Digital Topology Research Articles

Contractibility and fixed point property: the case of Khalimsky topological spaces

AbstractBased on the notions of both contractibility and local contractibility, many works were done in fixed point theory. The present paper concerns a relation between digital contractibility and the existence of fixed points of digitally continuous maps. In this paper, establishing a new digital homotopy named by a K-homotopy in the category of Khalimsky topological spaces, we prove that in digital topology, whereas contractibility implies local contractibility, the converse does not hold. Furthermore, we address the following problem, which remains open. Let X be a Khalimsky (K- for short) topological space with K-contractibility. Then we may pose the following question: does the space X have the fixed point property (FPP)? In this paper, we prove that not every K-topological space with K-contractibility has the FPP.

Banach fixed point theorem from the viewpoint of digital topology

Banach fixed point theorem from the viewpoint of digital topology

Connectivity Preserving Multivalued Functions in Digital Topology

We study connectivity preserving multivalued functions (Kovalevsky in A new concept for digital geometry, shape in picture, 1994) between digital images. This notion generalizes that of continuous multivalued functions (Escribano et al. in Discrete geometry for computer imagery, lecture notes in computer science, 2008; Escribano et al. in J Math Imaging Vis 42:76---91, 2012) studied mostly in the setting of the digital plane $${\mathbb {Z}}^2$$Z2. We show that connectivity preserving multivalued functions, like continuous multivalued functions, are appropriate models for digital morphological operations. Connectivity preservation, unlike continuity, is preserved by compositions, and generalizes easily to higher dimensions and arbitrary adjacency relations.

Properties of digital spaces on \(\mathbb{Z}^{2}\)

The two conditions $1^d$ and $2^d$ are so that any digital topology on \(\mathbb{Z}^d\) satisfies them is topologically connected whenever it is graphically connected. In this paper, we prove that the digital topologies on \(\mathbb{Z}^d\) are \(g\)-locally finite \(T_0\) Alexandroff spaces. We study the properties of the two digital topologies on \(\mathbb{Z}^2\) that satisfy \(1^2\) and \(2^2\). We describe the specialization orders of these topologies, and we determine the points in \(\mathbb{Z}^2\) that are minimal, maximal, and saddle points. We prove that, the summation topology is homeomorphic to the Khalimsky topology.

Path-set induced closure operators on graphs

Given a simple graph, we associate with every set of paths of the same positive length a closure operator on the (vertex set of the) graph. These closure operators are then studied. In particular, it is shown that the connectedness with respect to them is a certain kind of path connectedness. Closure operators associated with sets of paths in some graphs with the vertex set Z2 are discussed which include the well known Marcus-Wyse and Khalimsky topologies used in digital topology. This demonstrates possible applications of the closure operators investigated in digital image analysis.

Some properties of digital H-spaces

In this paper, we study certain properties of digital H-spaces. We prove that a digital image that has the same digital homotopy type with any digital H-space is also a digital H-space. We show that the digital fundamental group of a digital H-space is abelian. We give examples that are related to a digital homotopy associative H-space and a -contractible digital H-space. Several important applications of digital H-spaces are given in computer vision and image processing. Finally, we deal with the importance of digital H-space in digital topology and image processing. We conclude that any -contractible digital image is a digital H-space.

Properties of space set topological spaces

Since a locally finite topological structure plays an important role in the fields of pure and applied topology, the paper studies a special kind of locally finite spaces, so called a space set topology (for brevity, SST) and further, proves that an SST is an Alexandroff space satisfying the separation axiom T0. Unlike a point set topology, since each element of an SST is a space, the present paper names the topology by the space set topology. Besides, for a connected topological space (X,T) with |X| = 2 the axioms T0, semi-T1/2 and T1/2 are proved to be equivalent to each other. Furthermore, the paper shows that an SST can be used for studying both continuous and digital spaces so that it plays a crucial role in both classical and digital topology, combinatorial, discrete and computational geometry. In addition, a connected SST can be a good example showing that the separation axiom semi-T1/2 does not imply T1/2.

An extension problem of a connectedness preserving map between Khalimsky spaces

The goal of the present paper is to study an extension problem of a connected preserving (for short, CP-) map between Khalimsky (K-for brevity, if there is no ambiguity) spaces. As a generalization of a K-continuous map, for K-topological spaces the recent paper [13] develops a function sending connected sets to connected ones (for brevity, an A-map: see Definition 3.1 in the present paper). Since this map plays an important role in applied topology including digital topology, digital geometry and mathematical morphology, the present paper studies an extension problem of a CP-map in terms of both an A-retract and an A-isomorphism (see Example 5.2). Since K-topological spaces have been often used for studying digital images, this extension problem can contribute to a certain areas of computer science and mathematical morphology.

Homotopy based on Marcus–Wyse topology and its applications

The present paper establishes a homotopy based on Marcus–Wyse (for brevity M-) topology, which can contribute to the classification of 2D digital images in an M-topological approach. Since M-topology is considered in the Euclidean 2D space with integer coordinates, it can be used in studying 2D digital images from the viewpoints of both digital topology and digital geometry such as image processing, image analysis, computer graphics, mathematical morphology and so forth. To develop the homotopy, the present paper uses two maps such as an M-continuous map and Marcus–Wyse adjacency (for brevity MA-) map because they have their own features and merits. Besides, using this homotopy, the present paper proposes an MA-homotopy equivalence and further, MA-contractibility which can be also used in classifying digital images. Finally, the paper establishes an MA-homotopic thinning derived from the above homotopy, which can contribute to the compression of 2D digital images.

On topology preservation of mixed operators in triangular, square, and hexagonal grids

A crucial issue in digital topology is to ensure topology preservation of binary image operators. Several sufficient conditions have already been proposed (i.e., for operators that never change a white pixel to black one and a black pixel to a white one, respectively), however not much attention has been paid on mixed operators, which can modify both black and white pixels. In this paper, we intend to fill in this gap by presenting some sufficient criteria on the latter family of operators considering all the possible regular partitionings of the plane (i.e., triangular, square, and hexagonal grids).

Type synthesis of spatial 3-DoF parallel mechanisms with planar sub-chains using revised digital topological graphs and arrays

SUMMARYType synthesis of spatial 3-DoF (degree of freedom) parallel mechanisms (PMs) with planar sub-chains is studied using the revised digital topology graph (r-DTGs) and arrays. First, many DTGs for type synthesis of spatial 3-DoF PMs are derived from their contracted graphs (CGs). Second, a complicated derivation of the DTG and an identification of an isomorphic DTG are transformed into a simple derivation of an array and an identification of an isomorphic array using a compiled program. Third, one or more spatial closed-loop chains in the derived DTGs are changed into the planar closed-loop chains by modifying digits marked in closed-loop chains, and the DTGs are transformed into the r-DTGs for the type synthesis of the spatial 3-DoF PMs with planar sub-chains. Finally, the 52 novel spatial 3-DoF PMs with the planar sub-chains are synthesized and verified by simulation mechanisms.

Recent Research Discourses on <i>K</i>-T<sub>1/2</sub> Spaces (<i>K</i>=α,p,β,s & pr)

Framing of this paper is to project the recent research work on K–T1/2 spaces where K= α, p,β,s & pr and to overview their fundamental properties at a glance. However, the structure and characteristics possessed by these spaces are not always that easy to comprehend. These spaces are very useful in the study of certain objects in digital topology viz E.D. Khalinmsky et.al [23] showed that the digital line is a typical example of a T1/2-space in which the closed sets and g-closed sets coincide.

Digital Topology and Geometry in Medical Imaging: A Survey.

Digital topology and geometry refers to the use of topologic and geometric properties and features for images defined in digital grids. Such methods have been widely used in many medical imaging applications, including image segmentation, visualization, manipulation, interpolation, registration, surface-tracking, object representation, correction, quantitative morphometry etc. Digital topology and geometry play important roles in medical imaging research by enriching the scope of target outcomes and by adding strong theoretical foundations with enhanced stability, fidelity, and efficiency. This paper presents a comprehensive yet compact survey on results, principles, and insights of methods related to digital topology and geometry with strong emphasis on understanding their roles in various medical imaging applications. Specifically, this paper reviews methods related to distance analysis and path propagation, connectivity, surface-tracking, image segmentation, boundary and centerline detection, topology preservation and local topological properties, skeletonization, and object representation, correction, and quantitative morphometry. A common thread among the topics reviewed in this paper is that their theory and algorithms use the principle of digital path connectivity, path propagation, and neighborhood analysis.

A digitization method of subspaces of the Euclidean $$n$$ n D space associated with the Khalimsky adjacency structure

To digitize subspaces of the Euclidean \(n\)D space, the present paper uses the Khalimsky (for short \(K\)-, if there is no danger of ambiguity) topology, \(K\)-adjacency and \(K\)-localized neighborhoods of points in \(\mathbf{Z}^n\), where \(\mathbf{Z}^n\) represents the set of points in the Euclidean \(n\)D space with integer coordinates. Namely, given a point \(p \in \mathbf{Z}^n\), the paper first develops a \(K\)-localized neighborhood of \(p \in \mathbf{Z}^n\), denoted by \(N_K(p)\) in \(\mathbf{R}^n\), which is substantially used in digitizing subspaces of the Euclidean \(n\)D space. The recent paper Han and Sostak in (Comput Appl Math 32(3):521–536, 2013) proposes a connectedness preserving map (for short CP-map, e.g., an \(A\)-map in this paper) which need not be a continuous map under \(K\)-topology and further, develops a certain CP-isomorphism, e.g., an \(A\)-isomorphism in this paper. It turns out that an \(A\)-map overcomes some limitations of both a \(K\)-continuous map and a Khalimsky adjacency map (for brevity \(KA\)-map) so that both an \(A\)-map and an \(A\)-isomorphism can substantially contribute to applied topology including both digital topology and digital geometry Han and Sostak in (Comput Appl Math 32(3):521–536, 2013). Using both an \(A\)-map and a \(K\)-localized neighborhood, we further develop the notions of a lattice-based \(A\)-map (for short \(LA\)-map) and a lattice-based \(A\)-isomorphism (for brevity \(LA\)-isomorphism) which are used for digitizing subspaces of the Euclidean \(n\)D space in the \(K\)-topological approach. Thus, this approach can contribute to certain branches of applied topology and computer science such as image analysis, image processing, and mathematical morphology.

Combinatorial and discrete geometric problems in image analysis

Image analysis is a scientific discipline which provides theoretical foundations and methods for solving problems appearing in a range of areas as diverse as biology, medicine, physics, astronomy, chemistry, robotics, industrial manufacturing, and security. The very nature of the subject of image analysis determines its close relations to various facets of artificial intelligence. Unlike traditional approaches, which are based on continuous models requiring float arithmetic computations and rounding, "combinatorial" (or "discrete") approaches are based on studying combinatorial properties of discrete structures. They provide models and algorithms, which often appear to be more efficient and accurate than those based on continuous models. Some recent combinatorial approaches aim at constructing a self-contained digital topology and geometry, which might be of interest and importance not only for image analysis, but also as independent mathematical disciplines.

UTILITY OF DIGITAL COVERING THEORY

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.

REMARKS ON SIMPLY k-CONNECTIVITY AND k-DEFORMATION RETRACT IN DIGITAL TOPOLOGY

Abstract. To study a deformation of a digital space from the view-point of digital homotopy theory, we have often used the notions ofa weak k-deformation retract [20] and a strong k-deformation re-tract [10, 12, 13]. Thus the papers [10, 12, 13, 16] firstly developedthe notion of a strong k-deformation retract which can play an im-portant role in studying a homotopic thinning of a digital space.Besides, the paper [3] deals with a k-deformation retract and itshomotopic property related to a digital fundamental group. Thus,as a survey article, comparing among a k-deformation retract in [3],a strong k-deformation retract in [10, 12, 13], a weak deformationk-retract in [20] and a digital k-homotopy equivalence [5, 24], we ob-serve some relationships among them from the viewpoint of digitalhomotopy theory. Furthermore, the present paper deals with someparts of the preprint [10] which were not published in a journal (seeProposition 3.1). Finally, the present paper corrects Boxer’s paper[3] as follows: even though the paper [3] referred to the notion of adigital homotopy equivalence (or a same k-homotopy type) whichis a special kind of a k-deformation retract, we need to point outthat the notion was already developed in [5] instead of [3] and fur-ther corrects the proof of Theorem 4.5 of Boxer’s paper [3] (see theproof of Theorem 4.1 in the present paper). While the paper [4]refers some properties of a deck transformation group (or an auto-morphism group) of digital covering space without any citation, thestudy was early done by Han in his paper (see the paper [14]).Received May 15, 2014. Accepted June 10, 2014.2010 Mathematics Subject Classification. 55Q70, 52Cxx, 55P15, 68R10, 68U05.Key words and phrases. simply k-connected, digital isomorphism, strong k-deformation, weak k-deformation retract, k-homotopy equivalence (same k-homotopytype), digital topology.

Relative Homology Groups of Digital Images

In this paper we are interested in relative homology groups of digital images. Some properties of the Euler characteristics for digital images are given. We also present reduced homology groups for digital images. The main purpose is to obtain some differences between notions in digital topology and algebraic topology.

Characterization of the deletion of [formula omitted]-simple points as multivalued [formula omitted]-retractions

In previous papers we have introduced a notion of multivalued continuity in digital spaces which extends the usual notion of digital continuity and allows us to define topological notions, like retractions, in a far more realistic way than by using just single-valued digitally continuous functions. In particular, we have characterized the deletion of simple points in 2-D, one of the most important processing operations in digital topology, as a particular kind of retraction. In this work we extend this result to three-dimensional digital sets, characterizing the deletion of (26,6)-simple points in 3-D as (N,26)-retractions.

Thickness-based adaptive mesh refinement methods for multi-phase flow simulations with thin regions

In numerical simulations of multi-scale, multi-phase flows, grid refinement is required to resolve regions with small scales. A notable example is liquid-jet atomization and subsequent droplet dynamics. It is essential to characterize the detailed flow physics with variable length scales with high fidelity, in order to elucidate the underlying mechanisms. In this paper, two thickness-based mesh refinement schemes are developed based on distance- and topology-oriented criteria for thin regions with confining wall/plane of symmetry and in any situation, respectively. Both techniques are implemented in a general framework with a volume-of-fluid formulation and an adaptive-mesh-refinement capability. The distance-oriented technique compares against a critical value, the ratio of an interfacial cell size to the distance between the mass center of the cell and a reference plane. The topology-oriented technique is developed from digital topology theories to handle more general conditions. The requirement for interfacial mesh refinement can be detected swiftly, without the need of thickness information, equation solving, variable averaging or mesh repairing. The mesh refinement level increases smoothly on demand in thin regions. The schemes have been verified and validated against several benchmark cases to demonstrate their effectiveness and robustness. These include the dynamics of colliding droplets, droplet motions in a microchannel, and atomization of liquid impinging jets. Overall, the thickness-based refinement technique provides highly adaptive meshes for problems with thin regions in an efficient and fully automatic manner.