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A Heuristic for Short Homology Basis of Digital Objects

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Discrete Geometry and Mathematical Morphology (DGMM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13493))

Abstract

Finding the minimum homology basis of a simplicial complex is a hard problem unless one only considers the first homology group. In this paper, we introduce a general heuristic for finding a short homology basis of any dimension for digital objects (that is, for their associated cubical complexes) with complexity \(\mathcal {O}(m^3 + \beta _q \cdot n^3)\), where m is the size of the bounding box of the object, n is the size of the object and \(\beta _q\) is the rank of its qth homology group. Our heuristic makes use of the thickness-breadth balls, a tool for visualizing and locating holes in digital objects.

We evaluate our algorithm with a data set of 3D digital objects and compare it with an adaptation of the best current algorithm for computing the minimum radius homology basis by Dey, Li and Wang [10].

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References

  1. DGtal: Digital geometry tools and algorithms library. http://dgtal.org

  2. Bauer, U., Kerber, M., Reininghaus, J.: Clear and compress: computing persistent homology in chunks. In: Bremer, P.-T., Hotz, I., Pascucci, V., Peikert, R. (eds.) Topological Methods in Data Analysis and Visualization III. MV, pp. 103–117. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04099-8_7

    Chapter  MATH  Google Scholar 

  3. Bauer, U., Kerber, M., Reininghaus, J., Wagner, H.: PHAT - persistent homology algorithms toolbox. J. Symb. Comput. 78, 76–90 (2017). https://doi.org/10.1016/j.jsc.2016.03.008

    Article  MathSciNet  MATH  Google Scholar 

  4. Boissonnat, J., Dey, T.K., Maria, C.: The compressed annotation matrix: an efficient data structure for computing persistent cohomology. Algorithmica 73(3), 607–619 (2015). https://doi.org/10.1007/s00453-015-9999-4

    Article  MathSciNet  MATH  Google Scholar 

  5. Busaryev, O., Cabello, S., Chen, C., Dey, T.K., Wang, Y.: Annotating simplices with a homology basis and its applications. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 189–200. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31155-0_17

    Chapter  Google Scholar 

  6. Busaryev, O., Dey, T.K., Wang, Y.: Homology annotations via matrix reduction. Technical report, OSU-CISRC-4/12-TR04, Department of Computer Science and Engineering, the Ohio State University (2012). https://web.cse.ohio-state.edu/~dey.8/paper/annot/basis_TR.pdf

  7. Chen, C., Freedman, D.: Measuring and computing natural generators for homology groups. Comput. Geom. 43(2), 169–181 (2010). https://doi.org/10.1016/j.comgeo.2009.06.004

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, C., Freedman, D.: Hardness results for homology localization. Discret. Comput. Geom. 45(3), 425–448 (2011). https://doi.org/10.1007/s00454-010-9322-8

    Article  MathSciNet  MATH  Google Scholar 

  9. Coeurjolly, D., Klette, R.: A comparative evaluation of length estimators of digital curves. IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 252–257 (2004). https://doi.org/10.1109/TPAMI.2004.1262194

    Article  Google Scholar 

  10. Dey, T.K., Li, T., Wang, Y.: Efficient algorithms for computing a minimal homology basis. In: Bender, M.A., Farach-Colton, M., Mosteiro, M.A. (eds.) LATIN 2018. LNCS, vol. 10807, pp. 376–398. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-77404-6_28

    Chapter  Google Scholar 

  11. Dey, T.K., Sun, J., Wang, Y.: Approximating loops in a shortest homology basis from point data. In: Proceedings of the 26th ACM Symposium on Computational Geometry, Snowbird, Utah, USA, 13–16 June 2010, pp. 166–175 (2010). https://doi.org/10.1145/1810959.1810989

  12. Edelsbrunner, H., Harer, J.: Computational Topology - An Introduction. American Mathematical Society (2010)

    Google Scholar 

  13. Emmett, K.J., Schweinhart, B., Rabadan, R.: Multiscale topology of chromatin folding. In: Suzuki, J., Nakano, T., Hess, H. (eds.) Proceedings of the 9th EAI International Conference on Bio-inspired Information and Communications Technologies (formerly BIONETICS), BICT 2015, New York City, USA, 3–5 December 2015, pp. 177–180. ICST/ACM (2015). http://dl.acm.org/citation.cfm?id=2954838

  14. Erickson, J., Whittlesey, K.: Greedy optimal homotopy and homology generators. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, Vancouver, British Columbia, Canada, 23–25 January 2005, pp. 1038–1046 (2005). http://dl.acm.org/citation.cfm?id=1070432.1070581

  15. Escolar, E.G., Hiraoka, Y.: Optimal cycles for persistent homology via linear programming. In: Fujisawa, K., Shinano, Y., Waki, H. (eds.) Optimization in the Real World. MI, vol. 13, pp. 79–96. Springer, Tokyo (2016). https://doi.org/10.1007/978-4-431-55420-2_5

    Chapter  Google Scholar 

  16. Gonzalez-Lorenzo, A., Bac, A., Mari, J.-L., Real, P.: Two measures for the homology groups of binary volumes. In: Normand, N., Guédon, J., Autrusseau, F. (eds.) DGCI 2016. LNCS, vol. 9647, pp. 154–165. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-32360-2_12

    Chapter  MATH  Google Scholar 

  17. Gonzalez-Lorenzo, A., Bac, A., Mari, J.L., Real, P.: Allowing cycles in discrete Morse theory. Topol. Appl. 228, 1–35 (2017). https://doi.org/10.1016/j.topol.2017.05.008

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology, vol. 157, chap. 2, 7, pp. 255–258. Springer, Heidelberg (2004)

    Google Scholar 

  19. Milosavljevic, N., Morozov, D., Skraba, P.: Zigzag persistent homology in matrix multiplication time. In: Proceedings of the 27th ACM Symposium on Computational Geometry, Paris, France, 13–15 June 2011, pp. 216–225 (2011). https://doi.org/10.1145/1998196.1998229

  20. Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley (1984)

    Google Scholar 

  21. Otter, N., Porter, M.A., Tillmann, U., Grindrod, P., Harrington, H.A.: A roadmap for the computation of persistent homology. EPJ Data Sci. 6(1), 17 (2017). https://doi.org/10.1140/epjds/s13688-017-0109-5

    Article  Google Scholar 

  22. Rathod, A.: Fast algorithms for minimum cycle basis and minimum homology basis. In: Cabello, S., Chen, D.Z. (eds.) 36th International Symposium on Computational Geometry, SoCG 2020, Zürich, Switzerland, 23–26 June 2020. LIPIcs, vol. 164, pp. 64:1–64:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020). https://doi.org/10.4230/LIPIcs.SoCG.2020.64

  23. Zhang, X., Wu, P., Yuan, C., Wang, Y., Metaxas, D.N., Chen, C.: Heuristic search for homology localization problem and its application in cardiac trabeculae reconstruction. In: Kraus, S. (ed.) Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI 2019, Macao, China, 10–16 August 2019, pp. 1312–1318. ijcai.org (2019). https://doi.org/10.24963/ijcai.2019/182

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Authors and Affiliations

  1. Aix Marseille Univ, CNRS, LIS, Marseille, France

    Aldo Gonzalez-Lorenzo, Alexandra Bac & Jean-Luc Mari

Authors
  1. Aldo Gonzalez-Lorenzo
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  2. Alexandra Bac
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  3. Jean-Luc Mari
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Correspondence to Aldo Gonzalez-Lorenzo .

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Editors and Affiliations

  1. University of Strasbourg, Strasbourg, France

    Étienne Baudrier

  2. University of Strasbourg, Strasbourg, France

    Benoît Naegel

  3. University of Strasbourg, Strasbourg, France

    Adrien Krähenbühl

  4. University of Strasbourg, Strasbourg, France

    Mohamed Tajine

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Gonzalez-Lorenzo, A., Bac, A., Mari, JL. (2022). A Heuristic for Short Homology Basis of Digital Objects. In: Baudrier, É., Naegel, B., Krähenbühl, A., Tajine, M. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2022. Lecture Notes in Computer Science, vol 13493. Springer, Cham. https://doi.org/10.1007/978-3-031-19897-7_6

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  • DOI: https://doi.org/10.1007/978-3-031-19897-7_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-19896-0

  • Online ISBN: 978-3-031-19897-7

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