Morse Set Classification and Hierarchical Refinement Using Conley Index

Guoning Chen, Qingqing Deng, A. Szymczak, R. S. Laramee, E. Zhang

Research output: Contribution to journalArticlepeer-review

23 Scopus citations


Morse 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.
Original languageEnglish (US)
Pages (from-to)767-782
Number of pages16
JournalIEEE Transactions on Visualization and Computer Graphics
Issue number5
StatePublished - May 2012
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: We 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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