Remarks on fixed point assertions in digital topology, 10

Abstract. In image processing and computer graphics an object in the plane or 3-space is often approximated digitaly by a set of pixels or voxels. Digital topology studies properties of pixels or voxels that correspond to topological properties of the original object. In this paper, we discuss about digital space and digital picture from Rosenfeld's aspect of view and introduce regular and strongly normal digital picture space. Using these introductions, we impose restrictions on adjacency relation between points to establish some important theorem in digital space like as the Jordan Curve Theorem. Also, one can explore digital fundamental group in regular digital picture space but in this paper we do not deal with it. At the end, we express that the Jordan Curve Theorem in the strongly normal digital picture space is verified.

Compatible topologies on graphs: An application to graph isomorphism problem complexity

Digital topology is concerned with the topological characteristics of digital image pictures or objects. A peculiar arrangement of non-negative numbers configures digital images. Digital image processing is a technique of dismantling the picture into its fundamental components and analyzing its various features with respect to component parts. In analyzing the fundamental segments of image pictures, the connected segments are separated out to ascertain the relationship of adjacency. During this process of tracking, coding and thinning, it is kept in mind that the connectedness peculiarity of the object remains unchanged. The features of the component subsets and their relationships can be detailed when the image is decomposed into its constituents. Some of the characteristics of these constituent points or subsets are depending on their positions. Thus, the primary topological features of digital images like connectedness, adjacency, etc. can be the basic clues for their processing. Various kinds of contraction mappings and related fixed-point theorems can be applied in the field of science and technology, including mathematics, game theory, computer science, engineering, environmental science, etc. Fix point theorems are applied in computational techniques in engineering and science to explore the areas of parallel and distributed computation, simulation, modeling and image processing-digital images. In image processing fixed point theorems are applied to get digital contraction which would be a mathematical basis of contour filling, border following algorithm and thinning of a digital image. To broaden the applicability of contraction principle and associated fixed point theorem in image processing, we wish to explore some of them as significant applicable tools for digital image processing.

Topological Algorithms for Digital Image Processing

Digital topology is the study of the topological properties of digital images. In most of the literature, a digital image has been endowed with a graph structure; the vertices being the points of the image, and the edges giving the connectivity between the points. This has enabled the use of combinatorial methods to provide theorems and proofs for basic topological results. However, these methods have been shown to be inadequate for a full discussion of object thinning in three dimensions, and also for the development of a topological theory in dimensions higher than three. This has led to the investigation of algebraic topology as a means of providing results in digital topology; this paper surveys the results so far obtained, and shows how they relate to classical algebraic topology, and to digital topology as it has developed over the last two decades.

This paper continues a series discussing flaws in published assertions concerning fixed points in digital metric spaces.

In this paper, we recall some definitions and properties from digital topology, and we consider the simplices and simplicial complexes in digital images due to adjacency relations. Then we define the simplicial set and conclude that the simplicial identities are satisfied in digital images. Finally, we construct the group structure in digital images and define the simplicial groups in digital images. Consequently, we calculate the digital homology group of two-dimensional digital simplicial group. MSC:55N35, 68R10, 68U10, 18G30.

<p>In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology.</p><p>We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(C<sub>n</sub>) where C<sub>n</sub> is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including C<sub>n</sub>, in which F(X) does not equal {0, 1, . . . , #X}.</p><p>We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.</p>

In the study of topological properties of digital images, which is the central topic of digital topology, one is often interested in special operations on object boundaries and their properties. Examples are contour filling or border following. In classical topology there exists the strong concept of regularity: regular sets in R2 show no “exotic behaviour” and are extensively used in the theory of boundary value problems. In this paper we transfer the concept of regularity to digital topology within the framework of semi-topology. It is shown that regular open sets in (a special) semi-topology can be characterized graphically. A relationship between digital topology and image processing is established by showing that regular open digital sets, interpreted as digital pictures, are left unchanged when the cross-median filter is applied.

Abstract. Various properties of digital covering spaces have beensubstantially used in studying digital homotopic properties of dig-ital images. In particular, these are so related to the study of adigital fundamental group, a classi cation of digital images, an au-tomorphism group of a digital covering space and so forth. Thegoal of the present paper, as a survey article, to speak out utilityof digital covering theory. Besides, the present paper recalls thatthe papers [1, 4, 30] took their own approaches into the study ofa digital fundamental group. For instance, they consider the dig-ital fundamental group of the special digital image (X;4), whereX := SC 2;84 which is a simple closed 4-curve with eight elements inZ 2 , as a group which is isomorphic to an in nite cyclic group suchas (Z;+). In spite of this approach, they could not propose any dig-ital topological tools to get the result. Namely, the papers [4, 30]consider a simple closed 4 or 8-curve to be a kind of simple closedcurve from the viewpoint of a Hausdor topological structure, i.e. acontinuous analogue induced by an algebraic topological approach.However, in digital topology we need to develop a digital topologi-cal tool to calculate a digital fundamental group of a given digitalspace. Finally, the paper [9] rstly developed the notion of a digitalcovering space and further, the advanced and simpli ed version wasproposed in [21]. Thus the present paper refers the history and theprocess of calculating a digital fundamental group by using varioustools and some utilities of digital covering spaces. Furthermore, wedeal with some parts of the preprint [11] which were not publishedin a journal (see Theorems 4.3 and 4.4). Finally, the paper suggestsan ecient process of the calculation of digital fundamental groupsof digital images.Received August 18, 2014. Accepted August 28, 2014.2010 Mathematics Subject Classi cation. 55Q70, 52Cxx, 55P15, 68R10, 68U05.Key words and phrases. digital topology, digital product, k-homotopic thinning,normal adjacency, S-compatible adjacency, digital covering space, C-property, S-property.

Generic Axiomatized Digital Surface-Structures

Generic axiomatized digital surface-structures

Consider two digital spaces $ (X_i, k_i), i \in \{1, 2\} $, (in the sense of <i>Rosenfeld model</i>) satisfying the almost fixed point property(<i>AFPP</i> for brevity). Then, the problem of whether the <i>AFPP</i> for the digital spaces is, or is not necessarily invariant under Cartesian products plays an important role in digital topology, which remains open. Given a Cartesian product $ (X_1 \times X_2, k) $ with a certain $ k $-adjacency, after proving that the <i>AFPP</i> for digital spaces is not necessarily invariant under Cartesian products, the present paper proposes a certain condition of which the <i>AFPP</i> for digital spaces holds under Cartesian products. Indeed, we find that the product property of the <i>AFPP</i> is strongly related to both the sets $ X_i $ and the $ k_i $-adjacency, $ i \in \{1, 2\} $. Eventually, assume two $ k_i $-connected digital spaces $ (X_i, k_i), i \in \{1, 2\} $, and a digital product $ X_1 \times X_2 $ with a normal $ k $-adjacency such that $ N_k^\star(p, 1) = N_k(p, 1) $ for each point $ p \in X_1 \times X_2 $ (see Remark 4.2(1)). Then we obtain that each of $ (X_i, k_i), i \in \{1, 2\} $, has the <i>AFPP</i> if and only if $ (X_1 \times X_2, k) $ has the <i>AFPP</i>

Digital homotopy with obstacles

Concepts of digital topology