Elsevier

Pattern Recognition Letters

Volume 135, July 2020, Pages 321-328
Pattern Recognition Letters

Decomposition and construction of higher-dimensional neighbourhood operations

https://doi.org/10.1016/j.patrec.2020.04.015Get rights and content

Highlights

  • The neighbourhood in an n-space is decomposed into neighbourhoods in (n1)-space.

  • Morphological operations in an n-space is the union of those on lines.

  • The object boundary is constructed from those in lower dimensional space.

  • The distance transform is computed from those in lower-dimensional space.

Abstract

We prove that the 2n-neighbourhood in an n-dimensional digital space is decomposed into the 2(n1)-neighbourhoods in the mutually orthogonal (n1)-dimensional digital spaces. This decomposition and construction relation of the neighbourhoods and objects implies that morphological operations in an n-dimensional digital space can be computed as the union of one- and two-dimensional morphological operations on isothetic digital lines and planes intersecting with the digital object in the digital space.

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 x=(x1,x2,,xn). 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 isV(y)={x||xy|12}.

In this paper, we deal with the connectivity and adjacency of the centroids of the voxels [7], which are elements of Zn.

The sets FG and FG such

Neighbourhood operations

The 2n-neighbourhood of the origin in Zn isNn={x||xi|=1,x=(x1,x2,,xn)}.Let N(x)=Nn{x} 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 yN(x), if there exists at least one sequence pi+1N(pi) and piN(pi+1) for i=1,2,k1, the string {p}i=1k 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 formS(0;n)={s0n|s0n={0}},S(k;n)=U({skn|skn=sk1n(sk1nei),eisk1n})constructs digital k-simplex skn containing the origin 0 for k ≥ 1, where the operation U({ · }) removes redundant elements in the set { · }.

Proof

The elements of S(1;n)={{0

Digital boundary manifold

We define the boundary of a point set in Zn.

Definition 21

For a point set F, we call F=F(FNn) and +F=(FNn)F the internal and external boundaries of F, respectively.

For the internal and external boundaries, we have the following relations.

Lemma 4

F(FNn)=k=1nαN(k)(Fkα(FkαNkn1)),(FNn)F=k=1nαN(k)((FkαNkn1)Fkα).

This lemma derives the following theorem.

Theorem 4

The boundary±F of an n-dimensional digital object F is the union of its (n1)-dimensional boundaries.

Theorem 4 allows us to construct ∂±F from ∂±F

Distance transform, skeleton set and thinning

For integer d, the distance set F(d) of F ∈ Zn is defined as following.

Definition 26

F(d)={{x|minxF,yF¯|xy|=d},d>0,{x|minyF,xF¯|xy|=d},d<0.

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 F(1)=F and F(1)=+F. the recursive formsF(d+1)=F(d)(F(d)Nn),F(0):=F,d0,F(d1)=(F(d)Nn)F(d),F(0):=F,d0.compute 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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