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A Mathematical Approach to Research Problems of Science and Technology pp 119–130Cite as

Singularity Theory of Differentiable Maps and Data Visualization

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Part of the book series: Mathematics for Industry ((MFI,volume 5))

Abstract

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.

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References

  1. J. Callahan, Singularities and plane maps. Amer. Math. Mon. 81, 211–240 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Carr, J. Snoeyink, U. Axen, Computing contour trees in all dimensions. Comput. Geom. Theor. Appl. 24, 75–94 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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. 837–845

    Google Scholar 

  6. R. Gilmore, Catastrophe theory for scientists and engineers (Wiley, New York, 1981)

    MATH  Google 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 

  12. G. Reeb, 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)

    MathSciNet  MATH  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 

  15. S. Takahashi, Y. Takeshima, I. Fujishiro, Topological volume skeletonization and its application to transfer function design. Graph. Models 66, 24–49 (2004)

    Article  MATH  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 

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Acknowledgments

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.

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Authors and Affiliations

  1. Institute of Mathematics for Industry, Kyushu University, 744, Motooka, Nishiku, Fukuoka, 819-0395, Japan

    Osamu Saeki

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  1. Osamu Saeki
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Correspondence to Osamu Saeki .

Editor information

Editors and Affiliations

  1. Kyushu University, Fukuoka, Japan

    Ryuei Nishii

  2. Department of Mathematics, Hokkaido University, Sapporo, Japan

    Shin-ichiro Ei

  3. Kyushu University Institute of Mathematics for Industry, Fukuoka, Japan

    Miyuki Koiso

  4. Kyushu University Institute of Mathematics for Industry, Fukuoka, Japan

    Hiroyuki Ochiai

  5. Kyushu University Institute of Mathematics for Industry, Fukuoka, Japan

    Kanzo Okada

  6. Faculty of Arts and Science, Kyushu University, Fukuoka, Japan

    Shingo Saito

  7. Kyushu University Institute of Mathematics for Industry, Fukuoka, Japan

    Tomoyuki Shirai

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Saeki, O. (2014). Singularity Theory of Differentiable Maps and Data Visualization. In: Nishii, R., et al. A Mathematical Approach to Research Problems of Science and Technology. Mathematics for Industry, vol 5. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55060-0_9

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  • DOI: https://doi.org/10.1007/978-4-431-55060-0_9

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  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-55059-4

  • Online ISBN: 978-4-431-55060-0

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