Powerful Parallel and Symmetric 3D Thinning Schemes Based on Critical Kernels

Abstract

The main contribution of the present article consists of new 3D parallel and symmetric thinning schemes which have the following qualities:

• They are effective and sound, in the sense that they are guaranteed to preserve topology. This guarantee is obtained thanks to a theorem on critical kernels;

• They are powerful, in the sense that they remove more points, in one iteration, than any other symmetric parallel thinning scheme;

• They are versatile, as conditions for the preservation of geometrical features (e.g., curve extremities or surface borders) are independent of those accounting for topology preservation;

• They are efficient: we provide in this article a small set of masks, acting in the grid ℤ³, that is sufficient, in addition to the classical simple point test, to straightforwardly implement them.

Appendix

Proof of Theorem 5

The following result is a consequence of Theorem 4.3(iii) of [8], note that this theorem holds for complexes of arbitrary dimensions.

Let $S \in \mathbb{X}^{3}$ , let R⊆S, and let T such that R⊆T⊆S. If R ⁻ contains the critical kernel of S, then T ⁻ collapses onto R ⁻ .

Now let $X \in \mathbb{X}^{3}$ and let Y⊆X such that Y ⁻ contains the critical kernel of X.

Let X∖Y={x ₁,…,x _k }. Thus the faces of X∖Y are ordered according to their indices in an arbitrary way.

We set X ₀=X, X _i =X∖{x ₁,…,x _i }, i∈[1,k].

Let i∈[1,k]. The complex X _i contains Y, thus $X_{i}^{-}$ contains all the faces which are critical for X. By the above result $X_{i-1}^{-}$ collapses onto $X_{i}^{-} = [X_{i-1} \setminus \{ x_{i}\}]^{-}$, which means that x _i is simple for X _i−1 and that X _i is an elementary thinning of X _i−1 (Definition 1). Thus, the xel complex Y=X _k is a thinning of X=X ₀. □

Proof of Theorem 7

Let $X \in \mathbb{X}^{3}$ and let Y⊆X.

• (i) Suppose Y ⁻ contains the critical kernel of X. Let Z such that Y⊆Z⊆X. Since Z ⁻ contains the critical kernel of X, by Theorem 5, Z is a thinning of X.

• (ii) Suppose Y ⁻ does not contain the critical kernel of X. Then, there exists a face which is critical for X in X ⁻∖Y ⁻. There exists also a face x in X ⁻∖Y ⁻ which is $\mathcal{M}$-critical for X. Then, the $\mathcal{M}$-crucial clique $C = x^{+}_{X}$ is non-simple for X (see [12], Theorem 28, and Remark 19), i.e., the set Z=X∖C is not a thinning of X. We observe that Y⊆Z. Thus, there exists Z such that Y⊆Z⊆X, and such that Z is not a thinning of X.

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Proof of Proposition 8

We proved the proposition with the help of a computer program. All 2²⁶ possible configurations of the neighborhood of a point x in X were examined, and for each of them the equivalence between Definition 1 and conditions (1) and (2) was successfully tested. □

Proof of Proposition 10

We proved the proposition with the help of a computer program. All 2¹⁶ possible configurations of the $\mathcal{K}$-neighborhood of a 2-clique C in X were examined, and for each of them the equivalence between Definition 3 and conditions (1) and (2) was successfully tested. □

Proof of Proposition 11

We proved the proposition with the help of a computer program. All 2⁸ possible configurations of the $\mathcal{K}$-neighborhood of a 1-clique C in X were examined, and for each of them the equivalence between Definition 3 and the condition was successfully tested. □

Proof of Theorem 16 and Theorem 15

We proved the proposition with the help of a computer program. The condition “$\mathcal{K}^{*}(C) \cap X$ is reducible” could not be checked directly because of combinatorial explosion, so we proved the property recursively with respect to the cardinality of $S = \mathcal{K}^{*}(C) \cap X$. More precisely, knowing that the proposition is trivially true for |S|=0, we checked it for all possible configurations of n elements of S, for n=1,…,N (with N=26,16,8 for d=3,2,1 respectively, C being a d-clique), based on the fact that the proposition was already proved for n−1. For simplicity, the configurations of n elements out of N were generated by scanning all possible 2^N configurations and selecting those with precisely n elements. □

Proof of Theorem 17

We proved the theorem with the help of a computer program. It is trivially true when |S|=0 (case of a 0-clique). We checked it for all possible configurations of N elements of S, (with N=26,16,8 for d=3,2,1 respectively, C being a d-clique). For each of these configurations, we tested for each simple voxel x of S the reducibility of S∖{x}, thanks to Theorem 16 and to characterizations of regular cliques (Proposition 13) and simple points (Proposition 8). □

Proof of Theorem 21

Let $X \in \mathbb{V}^{3}$, let C be a d-clique which is critical for X, and let x=⋂{x∈C}.

Suppose C is not $\mathcal{M}$-crucial for X. Then there exists a d′-clique D which is critical for X, and such that x is a proper face of the face y=⋂{x∈D}. Thus, we have d<d′ and D is a proper subset of C.

• (i) Suppose D is $\mathcal{D}$-crucial for X. It means that C contains a voxel belonging to a d′-clique which is $\mathcal{D}$-crucial for X, with d′>d. Thus, C is not $\mathcal{D}$-crucial for X.

• (ii) Suppose D is not $\mathcal{D}$-crucial for X. It means that D (hence also C) contains a voxel belonging to a d″-clique which is $\mathcal{D}$-crucial for X, with d″>d′>d. Again, C cannot be $\mathcal{D}$-crucial for X.

Thus, a clique which is $\mathcal{D}$-crucial for X is necessarily $\mathcal{M}$-crucial for X. It follows that the $\mathcal{D}$-crucial kernel of X is a subset of its $\mathcal{M}$-crucial kernel. □