Gradient Vector Fields of Discrete Morse Functions and Watershed-cuts

TL;DR

This paper explores the relationship between discrete Morse functions, gradient vector fields, and watershed cuts, revealing that the gradient vector field encapsulates essential information for analyzing simplicial stacks and deriving watershed cuts.

Contribution

It establishes that the gradient vector field of a simplicial stack fully characterizes the structure and enables computation of watershed cuts from the dual graph's minimum spanning forest.

Findings

Gradient vector field captures all relevant information of a simplicial stack.

Minimum Spanning Forest of the dual graph is induced by the gradient vector field.

Watershed-cut can be computed directly from the gradient vector field.

Abstract

In this paper, we study a class of discrete Morse functions, coming from Discrete Morse Theory, that are equivalent to a class of simplicial stacks, coming from Mathematical Morphology. We show that, as in Discrete Morse Theory, we can see the gradient vector field of a simplicial stack (seen as a discrete Morse function) as the only relevant information we should consider. Last, but not the least, we also show that the Minimum Spanning Forest of the dual graph of a simplicial stack is induced by the gradient vector field of the initial function. This result allows computing a watershed-cut from a gradient vector field.