ITR: Procedures for the Rigorous Comparison of Vector and Tensor Fields

Stanford University, Stanford CA

Investigators

Abstract

During the last decade, researchers in our Scientific Visualization Group at Stanfor University have been developing general data analysis techniques for vector and tensor fields based on rigorous mathematical approaches. The work has systematically explored direct visualizations and feature extractions of both vector and tensor fields in two and three dimensions using computers. Recently, we have been developing the next stage of our visualization efforts: automated atad comparisons. Traditional techniques for vector field comparison fall into three basic categories: image, data, and feature based comparison. In most instances, comparisons are made visually, not automatically. In addition, there are fundamental limitations with these existing comparison techniques. Image base comparisons suffer from difficulty in representing datasets beyond two-dimensional vector fields, data based comparisons suffer from grid alignment problems, and feature base comparisons, while providing excellent location of specific features, may not show all the global information in the field. Our new approach to this problem is essentially a feature-based comparison technique, with the important stipulation that our features attempt to represent the topological structure of the field. This ensures that we do not overlook any important structures in the field. Our paradigm is to analytically study vector and tensor fields to extract topologically critical information, transfer this knowledge into effective computer programs, and then to visualize the fields using the results of our analysis. We have successfully implemented our ideas for two-dimensional vector fields and for three-dimensional vector fields. We now intend to study tensor fields associated with white matter brain functions. Our fundamental technique for the comparison of vector and tensord ata would be applicable to almost every field of science and engineering, ranging from the magnetic field of the sun to airflow over a wing. In addition, time varying fields can be studied by comparing a field at a fixed time with its state at later times. Our approach is unique in the field of scientific visualization because it is based on rigorous mathematical analysis, and it is the only approach available to quantitatively study NMR tensor brain data. Our methodology in the analysis of tensor and vector datasets allows better design of air vehicles, better understanding of electromagnetic problems, and provides substantial time savings when dealing with large experimental datasets. Quantitative understanding of white matter brain functions has the potential to open up a whole new area of medical research.