Persistent 1-Cycles: Definition, Computation, and Its Application

  • Tamal K. Dey
  • Tao HouEmail author
  • Sayan Mandal
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11382)


Persistence diagrams, which summarize the birth and death of homological features extracted from data, are employed as stable signatures for applications in image analysis and other areas. Besides simply considering the multiset of intervals included in a persistence diagram, some applications need to find representative cycles for the intervals. In this paper, we address the problem of computing these representative cycles, termed as persistent 1-cycles. The definition of persistent cycles is based on the interval module decomposition of persistence modules, which reveals the structure of persistent homology. After showing that the computation of the optimal persistent 1-cycles is NP-hard, we propose an alternative set of meaningful persistent 1-cycles that can be computed with an efficient polynomial time algorithm. We also inspect the stability issues of the optimal persistent 1-cycles and the persistent 1-cycles computed by our algorithm with the observation that the perturbations of both cannot be properly bounded. We design a software which applies our algorithm to various datasets. Experiments on 3D point clouds, mineral structures, and images show the effectiveness of our algorithm in practice.


Persistent homology Persistent cycle Minimal cycle NP-hardness Image segmentation 


  1. 1.
    Awodey, S.: Category Theory. Oxford University Press, Oxford (2010)zbMATHGoogle Scholar
  2. 2.
    Boissonnat, J., Dey, T.K., Maria, C.: The compressed annotation matrix: an efficient data structure for computing persistent cohomology. CoRR abs/1304.6813 (2013). Scholar
  3. 3.
    Boissonnat, J.D., Maria., C.: The simplex tree: an efficient data structure for general simplicial complexes. In: 20th Annual European Symposium, Ljubljana, Slovenia, vol. 2, pp. 731–742 (2012)CrossRefGoogle Scholar
  4. 4.
    Bubenik, P., Scott, J.A.: Categorification of persistent homology. Discret. Comput. Geom. 51(3), 600–627 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cardona, A., Saalfeld, S., Preibisch, S., Schmid, B., Cheng, A., Pulokas, J., et al.: An integrated micro- and macroarchitectural analysis of the drosophila brain by computer-assisted serial section electron microscopy. PLoS Biol. 8, e1000502 (2010)CrossRefGoogle Scholar
  6. 6.
    Carlsson, G., Ishkhanov, T., de Silva, V., Zomorodian, A.: On the local behavior of spaces of natural images. Int. J. Comput. Vis. 76(1), 1–12 (2008). Scholar
  7. 7.
    Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules. Springer, Cham (2016). Scholar
  8. 8.
    Chen, C., Freedman, D.: Quantifying homology classes II: localization and stability. arXiv preprint arXiv:0709.2512 (2007)
  9. 9.
    Chen, C., Freedman, D.: Hardness results for homology localization. Discret. Comput. Geom. 45(3), 425–448 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. In: Proceedings of the Twenty-First Annual Symposium on Computational Geometry, pp. 263–271. ACM (2005)Google Scholar
  11. 11.
    Derksen, H., Weyman, J.: Quiver representations. Not. AMS 52(2), 200–206 (2005)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dey, T.K., Hirani, A., Krishnamoorthy, B.: Optimal homologous cycles, total unimodularity, and linear programming. SIAM J. Comput. 40(4), 1026–1044 (2011). Scholar
  13. 13.
    Dey, T.K., Fan, F., Wang, Y.: Computing topological persistence for simplicial maps. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, p. 345. ACM (2014)Google Scholar
  14. 14.
    Dey, T.K., Sun, J., Wang, Y.: Approximating loops in a shortest homology basis from point data. In: Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry, pp. 166–175. ACM (2010)Google Scholar
  15. 15.
    Dey, T.K., Mandal, S., Varcho, W.: Improved image classification using topological persistence. In: Hullin, M., Klein, R., Schultz, T., Yao, A. (eds.) Vision, Modeling & Visualization. The Eurographics Association (2017).
  16. 16.
    Edelsbrunner, H.: Weighted alpha shapes. Technical report, Champaign, IL, USA (1992)Google Scholar
  17. 17.
    Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
  18. 18.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 454–463. IEEE (2000)Google Scholar
  19. 19.
    Emmett, K., Schweinhart, B., Rabadan, R.: Multiscale topology of chromatin folding. In: Proceedings of the 9th EAI International Conference on Bio-inspired Information and Communications Technologies (formerly BIONETICS), pp. 177–180. ICST (Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering) (2016)Google Scholar
  20. 20.
    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, vol. 13, pp. 79–96. Springer, Tokyo (2016). Scholar
  21. 21.
    Hoover, A., Goldbaum, M.: Locating the optic nerve in a retinal image using the fuzzy convergence of the blood vessels. IEEE Trans. Med. Imaging 22(8), 951–958 (2003). Scholar
  22. 22.
    Kiehart, D.P., Galbraith, C.G., Edwards, K.A., Rickoll, W.L., Montague, R.A.: Multiple forces contribute to cell sheet morphogenesis for dorsal closure in Drosophila. J. Cell Biol. 149(2), 471–490 (2000). Scholar
  23. 23.
    Obayashi, I.: Volume optimal cycle: tightest representative cycle of a generator on persistent homology. arXiv preprint arXiv:1712.05103 (2017)
  24. 24.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43(3), 425–440 (1991)MathSciNetCrossRefGoogle Scholar
  25. 25.
    The GUDHI Project: GUDHI User and Reference Manual. GUDHI Editorial Board (2015).
  26. 26.
    Wu, P., et al.: Optimal topological cycles and their application in cardiac trabeculae restoration. In: Niethammer, M., et al. (eds.) IPMI 2017. LNCS, vol. 10265, pp. 80–92. Springer, Cham (2017). Scholar

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Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringThe Ohio State UniversityColumbusUSA

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