Topological Analysis of Asymmetric Tensor Fields for Flow Visualization

Most existing flow visualization techniques focus on the analysis and visualization of the vector field that describes the flow. In this paper, we employ a rather different approach by performing tensor field analysis and visualization on the gradient of the vector field, which can provide additional and complementary information to the direct analysis of the vector field. Our techniques focus on the topological analysis of the eigenvector and eigenvalues of 2×2 tensors. At the core of our analysis is a reparameterization of the tensor space, which allows us to understand the topology of tensor fields by studying the manifolds of eigenvalues and eigenvectors. We present a partition of the eigenvalue manifold using a Voronoi diagram, which allows the segmentation of a tensor field based on relative strengths with respect to isotropic scaling, rotation, and anisotropic stretching. Our analysis of eigenvectors is based on two observations. First, the dual-eigenvectors of a tensor depend solely on the symmetric constituent of the tensor. The anti-symmetric component acts on the eigenvectors by rotating them either clockwise or counterclockwise towards the nearest dual-eigenvector. The orientation and the amount of the rotation are derived from the ratio between the symmetric and anti-symmetric components. Second, The boundary between regions of clockwise rotation and counterclockwise rotation is located where the tensor field is purely symmetric. Crossing such a boundary results in discontinuities in the dual eigenvectors. Thus we define symmetric tensors as part of tensor field topology in addition to degenerate tensors. These observations inspire our visualization techniques in which the topology of the symmetric component and anti-symmetric component are shown simultaneously. We also provide physical interpretations of our analysis, and demonstrate the utility of our visualization techniques on two applications from computational fluid dynamics, namely, engine simulation and cooling jacket design.