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Towards Morse Theory for Point Cloud Data

By F. Cazals, C. Mueller, C. Robert, A. Roth, F. Cazals, C. Mueller, C. Robert, A. Roth and Inria Sophia-antipolis

Abstract

Morse theory provides a powerful framework to study the topology of a manifold from a function de ned on it, but discrete constructions have remained elusive due to the di culty of translating smooth concepts to the discrete setting. Consider the problem of approximating the Morse-Smale (MS) complex of a Morse function from a point cloud and an associated nearest neighbor graph (NNG). While following the constructive proof of the Morse homology theorem, we present novel concepts for critical points of any index, and the associated Morse-Smale diagram. Our framework has three key advantages. First, it requires elementary data structures and operations, and is thus suitable for high-dimensional data processing. Second, it is gradient free, which makes it suitable to investigate functions whose gradient is unknown or expensive to compute. Third, in case of under-sampling and even if the exact (unknown) MS diagram is not found, the output conveys information in terms of ambiguous ow, and the Morse theoretical version of topological persistence, which consists in canceling critical points by ow reversal, applies. On the experimental side, we present a comprehensive analysis of a large panel of bi-variate an

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.372.242
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