Visualizing second-order tensor fields

The discipline of second-order tensor visualization covers the study of providing visual representation of tensor fields obtained in various scientific applications. Due to the large amount of information in a tensor field, it is difficult to design one technique that fits the demands of all applications. Several techniques specially designed to visualize different aspects of a tensor field are proposed in this dissertation. Some of them are capable of providing simple and powerful presentation of the tensor fields, while others can display the tensor fields in an intuitive manner. All these different techniques complement each other in helping users understand the underlying physics behind the data. They can be classified into four categories: topology based, texture based and deformation based methods for symmetric tensors, and composite eigenvectors based methods for asymmetric tensors analysis.Topology based method has proven to be successful in generating simple yet powerful visualizations of a data field, but 3D tensor topology remains largely unexplored in real applications. This dissertation is the first extensive study on 3D tensor topology including 3D degenerate lines and their separating surfaces. The classic works neither point out that the degenerate tensors form lines in 3D, nor do they lead to a stable numerical algorithm to extract them. In this dissertation, we show that 3D degenerate tensors form lines in their most basic forms, and propose several algorithms to extract them stably. We also further discuss the generation and property of separating surfaces around the extracted degenerate lines. The separating surfaces comprise hyperstreamlines emanating from the degenerate lines. Together they form a complete topological structure of the 3D tensor field.A weakness of topological methods is that the presentation is hard to understand for inexperienced users. This issue can be addressed by our dense texture synthesis technique, HyperLIC. This technique efficiently emulates Brownian motions in a diffusion process and produces images and animations reminiscent of line integral convolution (LIC). The anisotropic areas are represented with high contrast textures that point to the principal directions, while the isotropic areas are displayed with blurred textures that show no preference of direction at all. We demonstrate this technique using data sets from computational fluid dynamics as well as diffusion-tensor MRI.Since deformation is a very common source of tensors we encounter in our daily life, we also develop deformation-based methods to visualize the strain introduced by tensors. By observing primitives such as geometric objects or photons deformed under the influence of the 3D tensor fields, users perceive the underlying tensor properties in a natural way.Most of the previous works focus on symmetric tensor fields. When analyzing asymmetric tensor fields such as Jacobians in a continuous flow, the state-of-art is to decompose them into symmetric tensors. This dissertation proposes a technique that can visualize the total effect of an asymmetric tensor as a whole entity without resorting to decomposition. This is based on a novel idea of composite eigenvector that is well-defined and continuous all across a 2D asymmetric tensor field.We have applied the techniques developed in this dissertation on several synthetic and real datasets: single/double point load, randomly generated, hemisphere and delta wing.