Discrete Morse Functions and Watersheds

Any watershed, when defined on a stack on a normal pseudomanifold of dimension d, is a pure (d - 1)-subcomplex that satisfies a drop-of-water principle. In this paper, we introduce Morse stacks, a class of functions that are equivalent to discrete Morse functions. We show that the watershed of a Morse stack on a normal pseudomanifold is uniquely defined and can be obtained with a linear-time algorithm relying on a sequence of collapses. Last, we prove that such a watershed is the cut of the unique minimum spanning forest, rooted in the minima of the Morse stack, of the facet graph of the pseudomanifold.