Multi-Scale 3D Morse Complexes

Morse theory studies the relationship between the topology of a manifold M and the critical points of a scalar function f defined over M. Morse and Morse-Smale complexes, defined by critical points and integral lines of f, induce a subdivision of M into regions of uniform gradient flow, representing the morphology of M in a compact way. Function f can be simplified by canceling its critical points in pairs, thus simplifying the morphological representation of M, given by Morse and Morse-Smale complexes of f. Here, we propose a compact representation for the two Morse complexes in 3D, which is based on encoding the incidence relations of their cells, and on exploiting the duality among the complexes. We define cancellation operations, and their inverse expansion operations, on the Morse complexes and on their dual representation. We propose a multi-scale representation of the Morse complexes which provides a description of such complexes, and thus of the morphology of a 3D scalar field, at different levels of abstraction. This representation allows us also to perform selective refinement operations to extract description of the complexes which varies in different parts of the domain, thus improving efficiency on large data sets, and eliminating the noise in the data through topology simplification.