Singularity Theory of Differentiable Maps and Data Visualization
In many scientific situations, a given set of large data, obtained through simulation or experiment, can be considered to be a discrete set of sample values of a differentiable map between Euclidean spaces or between manifolds. From such a viewpoint, this article explores how the singularity theory of differentiable maps is useful in the visualization of such data. Special emphasis is put on Reeb graphs for scalar functions and on singular fibers of multi-variate functions.
KeywordsData visualization Multi-variate function Differential topology Singularity theory Reeb graph Singular fiber
Some of the contents of this article are the results of joint work with Professor Shigeo Takahashi (Graduate School of Frontier Sciences, the University of Tokyo), who kindly provided important materials for this article. The author would like to take this opportunity to express his sincere gratitude to him.
- 3.H. Edelsbrunner, J. Harer, Jacobi sets of multiple Morse functions. Foundations of Computational Mathematics, Minneapolis, 2002, pp. 37–57, London Mathematical Society Lecture Note Series, vol. 312 (Cambridge University Press, Cambridge, 2004)Google Scholar
- 4.I. Fujishiro, R. Otsuka, S. Takahashi, Y. Takeshima, T-Map: A topological approach to visual exploration of time-varying volume data, High-Performance Computing, ed. by J. Labarta, K. Joe, T. Sato, pp. 176–190, Lecture Notes in Computer Science, vol. 4759 (Springer, Berlin, Heidelberg, 2008)Google Scholar
- 5.X. Ge, I. Safa, M. Belkin, Y. Wang, Data skeletonization via Reeb graphs, in Twenty-Fifth Annual Conference on Neural Information Processing Systems 2011, pp. 837–845Google Scholar
- 7.M. Golubitsky, V. Guillemin, Stable Mappings and Their Singularities. Graduate Texts in Mathematics, vol. 14 (Springer, New York, Heidelberg, 1973)Google Scholar
- 8.J.N. Mather, Stability of \(C^\infty \) mappings. VI: The nice dimensions, in Proceedings of the Liverpool Singularities-Symposium, I (1969/70), pp. 207–253, Lecture Notes in Mathematics, vol. 192 (Springer, Berlin, 1971)Google Scholar
- 9.Y. Matsumoto, An introduction to Morse theory, Translated from the 1997 Japanese original by Kiki Hudson and Masahico Saito, Translations of Mathematical Monographs, vol. 208, American Mathematical Society, Providence, RI, Iwanami Series in Modern Mathematics (2002)Google Scholar
- 10.J. Milnor, Morse theory, based on lecture notes by M. Spivak and R. Wells, Ann. of Math. Stud. 51 (Princeton University Press, Princeton, NJ, 1963)Google Scholar
- 11.V. Pascucci, G. Scorzelli, P.-T. Bremer, A. Mascarenhas, Robust on-line computation of Reeb graphs: simplicity and speed. ACM Trans. Graph. 26(3), Article 58, 58.1–58.9 (2007)Google Scholar
- 13.O. Saeki, Topology of singular fibers of differentiable maps. vol. 1854, Lecture Notes in Mathematics (Springer-Verlag, Berlin, 2004)Google Scholar
- 14.S. Takahashi, T. Ikeda, Y. Shinagawa, T.L. Kunii, M. Ueda, Algorithms for extracting correct critical points and constructing topological graphs from discrete geographical elevation data. Comput. Graph. Forum 14, 181–192 (1995)Google Scholar
- 16.Y. Takeshima, S. Takahashi, I. Fujishiro, G.M. Nielson, Introducing topological attributes for objective-based visualization of simulated datasets, in Proceedings of the Volume Graphics, 2005, pp. 137–145 (2005)Google Scholar
- 17.H. Whitney, On singularities of mappings of euclidean spaces: mappings of the plane into the plane. Ann. of Math. (2) 62, 374–410 (1955)Google Scholar