Abstract
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.
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
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.
D. Hilbert. Über die Darstellung definiter Formen als Summen von Formenquadraten. Math. Annalen, 32:342–350, 1888.
P. D. Lax. On the discriminant of real symmetric matrices. Communications on Pure and Applied Mathematics, 51(11–12):1387–1396, 1998.
Peter Lax. Linear Algebra. Wiley, 1996.
B. N. Parlett. The (matrix) discriminant as a determinant. Linear Algebra and its Applications, 355:85–101, 2002.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Zheng, X., Tricoche, X., Pang, A. (2006). Degenerate 3D Tensors. In: Weickert, J., Hagen, H. (eds) Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31272-2_14
Download citation
DOI: https://doi.org/10.1007/3-540-31272-2_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25032-6
Online ISBN: 978-3-540-31272-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)