Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Aldroubi and P. Basser. Reconstruction of vector and tensor fields from sampled discrete data. Contemp. Math., 247:1–15, 1999.
P. Alliez, D. Cohen-Steiner, O. Devillers, B. Levy, and M. Desbrun. Anisotropic polygonal remeshing. Siggraph’03, 22(3):485–493, Jul 2003.
A. Bhalerao and C.-F. Westin. Tensor splats: Visualising tensor fields by texture mapped volume rendering. In MICCAI’03, pages 294–901, 2003.
O. Coulon, D. C. Alexander, and S. Arridge. Tensor field regularisation for dt-mr images. In MIUA01, pages 21–24, 2001.
T. Delmarcelle. The Visualization of Second-order Tensor Fields. PhD thesis, Stanford University, 1994.
L. Feng, I. Hotz, B. Hamann, and K. Joy. Anisotropic noise samples. IEEE TVCG, 14(2):342–354, 2008.
R. B. Haber. Visualization techiques for engineering mechanics. Computing Systems in Engineering, 1(1):37–50, 1990.
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.
G. Kindlmann. Superquadric tensor glyphs. In Eurographics Symposium on Visualization, pages 147–154, May 2004.
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.
G. Kindlmann and C.-F. Westin. Diffusion tensor visualization with glyph packing. IEEE TVCG, 12(5):1329–1336, 2006.
Y. Lavin, R. Batra, L. Hesselink, and Y. Levy. The topology of symmetric tensor fields. AIAA 13th Computational Fluid Dynamics Conference, page 2084, 1997.
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.
M. Moakher and P. G. Batchelor. Symmetric positive-definite matrices. In Visualization and Image Processing of Tensor Fields, pages 285–297. Springer, 2006.
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.
X. Tricoche. Vector and Tensor Field Topology Simplification, Tracking and Visualization. PhD thesis, TU Kaiserslautern, 2002.
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.
J. Weickert and M. Welk. Tensor field interpolation with pdes. In Visualization and Processing of Tensor Fields, pages 315–324. Springer, 2005.
X. Zheng and A. Pang. Hyperlic. In IEEE Visualization’03, pages 249–256, 2003.
X. Zheng and A. Pang. Topological lines in 3d tensor fields. In IEEE Visualization’04, 2004.
Acknowledgements
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Berlin Heidelberg
About this chapter
Cite this chapter
Sreevalsan-Nair, J., Auer, C., Hamann, B., Hotz, I. (2011). Eigenvector-based Interpolation and Segmentation of 2D Tensor Fields. In: Pascucci, V., Tricoche, X., Hagen, H., Tierny, J. (eds) Topological Methods in Data Analysis and Visualization. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15014-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-15014-2_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15013-5
Online ISBN: 978-3-642-15014-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)