Beyond Topology: A Lagrangian Metaphor to Visualize the Structure of 3D Tensor Fields

  • Xavier Tricoche
  • Mario Hlawitschka
  • Samer Barakat
  • Christoph Garth
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

Topology was introduced in the visualization literature some 15 years ago as a mathematical language to describe and capture the salient structures of symmetric second-order tensor fields. Yet, despite significant theoretical and algorithmic advances, this approach has failed to gain wide acceptance in visualization practice over the last decade. In fact, the very idea of a versatile visualization methodology for tensor fields that could transcend application domains has been virtually abandoned in favor of problem-specific feature definitions and visual representations. We propose to revisit the basic idea underlying topology from a different perspective. To do so, we introduce a Lagrangian metaphor that transposes to the structural analysis of eigenvector fields a perspective that is commonly used in the study of fluid flows. Indeed, one can view eigenvector fields as the local superimposition of two vector fields, from which a bidirectional flow field can be defined. This allows us to analyze the structure of a tensor field through the behavior of fictitious particles advected by this flow. Specifically, we show that the separatrices of 3D tensor field topology can in fact be captured in a fuzzy and numerically more robust setting as ridges of a trajectory coherence measure. As a result, we propose an alternative structure characterization strategy for the visual analysis of practical 3D tensor fields, which we demonstrate on several synthetic and computational datasets.

References

  1. 1.
    Barakat, S., Tricoche, X.: An image-based approach to interactive crease extraction and rendering. Procedia Comput. Sci. 1(1), 1709–1718 (2010). ICCS 2010Google Scholar
  2. 2.
    Criscione, J.C., Humphrey, J.D., Douglas, A.S., Hunter, W.C.: An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity. J. Mech. Phys. Solids 48, 2445–2465 (2000)Google Scholar
  3. 3.
    de Saint-Venant, M.: Surfaces à plus grande pente constituées sur des lignes courbes. Bulletin de la Société Philomathématique de Paris 24–30 (1852)Google Scholar
  4. 4.
    Delmarcelle, T., Hesselink, L.: The topology of symmetric, second-order tensor fields. In: Proceedings of IEEE Visualization 1994, pp. 140–147. IEEE Computer Society, Los Alamitos (1994)Google Scholar
  5. 5.
    Eberly, D.: Ridges in Image and Data Analysis. Kluwer, Boston (1996)Google Scholar
  6. 6.
    Eberly, D., Gardner, R., Morse, B., Pizer, S.: Ridges for image analysis. J. Math. Imaging Vis. 4, 351–371 (1994)Google Scholar
  7. 7.
    Furst, J.D., Pizer, S.M., Eberly, D.H.: Marching cores: a method for extracting cores from 3d medical images. In: Proceedings of IEEE Workshop on Mathematical Methods in Biomedical Image Analysis, pp. 124–130. IEEE Computer Society, Los Alamitos (1996)Google Scholar
  8. 8.
    Furst, J.D., Pizer, S.M.: Marching ridges. In: Proceedings of 2001 IASTED International Conference on Signal and Image Processing. IASTED/ACTA Press, Anaheim/Calgary (2001)Google Scholar
  9. 9.
    Globus, A., Levit, C., Lasinski, T.: A tool for visualizing the topology if three-dimensional vector fields. In: IEEE Visualization Proceedings, pp. 33–40. IEEE Computer Society, Los Alamitos, CA, October 1991Google Scholar
  10. 10.
    Haimes, R.: Using residence time for the extraction of recirculation regions. In: AIAA Paper 99–3291. Norfolk, VA (1999)Google Scholar
  11. 11.
    Haller, G.: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Phys. Fluids 13(11), 3365–3385 (2001)Google Scholar
  12. 12.
    Helman J., Hesselink, L.: Surface representation of two and three-dimensional fluid flow topology. In: Proceedings of IEEE Visualization ’90 Conference, pp. 6–13. IEEE Computer Society, Los Alamitos, CA (1990)Google Scholar
  13. 13.
    Hesselink, L., Levy, Y., Lavin, Y.: The topology of symmetric, second-order 3D tensor fields. IEEE Trans. Vis. Comput. Graph. 3(1), 1–11 (1997)Google Scholar
  14. 14.
    Hlawitschka, M., Garth, C., Tricoche, X., Kindlmann, G., Scheuermann, G., Joy, K., Hamann, B.: Direct visualization of fiber information by coherence. Int. J. Comput. Assist. Radiol. Surg. 5(2), 125ff (2010)Google Scholar
  15. 15.
    Kindlmann, G., Tricoche, X., Westin, C.-F.: Delineating white matter structure in diffusion tensor MRI with anisotropy creases. Med. Image Anal. 11(5), 492–502 (2007)Google Scholar
  16. 16.
    Kindlmann, G., San Jose Estepar, R., Smith, S.M., Westin, C.-F.: Sampling and visualizing creases with scale-space particles. IEEE Trans. Vis. Comput. Graph. 15(6), 1415–1424 (2009)Google Scholar
  17. 17.
    Koenderink, J.J., van Doorn, A.J.: Local features of smooth shapes: ridges and courses. In: Vemuri, B.C. (ed.) Proceedings of Geometric Methods in Computer Vision II, B.C. Vemuri (ed.), vol. 2031, pp. 2–13 (1993)Google Scholar
  18. 18.
    Laramee, R.S., Hauser, H., Zhao, L., Post, F.H.: Topology-based flow visualization, the state of the art. In: Hege, H.-C., Polthier, K., Scheuermann, G. (eds.) Topology-Based Methods in Visualization II, Mathematics and Visualization, pp. 1–19. Springer, Berlin (2007)Google Scholar
  19. 19.
    Lorensen, W.E., Cline, H.E.: Marching cubes: a high resolution 3d surface construction algorithm. Comput. Graph. 21(4), 163–169 (1987)Google Scholar
  20. 20.
    Marsden, J.E., Tromba, A.J.: Vector Calculus. W.H. Freeman, New York (1996)Google Scholar
  21. 21.
    Morse, B.S.: Computation of object cores from grey-level images. PhD thesis, University of North Carolina at Chapel Hill, Chapel Hill (1994)Google Scholar
  22. 22.
    Peikert, R., Roth, M.: The Parallel Vectors operator – a vector field visualization primitive. In: IEEE Visualization Proceedings ’00, pp. 263–270. IEEE Computer Society, Los Alamitos, CA (2000)Google Scholar
  23. 23.
    Roth, M.: Automatic extraction of vortex core lines and other line-type features for scientific visualization. PhD thesis, ETH Zürich (2000)Google Scholar
  24. 24.
    Sadlo, F., Peikert, R.: Efficient visualization of lagrangian coherent structures by filtered amr ridge extraction. IEEE Trans. Vis. Comput. Graph. 13(6), 1456–1463 (2007)Google Scholar
  25. 25.
    Sahner, J., Weinkauf, T., Hege, H.-C.: Galilean invariant extraction and iconic representation of vortex core lines. In: Brodlie, K.J.K., Duke, D. (eds.) Proceedings of Eurographics/IEEE VGTC Symposium on Visualization (EuroVis ’05), pp. 151–160. Eurographics Association, Leeds, UK, June 2005Google Scholar
  26. 26.
    Scheuermann, G., Tricoche, X.: Topological methods in flow visualization. In: Johnson, C., Hansen, C. (eds.) Visualization Handbook, pp. 341–356. Academic Press, Copyright Elsevier Butterworth-Heinemann, Oxford, UK (2004)Google Scholar
  27. 27.
    Schultz, T., Theisel, H., Seidel, H.-P.: Topological visualization of brain diffusion MRI data. IEEE Trans. Vis. Comput. Graph. 13(6), 1496–1503 (2007)Google Scholar
  28. 28.
    Schultz, T., Theisel, H., Seidel, H.-P.: Crease surfaces: from theory to extraction and application to diffusion tensor MRI. IEEE Trans. Vis. Comput. Graph. (2009). RapidPost, accepted 16 April 2009. doi:10.1109/TVCG.2009.44Google Scholar
  29. 29.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. III. Publish or Perish Incs., Berkeley (1979)Google Scholar
  30. 30.
    Stetten, G.D.: Medial-node models to identify and measure objects in real-time 3-d echocardiography. IEEE Trans. Med. Imaging 18(10), 1025–1034 (1999)Google Scholar
  31. 31.
    Theisel, H., Seidel, H.-P.: Feature flow fields. In: Proceedings of Joint Eurographics – IEEE TCVG Symposium on Visualization (VisSym ’03), pp. 141–148. Eurographics Association Aire-la-Ville, Switzerland (2003)Google Scholar
  32. 32.
    Tricoche, X., Zheng, X., Pang, A.: Visualizing the topology of symmetric, second-order, time-varying two-dimensional tensor fields. In: Weickert, J., Hagen, H. (eds.) Visualization and Processing of Tensor Fields, pp. 225–240. Springer, Berlin (2006)Google Scholar
  33. 33.
    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)Google Scholar
  34. 34.
    Zheng, X., Pang, A.: Topological lines in 3D tensor fields. In: VIS ’04: Proceedings of the Conference on Visualization ’04, Washington, DC, USA, pp. 313–320. IEEE Computer Society, Los Alamitos (2004)Google Scholar
  35. 35.
    Zheng, X., Parlett, B., Pang, A.: Topological lines in 3D tensor fields and discriminant hessian factorization. IEEE Trans. Vis. Comput. Graph. 11(4), 395–407 (2005)Google Scholar
  36. 36.
    Zheng, X., Parlett, B., Pang, A.: Topological structures of 3d tensor fields. In: Proceedings of IEEE Visualization, pp. 551–558. IEEE Computer Society, Los Alamitos, CA (2005)Google Scholar
  37. 37.
    Zheng, X., Tricoche, X., Pang, A.: Degenerate 3d tensors. In: Visualization and Processing of Tensor Fields, pp. 241–256. Springer, Berlin (2006)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2012

Authors and Affiliations

  • Xavier Tricoche
    • 1
  • Mario Hlawitschka
    • 2
  • Samer Barakat
    • 1
  • Christoph Garth
    • 2
  1. 1.Purdue UniversityLafayetteUSA
  2. 2.University of California at DavisDavisUSA

Personalised recommendations