Discrete morse gradient fields with stable matching
Ao Paix, Thomas Lewiner · 2013
Forman's discrete Morse theory for cell complexes essentially relies on a discrete analogue of gradient fields, defined as an acyclic matching between incident cells. One can easily extract from this object several pieces of topological and geometrical information of a cell complex such as homology or the Morse-Smale decomposition. Most constructions of this discrete gradient are based on some variant of greedy pairing of adjacent cells, similar to a steepest descent algorithm. In this work, an equivalent formulation in terms of stable matching is proposed to study the geometric accuracy of the construction. It further simplifies the proofs of previous results regarding the behavior of the gradient field and the position of critical points.