Gradient Vector Fields of Discrete Morse Functions and Watershed-cuts

Nicolas Boutry (1) , Gilles Bertrand (2) , Laurent Najman (2)

1 LRDE - Laboratoire de Recherche et de Développement de l'EPITA
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

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.

Mots clés: Topological Data Analysis; Mathematical Morphology; Discrete Morse Theory; Simplicial Stacks; Minimum Spanning Forest

Domaines: Topologie algébrique [math.AT]; Topologie géométrique [math.GT]; Mathématique discrète [cs.DM]

Nicolas Boutry, Gilles Bertrand, Laurent Najman. Gradient Vector Fields of Discrete Morse Functions and Watershed-cuts. DGMM 2022 -- IAPR Second International Conference on Discrete Geometry and Mathematical Morphology, Étienne Baudrier; Benoît Naegel; Adrien Krähenbühl; Mohamed Tajine, Oct 2022, Strasbourg, France. pp.1--13, ⟨10.1007/978-3-031-19897-7_4⟩. ⟨hal-03614850v2⟩