Propagate and Pair: A Single-Pass Approach to Critical Point Pairing in Reeb Graphs

  • Junyi Tu
  • Mustafa Hajij
  • Paul RosenEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11844)


With the popularization of Topological Data Analysis, the Reeb graph has found new applications as a summarization technique in the analysis and visualization of large and complex data, whose usefulness extends beyond just the graph itself. Pairing critical points enables forming topological fingerprints, known as persistence diagrams, that provides insights into the structure and noise in data. Although the body of work addressing the efficient calculation of Reeb graphs is large, the literature on pairing is limited. In this paper, we discuss two algorithmic approaches for pairing critical points in Reeb graphs, first a multipass approach, followed by a new single-pass algorithm, called Propagate and Pair.


Topological Data Analysis Reeb graph Critical point pairing 



This project was supported in part by National Science Foundation (IIS-1513616 and IIS-1845204). Mesh data are provided by AIM@SHAPE Repository.


  1. 1.
    Agarwal, P.K., Edelsbrunner, H., Harer, J., Wang, Y.: Extreme elevation on a 2-manifold. Discrete Comput. Geom. 36(4), 553–572 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Attene, M., Biasotti, S., Spagnuolo, M.: Shape understanding by contour-driven retiling. Vis. Comput. 19(2), 127–138 (2003)zbMATHGoogle Scholar
  3. 3.
    Bajaj, C.L., Pascucci, V., Schikore, D.R.: The contour spectrum. In: Proceedings of the 8th IEEE Visualization, p. 167-ff (1997)Google Scholar
  4. 4.
    Bauer, U., Ge, X., Wang, Y.: Measuring distance between Reeb graphs. In: Symposium on Computational Geometry, p. 464 (2014)Google Scholar
  5. 5.
    Boyell, R.L., Ruston, H.: Hybrid techniques for real-time radar simulation. In: Proceedings of the November 12–14, Fall Joint Computer Conference, pp. 445–458 (1963)Google Scholar
  6. 6.
    Carr, H., Snoeyink, J., Axen, U.: Computing contour trees in all dimensions. Comput. Geom. Theory Appl. 24(2), 75–94 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Carr, H., Snoeyink, J., van de Panne, M.: Simplifying flexible isosurfaces using local geometric measures. In: Proceedings of the 15th IEEE Visualization, pp. 497–504 (2004)Google Scholar
  8. 8.
    Cole-McLaughlin, K., Edelsbrunner, H., Harer, J., Natarajan, V., Pascucci, V.: Loops in Reeb graphs of 2-manifolds. In: Symposium on Computational Geometry, pp. 344–350 (2003)Google Scholar
  9. 9.
    Doraiswamy, H., Natarajan, V.: Efficient output-sensitive construction of Reeb graphs. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 556–567. Springer, Heidelberg (2008). Scholar
  10. 10.
    Doraiswamy, H., Natarajan, V.: Efficient algorithms for computing Reeb graphs. Comput. Geom. 42(6), 606–616 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Doraiswamy, H., Natarajan, V.: Computing Reeb graphs as a union of contour trees. IEEE Trans. Visual Comput. Graphics 19(2), 249–262 (2013)CrossRefGoogle Scholar
  12. 12.
    Edelsbrunner, H., Harer, J., Mascarenhas, A., Pascucci, V.: Time-varying Reeb graphs for continuous space-time data. In: Symposium on Computational Geometry, pp. 366–372 (2004)Google Scholar
  13. 13.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Symposium on Foundations of Computer Science, pp. 454–463 (2000)Google Scholar
  14. 14.
    Harvey, W., Wang, Y., Wenger, R.: A randomized O (m log m) time algorithm for computing Reeb graphs of arbitrary simplicial complexes. In: Symposium on Computational Geometry, pp. 267–276 (2010)Google Scholar
  15. 15.
    Hilaga, M., Shinagawa, Y.: Topology matching for fully automatic similarity estimation of 3D shapes. In: SIGGRAPH, pp. 203–212 (2001)Google Scholar
  16. 16.
    Kweon, I.S., Kanade, T.: Extracting topographic terrain features from elevation maps. CVGIP Image Underst. 59(2), 171–182 (1994)CrossRefGoogle Scholar
  17. 17.
    Munch, E.: A user’s guide to topological data analysis. J. Learn. Analytics 4(2), 47–61 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Parsa, S.: A deterministic \(O(m \log m)\) time algorithm for the Reeb graph. In: ACM Symposium on Computational Geometry (SoCG), pp. 269–276 (2012)Google Scholar
  19. 19.
    Pascucci, V., Cole-McLaughlin, K., Scorzelli, G.: Multi-resolution computation and presentation of contour trees. In: IASTED Conference on Visualization, Imaging, and Image Processing, pp. 452–290 (2004)Google Scholar
  20. 20.
    Pascucci, V., Scorzelli, G., Bremer, P.T., Mascarenhas, A.: Robust on-line computation of Reeb graphs: simplicity and speed. ACM Trans. Graph. 26(3), 58.1–58.9 (2007)CrossRefGoogle Scholar
  21. 21.
    Reeb, G.: Sur les points singuliers dune forme de pfaff completement intgrable ou dune fonction numrique. CR Acad. Sci. Paris 222, 847–849 (1946)zbMATHGoogle Scholar
  22. 22.
    Rosen, P., et al.: Using contour trees in the analysis and visualization of radio astronomy data cubes. In: TopoInVis (2019)Google Scholar
  23. 23.
    Singh, G., Mémoli, F., Carlsson, G.E.: Topological methods for the analysis of high dimensional data sets and 3D object recognition. In: Eurographics SPBG, pp. 91–100 (2007)Google Scholar
  24. 24.
    Takahashi, S., Takeshima, Y., Fujishiro, I.: Topological volume skeletonization and its application to transfer function design. Graph. Models 66(1), 24–49 (2004)CrossRefGoogle Scholar
  25. 25.
    Tierny, J., Vandeborre, J.P., Daoudi, M.: Partial 3D shape retrieval by Reeb pattern unfolding. Comput. Graphics Forum 28(1), 41–55 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of South FloridaTampaUSA
  2. 2.The Ohio State UniversityColumbusUSA

Personalised recommendations