The Visualization of Second-Order Tensor Fields

The field of scientific visualization covers the study of visual representations of sci- entific data. These data often consist of continuous scalar, vector, or second-order tensor fields extending across a n-dimensional space, with n less than 1. While research efforts aimed at developing efficient representations of scalar and vector data have led to important achievements, this dissertation is the first extensive study of visualization techniques for second-order tensor fields. Tensor fields are multivariate; they can embed as much information as n squared scalar fields, or equivalently, n vector fields. Visualizing continuous tensor data is, therefore, difficult mainly because the underlying continuity must be rendered while visual clutter has to be avoided. We develop a theoretical ground work for the visualization of symmetric tensor fields by studying their geometry and their topological structure, and by designing at each step appropriate icons to represent the information. We also extend some of our concepts to asymmetric tensor data. First, we design icons that emphasize the continuity of the tensor data, overcoming some of the limitations of discrete point icons. A n-dimensional, symmetric tensor field is equivalent to n orthogonal families of smooth and continuous curves that are tangent to the eigenvector fields. For n = 2 we generate textures to render these trajectories, and for n = 3 we use numerical integration. To fully represent the tensor data, we surround the resulting trajectories by tubular surfaces that represent the transverse eigenvectors-we call these surfaces hyper-streamlines. We also define the concept of a solenoidal tensor field, and we show that its hyperstreamlines possess geometric properties similar to the streamlines of solenoidal vector fields. Then, we analyze the topology of symmetric tensor fields by using a formalism which is analogous to the phase-space analysis of dynamical systems.