Morse Set Classification and Hierarchical Refinement Using Conley Index

Handle URI:
http://hdl.handle.net/10754/598894
Title:
Morse Set Classification and Hierarchical Refinement Using Conley Index
Authors:
Guoning Chen,; Qingqing Deng,; Szymczak, A.; Laramee, R. S.; Zhang, E.
Abstract:
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.
Citation:
Guoning 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.
Publisher:
Institute of Electrical and Electronics Engineers (IEEE)
Journal:
IEEE Transactions on Visualization and Computer Graphics
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
May-2012
DOI:
10.1109/TVCG.2011.107
PubMed ID:
21690641
Type:
Article
ISSN:
1077-2626
Sponsors:
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.
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Full metadata record

DC FieldValue Language
dc.contributor.authorGuoning Chen,en
dc.contributor.authorQingqing Deng,en
dc.contributor.authorSzymczak, A.en
dc.contributor.authorLaramee, R. S.en
dc.contributor.authorZhang, E.en
dc.date.accessioned2016-02-25T13:43:15Zen
dc.date.available2016-02-25T13:43:15Zen
dc.date.issued2012-05en
dc.identifier.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.en
dc.identifier.issn1077-2626en
dc.identifier.pmid21690641en
dc.identifier.doi10.1109/TVCG.2011.107en
dc.identifier.urihttp://hdl.handle.net/10754/598894en
dc.description.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.en
dc.description.sponsorshipWe 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.en
dc.publisherInstitute of Electrical and Electronics Engineers (IEEE)en
dc.subjecthierarchical refinementen
dc.subjectMorse decompositionen
dc.subjecttopology refinementen
dc.subjectupper bound of Conley indexen
dc.subjectvector field topologyen
dc.titleMorse Set Classification and Hierarchical Refinement Using Conley Indexen
dc.typeArticleen
dc.identifier.journalIEEE Transactions on Visualization and Computer Graphicsen
dc.contributor.institutionUniversity of Utah, Salt Lake City, United Statesen
dc.contributor.institutionOregon State University, Corvallis, United Statesen
dc.contributor.institutionColorado School of Mines, Golden, United Statesen
dc.contributor.institutionSwansea University, Swansea, United Kingdomen
kaust.grant.numberKUS-C1-016-04en

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