Morse homology descriptor for shape characterization

We propose a new topological method for shape description that is suitable for any multi-dimensional data set that can be modelled as a manifold. The description is obtained for all pairs (M, f), where M is a closed smooth manifold and f a Morse function defined on M. More precisely, we characterize the topology of all pairs of lower level sets (M_y, M_x) of f, where M_a = f^-1((-∞,a]), for all a ∈ R. Classical Morse theory is used to establish a link between the topology of a pair of lower level sets of f and its critical points lying between the two levels.