Discrete Morse Functions and Watersheds

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.

Appendices

Appendix A Normal pseudomanifolds

A normal pseudomanifold is usually defined as a pseudomanifold that satisfies a certain link condition, which corresponds to a local property [29,30,31]. In this section, we show that this definition is equivalent to the one given in Definition 6.

Let S be a finite set of simplexes. If x and y are facets of S, a p-chain (in S) from x to y is a sequence $\langle x = x_0,...,x_k = y \rangle$ of facets of S such that, for each $i \in [0,k-1]$, $x_i \cap x_{i+1}$ is a q-face of S, with $q \ge p$. The set S is p-connected if, for any two facets x, y in S, there is a p-chain in S from x to y.

We observe that:

• A complex is connected if and only if it is 0-connected.

• A d-pure complex is strongly connected if and only if it is $(d-1)$-connected.

Let X be a complex. Two faces $x,y \in X$ are adjacent if $x \cup y \in X$. The link of $x \in X$ in X is the complex $lk(x,X) = \{y \in X \; \mid \; x \cap y = \emptyset$ and $x \cup y \in X \}$. The star of $x \in X$ in X is the set $st(x,X) = \{y \in X \; \mid \; x \subseteq y \}$.

Let X be a d-pseudomanifold. We say that X satisfies the link condition if lk(x, X) is connected whenever x is a p-face of X and $p \le d-2$.

Let X be a complex and x be a p-face of X. Let $st^*(x,X) = st(x,X) {\setminus } \{ x \}$. We have $lk(x,X) = \{y {\setminus } x \; \mid \; y \in st^*(x,X) \}$ and $st^*(x,X) = \{z \cup x \; \mid \; z \in lk(x,X) \}$.

We note that there is a set isomorphism between lk(x, X) and $st^*(x,X)$, which preserves set inclusion. If $y \in st^*(x,X)$, the corresponding face $y \setminus x$ of lk(x, X) is such that $dim(y \setminus x) = dim(y) - (p + 1)$. Thus, $dim(y {\setminus } x) = dim(y) - p^+$, where $p^+ = p +1$ is the number of elements in x.

Let X be a d-pseudomanifold and x be a p-face of X. Let $p^+ = p+1$ and $d' = d - p^+$. The following facts are a direct consequence of the above isomorphism:

• The complex lk(x, X) is $d'$-pure.

• The complex lk(x, X) is non-branching.

• The set $st^*(x,X)$ is q-connected if and only if lk(x, X) is $q'$-connected, with $q' = q - p^+$.

Proposition 32

A pseudomanifold is normal if and only if it satisfies the link condition.

Proof

Let X be a d-pseudomanifold.

• 1. Suppose X satisfies the link condition and let S be a connected open subset of X. Let x and y be two d-faces of S. By Remark 2, there exists a p-chain $\pi$ in S from x to y. Thus, $\pi = \langle x = x_0,\ldots ,x_k = y \rangle$ is a sequence of facets of S such that, for each $i \in [0,k-1]$, $x_i \cap x_{i+1}$ is a q-face of S, with $q \ge p$. We choose $\pi$ such that p is maximal and, if p is maximal, such that the number $K(\pi )$ of p-faces $x_i \cap x_{i+1}$, with $i \in [0,k-1]$, is minimal. If $p = d-1$, it means that S is strongly connected; then, we are done. Suppose $p < d-1$ and let $x_i, x_{i+1}$ such that $z = x_i \cap x_{i+1}$ is a p-face. Since X satisfies the link condition, lk(z, X) is connected. By the isomorphism between lk(z, X) and $st^*(z,X)$, it follows there is a q-chain $\langle x_i = w_0,\ldots ,w_l = x_{i+1} \rangle$ in $st^*(z,X)$ with $q >p$. Therefore, $\pi ' = \langle x = x_0,\ldots ,x_i = w_0,\ldots ,w_l = x_{i+1},\ldots ,x_k = y \rangle$ is a p-chain in S from x to y. But we have $K(\pi ') < K(\pi )$, a contradiction. Thus, each connected open subset of X is strongly connected.

• 2. Suppose X is strictly connected. That is, any connected open subset of X is $(d-1)$-connected. Let x be a p-face of X with $p \le d-2$. The set st(x, X) is a connected open subset of X, thus it is $(d-1)$-connected. Since $p < d-1$, it means that $st^*(x,X)$ is $(d-1)$-connected. Therefore, $st^*(x,X)$ is strongly connected. By the isomorphism between lk(x, X) and $st^*(x,X)$, it follows that lk(x, X) is strongly connected. Thus, lk(x, X) is connected.

$\square$

In the second part of the proof of Proposition 32, we showed that lk(x, X) is strongly connected. Consequently, we have the following characterization of a normal pseudomanifold.

Proposition 33

A pseudomanifold X is normal if and only if, for each p-face x of X, with $p \le d-2$, the complex lk(x, X) is a pseudomanifold.

Appendix B Discrete Morse functions

Let us consider the following definition of a discrete Morse function:

Definition 34

(Morse function) Let X be a complex and let F be a map from X to $\mathbb {Z}$. We say that F is a discrete Morse function on X if any face of X is in at most one covering pair (x, y) in X such that $F(x) \ge F(y)$. If F is a discrete Morse function, we say that such a pair is a regular pair of F.

It may be checked that this definition is equivalent to the classical one given by Forman (See Def. 2.1 and Lemma 2.5 of [43]).

In this way, the gradient vector field of a discrete Morse function F, written $\overrightarrow{\textrm{grad}} (F)$, is the set composed of all regular pairs of F.

The following restriction of a discrete Morse function will lead us to Morse stacks.

We say that a discrete Morse function F on X is flat if we have $F(x) = F(y)$ whenever (x, y) is a regular pair of F, that is, if each regular pair of F is a flat pair of F.

We can check that a map F from X to $\mathbb {Z}$ is a flat discrete Morse function if and only if:

• 1. Each covering pair (x, y) in X is such that $F(x) \le F(y)$;

• 2. Each face of X is in at most one flat pair of F.

Therefore, if we consider the function $-F$, we obtain the following:

Proposition 35

Let X be a complex and let F be a map from X to $\mathbb {Z}$. The map F is a Morse stack on X if and only if the map $-F$ is a flat discrete Morse function on X.

The following proposition claims that, up to an equivalence, we may assume that any discrete Morse function is flat (see Def. 2.27 and Prop. 4.16 of [10]).

Proposition 36

(from [10]) If F is a discrete Morse function on X, then there exists a flat discrete Morse function G on X such that, for every covering pair (x, y) in X, we have $F(x) \ge F(y)$ if and only if $G(x) \ge G(y)$. In other words, the function G is such that $\overrightarrow{\textrm{grad}} (G) = \overrightarrow{\textrm{grad}} (F)$.