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Morse Set Classification and Hierarchical Refinement Using Conley Index
May 2012 (vol. 18 no. 5)
pp. 767-782
ASCII Text
x 
 
Guoning Chen, Qingqing Deng, A. Szymczak, R. S. Laramee, E. Zhang, "Morse Set Classification and Hierarchical Refinement Using Conley Index," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 5, pp. 767-782, May, 2012.
 
 
BibTex
x
 
@article{ 10.1109/TVCG.2011.107,
author = { Guoning Chen and Qingqing Deng and A. Szymczak and R. S. Laramee and E. Zhang},
title = {Morse Set Classification and Hierarchical Refinement Using Conley Index},
journal ={IEEE Transactions on Visualization and Computer Graphics},
volume = {18},
number = {5},
issn = {1077-2626},
year = {2012},
pages = {767-782},
doi = {http://doi.ieeecomputersociety.org/10.1109/TVCG.2011.107},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Visualization and Computer Graphics
TI - Morse Set Classification and Hierarchical Refinement Using Conley Index
IS - 5
SN - 1077-2626
SP767
EP782
EPD - 767-782
A1 - Guoning Chen,
A1 - Qingqing Deng,
A1 - A. Szymczak,
A1 - R. S. Laramee,
A1 - E. Zhang,
PY - 2012
KW - vectors
KW - data visualisation
KW - graph theory
KW - mathematics computing
KW - numerical stability
KW - pattern classification
KW - set theory
KW - topology
KW - real-world simulation data
KW - Morse set classification
KW - hierarchical refinement
KW - Conley index
KW - Morse decomposition
KW - vector field
KW - numerically stable topological representation
KW - Morse connection graph
KW - flow combinatorialization graph
KW - Poincare index
KW - Betti numbers
KW - visualization technique
KW - synthetic data
KW - Indexes
KW - Topology
KW - Orbits
KW - Approximation methods
KW - Upper bound
KW - Trajectory
KW - Electrocardiography
KW - hierarchical refinement.
KW - Morse decomposition
KW - vector field topology
KW - upper bound of Conley index
KW - topology refinement
VL - 18
JA - IEEE Transactions on Visualization and Computer Graphics
ER -
 
Guoning Chen, Sci. Comput. & Imaging Inst., Univ. of Utah, Salt Lake City, UT, USA
Qingqing Deng, Sch. of Electr. Eng. & Comput. Sci, Oregon State Univ., Corvallis, OR, USA
A. Szymczak, Dept. of Math. & Comput. Sci., Colorado Sch. of Mines, Golden, CO, USA
R. S. Laramee, Dept. of Comput. Sci., Swansea Univ., Swansea, UK
E. Zhang, Sch. of Electr. Eng. & Comput. Sci, Oregon State Univ., Corvallis, OR, USA
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 Poincaré 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.

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Index Terms:
vectors,data visualisation,graph theory,mathematics computing,numerical stability,pattern classification,set theory,topology,real-world simulation data,Morse set classification,hierarchical refinement,Conley index,Morse decomposition,vector field,numerically stable topological representation,Morse connection graph,flow combinatorialization graph,Poincare index,Betti numbers,visualization technique,synthetic data,Indexes,Topology,Orbits,Approximation methods,Upper bound,Trajectory,Electrocardiography,hierarchical refinement.,Morse decomposition,vector field topology,upper bound of Conley index,topology refinement
Citation:
Guoning Chen, Qingqing Deng, A. Szymczak, R. S. Laramee, E. Zhang, "Morse Set Classification and Hierarchical Refinement Using Conley Index," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 5, pp. 767-782, May 2012, doi:10.1109/TVCG.2011.107
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