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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 675373, 6 pages
Computing Hypercrossed Complex Pairings in Digital Images
Mathematics Department, Faculty of Science and Arts, Celal Bayar University, 45047 Manisa, Turkey
Received 3 October 2013; Accepted 9 November 2013
Academic Editor: Abdon Atangana
Copyright © 2013 Simge Öztunç et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider an additive group structure in digital images and introduce the commutator in digital images. Then we calculate the hypercrossed complex pairings which generates a normal subgroup in dimension 2 and in dimension 3 by using 8-adjacency and 26-adjacency.
In this paper we denote the set of integers by . Then represents the set of lattice points in Euclidean -dimensional spaces. A finite subset of with an adjacency relation is called a digital image.
Definition 1 (see [1, 2]). Consider the following.(1)Two points and in are 2-adjacent if .(2)Two points and in are 8-adjacent if they are distinct and differ by at most 1 in each coordinate.(3)Two points and in are 4-adjacent if they are 8-adjacent and differ by exactly one coordinate.(4)Two points and in are 26-adjacent if they are distinct and differ by at most 1 in each coordinate.(5)Two points and in are 18-adjacent if they are 26-adjacent and differ in at most two coordinates.(6)Two points and in are 6-adjacent if they are 18-adjacent and differ by exactly one coordinate.
Definition 2. Let be a subset of a digital image. A simplicial group in digital images consists of a sequence of groups and collections of group homomorphisms and , , that satisfies the following axioms:
Definition 3. Given a simplicial group with -adjacency, the Moore complex of is the chain complex defined by
with induced from by restriction.
The th homology group of the Moore complex of is
2. Hypercrossed Complex Pairings in Digital Images
First of all we adapt ideas from Carrasco and Cegarra [3–5] to get the construction in digital images. We define a set consisting of pairs of elements from with and , with respect to lexicographic ordering in where and .
Consider the following diagram: (4) where and define and as and . Since a digital image has the additive group structure, define the commutator as Thus The normal subgroup of is generated by the elements of the form where and .
Theorem 4. 2-dimensional normal subgroup with 8-adjacency is generated by the elements of the form
Proof. Let and for . For and , Thus and this is the element generating normal subgroups.
Proposition 5. 3-dimensional normal subgroup with 26-adjacency is generated by the elements of the following forms:(i),(ii),(iii),(iv),(v),(vi).
Proof. For the possible pairings are the following:(i),(ii),(iii),(iv),(v),(vi).For all and the corresponding generators of are the following with and for , :
(ii) For all and and considering the map , the corresponding generator of is
(iii) For all , and for the corresponding generators of are
Theorem 6. Let be a 2-dimensional Moore complex of a simplicial group . Then where is induced from by restriction.
Proof. For , assume that , , and , . Now calculate .
Since , from Proposition 5
At first we investigate whether is in or not. therefore .
Secondly we examine whether is in or not.
From the assumption , we get
Theorem 7. Let be a 3-dimensional Moore complex of a simplicial group with 26-adjacency. Then where is induced from by restriction.
Proof. For investigate where and .
From Proposition 5 we have . Then applying to , we get the following: Firstly, examine whether is in or not: So .
Secondly we investigate whether is in or not: Therefore .
Finally we check whether is in or not.
Since therefore .
We get since .
At first we check whether is in , , and or not. Thus .
Next, since and, finally,
Now examine whether is in , , and or not: Therefore . We have the following: So .
For all and if then Firstly investigate whether is in , , and or not: Thereby . We have For this reason . We also have Hence .
Later on we research whether is in , , and or not.
Since , .
Since , .
Since , .
For all , since , By using properties of the commutator we have Thus If , then Firstly we check whether is in , , and or not: Therefore . Since . We have Hence .
Because of the case , If , then Consider the following commutator: and code the terms of this commutator such as in order to simplify the algebraic operations. Thus, by using the properties and definition of the commutator we obtain the following: Consider the following cases: And from the remaining terms we get Consequently for we have
Corollary 8. Let be a 3-dimensional Moore complex of a simplicial group with 26-adjacency. Then
In this paper for dimension 2 and dimension 3, we obtained the Moore complex of simplicial groups generated by hypercrossed complex pairings in digital images.
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