Collapses and Watersheds in Pseudomanifolds of Arbitrary Dimension

Abstract

This work is settled in the framework of abstract simplicial complexes. We propose a definition of a watershed and of a collapse (i.e., a homotopic retraction) for maps defined on pseudomanifolds of arbitrary dimension. Then, we establish two important results linking watersheds and homotopy. The first one generalizes a property known for distance transforms in a continuous setting to any map on pseudomanifolds: a watershed of any map is a subset of an ultimate collapse of the support of this map. The second result establishes, through an equivalence theorem, a deep link between watershed and collapse of maps: any watershed of any map can be straightforwardly obtained from an ultimate collapse of this map, and conversely any ultimate collapse of the initial map straightforwardly induces a watershed.

Appendix 1: Local cycles (proof of Lemma 5)

This appendix section is devoted to a property of pseudomanifolds that allows Lemma 5 to be established. Let us first illustrate this property on an example. In Fig. 11, the set $\{x_0, \ldots , x_7\}$ is a cycle. This cycle is said “local to the point $x$” in the sense that any of its elements belongs to $star({x})$. We prove in this appendix section that there exists a cycle local to each $(n-2)$-face, in any $n$-pseudomanifold. More remarkably, Theorem 44 states, for any $(n-2)$-face $x$ of any $n$-pseudomanifold, that the set of $n$-faces of any connected component of $star({x})$ is a cycle. From this result, the proof of Lemma 5 will be easily derived.

Let $X$ be a nonempty set of $n$-faces of $\mathbb {M}$. We say that $X$ is a cycle (for $\mathbb {M})$, if there exists a simple path $\pi = \langle x_0, \ldots , x_\ell \rangle$ in $\mathbb {M}$ such that $X = \{x_0, \ldots , x_\ell \}$ and such that $x_0 \cap x_\ell$ is a $(n-1)$-face of $\mathbb {M}$.

Theorem 44

Let $x$ be an $(n-2)$-face of $\mathbb {M}$. The set of $n$-faces of any component of $star({x})$ is a cycle for $\mathbb {M}$.

In order to prove Theorem 44, we first state Lemma 45 and Corollary 46.

By its very definition, any $k$-face (with $k \in \{0,\ldots ,n\}$) contains $k+1$ elements. Using this fact the following result can be proved easily.

Lemma 45

Let $x$ be an $(n-2)$-face of $\mathbb {M}$, and let $x_0$ be an $n$-face in $star({x})$. Then, there exist exactly two distinct $(n-1)$-faces in $star({x})$ that are included in $x_0$.

Corollary 46

Let $x$ be an $(n-2)$-face of $\mathbb {M}$, and let $x_1$ be an $n$-face in $star({x})$. Then, there exist exactly two distinct $n$-faces $x_0$ and $x_2$ in $star({x})$ whose intersections with $x_1$ are $(n-1)$-faces.

Proof

(of Theorem 44) Since any component of $star({x})$ is a star, and since any nonempty star contains an $n$-face of $\mathbb {M}$ (Remark 7), to study all components of $star({x})$, it is sufficient to consider, for any $n$-face $x_0$ in $star({x})$, the component of $star({x})$ that contains $x_0$. Let $x_0$ be any $n$-face in $star({x})$, and let $X$ be the set of all $n$-faces of the component of $star({x})$ that contains $x_0$.

• (1) Let us first prove that $X$ includes a cycle. As a consequence of Corollary 46, we may always find $x_1 \in X$ such that $\langle x_0, x_1 \rangle$ is a simple path in $star({x})$. Using again Corollary 46, we can construct, by induction, a simple path $\pi = \langle x_0, \ldots , x_\ell \rangle$ in $star({x})$ such that the only two $n$-faces in $star({x})$ whose intersections with $x_\ell$ are $(n-1)$-faces of $\mathbb {M}$ both belong to $\{x_0, \ldots x_\ell \}$. By construction, one of these two $n$-faces is $x_{\ell -1}$. Let us denote the other one by $z$. Necessarily $z = x_i$, for some $i \in \{0, \ldots , \ell - 2\}$. If $i > 0$, then $x_{i-1}$, $x_{i+1}$ and $x_\ell$ are three distinct faces in $star({x})$ whose intersections with $x_i$ are $(n-1)$-faces of $\mathbb {M}$, which constitutes a contradiction with Corollary 46. Thus, we necessarily have $z = x_0$. Hence, the set $\{x_0, \ldots , x_\ell \} \subseteq X$ is a cycle.

• (2) Let us now prove, by contradiction, that $X = \{x_0, \ldots , x_\ell \}$, hence, by (1), that $X$ is a cycle. Suppose that there exists an element $z$ in $X$ such that $z \notin \{x_0, \ldots , x_\ell \}$. By definition of $X$, there exists, in $star({x})$, a simple path $\pi ' = \langle y_0, \ldots y_m \rangle$ from $x_0 = y_0$ to $z = y_m$. Let $k \in \{1, \ldots , m\}$ be the lowest index such that $y_k \notin \{x_0, \ldots , x_\ell \}$. Hence, $y_{k-1} \in \{x_0, \ldots , x_\ell \}$. By Corollary 46, there exist exactly two $n$-faces in $star({x})$, whose intersections with $y_{k-1}$ are $(n-1)$-faces of $\mathbb {M}$. By construction of $\pi$, these two $n$-faces belong to $\{x_0, \ldots , x_\ell \}$. Hence, we have $y_k \in \{x_0, \ldots , x_\ell \}$, a contradiction. Thus, since by (1), we have $\{x_0, \ldots , x_\ell \} \subseteq X$, we deduce that $X = \{ x_0, \ldots , x_\ell \}$. $\square$

Theorem 44 can be easily verified on Fig. 11. It can also be verified on the 2-pseudomanifold shown in Fig. 9b. In particular, let $x$ denote the 0-face represented by a light gray dot. It can be seen that $star({x})$ includes two components: one is made of the triangles and edges at the left of $x$, and the other is made of the triangles and edges at the right of $x$. It can be easily seen that the sets of triangles associated to these two components are cycles for the considered $2$-pseudomanifold.

Remark also that, if $\mathbb {M}$ is a not a pseudomanifold, then Theorem 44 is, in general, not true. For instance, let us consider the complex of Fig. 9c, which is not a pseudomanifold, and let $x$ be any of the two points that belong to the edge depicted in light gray (i.e., the pinching). The set $star({x})$ itself is the only component of $star({x})$. However, it can be verified that the set of all triangles in $star({x})$ is not a cycle.

Proof

(of Lemma 5) Clearly, the two $n$-faces $x_0$ and $x_1$ belong to the same component $X$ of $star({x})$. The set $X_n$ of all $n$-faces of $X$ is, by Theorem 44, a cycle. Thus, as $x_0$ and $x_1$ belong to $X$, it can be seen that there exists, in $star({x})$, two distinct simple paths $\pi = \langle y_0 = x_0, \ldots , y_\ell =x_1 \rangle$ and $\pi ' = \langle z_0 = x_0, \ldots , z_m =x_1 \rangle$ from $x_0$ to $x_1$ such that $\{y_1, \ldots , x_\ell \} \cap \{z_1, \ldots , z_{m-1}\} = \emptyset$. At least one of $\pi$ and $\pi '$ is a path in $star({X}) \setminus \{y\}$. Therefore, $x_0$ and $x_1$ are linked for $star({x}) \setminus \{y\}$. $\square$