Decomposition and construction of higher-dimensional neighbourhood operations
Graphical abstract
Introduction
This paper develops a method to construct higher-dimensional digital morphological operations from a collection of one- and two-dimensional set operations along digital isothetic lines and on digital planes, respectively, in a space. Together with decomposition of digital objects, the decomposition of the neighbourhood shows that the neighbourhood-based operation in the higher-dimensional digital spaces can be decomposed into the union of neighbourhood-based operations in the lower-dimensional digital spaces [1].
Decompositions of morphological operations, such as closing, opening, the hit-or-miss transform, distance transform, boundary detection, skeletonisation [2], [3], [4] and thinning, clarify that the higher-dimensional operations are hierarchically constructed from those in the lower-dimensional spaces. Mathematically reformulations of algorithms based on decomposition properties of morphological operations bring theoretical bridge between mathematical descriptions and programme developments of morphological operations [5], [6].
Section snippets
Mathematical preliminaries
Setting Rn to be an n-dimensional Euclidean space, we express vectors in Rn as . Let Z be the set of all integers. The n-dimensional digital space Zn is set of all x for which all xi are integers. Then, we define the voxel centred at points in Zn as a unit hypercube in Rn. Definition 1 The voxels centred at the point y ∈ Zn in Rn is
In this paper, we deal with the connectivity and adjacency of the centroids of the voxels [7], which are elements of Zn.
The sets F⊕G and F⊖G such
Neighbourhood operations
The 2n-neighbourhood of the origin in Zn isLet for x ∈ Zn. Using the neighbourhood, connectivity of a pair of points a path between a pair of points in point set connectivity of a pair of in a point set are defined. Definition 7 If y ∈ N(x) and x ∈ N(y), x and y are connected to each other. Definition 8 For if there exists at least one sequence and for the string is a path from p1 ≔ x to p2 ≔ y. Definition 9 For a pair of points x and y, if
Digital simplex, complex and object
We first define digital simplices in Zn assuming 2n-connectivity of points. Definition 11 A digital k-simplex in Zn is the union of vertices of unit k-cube in Zn for 1 ≤ k ≤ n assuming 2n-connectivity of points.
The following procedure constructs digital simplices. Lemma 2 The recursive formconstructs digital k-simplex containing the origin 0 for k ≥ 1, where the operation U({ · }) removes redundant elements in the set { · }. Proof The elements of
Digital boundary manifold
We define the boundary of a point set in Zn. Definition 21 For a point set F, we call and the internal and external boundaries of F, respectively.
For the internal and external boundaries, we have the following relations. Lemma 4
This lemma derives the following theorem. Theorem 4 The boundary ∂±F of an n-dimensional digital object F is the union of its -dimensional boundaries.
Theorem 4 allows us to construct ∂±F from ∂±Fkα
Distance transform, skeleton set and thinning
For integer d, the distance set F(d) of F ∈ Zn is defined as following. Definition 26
The negative value of the distance in Eq. (52) expresses the distance between an object, which is a connected component, and a point which are not connected to the object. Since Definitions 21 and 26 imply the relations and . the recursive formscompute the distance set F(d) of F
Conclusions
We showed an explicit decomposition geometry of the neighbourhood in higher-dimensional digital space. This decomposition property clarifies that morphological operations in an n-dimensional digital space can be computed as the union of lower-dimensional morphological operations on isothetic digital subspaces intersecting with the digital objects. Decomposition procrdures of motphological operations provide us to construct new operations by the combination of the well-established operations in
Acknowledgements
This research was supported by the “Multidisciplinary Computational Anatomy and Its Application to Highly Intelligent Diagnosis and Therapy” project funded by a Grant-in-Aid for Scientific Research on Innovative Areas from MEXT, Japan, and by Grants-in-Aid for Scientific Research funded by the Japan Society for the Promotion of Science (Grant no. 17K00226 and 20K11881).
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