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 CWspaces.
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 EilenbergSteenrod 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
in which 2adjacency 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.,
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
The following result shows that the ‘’ product operation of digital loop classes is welldefined.
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

;

;

; and

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
is welldefined 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
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
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
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
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
Lemma 3.7
If , then
Proof We first consider
Secondly,
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
The lefthand term and the righthand 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
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
defined by
where .
Lemma 3.11
If is a continuous function, then for every there is a commutative diagram
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