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Strategies for Direct Visualization of Second-Rank Tensor Fields

  • Werner Benger
  • Hans-Christian Hege
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

Tensor field visualization aims either at depiction of the full information contained in the field or at extraction and display of specific features. Here, we focus on the first task and evaluate integral and glyph based methods with regard to their power of providing an intuitive visual representation. Tensor fields are considered in a differential geometric context, using a coordinate-free notation when possible. An overview and classification of glyph-based methods is given and selected innovative visualization techniques are presented in more detail. The techniques are demonstrated for applications from medicine and relativity theory.

Keywords

View Plane Initial Seed Deviation Vector Visualization Method Riemann Tensor 
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.

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References

  1. [BAS02]
    M. Bondarescu, M. Alcubierre, and E. Seidel, Isometric embeddings of black hole horizons in three-dimensional flat space, Class. Quant. Grav. 19 (2002).Google Scholar
  2. [Ben04]
    W. Benger, Visualization of general relativistic tensor fields via a fiber bundle data model, Ph.D. thesis, Free University Berlin, August 2004, published by Lehmanns Media, Berlin, 2005.Google Scholar
  3. [BH]
    W. Benger and H.-C. Hege, Analysing curved spacetimes with tensor splats, Proc. 10th Marcel Grossmann Meeting on General Relativity, Rio de Janeiro, July 20–26, 2003, to appear.Google Scholar
  4. [BH04]
    _____ Tensor splats, Conference on Visualization and Data Analysis 2004 (R. Erbacher et al., ed.), Proc. of SPIE, Vol. 5295, 2004, IS&T/SPIE Electronic Imaging Symposium, San Jose, CA.Google Scholar
  5. [DH93]
    T. Delmarcelle and L. Hesselink, Visualizing second order tensor fields with hyperstream lines, IEEE Computer Graphics and Applications 13 (1993), 25–33.CrossRefGoogle Scholar
  6. [GTS+04]
    C. Garth, X. Tricoche, T. Salzbrunn, T. Bobach, and G. Scheuermann, Surface techniques for vortex visualization, VisSym 2004, Symposium on Visualization, Konstanz, Germany, May 19–21, 2004, Eurographics Association, 2004, pp. 155–164, 346.Google Scholar
  7. [Hab90]
    R.B. Haber, Visualization techniques for engineering mechanics, Comp. Sys. in Engineering 1 (1990), no. 1, 37–50.CrossRefGoogle Scholar
  8. [Hot02]
    I. Hotz, Isometric embedding by surface reconstruction from distances, IEEE Visualization 2002, 2002, pp. 251–257.Google Scholar
  9. [Hul92]
    J.P.M. Hultquist, Constructing stream surfaces in steady 3d vector fields, Visualization, IEEE Computer Society, 1992, pp. 171–178.Google Scholar
  10. [Kin04]
    G.L. Kindlmann, Superquadric tensor glyphs., VisSym 2004, Symposium on Visualization, Konstanz, Germany, May 19–21, 2004, Eurographics Association, 2004, pp. 147–154.Google Scholar
  11. [LAK+98]_D.H. Laidlaw, E.T. Ahrens, D. Kremers, M.J. Avalos, R.E. Jacobs, and C. Readhead, Visualizing diffusion tensor images of the mouse spinal cord., IEEE Visualization, 1998, pp. 127–134.Google Scholar
  12. [MSM95]
    J.G. Moore, S.A. Schorn, and J. Moore, Methods of classical mechanics applied to turbulence stresses in a tip leakage vortex, Proc. ASME Gas Turbine Conf., Houston, Texas, no. 95-GT-220, 1995.Google Scholar
  13. [OHW02]
    L. O’Donnell, S. Haker, and C.-F. Westin, New approaches to estimation of white matter connectivity in diffusion tensor MRI: Elliptic PDEs and geodesics in a tensor-warped space, Fifth Int. Conf. Medical Image Computing and Computer-Assisted Intervention (MICCAI’02) (Tokyo, Japan), 2002, pp. 459–466.Google Scholar
  14. [O’N83]
    B. O’Neill, Semi-riemannian geometry, with applications to relativity, Academic Press, Inc., 1983.Google Scholar
  15. [SEF99]
    P. Schneider, J. Ehlers, and E.E. Falco, Gravitational lenses, Springer Verlag Berlin Heidelberg, 1999.Google Scholar
  16. [Set01]
    G.S. Settles, Schlieren and shadowgraph techniques: Visualizing phenomena in transparent media, Springer Verlag, 2001.Google Scholar
  17. [SSE94]
    S. Seitz, P. Schneider, and J. Ehlers, Light propagation in arbitrary spacetimes and the gravitational lens approximation, Class. Quant. Grav (1994).Google Scholar
  18. [Sta98]
    D. Stalling, Fast texture-based algorithms for vector field visualization, Ph.D. thesis, Free University Berlin, 1998.Google Scholar
  19. [TWHS05]
    H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel, Topological methods for 2d time-dependent vector fields based on stream lines and path lines, IEEE Trans. Visual. Comp. Graph. (TVCG) 11 (2005), no. 4.Google Scholar
  20. [WKL99]
    D. Weinstein, G. Kindlmann, and E. Lundberg, Tensorlines: Advection-diffusion based propagation through diffusion tensor fields, IEEE Visualization 1999, IEEE Computer Society Press, 1999, pp. 249–253.Google Scholar
  21. [WMM+02]_C.F. Westin, S.E. Maier, H. Mamata, A. Nabavi, F.A. Jolesz, and R. Kikinis, Processing and visualization for diffusion tensor MRI, Medical Image Analysis 2 (2002), no. 6, 93–108.CrossRefGoogle Scholar
  22. [WPG+97]_C.F. Westin, S. Peled, H. Gudbjartsson, R. Kikinis, and FA. Jolesz, Geometrical diffusion measures for MRI from tensor basis analysis, Proceedings of ISMRM, Fifth Meeting, Vancouver, Canada, April 1997, p. 1742.Google Scholar
  23. [ZP03a]
    X. Zheng and A. Pang, HyperLIC, IEEE Visualization 2003, 2003, pp. 249–256.Google Scholar
  24. [ZP03b]
    _____ Interaction of light and tensor fields, VisSym’ 03, 2003, pp. 157–166.Google Scholar
  25. [ZSH96]
    M. Zöckler, D. Stalling, and H.-C. Hege, Interactive visualization of 3d-vector fields using illuminated streamlines, IEEE Visualization’ 96, Oct./Nov. 1996, pp. 107–113.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Werner Benger
    • 1
    • 2
  • Hans-Christian Hege
    • 1
  1. 1.Zuse-Institute BerlinBerlin-DahlemGermany
  2. 2.Max-Planck Institute for Gravitational Physics (Albert-Einstein Institute)Golm/PotsdamGermany

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