A novel approach is introduced to define a quantitative measure of closeness between vector fields. The usefulness of this measurement can be seen when comparing computational and experimental flow fields under the same conditions. Furthermore, its applicability can be extended to more cumbersome tasks, such as navigating through a large database, searching for similar topologies. This new measure relies on the use of critical points, which are a key feature in vector field topology. In order to characterize critical points, /spl alpha/ and /spl beta/ parameters are introduced. They are used to form a closed set of eight unique patterns for simple critical points. These patterns are also basic building blocks for higher-order nonlinear vector fields. In order to study and compare a given set of vector fields, a measure of distance between different patterns of critical points is introduced. The basic patterns of critical points are mapped onto a unit circle in /spl alpha/-/spl beta/ space. The concept of the "Earth mover's distance" is used to compute the closeness between various pairs of vector fields, and a nearest-neighbor query is thus produced to illustrate the relationship between the given set of vector fields. This approach quantitatively measures the similarity and dissimilarity between vector fields. It is ideal for data compression of a large flow field, since only the number and types of critical points along with their corresponding /spl alpha/ and /spl beta/ parameters are necessary to reconstruct the whole field. It can also be used to better quantify the changes in time-varying data sets.

The authors study the topology of symmetric, second-order tensor fields. The results of the study can be readily extended to include general tensor fields through linear combination of symmetric tensor fields and vector fields. The goal is to represent their complex structure by a simple set of carefully chosen points, lines, and surfaces analogous to approaches in vector field topology. They extract topological skeletons of the eigenvector fields and use them for a compact, comprehensive description of the tensor field. Their approach is based on the premise: "analyze, then visualize". The basic constituents of tensor topology are the degenerate points, or points where eigenvalues are equal to each other. Degenerate points play a similar role as critical points in vector fields. In tensor fields they identify two kinds of elementary degenerate points, which they call wedge points and trisector points. They can combine to form more familiar singularities-such as saddles, nodes, centers, or foci. However, these are generally unstable structures in tensor fields. Based on the notions developed for 2D tensor fields, they extend the theory to include 3D degenerate points. Examples are given on the use of tensor field topology for the interpretation of physical systems.