The paper introduces the notion of $$A$$
 -retract to compress digital images derived from a Khalimsky topological structure. Compared with a Khalimsky retract, an $$A$$
 -retract is very flexible and efficient for the process of compressing (or thinning) (see Fig. 10). More precisely, to study digital images (or spaces) derived from a Khalimsky topological structure, a Khalimsky continuous map and a Khalimsky homeomorphism have been often used in digital topology. However, these maps are so rigid that they have some limitations of both a rotation and a parallel translation from the viewpoint of geometry (see Examples 3.4, 4.1 and 5.1 as motivating examples). To overcome these limitations, the paper generalizes a Khalimsky continuous map, a Khalimsky adjacency map and a homeomorphism between Khalimsky topological spaces so that we can establish new maps such as an $$A$$
 -map and an $$A$$
 -isomorphism. Using these notions, we can substantially study and classify digital images induced by a Khalimsky topological structure (see the table in Fig. 5 explaining some merits of an $$A$$
 -map compared with both a Khalimsky continuous map and a Khalimsky adjacency map). As an application, for digital images derived from a Khalimsky topological structure $$X$$
 , the paper proposes a method of thinning or compressing by using an $$A$$
 -retract of $$X$$
 , which shows that this approach can contribute to computer science such as image analysis, image processing, computer graphics, mathematical morphology and so forth.

A compression of digital images derived from a Khalimsky topological structure

Generalizations of continuity of maps and homeomorphisms for studying 2D digital topological spaces and their applications

Let SC ni,li ki be a simple closed k i -curve in Z ni with l i elements, i 6 {1,2}. After doing a (3 n 1 +n 2 - 1)-homotopic thinning of SC n1,l1 k1 × SC n2,l2 k2 to obtain a closed (3 n1+n2 - 1)-surface, we calculate the (3 n1+n2 - 1)-fundamental group of SC n1,l1 k1 × SC n2,l2 k2 by the use of some properties of an (8, 3 n1+n2 - 1)-covering.

Discrete homotopy of a closed k-surface

When digitizing subspaces in the Euclidean subspace in a certain digital topological approach, we are strongly required to preserve topological properties of the given spaces such as connectedness. Thus the present paper studies four kinds of local rules associated with lower limit, upper limit, Khalimsky and Marcus Wyse (for short, L-, U-, K- and M-, respectively) topologies which are used for L-, U-, K- and M-digitizing subspaces of the Euclidean nD space into digital topological spaces. While the L-, U- and K-digitizations are proved to preserve connectedness of objects, the M-digitization has some limitation of having the connectedness preserving (for short, CP-) property. This approach can be substantially used for studying applied topology and computer science.

Digitizations associated with several types of digital topological approaches

The study of digital covering transformation groups (or automorphism groups, discrete deck transformation groups) plays an important role in the classification of digital spaces (or digital images). In particular, the research into transitive or nontransitive actions of automorphism groups of digital covering spaces is one of the most important issues in digital covering and digital homotopy theory. The paper deals with the problem: Is there a digital covering space which is not ultra regular and has an automorphism group which is not trivial? To solve the problem, let us consider a digital wedge of two simple closed ki-curves with a compatible adjacency, i ∈ {1,2}, denoted by (X,k). Since the digital wedge (X,k) has both infinite or finite fold digital covering spaces, in the present paper some of these infinite fold digital covering spaces were found not to be ultra regular and further, their automorphism groups are not trivial, which answers the problem posed above. These findings can be substantially used in classifying digital covering spaces and digital images so that the paper improves on the research in Section 4 of (3) (compare Figure 2 of the present paper with Figure 2 of (3)), which corrects an error that appears in the Boxer and Karaca's paper (3) (see the points (0,0),(0,8),(6, −1) and (6,7) in Figure 2 of (3)).

/pdf/non-ultra-regular-digital-covering-spaces-with-nontrivial-1szcerzs4b.pdf

Non-ultra regular digital covering spaces with nontrivial automorphism groups

Aiming at the study of the compression of Khalimsky topological spaces which is an interesting field in digital geometry and computer science, the present paper develops a new homotopy thinning suitable for the work. Since Khalimsky continuity of maps between Khalimsky topological spaces has some limitations of performing a discrete geometric transformation, the paper uses another continuity (see Definition 3.4) that can support the discrete geometric transformation and a homotopic thinning suitable for studying Khalimsky topological spaces. By using this homotopy, we can develop a new homotopic thinning for compressing the spaces and can write an algorithm for compressing 2D Khalimsky topological spaces.

/pdf/compression-of-khalimsky-topological-spaces-3xc2ousdty.pdf

Min Jeang Kang

Papers

Compression of Khalimsky topological spaces