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.

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Notes

  1. 1.

    Note that some of these properties were first presented in a conference article [24] without proof.

  2. 2.

    Note that all simplicial complexes considered in this paper are finite. Indeed, in general, the extension of the proposed notions to the case of infinite complexes is not direct and is beyond the scope of this paper.

  3. 3.

    A property similar to Property 10 was proposed in [52] (Lemma 3.1) for the so-called \(k\)-deletable sets in the 2D square grid.

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Acknowledgments

This work received funding from the Agence Nationale de la Recherche, contract ANR-2010-BLAN-0205-03.

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Correspondence to Jean Cousty.

Appendix 1: Local cycles (proof of Lemma 5)

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. (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. (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 \(star({x}) \setminus \{y\}\). \(\square \)

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Cousty, J., Bertrand, G., Couprie, M. et al. Collapses and Watersheds in Pseudomanifolds of Arbitrary Dimension. J Math Imaging Vis 50, 261–285 (2014). https://doi.org/10.1007/s10851-014-0498-z

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Keywords

  • Watershed
  • Segmentation
  • Collapse
  • Topology preservation
  • Simplicial complex
  • Pseudomanifold