Topological analysis of shapes using Morse theory

In this paper, we propose a novel method for shape analysis that is suitable for any multi-dimensional data set that can be modelled as a manifold. The descriptor is obtained for any pair (M, φ), where M is a closed smooth manifold and φ is a Morse function defined on M. More precisely, we characterize the topology of all pairs of sub-level sets (My, Mx) of φ, where Ma = φ−1((−∞, a  ]), for all a∈Ra∈R. Classical Morse theory is used to establish a link between the topology of a pair of sub-level sets of φ and its critical points lying between the two levels.