Discrete Morse Functions and Watersheds

TL;DR

This paper introduces Morse stacks, a class of functions equivalent to discrete Morse functions, and demonstrates their use in efficiently computing watersheds on pseudomanifolds, linking them to minimum spanning forests.

Contribution

It defines Morse stacks, proves their properties for watersheds, and provides a linear-time algorithm for watershed computation on pseudomanifolds.

Findings

Watersheds of Morse stacks are uniquely defined.

Watersheds can be computed in linear time.

Watersheds correspond to cuts in minimum spanning forests.

Abstract

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.