Visualizing Multivariate Data Using Singularity Theory

  • Osamu Saeki
  • Shigeo Takahashi
  • Daisuke Sakurai
  • Hsiang-Yun Wu
  • Keisuke  Kikuchi
  • Hamish Carr
  • David Duke
  • Takahiro Yamamoto
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 1)


This is a survey article on recent developments in visualization of large data, especially that of multivariate volume data. We present two essential ingredients. The first one is the mathematical background, especially the singularity theory of differentiable mappings, which enables us to capture topological features of given multivariate data in a mathematically rigorous way. The second one is a new development in computer science, called the joint contour net, which can encode topological structures of a given set of multivariate data in an efficient and robust way. Some applications to real data analysis are also presented.


Multivariate data Singular fiber Differential topology Joint contour net Reeb space Data visualization Jacobi set 3-Manifold with boundary 



The “Hurricane Isabel” data set was produced by the Weather Research and Forecast (WRF) model, courtesy of NCAR and the U.S. National Science Foundation (NSF). This research has been partially supported by JSPS KAKENHI Grant Number 25540041, and EPSRC EP/J013072/1.


  1. 1.
    Bachthaler, S., Weiskopf, D.: Continuous scatterplots. IEEE Trans. Vis. Comput. Graph. 14(6), 1428–1435 (2008)CrossRefGoogle Scholar
  2. 2.
    Carr, H., Duke, D.: Joint contour nets. to appear in IEEE Transactions on Visualization and Computer Graphics (2013)Google Scholar
  3. 3.
    Carr, H., Snoeyink, J., Axen, U.: Computing contour trees in all dimensions. Comput. Geom. Theory Appl. 24, 75–94 (2003)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Duke, D., Carr, H., Knoll, A., Schunck, N., Namh, A., Staszczak, A.: Visualizing nuclear scission through a multifield extension of topological analysis. IEEE Trans. Vis. Comput. Graph. 18(12), 2033–2040 (2012)CrossRefGoogle Scholar
  5. 5.
    Edelsbrunner, H., Harer, J.: Jacobi sets of multiple Morse functions. Foundations of computational mathematics: Minneapolis, 2002, London Mathematical Society Lecture Note Series, vol. 312, pp. 37–57, Cambridge Univ. Press, Cambridge (2004)Google Scholar
  6. 6.
    Edelsbrunner, H., Harer, J., Patel, A.K.: Reeb spaces of piecewise linear mappings. Proceedings of the twenty-fourth annual symposium on computational geometry, pp. 242–250 (2008)Google Scholar
  7. 7.
    Fuchs, R., Hauser, H.: Visualization of multi-variate scientific data. Comput. Graph. Forum 28(6), 1670–1690 (2009)CrossRefGoogle Scholar
  8. 8.
    Fujishiro, I., Otsuka, R., Takahashi, S., Takeshima, Y.: T-Map: A topological approach to visual exploration of time-varying volume data. In: Labarta, J., Joe, K., Sato, T. (eds.) High-Performance Computing. Lecture Notes in Computer Science, vol. 4759, pp. 176–190, Springer, Berlin (2008)Google Scholar
  9. 9.
    Ge, X., Safa, I., Belkin, M., Wang, Y.: Data skeletonization via Reeb graphs. Twenty-Fifth Annual Conference on Neural Information Processing Systems, pp. 837–845 (2011)Google Scholar
  10. 10.
    Golubitsky, M., Guillemin, V.: Stable mappings and their singularities. Graduate Texts in Mathematics, vol. 14, Springer (1973)Google Scholar
  11. 11.
    Ikegami, K., Saeki, O.: Cobordism of Morse maps and its application to map germs. Math. Proc. Camb. Phil. Soc. 147, 235–254 (2009)Google Scholar
  12. 12.
    Lehmann, D.J., Theisel, H.: Discontinuities in continuous scatterplots. IEEE Trans. Vis. Comput. Graph. 16(6), 1291–1300 (2010)CrossRefGoogle Scholar
  13. 13.
    Levine, H.: Classifying immersions into \(\mathbf{R}^4\) over stable maps of \(3\)-manifolds into \(\mathbf{R}^2\). Lecture Notes in Math, vol. 1157, Springer, Berlin (1985)Google Scholar
  14. 14.
    Martins, L.F., Nabarro, A.C.: Projections of hypersurfaces in \(\mathbf{R}^4\) with boundary to planes. Glasgow Math. J. 56(1), 149–167 (2014)Google Scholar
  15. 15.
    Pascucci, V., Scorzelli, G., Bremer, P.T., Mascarenhas, A.: Robust on-line computation of Reeb graphs: Simplicity and speed. ACM Trans. Graph. 26, No. 3, (2007), Article 58, 58.1–58.9Google Scholar
  16. 16.
    Reeb, G.: Sur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numérique. C. R. Acad. Sci. Paris 222, 847–849 (1946)MATHMathSciNetGoogle Scholar
  17. 17.
    Saeki, O.: Topology of singular fibers of differentiable maps. Lecture Notes in Math, vol. 1854, Springer (2004)Google Scholar
  18. 18.
    Saeki, O.: Cobordism of Morse functions on surfaces, universal complex of singular fibers, and their application to map germs. Algebr. Geom. Topol. 6, 539–572 (2006)Google Scholar
  19. 19.
    Saeki, O., Yamamoto, T.: Singular fibers of stable maps of 3-manifolds with boundary into surfaces and their applications. preprint (2014)Google Scholar
  20. 20.
    Shibata, N.: On non-singular stable maps of \(3\)-manifolds with boundary into the plane. Hiroshima Math. J. 30, 415–435 (2000)MATHMathSciNetGoogle Scholar
  21. 21.
    Takahashi, S., Takeshima, Y., Fujishiro, I.: Topological volume skeletonization and its application to transfer function design. Graph. Models 66, 24–49 (2004)CrossRefMATHGoogle Scholar
  22. 22.
    Takeshima, Y., Takahashi, S., Fujishiro, I., Nielson, G.M.: Introducing topological attributes for objective-based visualization of simulated datasets. In: Proceedings of the Volume Graphics 2005, pp. 137–145 (2005)Google Scholar
  23. 23.
    Weber, G., Dillard, S., Carr, H., Pascucci, V., Hamann, B.: Topology-controlled volume rendering. IEEE Trans. Vis. Comput. Graph. 13(2), 330–341 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Osamu Saeki
    • 1
  • Shigeo Takahashi
    • 2
  • Daisuke Sakurai
    • 2
  • Hsiang-Yun Wu
    • 2
  • Keisuke  Kikuchi
    • 2
  • Hamish Carr
    • 3
  • David Duke
    • 3
  • Takahiro Yamamoto
    • 4
  1. 1.Institute of Mathematics for IndustryKyushu UniversityNishi-kuJapan
  2. 2.Graduate School of Frontier SciencesThe University of TokyoKashiwaJapan
  3. 3.School of ComputingUniversity of LeedsLeedsUK
  4. 4.Faculty of EngineeringKyushu Sangyo UniversityFukuokaJapan

Personalised recommendations