Glyphs for Non-Linear Vector Field Singularities

  • Alexander WiebelEmail author
  • Stefan Koch
  • Gerik Scheuermann
Part of the Mathematics and Visualization book series (MATHVISUAL)


Glyphs are a widespread technique to depict local properties of different kinds of fields. In this paper we present a new glyph for singularities in non-linear vector fields. We do not simply show the properties of the derivative at the singularities as most previous methods do, but instead illustrate the behavior that goes beyond the local linear approximation. We improve the concept of linear neighborhoods to determine the size of the vicinity from which we derive the data for the glyph. To obtain data from outside this neighborhood we use integration in the vector field. The gathered information is used to depict convergence and divergence of the flow, and non-linear behavior in general. These properties are communicated by color, radius, the overall shape of the glyphs and streamlets on their surface. This way we achieve a depiction of the non-linear behavior of the flow around the singularities.


Diffusion Tensor Imaging IEEE Computer Society Binary Search Lagrangian Coherent Structure Iconic Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



First of all the authors would like to thank the reviewers for their many constructive remarks and suggestions which greatly helped to improve the paper. The authors would like to thank Markus Rütten from DLR in Göttingen for providing the datasets. Thanks also go to Wieland Reich and Roxana Bujack for helpful discussions. Special thanks go to the FAnToM development group for providing the environment for the implementation of the presented work. This work was partially supported by DFG grant SCHE 663/3-8. During the course of this work Alexander Wiebel was hired by the Max Planck Institute for Human Cognitive and Brain Sciences in Leipzig. The first two authors contributed equally to this work.


  1. 1.
    Alexander, D.C., Barker, G.J., Arridge, S.R.: Detection and modeling of non–gaussian apparent diffusion coefficient profiles in human brain data. Magn. Reson. Med. 48(2), 331–340 (2002)CrossRefGoogle Scholar
  2. 2.
    Becker, B.G., Lane, D.A., Max, N.L.: Unsteady flow volumes. In: Nielson, G.M., Silver, D. (eds.) Proceedings of the 6th Conference on Visualization ’95, pp. 329–335. IEEE Computer Society, Washington (1995)Google Scholar
  3. 3.
    de Leeuw, W.C., van Wijk, J.J.: A probe for local flow field visualization. In: Proceedings of the 4th Conference on Visualization ’93, VIS ’93, pp. 39–45. IEEE Computer Society, Washington (1993)Google Scholar
  4. 4.
    Domin, M., Langner, S., Hosten, N., Linsen, L.: Direct glyph-based visualization of diffusion mr data using deformed spheres. In: Linsen, L., Hagen, H., Hamann, B. (eds) Visualization in Medicine and Life Sciences, pp. 177–195, 321. Springer-Verlag, New York (2007)Google Scholar
  5. 5.
    Frank, L.R.: Characterization of anisotropy in high angular resolution diffusion-weighted MRI. Magn. Reson. Med. 47, 1083–1099 (2002)CrossRefGoogle Scholar
  6. 6.
    Fuchs, R., Kemmler, J., Schindler, B., Waser, J., Sadlo, F., Hauser, H., Peikert, R.: Toward a lagrangian vector field topology. Comput. Graphics Forum 29(3), 1163–1172 (2010)CrossRefGoogle Scholar
  7. 7.
    Garth, C., Tricoche, X., Scheuermann, G.: Tracking of vector field singularities in unstructured 3D time-dependent datasets. In: Rushmeier, H., Turk, G., van Wijk, J.J. (eds.) Proceedings of the IEEE Visualization 2004 (VIS’04), pp. 329–336. IEEE Computer Society, Washington (October 2004)CrossRefGoogle Scholar
  8. 8.
    Garth, C., Wiebel, A., Tricoche, X., Joy, K., Scheuermann, G.: Lagrangian visualization of flow-embedded surface structures. Comput. Graphics Forum 27(3), 1007–1014 (2008)CrossRefGoogle Scholar
  9. 9.
    Haller, G.: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Phys. Fluids 13(11) (2001)Google Scholar
  10. 10.
    Helman, J.L., Hesselink, L.: Surface representations of two- and three-dimensional fluid flow topology. In: VIS ’90: Proceedings of the 1st Conference on Visualization ’90, pp. 6–13. IEEE Computer Society Press, Los Alamitos (1990)Google Scholar
  11. 11.
    Hentschel, B., Tedjo, I., Probst, M., Wolter, M., Behr, M., Bischof, C.H., Kuhlen, T.: Interactive blood damage analysis for ventricular assist devices. IEEE TVCG 14(6), 1515–1522 (2008)Google Scholar
  12. 12.
    Hlawitschka, M., Scheuermann, G.: Hot-lines: Tracking lines in higher order tensor fields. In: IEEE Visualization, p. 4. IEEE Computer Society, Washington (2005)Google Scholar
  13. 13.
    Hyslop, J.: Linear transformations and geometry. Edinburgh Math. Notes 25, iv–x (1930)Google Scholar
  14. 14.
    Krishnan, H., Garth, C., Joy, K.: Time and Streak Surfaces for Flow Visualization in Large Time-Varying Data Sets. IEEE Transactions on Visualization and Computer Graphics 15(6): pp. 1267–1274 Nov.-Dec. 2009Google Scholar
  15. 15.
    Löffelmann, H., Doleisch, H., Gröller, E.: Visualizing dynamical systems near critical points. In: Proceedings of the Spring Conference on Computer Graphics and its Applications 1998, pp. 175–184. Budmerice, Slovakia (April 1998)Google Scholar
  16. 16.
    Löffelmann, H., Gröller, E.: Enhancing the visualization of characteristic structures in dynamical systems. In: Bartz, D. (ed.) Visualization in Scientific Computing ’98, pp. 95–68. Eurographics, Springer-Verlag, New York (1998)Google Scholar
  17. 17.
    Özarslan, E., Mareci, T.H.: Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. Magn. Reson. Med. 50, 955–965 (2003)CrossRefGoogle Scholar
  18. 18.
    Ropinski, T., Viola, I., Biermann, M., Hauser, H., Hinrichs, K.: Multimodal Visualization with Interactive Closeups. In: Tang, W., Collomosse, J. P. (eds.), EG UK Theory and Practice of Computer Graphics, Cardiff University, United Kingdom, 2009. Proceedings. Pages 17–24, Eurographics Association, 2009.Google Scholar
  19. 19.
    Schneider, D., Reich, W., Wiebel, A., Scheuermann, G.: Topology aware stream surfaces. Comput Graphics Forum 23(3), 1153–1161 (2010)CrossRefGoogle Scholar
  20. 20.
    Shadden, S.C., Lekien, F., Marsden, J.E.: Definition and properties of lagrangian coherent structures from finite-time lyapunov exponents in two-dimensional aperiodic flows. Physica D 212, 271–304 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Theisel, H., Weinkauf, T., Hege, H.-C., Seidel, H.-P.: Saddle Connectors – An Approach to Visualizing the Topological Skeleton of Complex 3D Vector Fields. In: Proceedings of IEEE Visualization 03. pp. 225–232, IEEE Computer Society Press, 2003.Google Scholar
  22. 22.
    Weinkauf, T., Theisel, H., Shi, K., Hege, H.-C., Seidel, H.-P.: Extracting higher order critical points and topological simplification of 3D vector fields. In: Proceedings of IEEE Visualization 2005, pp. 559–566. Minneapolis (October 2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alexander Wiebel
    • 1
    Email author
  • Stefan Koch
    • 2
  • Gerik Scheuermann
    • 2
  1. 1.Zuse-Institut BerlinBerlin-DahlemDeutschland
  2. 2.Institut für InformatikUniversität LeipzigLeipzigDeutschland

Personalised recommendations