Eigenvector-based Interpolation and Segmentation of 2D Tensor Fields

  • Jaya Sreevalsan-Nair
  • Cornelia Auer
  • Bernd Hamann
  • Ingrid Hotz
Part of the Mathematics and Visualization book series (MATHVISUAL)


We propose a topology-based segmentation of 2D symmetric tensor fields, which results in cells bounded by tensorlines. We are particularly interested in the influence of the interpolation scheme on the topology, considering eigenvector-based and component-wise linear interpolation. When using eigenvector-based interpolation the most significant modification to the standard topology extraction algorithm is the insertion of additional vertices at degenerate points. A subsequent Delaunay re-triangulation leads to connections between close degenerate points. These new connections create degenerate edges and tri angles.When comparing the resulting topology per triangle with the one obtained by component-wise linear interpolation the results are qualitatively similar, but our approach leads to a less “cluttered” segmentation.


Topological Structure Radial Line Edge Label Degenerate Point IEEE Visualization 
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.



This work was supported in part by the German Research Foundation (DFG) through a Junior Research Group Leader award (Emmy Noether Program), and in part by the the National Science Foundation under contract CCF-0702817. We thank our colleagues at the Zuse Institute Berlin and the Institute for Data Analysis and Visualization (IDAV), UC Davis.


  1. 1.
    A. Aldroubi and P. Basser. Reconstruction of vector and tensor fields from sampled discrete data. Contemp. Math., 247:1–15, 1999.MathSciNetGoogle Scholar
  2. 2.
    P. Alliez, D. Cohen-Steiner, O. Devillers, B. Levy, and M. Desbrun. Anisotropic polygonal remeshing. Siggraph’03, 22(3):485–493, Jul 2003.Google Scholar
  3. 3.
    A. Bhalerao and C.-F. Westin. Tensor splats: Visualising tensor fields by texture mapped volume rendering. In MICCAI’03, pages 294–901, 2003.Google Scholar
  4. 4.
    O. Coulon, D. C. Alexander, and S. Arridge. Tensor field regularisation for dt-mr images. In MIUA01, pages 21–24, 2001.Google Scholar
  5. 5.
    T. Delmarcelle. The Visualization of Second-order Tensor Fields. PhD thesis, Stanford University, 1994.Google Scholar
  6. 6.
    L. Feng, I. Hotz, B. Hamann, and K. Joy. Anisotropic noise samples. IEEE TVCG, 14(2):342–354, 2008.Google Scholar
  7. 7.
    R. B. Haber. Visualization techiques for engineering mechanics. Computing Systems in Engineering, 1(1):37–50, 1990.MathSciNetCrossRefGoogle Scholar
  8. 8.
    I. Hotz, L. Feng, H. Hagen, B. Hamann, B. Jeremic, and K. I. Joy. Physically based methods for tensor field visualization. In IEEE Visualization 2004, pages 123–130, 2004.Google Scholar
  9. 9.
    G. Kindlmann. Superquadric tensor glyphs. In Eurographics Symposium on Visualization, pages 147–154, May 2004.Google Scholar
  10. 10.
    G. Kindlmann, R. S. J. Estepar, M. Niethammer, S. Haker, and C.-F. Westin. Geodesic-loxodromes for diffusion tensor interpolation and difference measurement. In MICCAI’07, pages 1–9, 2007.Google Scholar
  11. 11.
    G. Kindlmann and C.-F. Westin. Diffusion tensor visualization with glyph packing. IEEE TVCG, 12(5):1329–1336, 2006.Google Scholar
  12. 12.
    Y. Lavin, R. Batra, L. Hesselink, and Y. Levy. The topology of symmetric tensor fields. AIAA 13th Computational Fluid Dynamics Conference, page 2084, 1997.Google Scholar
  13. 13.
    M. Martin-Fernandez, C.-F. Westin, and C. Alberola-Lopez. 3d bayesian regularization of diffusion tensor MRI using multivariate gaussian markov random fields. In MICCAI’04, pages 351–359, 2004.Google Scholar
  14. 14.
    M. Moakher and P. G. Batchelor. Symmetric positive-definite matrices. In Visualization and Image Processing of Tensor Fields, pages 285–297. Springer, 2006.Google Scholar
  15. 15.
    G. M. Nielson and I.-H. Jung. Tools for computing tangent curves for linearly varying vector fields over tetrahedral domains. IEEE TVCG, 5(4):360–372, 1999.Google Scholar
  16. 16.
    X. Tricoche. Vector and Tensor Field Topology Simplification, Tracking and Visualization. PhD thesis, TU Kaiserslautern, 2002.Google Scholar
  17. 17.
    X. Tricoche, G. Scheuermann, H. Hagen, and S. Clauss. Vector and tensor field topology simplification on irregular grids. In VisSym ’01, pages 107–116, 2001.Google Scholar
  18. 18.
    J. Weickert and M. Welk. Tensor field interpolation with pdes. In Visualization and Processing of Tensor Fields, pages 315–324. Springer, 2005.Google Scholar
  19. 19.
    X. Zheng and A. Pang. Hyperlic. In IEEE Visualization’03, pages 249–256, 2003.Google Scholar
  20. 20.
    X. Zheng and A. Pang. Topological lines in 3d tensor fields. In IEEE Visualization’04, 2004.Google Scholar

Copyright information

© Springer Berlin Heidelberg 2011

Authors and Affiliations

  • Jaya Sreevalsan-Nair
    • 1
  • Cornelia Auer
    • 2
  • Bernd Hamann
    • 3
  • Ingrid Hotz
    • 2
  1. 1.Texas Advanced Computing CenterUniversity of Texas at AustinAustinUSA
  2. 2.Visualization and Data AnalysisZuse Institute BerlinBerlinGermany
  3. 3.Dept. of CSInstitute for Data Analysis and VisualizationUC DavisUSA

Personalised recommendations