Skip to main content
SpringerLink
Log in

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

  • Conference paper
  • First Online:
New Developments in the Visualization and Processing of Tensor Fields

Part of the book series: Mathematics and Visualization ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Barakat, S., Tricoche, X.: An image-based approach to interactive crease extraction and rendering. Procedia Comput. Sci. 1(1), 1709–1718 (2010). ICCS 2010

    Google Scholar 

  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. 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. 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. Eberly, D.: Ridges in Image and Data Analysis. Kluwer, Boston (1996)

    Google Scholar 

  6. Eberly, D., Gardner, R., Morse, B., Pizer, S.: Ridges for image analysis. J. Math. Imaging Vis. 4, 351–371 (1994)

    Google Scholar 

  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. 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. 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 1991

    Google Scholar 

  10. Haimes, R.: Using residence time for the extraction of recirculation regions. In: AIAA Paper 99–3291. Norfolk, VA (1999)

    Google Scholar 

  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. 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. 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. 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. 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. 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. 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. 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. 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. Marsden, J.E., Tromba, A.J.: Vector Calculus. W.H. Freeman, New York (1996)

    Google Scholar 

  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. 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. Roth, M.: Automatic extraction of vortex core lines and other line-type features for scientific visualization. PhD thesis, ETH Zürich (2000)

    Google Scholar 

  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. 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 2005

    Google Scholar 

  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. 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. 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.2009.44

    Google Scholar 

  29. Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. III. Publish or Perish Incs., Berkeley (1979)

    Google Scholar 

  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. 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. 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. 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. 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. 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. 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. Zheng, X., Tricoche, X., Pang, A.: Degenerate 3d tensors. In: Visualization and Processing of Tensor Fields, pp. 241–256. Springer, Berlin (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Purdue University, Lafayette, IN, USA

    Xavier Tricoche & Samer Barakat

  2. University of California at Davis, Davis, CA, USA

    Mario Hlawitschka & Christoph Garth

Authors
  1. Xavier Tricoche
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mario Hlawitschka
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Samer Barakat
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Christoph Garth
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xavier Tricoche .

Editor information

Editors and Affiliations

  1. Dept. Computer Science, Brown University, Providence, Rhode Island, USA

    David H. Laidlaw

  2. Department of Biomedical Engineering, Einhoven University of Technology, Eindhoven, 5600 MB, Netherlands

    Anna Vilanova

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Berlin Heidelberg

About this paper

Cite this paper

Tricoche, X., Hlawitschka, M., Barakat, S., Garth, C. (2012). Beyond Topology: A Lagrangian Metaphor to Visualize the Structure of 3D Tensor Fields. In: Laidlaw, D., Vilanova, A. (eds) New Developments in the Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27343-8_5

Download citation

Publish with us

Policies and ethics

Search

Navigation