Search

Topology preservation is a crucial issue in parallel reductions that transform binary pictures by changing only a set of black points to white at a time. In this paper, we present sufficient conditions for topology-preserving parallel reductions on the three types of pictures of the unconventional 3D face-centered cubic (FCC) grid. Some conditions provide methods of verifying that a given parallel reduction always preserves the topology, and the remaining ones directly provide deletion rules of topology-preserving parallel reductions, and make us possible to generate topologically correct thinning algorithms. We give local characterizations of P-simple points, whose simultaneous deletion preserves the topology, and the relationships among the existing universal sufficient conditions for arbitrary types of binary pictures and our new FCC-specific results are also established. [ABSTRACT FROM AUTHOR]

After 1922 definition of contraction mapping, its applicability and usefulness were worked upon by several researchers through their various research papers. Different fields of engineering, science and technology could not remain untouched by its use. The notion of compatible mapping and compatible mapping of type (A) is further transformed with the concept of intimate mapping. The development of operator theory was brought forward by the idea of fixed points impending upon contraction mappings. Apart from general metric space, metric spaces like b-metric space, G-metric space, F-metric space and digital space enhanced the applicability of fixed-point theory. We wish to put forward the notion of intimate mapping in digital metric space through some consequences on it. [ABSTRACT FROM AUTHOR]

As opposed to the 3D cubic grid, the body-centered cubic (BCC) grid has some favorable topological properties: each set of voxels in the grid is a 3-manifold, with 2-manifold boundary. Thus, the Euler characteristic of an object O in this grid can be computed as half of the Euler characteristic of its boundary ∂ O . We propose three new algorithms to compute the Euler characteristic in the BCC grid with this surface-based approach: one based on (critical point) Morse theory and two based on the discrete Gauss–Bonnet theorem. We provide a comparison between the three new algorithms and the classic approach based on counting the number of cells, either of the 3D object or of its 2D boundary surface. [ABSTRACT FROM AUTHOR]

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

This paper continues a series discussing aws in published assertions concerning fixed points in digital metric spaces. [ABSTRACT FROM AUTHOR]

Freezing sets and cold sets have been introduced as part of the theory of fixed points in digital topology. In this paper, we introduce a generalization of these notions, the limiting set, and examine properties of limiting sets. [ABSTRACT FROM AUTHOR]

As in [6, 3, 4, 5], we discuss published assertions concerning fixed points in “digital metric spaces” - assertions that are incorrect or incorrectly proven, or reduce to triviality.

The paper initially develops the semi-Jordan curve theorem on the digital plane with the Marcus-Wyse topology, i.e., $ MW $-topological plane or $ ({\mathbb Z}^2, \gamma) $ for brevity. We first prove that while every simple closed $ MW $-curve is semi-open in $ ({\mathbb Z}^2, \gamma) $, it may not be semi-closed. Given a simple closed $ MW $-curve with $ l $ elements, denoted by $ SC_{\gamma}^l $, after establishing a continuous analog of $ SC_{\gamma}^l $ denoted by $ \mathcal{A}(SC_{\gamma}^l) $, we initially show that $ \mathcal{A}(SC_{\gamma}^l) $ is both semi-open and semi-closed in $ ({\mathbb R}^2, \mathcal{U}) $, where $ ({\mathbb R}^2, \mathcal{U}) $ is the $ 2 $-dimensional real plane $ {\mathbb R}^2 $ with the usual topology $ \mathcal{U} $. Furthermore, we find a condition for $ \mathcal{A}(SC_{\gamma}^l) $ to separate $ ({\mathbb R}^2, \mathcal{U}) $ into exactly two non-empty components, compared to a typical Jordan curve theorem on $ ({\mathbb R}^2, \mathcal{U}) $. Since not every $ SC_{\gamma}^l $ always separates ($ {\mathbb Z}^2, \gamma) $ into two nonempty components, we find a condition for $ SC_{\gamma}^l, l\neq 4, $ to separate $ ({\mathbb Z}^2, \gamma) $ into exactly two components. The semi-Jordan curve theorem on the $ MW $-topological plane plays an important role in applied topology such as digital topology, mathematical morphology as well as computer science.

In this paper, using the notion of subdivision, the authors generalize the definition of cofibration in digital topology and show that this kind of cofibration is injective in the sense of subdivision. Meanwhile, they give the necessary condition under which a digital map is a cofibration. Furthermore, they consider the Lusternik-Schnirelmann category of digital maps in the sense of subdivision and give several fundamental homotopy properties about it. [ABSTRACT FROM AUTHOR]

We introduce digital projective product spaces based on Davis' projective product spaces. We determine an upper bound for the digital LS-category of digital projective product spaces. In addition, we obtain an upper bound for the digital topological complexity of these spaces through an explicit motion planning construction, which shows digital perspective validity of results given by S. Fişekci and L. Vandembroucq. We apply our outcomes on specific spaces in order to be more clear. [ABSTRACT FROM AUTHOR]

Let $ (X, T) $ be an Alexandroff space. We define the adjacency relation $ AR_T $ on $ X $ induced by $ T $ as the irreflexive relation defined for $ x \neq y $ in $ X $ by: $ (x,y) \in AR_T\,\,{\rm{if \;and\; only\; if}}\,\, x \in SN_T(y)\,\,{\rm{or}}\,\, y \in SN_T(x), $ where $ SN_T(z) $ is the smallest open set containing $ z $ in $ (X, T) $ and $ z \in \{x, y\} $. Two families of Alexandroff topologies $ (T_k, k \in {\mathbb Z}) $ and $ (T_k^\prime, k \in {\mathbb Z}) $ have been recently introduced on $ {\mathbb Z} $. The aim of this paper is to show that for each nonzero integers $ k $, the topologies $ T_k, T_k^\prime $, $ T_{-k} $, and $ T_{-k}^\prime $ are homeomorphic. The adjacency relations induced by the product topologies $ (T_k)^n $ and $ (T_k^\prime)^n $ are studied and compared with classical ones. We also show that the adjacency relations induced by $ T_k, T_k^\prime $, $ T_{-k} $, and $ T_{-k}^\prime $ are isomorphic. Then, note that the adjacency relations on $ {\mathbb Z} $ induced by these topologies, $ k \neq 0 $, are different from each other.

Two objects may be close in the Hausdorff metric, yet have very different geometric and topological properties. We examine other methods of comparing digital images such that objects close in each of these measures have some similar geometric or topological property. Such measures may be combined with the Hausdorff metric to yield a metric in which close images are similar with respect to multiple properties.

The purpose of the paper is to study digital topological versions of typical topological groups. In relation to this work, given a digital image $ (X, k), X \subset {\mathbb Z}^n $, we are strongly required to establish the most suitable adjacency relation in a digital product $ X \times X $, say $ G_{k^\ast} $, that supports both $ G_{k^\ast} $-connectedness of $ X \times X $ and $ (G_{k^\ast}, k) $-continuity of the multiplication $ \alpha: (X \times X, G_{k^\ast}) \to (X, k) $ for formulating a digitally topological $ k $-group (or $ DT $-$ k $-group for brevity). Thus the present paper studies two kinds of adjacency relations in a digital product such as a $ C_{k^\ast} $- and $ G_{k^\ast} $-adjacency. In particular, the $ G_{k^\ast} $-adjacency relation is a new adjacency relation in $ X \times X\subset {\mathbb Z}^{2n} $ derived from $ (X, k) $. Next, the paper initially develops two types of continuities related to the above multiplication $ \alpha $, e.g., the $ (C_{k^\ast}, k) $- and $ (G_{k^\ast}, k) $-continuity. Besides, we prove that while the $ (C_{k^\ast}, k) $-continuity implies the $ (G_{k^\ast}, k) $-continuity, the converse does not hold. Taking this approach, we define a $ DT $-$ k $-group and prove that the pair $ (SC_k^{n, l}, \ast) $ is a $ DT $-$ k $-group with a certain group operation $ \ast $, where $ SC_k^{n, l} $ is a simple closed $ k $-curve with $ l $ elements in $ {\mathbb Z}^{n} $. Also, the $ n $-dimensional digital space $ ({\mathbb Z}^n, 2n, +) $ with the usual group operation "$ + $" on $ {\mathbb Z}^n $ is a $ DT $-$ 2n $-group. Finally, the paper corrects some errors related to the earlier works in the literature.

In this paper, we define a new flavour of well-composedness, called strong Euler well-composedness. In the general setting of regular cell complexes, a regular cell complex of dimension n is strongly Euler well-composed if the Euler characteristic of the link of each boundary cell is 1, which is the Euler characteristic of an (n - 1) -dimensional ball. Working in the particular setting of cubical complexes canonically associated with n D pictures, we formally prove in this paper that strong Euler well-composedness implies digital well-composedness in any dimension n ≥ 2 and that the converse is not true when n ≥ 4 . [ABSTRACT FROM AUTHOR]

Thinning is a frequently used technique capable of producing all kinds of skeleton-like shape features in a topology-preserving way. It is an iterative object reduction: some border points of binary objects that satisfy some topological and geometrical constraints are deleted, and the entire process is repeated until stability is reached. In the conventional implementation of thinning algorithms, the deletability of all border points in the actual picture is to be investigated. That is why, we introduced the concept of k-attempt thinning ( k ≥ 1 ) in our previous work (presented in the 20th International Workshop on Combinatorial Image Analysis, IWCIA 2020). In the case of a k-attempt algorithm, if a border point 'survives' at least k successive iterations, it is 'immortal' (i.e., it cannot be deleted later). In this paper, we give a computationally efficient implementation scheme for 1-attempt thinning, and a 1-attempt 2D parallel thinning algorithm is reported. The advantage of the new implementation scheme over the conventional one is also illustrated. [ABSTRACT FROM AUTHOR]

The present paper intensively studies various properties of certain topologies on the set of integers $ {\mathbb Z} $ (resp. $ {\mathbb Z}^n $) which are either homeomorphic or not homeomorphic to the typical Khalimsky line topology (resp. $ n $-dimensional Khalimsky topology). This finding plays a crucial role in addressing some problems which remain open in the field of digital topology.

As in [6, 3, 4, 5], we discuss published assertions concerning fixed points in "digital metric spaces" - assertions that are incorrect or incorrectly proven, or reduce to triviality. [ABSTRACT FROM AUTHOR]

Let (X, T) be an Alexandroff space. We define the adjacency relation ART on X induced by T as the irreflexive relation defined for x, y in X by: (x, y) ∈ ART if and only if x ∈ S NT (y) or y ∈ S NT (x), where S NT (z) is the smallest open set containing z in (X, T) and z ∈ {x, y}. Two families of Alexandroff topologies (Tk, k ∈ Z) and (T′ k, k ∈ Z) have been recently introduced on Z. The aim of this paper is to show that for each nonzero integers k, the topologies Tk, T′ k, T-k, and T′ -k are homeomorphic. The adjacency relations induced by the product topologies (Tk)n and (T′ k)n are studied and compared with classical ones. We also show that the adjacency relations induced by Tk, T′ k, T-k, and T′ -k are isomorphic. Then, note that the adjacency relations on Z induced by these topologies, k, 0, are different from each other. [ABSTRACT FROM AUTHOR]

This paper is devoted to the notion digital pseudocovering map introduced by Han [9]. We show that, according to Han, s definition, a function is a pseudocovering map if and only if it is a covering map. We give a modified definition of pseudocovering in order to obtain the results Han sought. [ABSTRACT FROM AUTHOR]

In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images. We introduce a new type of homotopy relation for digitally continuous functions which we call ``strong homotopy.'' Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane. We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by D. W. Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca \& Ege. We show that the two simplicial homology theories are isomorphic to each other, but distinct from the two cubical theories. We also show that homotopic maps have the same induced homomorphisms in the cubical homology theory, and strong homotopic maps additionally have the same induced homomorphisms in the simplicial theory.

A digital image is a finite set of integer lattice points in an ambient Euclidean space together with a suitable adjacency relation between points. Subdivision, which is a process of enlarging a digital image in the photographic sense, provides a basic tool for operating with digital images. But given a map of digital images, there is as yet no general way to define a map of their subdivisions that might reasonably be called a subdivision of the map. In this paper, we construct such maps of subdivisions when the map of digital images has a 1- or 2-dimensional domain. From our constructions we deduce path covering and homotopy covering results that play a role in our development of the digital fundamental group. [ABSTRACT FROM AUTHOR]

We continue the study of freezing sets in digital topology, introduced in [4]. We show how to find a minimal freezing set for a "thick" convex disk X in the digital plane Z^2. We give examples showing the significance of the assumption that X is convex.

In this paper, we define digital homotopic distance and give its relation with LS category of a digital function and of a digital image. Moreover, we introduce some properties of digital homotopic distance such as being digitally homotopy invariance.

In this paper, we prove that when a n-D cubical set is continuously well-composed (CWC), that is, when the boundary of its continuous analog is a topological (n - 1) -manifold, then it is digitally well-composed (DWC), which means that it does not contain any critical configuration. We prove this result thanks to local homology. This paper is the sequel of a previous paper where we proved that DWCness does not imply CWCness in 4D. [ABSTRACT FROM AUTHOR]

Digital topology is part of the ongoing endeavor to understand and analyze digitized images. With a view to supporting this endeavor, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik–Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics. [ABSTRACT FROM AUTHOR]

We study the consequences of some restrictions on digitally continuous functions. One of our results modififies easily to yield an analogous result for topological spaces. [ABSTRACT FROM AUTHOR]

Digital topological methods are often used in computing the topological complexity of digital images. We give new results on the relation between reducibility and digital contractibility in order to determine the topological complexity of a digitally connected finite digital image. We present all possible cases of the topological complexity TC of a finite digital image in Z and Z2. Finally, we determine the higher topological complexity TCn of finite irreducible digital images independently of the number of points for n > 1. [ABSTRACT FROM AUTHOR]

Two objects may be close in the Hausdorff metric, yet have very different geometric and topological properties. We examine other methods of comparing digital images such that objects close in each of these measures have some similar geometric or topological property. Such measures may be combined with the Hausdorff metric to yield a metric in which close images are similar with respect to multiple properties. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a topological method for constructing discretizations (tessellations) of topological conceptual spaces. The method works for a class of topological spaces that the Russian mathematician Pavel Alexandroff defined more than 80 years ago. The aim of this paper is to show that Alexandroff spaces, as they are called today, have many interesting properties that can be used to explicate and clarify a variety of problems in philosophy, cognitive science, and related disciplines. For instance, recently, Ian Rumfitt used a special type of Alexandroff spaces to elucidate the logic of vague concepts in a new way. Moreover, Rumfitt's class of Alexandroff spaces can be shown to provide a natural topological semantics for Susanne Bobzien's "logic of clearness". Mainly due to the work of Peter Gärdenfors and his collaborators, conceptual spaces have become an increasingly popular tool of dealing with a variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics and philosophy. For Gärdenfors's conceptual spaces, geometrically defined discretizations (so-called Voronoi tessellations) play an essential role. These tessellations can be shown to be extensionally equivalent to topological tessellations that can be constructed for Alexandroff spaces in general. Thereby, Rumfitt's and Gärdenfors's constructions turn out to be special cases of an approach that works for a more general class of spaces, namely, for weakly scattered Alexandroff spaces. The main aim of this paper is to show that this class of spaces provides a convenient framework for conceptual spaces as used in epistemology and related disciplines in general. Weakly scattered Alexandroff spaces are useful for elucidating problems related to the logic of vague concepts, in particular they offer a solution of the Sorites paradox (Rumfitt). Further, they provide a semantics for the logic of clearness (Bobzien) that overcomes certain problems of the concept of higher-order vagueness. Moreover, these spaces help find a natural place for classical syllogistics in the framework of conceptual spaces. The specialization order of Alexandroff spaces can be used to refine the all-or-nothing distinction between prototypical and non-prototypical stimuli in favor of a gradual distinction between more or less prototypical elements of conceptual spaces. The greater conceptual flexibility of the topological approach helps avoid some inherent inadequacies of the geometrical approach, for instance, the so-called "thickness problem". Finally, it is shown that the Alexandroff spaces offer an appropriate framework to deal with digital conceptual spaces that are gaining more and more importance in the areas of artificial intelligence, computer science and related disciplines. [ABSTRACT FROM AUTHOR]

Y. Rudyak develops the concept of the topological complexity TC(X) defined by M. Farber. We study this notion in digital images by using the fundamental properties of the digital homotopy. These properties can also be useful for the future works in some applications of algebraic topology besides topological robotics. Moreover, we show that the cohomological lower bounds for the digital topological complexity TC(X,κ) do not hold.

We continue the work of [4, 2, 3], in which we discuss published assertions that are incorrect or incorrectly proven; that are severely limited or reduce to triviality; or that we improve upon.

We examine the relationship between convexity and the approxi- mate fixed point property (AFPP) for digital images in Z. [ABSTRACT FROM AUTHOR]