On the digital homology groups of digital images

In this article we study the digital homology groups of digital images which are based on the singular homology groups of topological spaces in algebraic topology. Specifically, we define a digitally standard n-simplex, a digitally singular n-simplex, and the digital homology groups of digital images with k-adjacency relations. We then construct a covariant functor from a category of digital images and digitally continuous functions to the one of abelian groups and group homomorphisms, and investigate some fundamental and interesting properties of digital homology groups of digital images, such as the digital version of the dimension axiom which is one of the Eilenberg-Steenrod axioms.

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1 Introduction

The digital fundamental groups [KO, LB1] of pointed digital images may be thought of as the tool in order to characterize properties of pointed digital images in a fashion analogous to that of classical fundamental groups [M] of topological spaces. They are basically derived from a classical notion of homotopy classes of based loops in the pointed homotopy category of pointed topological spaces or pointed CW-spaces.

Even though the digital fundamental group is a nice tool to classify the digital images with

-adjacency relations, it does not yields information at all in a large class of obvious problems. This is hardly surprising when we recall that the digital fundamental group of a digital image completely depends on the -adjacency and the digital homotopy type of digital images, and even is difficult or fails to distinguish higher dimensions of pointed digital images. For example, as in the case of algebraic topology, and can not be classified by using the classical fundamental groups. But it is very easy to say that they do not have the same homotopy type when we use the singular (or simplicial) homology groups or (or )-dimensional homotopy groups. Motivated from the above statements, we need to set up a new algebraic device which is called the digital homology groups in order to classify the various digital images with -adjacency relations.

The homology groups and higher (or stable) homotopy groups are useful algebraic tools in a large number of topological problems, and are in practice the standard tools of algebraic topology. In the same lode, the digital homology group can be an important gadget to classify digital images from the point of view for the digital version of the homotopy type, mathematical morphology and image synthesis.

This paper is concerned with setting up more algebraic invariants for a digital image with -adjacency. The paper is organized as follows: In Section 2 we introduce the general notions of digital images with -adjacency relations. In Section 3 we define a digitally standard -simplex, a digitally singular -simplex, the digitally singular -chains, and the digital homology groups of digital images. We then construct a covariant functor from a category of digital images and digitally continuous functions to the one of abelian groups and group homomorphisms, and investigate some fundamental and interesting properties of digital homology groups of digital images. Moreover, we show that the digital version of the dimension axiom, one of the Eilenberg-Steenrod axioms in algebraic topology, is satisfied.

2 Preliminaries

Let and be the sets of all integers and real numbers, respectively. Let be the set of lattice points in the Euclidean -dimensional space . A (binary) digital image is a pair , where is a finite subset of and indicates some adjacency relation for the members of . The -adjacency relations are used in the study of digital images in . For a positive integer with , we define an adjacency relation of a digital image in as follows. Two distinct points and in are -adjacent [BK] if

• there are at most distinct indices such that ; and

• for all indices , if , then .

A -adjacency relation on may be denoted by the number of points that are -adjacent to a point . Moreover,

• the -adjacent points of are called -adjacent;

• the -adjacent points of are called -adjacent, and the -adjacent points in are called -adjacent;

• the -adjacent points of are called -adjacent, the -adjacent points of are called -adjacent, and the -adjacent points of are called -adjacent;

• the -, -, -, and -adjacent points of are called -adjacent, -adjacent, -adjacent, and -adjacent, respectively; and so forth.

We note that the above number is just the cardinality of the set of lattice points which have the -adjacency relations centered at in . We sometimes denote the -adjacency by -adjacency for short if there is no chance of ambiguity.

Definition 2.1

A -neighbor of a lattice point is a point of that is -adjacent to .

Definition 2.2

([LB1]) Let be an adjacency relation defined on . A digital image is said to be -connected if and only if for every pair of points with , there exists a set of distinct points such that , and and are -adjacent for . The length of the set is the number .

The following generalizes an earlier definition of digital continuity given in [R].

Definition 2.3

Let and be the digital images with -adjacent and -adjacent relations, respectively. A function is said to be a -continuous function if the image under of every -connected subset of the digital image is a -connected subset of .

The following is an easy consequence of the above definition: Let and be digital images with -adjacency and -adjacency, respectively. Then the function is a -continuous function if and only if for every such that and are -adjacent in , either or and are -adjacent in .

We note that if is -continuous and if is -continuous, then the composite is -continuous.

Definition 2.4

([LB0]) Two digital images and with adjacency relations and , respectively, are -homeomorphic if there is a bijective function that is -continuous such that the inverse function is -continuous. In this case, we call the function a digital -homeomorphism, and we denote it by .

Definition 2.5

([LB2]) Let . A digital interval is a set of the form

 [a,b]Z={z∈Z | a≤z≤b}

in which 2-adjacency is assumed. A digital -path in a digital image is a (2,)-continuous function . If , we call a digital -loop. If is a constant function, it is called a trivial loop.

Definition 2.6

([LB3]) Let and be digital images with -adjacent and -adjacent relations, respectively, and let be the -continuous functions. Suppose that there is a positive integer and a function such that

• for all and ;

• for all , the induced function defined by for all is -continuous; and

• for all , the induced function defined by for all is -continuous.

Then is called a digital -homotopy between and , written , and and are said to be digitally -homotopic in .

We use to denote the digital homotopy class of a -continuous function , i.e.,

 [f]={g:X→Y | g is (k1,k2)−continuous, and f≃(k1,k2)g}.

Similarly, we denote by the -loop class of a digital -loop in a digital image with -adjacency.

A pointed digital image is a pair , where is a digital image and ; is called the base point of . A pointed digitally continuous function is a digitally continuous function from to such that . A digital homotopy between and is said to be pointed digital homotopy between and if for all , . If a pointed digital homotopy between and exists, we say and belong to the same pointed digital homotopy class. It is not difficult to see that the (pointed) digital homotopy is an equivalence relation among the (pointed) digital homotopy classes of digitally continuous functions.

We consider the digital version of products just as in the case of products of paths (or loops) of homotopy classes in homotopy theory. If and are digital -paths in the digital image with , the product (see [LB1] and [LB3]) of and is the digital -path in defined by

 (f∗g)(t)={f(t)\hbox{if} t∈[0,m1]Z;g(t−m1)% \hbox{if} t∈[m1,m1+m2]Z.

The following result shows that the ‘’ product operation of digital loop classes is well-defined.

Proposition 2.7

([K]) Suppose and are digital loops in a pointed digital image with and . Then .

We now describe the notion of trivial extension [BK] which is used to allow a loop to stretch and remain in the same pointed homotopy class.

Definition 2.8

Let and be digital -loops in a pointed digital image . We say that is a trivial extension of if there are sets of -paths and in such that

1. ;

2. ;

3. ; and

4. there are indices such that

• ; and

• implies is a trivial loop.

Two digital loops and with the same base point belong to the same digital loop class (see [LB2]) if they have trivial extensions that can be joined by a homotopy that holds the endpoints fixed.

We end this section with the digital -fundamental group originally derived from a classical notion of homotopy theory (see [S, W]). Let be a pointed digital image with -adjacency. Consider the set of digital -loop classes in with base point . By Proposition 2.7, the product operation

 [f]+[g]=[f∗g]

is well-defined on . One can see that becomes a group under the ‘’ product operation which is called the digital -fundamental group of . As in the case of basic notions in algebraic topology, it is well known in [LB1, Theorem 4.14] that is a covariant functor from the category of pointed digital images and pointed digitally continuous functions to the category of groups and group homomorphisms.

3 Digital homology groups

We now consider the digital version of singular homology groups as follows: For , let denote the point in having coordinates all zeros except for 1 in the st position, i.e., , , , .

Definition 3.1

A digitally convex combination of points in is a point with

 x=t0e0+t1e1+⋯+tnen,

where , and or . The entries of are called the digitally barycentric coordinates of .

We denote by the set of all digitally convex combinations of points in , that is, . If we consider as the digital image with -adjacency, then it is -connected.

Definition 3.2

By using the -adjacent relation in the digital image , we define an orientation of by a linear ordering of its vertices, and call it a digitally standard -simplex with a linear order for its orientation.

Definition 3.3

Let be a digital image with -adjacency. A digitally singular -simplex in is a -continuous function

 σ:Δn→(X,k),

where is the digitally standard -simplex.

Definition 3.4

Let be a digital image with -adjacency. For each , define as the free abelian group with basis all digitally singular -simplexes in , and define . The elements of are called digitally singular -chains in .

The oriented boundary of a digitally singular -simplex have to be , where the symbol means that the vertex is to be deleted from the array in the digitally standard -simplex . Technically, we prefer that this is a digitally singular -chain in .

Definition 3.5

For each and , we define the th face function

 ϵi=ϵni:Δn−1→Δn

to be the function sending the ordered vertices to the ordered vertices preserving the displayed orderings as follows:

• ; and

• for .

For example, there are three face functions such as ; ; and .

Definition 3.6

Let be a digital image with -adjacency. If is a digitally singular -simplex, then the function defined by

 ∂nσ=⎧⎪⎨⎪⎩n∑i=0(−1)iσ∘ϵni\hbox{if}~{}~{} n≥1;0\hbox{if}~{}~{} n=0;

is called the digitally boundary operator of the digital image .

We note that is a homomorphism. We thus extend the above definition by linearity to the digitally singular -chains. In particular, if and is the identity, then

 ∂n(I)=n∑i=0(−1)iϵni.
Lemma 3.7

If , then

 ϵn+1j∘ϵnk=ϵn+1k∘ϵnj−1:Δn−1→Δn+1.

Proof  We first consider

 ϵn+1j∘ϵnk(t0,…,tk,…,tj,…,tn−1)=ϵn+1j(t0,…,tk−1,0,tk,…,tj,…,tn−1)=(t0,…,tk−1,0,tk,…,tj−2,0,tj−1,…,tn−1).

Secondly,

 ϵn+1k∘ϵnj−1(t0,…,tk,…,tj,…,tn−1)=ϵn+1k(t0,…,tk,…,tj−2,0,tj−1,…,tn−1)=(t0,…,tk−1,0,tk,…,tj−2,0,tj−1,…,tn−1),

as required.

Theorem 3.8

For all , we have .

Proof  It suffices to show that for each digitally singular -simplex . By Lemma 3.7, we have

 ∂n∘∂n+1(σ)=∂n(n+1∑j=0(−1)jσ∘ϵn+1j)=∂n(σ∘ϵn+10−σ∘ϵn+11+σ∘ϵn+12+⋯+(−1)n+1σ∘ϵn+1n+1)=n∑k=0(−1)kσ∘ϵn+10∘ϵnk−n∑k=0(−1)kσ∘ϵn+11∘ϵnk+                               ⋯+(−1)n+1n∑k=0(−1)kσ∘ϵn+1n+1∘ϵnk=n+1∑j=0(−1)j[n∑k=0(−1)kσ∘ϵn+1j∘ϵnk]=n+1∑j=0n∑k=0(−1)j+kσ∘ϵn+1j∘ϵnk=∑j≤k(−1)j+kσ∘ϵn+1j∘ϵnk+∑k

The left-hand term and the right-hand term in denote the upper triangular matrix and the lower triangular matrix in the -matrix. These terms cancel in pairs, and thus .

Definition 3.9

The kernel of is called the group of digitally singular -cycles in and denoted by . The image of is called the group of digitally singular -boundaries in and denoted by .

By Theorem 3.8, each digitally singular -boundary of digitally singular -chains is automatically a digitally singular -cycle, that is, is a normal subgroup of for each . Thus we can define

Definition 3.10

For each , the th digital homology group of a digital image with -adjacency is defined by

 ˇHn(X,k)=ˇZn(X,k)/ˇBn(X,k).

The coset is called the digital homology class of , where is a digitally singular -cycle.

We note that if is a -continuous function and if is a digitally singular -simplex in , then is a digitally singular -simplex in . By extending by linearity, we have a homomorphism

 f♯:ˇSn(X,k1)→ˇSn(Y,k2)

defined by

 f♯(Σmσσ)=Σmσ(f∘σ),

where .

Lemma 3.11

If is a -continuous function, then for every there is a commutative diagram