Morse Set Classification and Hierarchical Refinement Using Conley Index
KAUST Grant NumberKUS-C1-016-04
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AbstractMorse decomposition provides a numerically stable topological representation of vector fields that is crucial for their rigorous interpretation. However, Morse decomposition is not unique, and its granularity directly impacts its computational cost. In this paper, we propose an automatic refinement scheme to construct the Morse Connection Graph (MCG) of a given vector field in a hierarchical fashion. Our framework allows a Morse set to be refined through a local update of the flow combinatorialization graph, as well as the connection regions between Morse sets. The computation is fast because the most expensive computation is concentrated on a small portion of the domain. Furthermore, the present work allows the generation of a topologically consistent hierarchy of MCGs, which cannot be obtained using a global method. The classification of the extracted Morse sets is a crucial step for the construction of the MCG, for which the Poincar index is inadequate. We make use of an upper bound for the Conley index, provided by the Betti numbers of an index pair for a translation along the flow, to classify the Morse sets. This upper bound is sufficiently accurate for Morse set classification and provides supportive information for the automatic refinement process. An improved visualization technique for MCG is developed to incorporate the Conley indices. Finally, we apply the proposed techniques to a number of synthetic and real-world simulation data to demonstrate their utility. © 2006 IEEE.
CitationGuoning Chen, Qingqing Deng, Szymczak A, Laramee RS, Zhang E (2012) Morse Set Classification and Hierarchical Refinement Using Conley Index. IEEE Transactions on Visualization and Computer Graphics 18: 767–782. Available: http://dx.doi.org/10.1109/TVCG.2011.107.
SponsorsWe would like to thank Konstantin Mischaikow for the important discussions at the initial stage of this work. We are grateful for the valuable suggestions from Charles Hansen. We also appreciate Zhongzang Lin in helping preprocess the data and Edward Grundy and Timothy O'Keefe for proof-reading the paper. Finally, we wish to thank our anonymous reviewers for their constructive comments and suggestions. This work was supported by US National Scence Foundation (NSF) IIS-0546881 and CCF-0830808 awards, and in part, by EPSRC research grant EP/F002335/1. Guoning Chen was partially supported by King Abdullah University of Science and Technology (KAUST) Award No. KUS-C1-016-04.
CollectionsPublications Acknowledging KAUST Support
- Efficient morse decompositions of vector fields.
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- Visualization of Morse connection graphs for topologically rich 2D vector fields.
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- Issue date: 2013 Dec
- Robust Morse decompositions of piecewise constant vector fields.
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- Vector field editing and periodic orbit extraction using Morse decomposition.
- Authors: Chen G, Mischaikow K, Laramee RS, Pilarczyk P, Zhang E
- Issue date: 2007 Jul-Aug
- Squigraphs for fine and compact modeling of 3-D shapes.
- Authors: Aouada D, Krim H
- Issue date: 2010 Feb