symb. Comput. 78, 61–75 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allili, M., Kaczynski, T., Landi, C., Masoni, F.: Algorithmic construction of acyclic partial matchings for multidimensional persistence. In: Kropatsch, W.G., Artner, N.M., Janusch, I. (eds.) Discrete Geometry for Computer Imagery. Proceedings of the 20th IAPR International Conference, DGCI2017, pp. 375–387. Springer (2017)Google Scholar
  3. 3.
    Biasotti, S., Cerri, A., Frosini, P., Giorgi, D., Landi, C.: Multidimensional size functions for shape comparison. J. Math. Imaging Vis. 32(2), 161–179 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Biasotti, S., Cerri, A., Giorgi, D., Spagnuolo, M.: Phog: photometric and geometric functions for textured shape retrieval. Comput. Graphics Forum 32(5), 13–22 (2013)CrossRefGoogle Scholar
  5. 5.
    Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. In: Proceedings of the 23rd annual Symposium on Computational Geometry, SCG ’07, pp. 184–193. ACM, New York, NY, USA (2007). https://doi.org/10.1145/1247069.1247105
  7. 7.
    Cavazza, N., Ethier, M., Frosini, P., Kaczynski, T., Landi, C.: Comparison of persistent homologies for vector functions: from continuous to discrete and back. Comput. Math. Appl. 66, 560–573 (2013).  https://doi.org/10.1016/j.camwa.2013.06.004 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cerri, A., Frosini, P., Krospatch, W.G., Landi, C.: A global method for reducing multidimensional size graphs-graph-based representations in pattern recognition. Lect. Notes Comput. Sci. 6658, 1–11 (2011)CrossRefGoogle Scholar
  9. 9.
    Edelsbrunner, H., Harer, J.: Jacobi sets of multiple Morse functions. In: Foundations of Computational Mathematics: FoCM’02, Minneapolis 2002, pp. 37–57. Cambridge University Press, Cambridge (2004)Google Scholar
  10. 10.
    Edelsbrunner, H., Harer, J.: Persistent homology: a survey. In: Surveys on Discrete and Computational Geometry. Contemporary Mathematics, vol. 453, pp. 257–282. American Mathematical Society, Providence, RI (2008)Google Scholar
  11. 11.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Escolar, E.G., Hiraoka, Y.: Persistence modules on commutative ladders of finite type. In: Mathematical Software ICMS 2014, pp. 144–151. Lecture Notes in Computer Science 8592 (2014)Google Scholar
  13. 13.
    Forman, R.: Morse theory for cell complexes. Adv. Math. 134, 90–145 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Frosini, P.: Measuring shapes by size functions. In: Proceedings of the of SPIE, Intelligent Robots and Computer Vision X: Algorithms and Techniques, vol. 1607, pp. 122–133. SPIE, Boston, MA, USA (1991)Google Scholar
  15. 15.
    Gäfvert, O.: Algorithms for multidimensional persistence. Master’s thesis, KTH, Stockholm, Sweden (2016)Google Scholar
  16. 16.
    Iuricich, F., Scaramuccia, S., Landi, C., De Floriani, L.: A discrete morse-based approach to multivariate data analysis. In: SIGGRAPH ASIA 2016 Symposium on Visualization, SA ’16, pp. 5:1–5:8. ACM, New York, NY, USA (2016).  https://doi.org/10.1145/3002151.3002166
  17. 17.
    Kaczynski, T., Mrozek, M., Slusarek, M.: Homology computation by reduction of chain complexes. Comput. Math. Appl. 35(4), 59–70 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    King, H., Knudson, K., Mramor, N.: Generating discrete Morse functions from point data. Exp. Math. 14(4), 435–444 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    King, H., Knudson, K., Mramor, N.: Algebraic Topology. Colloquium Publications, vol. 27. American Mathematical Society, Providence (1942)Google Scholar
  20. 20.
    Mischaikow, K., Nanda, V.: Morse theory for filtrations and efficient computation of persistent homology. Discrete Comput. Geom. 50(2), 330–353 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mrozek, M., Batko, B.: Coreduction homology algorithm. Discrete Comput. Geom. 41, 96–118 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Patel, A.: Reeb spaces and the robustness of preimages. Ph.D. thesis, Duke University (2010)Google Scholar
  23. 23.
    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
  24. 24.
    Sedgewick, R., Wayne, K.: Algorithms. Addison-Wesley, Boston (2011)Google Scholar
  25. 25.
    Smale, S.: Global analysis and economics. I. Pareto optimum and a generalization of Morse theory. In: Dynamical systems. Proceedings of a Symposium Held at the University of Bahia, Salvador, 1971, pp. 531–544. Academic Press, New York (1973)Google Scholar
  26. 26.
    The GTS Library (2000). http://gts.sourceforge.net/samples.html. Accessed 26 July 2018
  27. 27.
    Turner, K., Murkherjee, A., Boyer, D.M.: Persistent homology transform for modeling shapes and surfaces. Inf. Inference 3(4), 310–344 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Xia, K., Wei, G.W.: Multidimensional persistence in biomolecular data. J. Comput. Chem. 36(20), 1502 (2015)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceBishop’s UniversitySherbrookeCanada
  2. 2.Département de mathématiquesUniversité de SherbrookeSherbrookeCanada
  3. 3.Dipartimento di Scienze e Metodi dell’IngegneriaUniversità di Modena e Reggio EmiliaReggioItaly

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