Morse set classification and hierarchical refinement using Conley index

Reliable analysis of vector elds is crucial for the rigorous interpretation of the ow data stemming from a wide range of\ud engineering applications. Morse decomposition of a vector field has proven a useful topological representation that is more numerically stable than previous vector field skeletons. In this paper, we enhance the procedure of Morse decomposition and propose an automatic refinement scheme to construct the Morse Connection Graph (MCG) of a given vector eld in a hierarchical fashion. Our framework allows a Morse set to be re ned through a local update of the flow combinatorialization, which leads to a more detailed MCG. This refined MCG has consistent topology with the original MCG because the refinement is conducted locally. The computation is faster than the original t-map approach because we reuse the previous tracing information and perform only local updates.\ud \ud The classification of the exetracted Morse sets is a crucial step for the construction of MCG. In this work, we advocate the use of Conley index for the classification. Conley index is a more general characteristic than Poincar´e index for the classi cation of flow dynamics. We present a framework to compute the Conley index of an isolating block in a flow. In addition, an efficient algorithm for computing an upper bound of the Conley index of any given Morse set is introduced to assist the automatic refinement process. Furthermore, an improved visualization technique for MCG is described which conveys the classification information of different Morse sets with the aid of the visualization of their Conley indices. Finally, we apply the proposed techniques to a number of synthetic and simulation data sets to demonstrate their utility