EAGER: III: Topological Analysis and Visualization of Time-Varying Symmetric Tensor Fields

This project proposes the study of novel data visualization techniques based on tensor fields, which are a mathematical approach to represent relationships between different data in a model. Tensor fields have a wide range of applications in science, engineering, and medicine. For instance, enabling the three dimensional visualization of the wind speed of tornadoes.

Despite these applications, traditional visualization of tensor fields is limited to static conditions; that is, the visualization of data at a fixed point in time. Thus, the visualization of the wind in a tornado would be at a fixed point in time not over the course of tornado. In contrast, this team will study the visualization based on tensor fields as the model is changing over time.

For this purpose, the project proposes creating mathematical models of two dimensions for visualization of a dynamic tensor field. The team will also investigate efficient techniques to enable domain scientists, engineers, and data stakeholders to gain critical insight into their data and the underlying physics of the movement. Such insight can be beneficial to tasks such as natural disaster modeling and management, structural stability in the nation’s critical infrastructure, and medicine.

The developed visualization can also be used in classroom teaching of advanced mathematical concepts to undergraduate and graduate students in both computer science and other fields.

The core activities of the research include the investigation of the following: (1) the set of all atomic bifurcations in 3D time-varying symmetric tensor fields; (2) a holistic view of all the structures in the tensor fields in the form of a bifurcation graph; (3) robust and efficient algorithms to extract bifurcations from such tensor fields; and, (4) a fast and effective visualization of both the tensor fields and their bifurcation graphs. The research leverages on existing research in topology-driven scalar and vector field visualization as well as two-dimensional tensor field analysis.

Moreover, modern mathematical machinery such as abstract algebra, differential geometry, and algebraic topology is used in the enumeration of all atomic bifurcations in the fields and their robust and efficient extraction. The research can help not only push the envelope of tensor field visualization in a significant way but also benefit related visualization topics such as scalar and vector field visualization.

The set of atomic bifurcations in tensor fields can serve as a dictionary to physicists and engineers who can benefit from fundamental understanding of Physics using tensor fields.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.