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Robustness for 2D Symmetric Tensor Field Topology

  • Bei Wang
  • Ingrid Hotz
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

Topological feature analysis is a powerful instrument to understand the essential structure of a dataset. For such an instrument to be useful in applications, however, it is important to provide some importance measure for the extracted features that copes with the high feature density and discriminates spurious from important structures. Although such measures have been developed for scalar and vector fields, similar concepts are scarce, if not nonexistent, for tensor fields. In particular, the notion of robustness has been proven to successfully quantify the stability of topological features in scalar and vector fields. Intuitively, robustness measures the minimum amount of perturbation to the field that is necessary to cancel its critical points.

This chapter provides a mathematical foundation for the construction of a feature hierarchy for 2D symmetric tensor field topology by extending the concept of robustness, which paves new ways for feature tracking and feature simplification of tensor field data. One essential ingredient is the choice of an appropriate metric to measure the perturbation of tensor fields. Such a metric must be well-aligned with the concept of robustness while still providing some meaningful physical interpretation. A second important ingredient is the index of a degenerate point of tensor fields, which is revisited and reformulated rigorously in the language of degree theory.

References

  1. 1.
    Alliez, P., Cohen-Steiner, D., Devillers, O., Levy, B., Desbrun, M.: Anisotropic polygonal remeshing. In: Siggraph ’03 - ACM Transactions on Graphics (TOG), vol. 22, issue 3, pp. 485–493 (2003)Google Scholar
  2. 2.
    Auer, C., Hotz, I.: Complete tensor field topology on 2D triangulated manifolds embedded in 3D. Comput. Graph. Forum 30(3), 831–840 (2011)CrossRefGoogle Scholar
  3. 3.
    Auer, C., Stripf, C., Kratz, A., Hotz, I.: Glyph- and texture-based visualization of segmented tensor fields. In: International Conference on Information Visualization Theory and Applications (IVAPP’12), (2012)Google Scholar
  4. 4.
    Basser, P.J., Pierpaoli, C.: Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. J. Magn. Reson. Imaging 111(3), 209–219 (1996)CrossRefGoogle Scholar
  5. 5.
    Berger, M.: Geometry Revealed: A Jacob’s Ladder to Modern Higher Geometry. Springer, Heidelberg (2010)CrossRefMATHGoogle Scholar
  6. 6.
    Carr, H., Snoeyink, J., Axen, U.: Computing contour trees in all dimensions. In: SODA - Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 918–926 (2000)Google Scholar
  7. 7.
    Chazal, F., Patel, A., Skraba, P.: Computing well diagrams for vector fields on R n. Appl. Math. Lett. 25(11), 1725–1728 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, G., Mischaikow, K., Laramee, R., Pilarczyk, P., Zhang, E.: Vector field editing and periodic orbit extraction using Morse decomposition. IEEE Trans. Vis. Comput. Graph. 13(4), 769–785 (2007)CrossRefGoogle Scholar
  9. 9.
    Delmarcelle, T.: The visualization of second-order tensor fields. Ph.D. thesis, Stanford University (1994)Google Scholar
  10. 10.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discret. Comput. Geom. 28, 511–533 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Edelsbrunner, H., Morozov, D., Patel, A.: The stability of the apparent contour of an orientable 2-manifold. In: Topological Methods in Data Analysis and Visualization, pp. 27–41. Springer, Heidelberg (2010)Google Scholar
  12. 12.
    Edelsbrunner, H., Morozov, D., Patel, A.: Quantifying transversality by measuring the robustness of intersections. Found. Comput. Math. 11, 345–361 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Guillemin, V., Pollack, A.: Differential Topology. Prentice-Hall, Englewood Cliffs (1974)MATHGoogle Scholar
  14. 14.
    Hopf, H.: Vektorfelder in n-dimensionalen Mannigfaltigkeiten. Math. Ann. 96, 225–250 (1926)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hotz, I., Sreevalsan-Nair, J., Hagen, H., Hamann, B.: Tensor field reconstruction based on eigenvector and eigenvalue interpolation. In: Hagen, H. (ed.) Scientific Visualization: Advanced Concepts. Dagstuhl Follow-Ups, vol. 1, pp. 110–123. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Wadern (2010)Google Scholar
  16. 16.
    Kälberer, F., Nieser, M., Polthier, K.: Quadcover–surface parameterization using branched coverings. Comput. Graph. Forum 26(3), 375–384 (2007)CrossRefMATHGoogle Scholar
  17. 17.
    Knöppel, F., Crane, K., Pinkall, U., Schröder, P.: Globally optimal direction fields. ACM Trans. Graph. 32(4) (2013)Google Scholar
  18. 18.
    Kratz, A., Auer, C., Stommel, M., Hotz, I.: Visualization and analysis of second-order tensors: moving beyond the symmetric positive-definite case. Comput. Graph. Forum–State Art Rep. 32(1), 49–74 (2013)CrossRefGoogle Scholar
  19. 19.
    McLoughlin, T., Laramee, R.S., Peikert, R., Post, F.H., Chen, M.: Over two decades of integration-based, geometric flow visualization. Comput. Graph. Forum 29(6), 1807–1829 (2010)CrossRefGoogle Scholar
  20. 20.
    Skraba, P., Rosen, P., Wang, B., Chen, G., Bhatia, H., Pascucci, V.: Critical point cancellation in 3D vector fields: Robustness and discussion. IEEE Trans. Vis. Comput. Graph. 22(6), 1683–1693 (2016)CrossRefGoogle Scholar
  21. 21.
    Skraba, P., Wang, B.: Interpreting feature tracking through the lens of robustness. In: Topological Methods in Data Analysis and Visualization III. Springer, Cham (2014)CrossRefMATHGoogle Scholar
  22. 22.
    Skraba, P., Wang, B., Chen, G., Rosen, P.: 2D vector field simplification based on robustness. In: Proceedings of IEEE Pacific Visualization Symposium (2014)CrossRefGoogle Scholar
  23. 23.
    Skraba, P., Wang, B., Chen, G., Rosen, P.: Robustness-based simplification of 2D steady and unsteady vector fields. IEEE Trans. Vis. Comput. Graph. 21(8), 930–944 (2015)CrossRefGoogle Scholar
  24. 24.
    Sreevalsan-Nair, J., Auer, C., Hamann, B., Hotz, I.: Eigenvector-based interpolation and segmentation of 2D tensor fields. In: Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications (TopoInVis’09). Mathematics and Visualization. Springer, Heidelberg (2011)Google Scholar
  25. 25.
    Tricoche, X.: Vector and tensor field topology simplification, tracking and visualization. Ph.D. thesis, University of Kaiserslautern (2002)Google Scholar
  26. 26.
    Tricoche, X., Kindlmann, G., Westin, C.F.: Invariant crease lines for topological and structural analysis of tensor fields. IEEE Trans. Vis. Comput. Graph. 14(6), 1627–1634 (2008)CrossRefGoogle Scholar
  27. 27.
    Tricoche, X., Scheuermann, G., Hagen, H., Clauss, S.: Vector and tensor field topology simplification on irregular grids. In: Ebert, D., Favre, J.M., Peikert, R. (eds.) VisSym ’01: Proceedings of the Symposium on Data Visualization, pp. 107–116. Springer, Berlin (2001)Google Scholar
  28. 28.
    Wang, B., Rosen, P., Skraba, P., Bhatia, H., Pascucci, V.: Visualizing robustness of critical points for 2D time-varying vector fields. Comput. Graph. Forum 32(2), 221–230 (2013)CrossRefGoogle Scholar
  29. 29.
    Zhang, C., Schultz, T., Lawonn, K., Eisemann, E., Vilanova, A.: Glyph-based comparative visualization for diffusion tensor fields. IEEE Trans. Vis. Comput. Graph. 22(1), 797–806 (2016)CrossRefGoogle Scholar
  30. 30.
    Zhang, E., Hays, J., Turk, G.: Interactive tensor field design and visualization on surfaces. IEEE Trans. Vis. Comput. Graph. 13(1), 94–107 (2007)CrossRefGoogle Scholar
  31. 31.
    Zhang, Y., Palacios, J., Zhang, E.: Topology of 3D linear symmetric tensor fields. In: Hotz, I., Schultz, T. (eds.) Visualization and Processing of Higher Order Descriptors for Multi-Valued Data (Dagstuhl’14), pp. 73–92. Springer, Berlin (2015)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of UtahSalt Lake CityUSA
  2. 2.Linköping UniversityNorrköpingSweden

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