Eigenvector-based Interpolation and Segmentation of 2D Tensor Fields
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.
KeywordsTopological Structure Radial Line Edge Label Degenerate Point IEEE Visualization
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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.
- 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.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.O. Coulon, D. C. Alexander, and S. Arridge. Tensor field regularisation for dt-mr images. In MIUA01, pages 21–24, 2001.Google Scholar
- 5.T. Delmarcelle. The Visualization of Second-order Tensor Fields. PhD thesis, Stanford University, 1994.Google Scholar
- 6.L. Feng, I. Hotz, B. Hamann, and K. Joy. Anisotropic noise samples. IEEE TVCG, 14(2):342–354, 2008.Google Scholar
- 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.G. Kindlmann. Superquadric tensor glyphs. In Eurographics Symposium on Visualization, pages 147–154, May 2004.Google Scholar
- 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.G. Kindlmann and C.-F. Westin. Diffusion tensor visualization with glyph packing. IEEE TVCG, 12(5):1329–1336, 2006.Google Scholar
- 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.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.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.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.X. Tricoche. Vector and Tensor Field Topology Simplification, Tracking and Visualization. PhD thesis, TU Kaiserslautern, 2002.Google Scholar
- 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.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.X. Zheng and A. Pang. Hyperlic. In IEEE Visualization’03, pages 249–256, 2003.Google Scholar
- 20.X. Zheng and A. Pang. Topological lines in 3d tensor fields. In IEEE Visualization’04, 2004.Google Scholar