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Efficient Morse Decompositions of Vector Fields
Guoning Chen, Konstantin Mischaikow, Robert S. Laramee, and Eugene Zhang,
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).
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).
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.