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Unformatted text preview: 2D Asymmetric Tensor Analysis Xiaoqiang Zheng * Alex Pang † Computer Science Department University of California, Santa Cruz, CA 95064 A BSTRACT Analysis of degenerate tensors is a fundamental step in finding the topological structures and separatrices in tensor fields. Previous work in this area have been limited to analyzing symmetric second order tensor fields. In this paper, we extend the topological analysis to 2D general (asymmetric) second order tensor fields. We show that it is not sufficient to define degeneracies based on eigenvalues alone, but one must also include the eigenvectors in the analysis. We also study the behavior of these eigenvectors as they cross from one topological region into another. CR Categories: I.3.6 [Computer Graphics]: Methodology and Techniques—Interaction Techniques; Keywords: critical points, general tensors, symmetric tensors, degenerate tensors, tensor topology, topological lines, hyperstream- lines, 1 I NTRODUCTION Many different physical processes can be described by 2nd order tensor fields. Two common examples are stress tensors in materials and geomechanics, and diffusion tensors in medical imaging. These types of tensors are generally symmetric tensors where there are no rotational components. For these, there are a few visualization tech- niques available such as tensor ellipsoids [9, 7], texture renderings [16, 6], volume rendering [8, 2, 14], and tensor topology [5, 13, 17]. However, there is a large class of mathematical and physical pro- cesses that cannot be adequately represented by symmetric tensors. This is particularly true in physical processes with strong rotational components such as general deformation tensors with both plastic and elastic deformations, and velocity gradient tensors in compress- ible flows. For these, there is a more limited set of tools available such as hyperstreamlines , and axis tripod glyphs . The state of the art in visualizing general asymmetric tensor fields is to decompose them into a symmetric tensor field and a rota- tional vector field and then try to visualize these simultaneously or separately either continuously or with discrete glyphs [3, 11]. How- ever, these approaches can hardly deliver the effect of the asym- metric tensor field as a whole entity. For example, the user has the daunting task of somehow integrating the rotational components de- picted by ribbons along the major, medium and minor hyperstream- lines over the spatial domain of the data set. The strategy proposed in this paper is to study the topology of asymmetric tensor fields directly without having to explicitly de- compose them first. This paper will focus on the analysis and visu- alization of 2D general tensors of rank two....
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This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.
- Spring '11