Efficient Morse Decompositions of Vector Fields

Existing topology-based vector field analysis techniques rely on the ability to extract the individual trajectories such as fixed points, periodic orbits and separatrices which are sensitive to noise and errors introduced by simulation and interpolation. This can make such vector field analysis unsuitable for rigorous interpretations. We advocate the use of Morse decompositions, which are robust with respect to perturbations, to encode the topological structures of a vector field in the form of a directed graph, called a Morse connection graph (MCG). While an MCG exists for every vector field, it need not be unique. Previous techniques for computing MCGs, while fast, are overly conservative and usually result in MCGs that are too coarse to be useful for the applications. To address this issue, we present a new technique for performing Morse decomposition based on the concept of τ-maps, which typically provides finer MCGs than existing techniques. Furthermore, the choice of τ provides a natural tradeoff between the neness of the MCGs and the computational costs. We provide efcient implementations of Morse decomposition based on τ-maps, which include the use of forward and backward mapping techniques and an adaptive approach in constructing better approximations of the images of the triangles in the meshes used for simulation. Furthermore, we propose the use of spatial τ-maps in addition to the original temporal τ-maps. These techniques provide additional tradeoffs between the quality of the MCGs and the speed of computation. We demonstrate the utility of our technique with various examples in the plane and on surfaces including engine simulation datasets