Degenerate 3D Tensors
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Topological analysis of 3D tensor fields starts with the identification of degeneracies in the tensor field. In this chapter, we present a new, intuitive and numerically stable method for finding degenerate tensors in symmetric second order 3D tensor fields. This method is based on a description of a tensor having an isotropic spherical component and a linear or planar component. As such, we refer to this formulation as the geometric approach. In this chapter, we also show that the stable degenerate features in 3D tensor fields form lines. On the other hand, degenerate features that form points, surfaces or volumes are not stable and either disappear or turn into lines when noise is introduced into the system. These topological feature lines provide a compact representation of the 3D tensor field and are useful in helping scientists and engineers understand their complex nature.
- T. Delmarcelle and L. Hesselink. The topology of second-order tensor fields. In R.D. Bergeron and A.E. Kaufman, editors, Proceedings IEEE Visualization’ 94, pp. 140–148. IEEE Computer Society Press, 1994.
- L. Hesselink, Y. Levy, and Y. Lavin. The topology of symmetric, second-order 3D tensor fields. IEEE Transactions on Visualization and Computer Graphics, 3(1):1–11, Jan–Mar 1997. CrossRef
- D. Hilbert. Über die Darstellung definiter Formen als Summen von Formenquadraten. Math. Annalen, 32:342–350, 1888. CrossRef
- P. D. Lax. On the discriminant of real symmetric matrices. Communications on Pure and Applied Mathematics, 51(11–12):1387–1396, 1998. CrossRef
- Peter Lax. Linear Algebra. Wiley, 1996.
- B. N. Parlett. The (matrix) discriminant as a determinant. Linear Algebra and its Applications, 355:85–101, 2002. CrossRef
- Wei Shen. Seeding strategries for hyperstreamlines. Master’s thesis, University of California, Santa Cruz, 2004.
- Xiaoqiang Zheng and Alex Pang. Topological lines in 3D tensor fields. In Proceedings of Visualization 04, pp. 313–320, Austin, 2004.
- Xiaoqiang Zheng and Alex Pang. Topological lines in 3D tensor fields and discriminant hessian factorization. IEEE Transactions on Visualization and Computer Graphics, 2005. to appear.
- Degenerate 3D Tensors
- Book Title
- Visualization and Processing of Tensor Fields
- Book Part
- Part III
- pp 241-256
- Print ISBN
- Online ISBN
- Series Title
- Mathematics and Visualization
- Series ISSN
- Springer Berlin Heidelberg
- Copyright Holder
- Springer-Verlag Berlin Heidelberg
- Additional Links
- Industry Sectors
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- Editor Affiliations
- 1. Faculty of Mathematics and Computer Science, Saarland University
- 2. Computer Graphics and Visualization Group, Technical University of Kaiserslautern
- Author Affiliations
- 3. University of California, Santa Cruz, CA, 95064, USA
- 4. University of Utah, Salt Lake City, UT, 84112, USA
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