Efficient Morse Decompositions of Vector Fields

Guoning Chen, Konstantin Mischaikow, Robert S. Laramee, and Eugene Zhang,
IEEE Transactions on Visualization and Computer Graphics, Vol 14(4), 2008, pp. 848-862.

Paper (PDF, 3.94 Mb).

This material is based upon work supported by the National Science Foundation under Grant No. CCF-0546881.

Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

Abstract

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 MCG's, while fast,
are overly conservative and usually result in MCG's 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 tau-maps, which typically provides finer MCG's than existing techniques. Furthermore, the choice of tau provides a natural tradeoff between the fineness of the MCG's and the computational costs.

We provide efficient implementations of Morse decomposition based on tau-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 tau-maps in addition to the original temporal tau-maps. These techniques provide additional tradeoffs between the quality of the MCG's 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.

Figures

1. Vector field topology defined using individual orbits (middle row: colored points and curves) are sensitive to: (a) discretization scheme, (b) noise in the data, and (c) numerical integration error. Topology based on Morse decomposition (bottom row: colored regions) are numerically stable with respect small purturbations.

 

2. Tau-map based Morse decomposition (middle and right) produces finer decomposition over the geometry-based approach (left). Larger tau value (right) tends to lead to finer decomposition. Compare the colored regions in the images in the middle and right. Also, compare the corresponding MCG graphs.

 

3. Morse decomposition using tau-maps (middle: temporary; right: spatial) are typically finer than geometry-based method (left). On the other hand, spatial-based method is typically faster with similar quality.