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Gromov–Hausdorff Approximation of Filamentary Structures Using Reeb-Type Graphs

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Abstract

In many real-world applications, data appear to be sampled around 1-dimensional filamentary structures that can be seen as topological metric graphs. In this paper, we address the metric reconstruction problem of such filamentary structures from data sampled around them. We prove that they can be approximated with respect to the Gromov–Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and provide an efficient and easy-to-implement algorithm to compute such approximations in almost linear time. We illustrate the performance of our algorithm on a few datasets.

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Notes

  1. See [28], Chap. 1] for the definition of the length of a continuous curve in a general metric space.

  2. Recall that a minimizing geodesic in \(X\) is any curve \(\gamma : I \rightarrow X\), where \(I\) is a real interval, such that \(d_X(\gamma (t),\gamma (t')) = |t - t'|\) for any \(t,t' \in I\).

  3. Strictly speaking we should call it the \(\alpha \)-Reeb graph associated to the covering \(\mathcal {I}\), but we assume in the sequel that some covering \(\mathcal {I}\) has been chosen and we omit it in notations.

References

  1. Aanjaneya, M., Chazal, F., Chen, D., Glisse, M., Guibas, L., Morozov, D.: Metric graph reconstruction from noisy data. Int. J. Comput. Geom. Appl. 22(04), 305–325 (2012)

    Article  MathSciNet  Google Scholar 

  2. Abraham, I., Balakrishnan, M., Kuhn F., Malkhi, D., Ramasubramanian, V., Talwar, K.: Reconstructing approximate tree metrics. In: Proceedings of the Twenty-Sixth Annual ACM Symposium on Principles of Distributed Computing, PODC, pp. 43–52. ACM Press, New York, NY (2007)

  3. Amenta, N., Bern, M., Eppstein, D.: The crust and the \(\beta \)-skeleton: combinatorial curve reconstruction. Graph. Models Image Process. 60(2), 125–135 (1998)

    Article  Google Scholar 

  4. Arias-Castro, E., Donoho, D., Huo, X.: Adaptive multiscale detection of filamentary structures in a background of uniform random points. Ann. Stat. 34(1), 326–349 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bǎdoiu, M., Indyk, P., Sidiropoulos, A.: Approximation algorithms for embedding general metrics into trees. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms, SODA ’07, pp. 512–521 (2007)

  6. Bauer, U., Ge, X., Wang, Y.: Measuring distance between Reeb graphs. arXiv:1307.2839 (2013)

  7. Belkin, M., Niyogi, P.: Laplacian Eigenmaps for dimensionality reduction and data representation. Neural Comput. 15(6), 1373–1396 (2003)

    Article  MATH  Google Scholar 

  8. Burago, D., Burago, Y.: A Course in Metric Geometry. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2001)

    Book  MATH  Google Scholar 

  9. Carlsson, G.: Topology and data. AMS Bull. 46, 255–308 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chazal, F., Sun, J.: Gromov–Hausdorff approximation of filament structure using Reeb-type graph. In: Proceedings of the Symposium on Computational Geometry, pp. 491–500. ACM Press, New York, NY (2014)

  11. Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The structure and stability of persistence modules. arXiv:1207.3674 (2012)

  12. Chen, D., Guibas, L., Hershberger, J., Sun, J.: Road network reconstruction for organizing paths. In: Proceedings 21st ACM-SIAM Symposium on Discrete Algorithms, pp. 1309–1320. SIAM, Philadelphia (2010)

  13. Chepoi, V., Dragan, F., Estellon, B., Habib, M., Vaxès, Y.: Notes on diameters, centers, and approximating trees of delta-hyperbolic geodesic spaces and graphs. Electron. Notes Discrete Math. 31, 231–234 (2008)

    Article  Google Scholar 

  14. Choi, E., Bond, N.A., Strauss, M.A., Coil, A.L., Davis, M., Willmer, C.N.A.: Tracing the filamentary structure of the galaxy distribution at \(z\sim 0.8\). Monthly Notices Royal Astron. Soc., 406(1), 320–328. arXiv:1003.3239 (2010)

  15. Dey, T., Wenger, R.: Reconstructing curves with sharp corners. Comput. Geom. Theory Appl. 19, 89–99 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dey, T., Mehlhorn, K., Ramos, E.: Curve reconstruction: connecting dots with good reason. In: Proceedings of the 15th Symposium on Computational Geometry, pp. 197–206 (1999)

  17. Dey, T.K., Fan, F., Wang, Y.: Graph induced complex on point data. In: Proceedings of the 29th Annual ACM Symposium on Computational Geometry, pp. 107–116. ACM Press, New Yok, NY (2013)

  18. Dhamdhere, K., Gupta, A., Räcke, H.: Improved embeddings of graph metrics into random trees. In: Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithm, SODA ’06, pp. 61–69 (2006)

  19. Dieudonné, J.: Foundations of Modern analysis. Pure and Applied Mathematics. Academic Press, New York (1969)

    MATH  Google Scholar 

  20. Earthquake search. http://earthquake.usgs.gov/earthquakes/eqarchives/epic/. Accessed Sep 2008

  21. Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence, RI (2010)

    Google Scholar 

  22. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological Persistence and Simplification. Discrete Comput. Geom. 28, 511–533 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. In: Proceedings of the 35th ACM Symposium on Theory of Computing, STOC ’03, pp. 448–455. ACM, New York, NY (2003)

  24. Ge, X., Safa, I., Belkin, M., Wang, Y.: Data skeletonization via Reeb graphs. Neural Inf. Process. Syst. 24, 837–845 (2011)

  25. Genovese, C., Perone-Pacifico, M., Verdinelli, I., Wasserman, L.: On the path density of a gradient field. Ann. Stat. 37(6A), 3236–3271 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Genovese, C., Perone-Pacifico, M., Verdinelli, I., Wasserman, L.: Nonparametric ridge estimation. arXiv:1212.5156 (2012)

  27. Genovese, C.R., Perone-Pacifico, M., Verdinelli, I., Wasserman, L.: The geometry of nonparametric filament estimation. J. Am. Stat. Assoc., 107(498), 788–799 (2012)

  28. Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Modern Birkhäuser Classics. Birkhäuser, Berlin (2002)

    Google Scholar 

  29. Harvey, W., Wang, Y., Wenger, R.: A randomized \(O(m\) log \(m)\) time algorithm for computing Reeb graph of arbitrary simplicial complexes. In: Proceedings of the 26th Annual ACM Symposium on Computational Geometry (2010). ACM, New York, NY

  30. Hatcher, A.: Algebraic Topology. Cambridge University Press, New York (2002)

    MATH  Google Scholar 

  31. Lafon, S.: Diffusion Maps and Geodesic Harmonics, PhD. Thesis, Yale University (2004). https://sites.google.com/site/stefansresearchpapers/

  32. Openstreetmap. http://www.openstreetmap.org/. Accessed Jun 2010

  33. Parsa, S.: A deterministic \(O(m\) log \(m)\) time algorithm for the Reeb graph. In: Proceedings of the 2012 Symposium on Computational Geometry, pp. 269–276 (2012)

  34. Singh, G., Mémoli, F., Carlsson, G.: Topological methods for the analysis of high dimensional data sets and 3D object recognition. In: Proceedings of the Eurographics Symposium on Point-Based Graphics, pp. 91–100 (2007)

  35. Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000)

    Article  Google Scholar 

  36. Tupin, F., Maitre, H., Mangin, J.-F., Nicolas, J.-M., Pechersky, E.: Detection of linear features in SAR images: application to road network extraction. IEEE Trans. Geosci. Remote Sens. 36, 434–453 (1998)

    Article  Google Scholar 

  37. Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33, 249–274 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors acknowledge Daniel Müllner and Gunnar Carlsson for fruitful discussions and for providing code for the Mapper algorithm. They acknowledge the European Project CG-Learning EC contract No. 255827; the ANR project TopData (ANR-13-BS01-0008); The National Basic Research Program of China (973 Program 2012CB825501); The NSF of China (11271011).

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Authors and Affiliations

  1. INRIA Saclay, Palaiseau, France

    Frédéric Chazal & Ruqi Huang

  2. Tsinghua University, Beijing, China

    Jian Sun

Authors
  1. Frédéric Chazal
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  2. Ruqi Huang
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  3. Jian Sun
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Correspondence to Frédéric Chazal.

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Editors in charge: Siu-Wing Cheng and Olivier Devillers

Preliminary version in Proceedings of the 30th Annual Symposium on Computational Geometry, 2014.

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Chazal, F., Huang, R. & Sun, J. Gromov–Hausdorff Approximation of Filamentary Structures Using Reeb-Type Graphs. Discrete Comput Geom 53, 621–649 (2015). https://doi.org/10.1007/s00454-015-9674-1

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