arXiv subjects

Gilles Bertrand

Publications and source records attributed to Gilles Bertrand.

7 recordsLinked to original sources

arXiv2026

Morse frames

In the context of discrete Morse theory, we introduce Morse frames, which are maps that associate a set of critical simplexes to all simplexes. The main example of Morse frames are the Morse references. In particular, these Morse references allow computing Morse complexes, an important tool for homology. We highlight the link between Morse references and gradient flows. We also propose a novel presentation of the Annotation algorithm for persistent cohomology, as a variant of a Morse frame. Finally, we propose another construction, that takes advantage of the Morse reference for computing the Betti numbers in mod 2 arithmetic.

cs.DM↗

arXiv2026

Morse sequences on stacks and flooding sequences

This paper builds upon the framework of \emph{Morse sequences}, a simple and effective approach to discrete Morse theory. A Morse sequence on a simplicial complex consists of a sequence of nested subcomplexes generated by expansions and fillings-two operations originally introduced by Whitehead. Expansions preserve homotopy, while fillings introduce critical simplexes that capture essential topological features. We extend the notion of Morse sequences to \emph{stacks}, which are monotonic functions defined on simplicial complexes, and define \emph{Morse sequences on stacks} as those whose expansions preserve the homotopy of all sublevel sets. This extension leads to a generalization of the fundamental collapse theorem to weighted simplicial complexes. Within this framework, we focus on a refined class of sequences called \emph{flooding sequences}, which exhibit an ordering behavior similar to that of classical watershed algorithms. Although not every Morse sequence on a stack is a flooding sequence, we show that the gradient vector field associated with any Morse sequence can be recovered through a flooding sequence. Finally, we present algorithmic schemes for computing flooding sequences using cosimplicial complexes.

cs.DM↗

arXiv2025

Computing gradient vector fields with Morse sequences

We rely on the framework of Morse sequences to enable the direct computation of gradient vector fields on simplicial complexes. A Morse sequence is a filtration from a subcomplex $L$ to a complex $K$ via elementary expansions and fillings, naturally encoding critical and regular simplexes. Maximal increasing and minimal decreasing schemes allow constructing these sequences, and are linked to algorithms like Random Discrete Morse and Coreduction. Extending the approach to cosimplicial complexes ($S=K\setminus L$) allows for efficient computation using reductions, perforations, coreductions, and coperforations. We further generalize to $F$-sequences, which are Morse sequences weighted by an arbitrary stack function $F$, and provide algorithms to compute maximal and minimal sequences. A particular case is when the stack function is given through a vertex map, common in topological data analysis. For injective maps, the complex decomposes into lower stars, recovering established methods and enabling parallel computation; for non-injective maps, our approach applies directly without requiring perturbations. Thus, the paper adopts Morse sequences as a framework that simplifies and connects some important existing propagation-based methods, while also introducing new schemes that extend their scope and practical applicability.

cs.DM↗

arXiv2025

Morse Sequences: A simple approach to discrete Morse theory

In this paper, we develop the notion of a Morse sequence, which provides an alternative approach to discrete Morse theory, and which is both simple and effective. A Morse sequence on a finite simplicial complex is a sequence composed solely of two elementary operations, that is, expansions (the inverse of a collapse), and fillings (the inverse of a perforation). In a dual manner, a Morse sequence may be obtained by considering only collapses and perforations. Such a sequence is another way to represent the gradient vector field of an arbitrary discrete Morse function. To each Morse sequence, we assign a reference map and an extension map. A reference map associates a set of critical simplexes to each simplex of a given complex, and an extension map associates a set of simplexes to each critical simplex. By considering the boundary of each critical simplex, we obtain a chain complex from these maps, which corresponds precisely to the Morse complex. We show that, when restricted to homology, an extension map is the inverse of a reference map. Also we show that these two maps allow us to recover directly the isomorphism theorem between the homology of an object and the homology of its Morse complex. At last, we introduce the notion of a flow complex, which is based solely on extension maps. We prove that this notion is equivalent to the classical one based on gradient flows.

cs.DM↗

arXiv2024

Morse sequences

We introduce the notion of a Morse sequence, which provides a simple and effective approach to discrete Morse theory. A Morse sequence is a sequence composed solely of two elementary operations, that is, expansions (the inverse of a collapse), and fillings (the inverse of a perforation). We show that a Morse sequence may be seen as an alternative way to represent the gradient vector field of an arbitrary discrete Morse function. We also show that it is possible, in a straightforward manner, to make a link between Morse sequences and different kinds of Morse functions. At last, we introduce maximal Morse sequences, which formalize two basic schemes for building a Morse sequence from an arbitrary simplicial complex.

cs.CV↗

arXiv2023

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.

cs.DM↗

arXiv2022

Gradient Vector Fields of Discrete Morse Functions and Watershed-cuts

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.

cs.DM↗