Issue No. 05 - May (2012 vol. 18)
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TVCG.2011.107
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.
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
A. Szymczak, Qingqing Deng, Guoning Chen, R. S. Laramee and E. Zhang, "Morse Set Classification and Hierarchical Refinement Using Conley Index," in IEEE Transactions on Visualization & Computer Graphics, vol. 18, no. , pp. 767-782, 2012.