Gradient vector fields of discrete Morse functions and watershed-cuts (Q6160786)

In the present paper the authors explore the recently discovered connections between Mathematical Morphology (MM) to Discrete Morse Theory (DMT). Their first important result is the fact that discrete Morse functions (DMF), are equivalent, under certain constraints, to a class of spaces called \textit{simplicial stacks}, which are a class of weighted simplicial complexes whose upper threshold sets are also complexes. They show that, in fact (again under some constraints) any DMF is the opposite of a simplicial stack, and vice versa. Furthermore, they prove that, as in Discrete Morse Theory, the gradient vector field of a simplicial stack (seen as a discrete Morse function) can be viewed as the only relevant information to be considered. Moreover, the authors 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, a result that allows for the computation of a watershed-cut from a gradient vector field. For the entire collection see [Zbl 1511.68008].