Analysis and comparison of vector fields can be done via a segmentation of the flow field into regions of similar behavior and a subsequent extraction of the topological skeleton. Our goals are a numerically stable computation of the topological skeleton including the extraction and classification of all closed streamlines, and a simple but consistent simplification of the topological skeleton to allow for a multi-scale vector field analysis. To achieve these goals we are investigating the applicability of a combinatorial approach to vector field topology.
In this project we develop discrete algorithms for the computation of vector field topology. Robin Forman has described a discrete Morse theory for general vector fields, our theoretical foundation. Standard algorithms for the extraction of the topological skeleton of a vector field involve many numerical challenges: finding all zeros of a vector field, integrating streamlines and streamsurfaces, the intersection of those, and the extraction of periodic orbits. While there are stable numerical algorithms to do this, the overall resulting framework has many parameters that may strongly influence the result.
Our approach to computational vector field topology avoids this problem by first computing a discrete representative of the vector field and then extracting the topological skeleton in this combinatorial setting. The main advantages of this approach are: