scholarly journals Optimization of WSNs Flooding Rates by Khalimsky Topology

2014 ◽  
Vol 2 (6) ◽  
Author(s):  
Mahmoud Abdellaoui ◽  
Riyadh Gargouri ◽  
Mahmoud Mezghani
Keyword(s):  
Khalimsky Topology

scholarly journals REMARKS ON HOMOTOPIES ASSOCIATED WITH KHALIMSKY TOPOLOGY

2015 ◽  
Vol 37 (4) ◽  
pp. 577-593
Author(s):  
SANG-EON HAN ◽  
SIK LEE
Keyword(s):  
Khalimsky Topology

scholarly journals Remarks on topological spaces on $ {\mathbb Z}^n $ which are related to the Khalimsky $ n $-dimensional space

2021 ◽  
Vol 7 (1) ◽  
pp. 1224-1240
Author(s):  
Sang-Eon Han ◽  
◽  
Saeid Jafari ◽  
Jeong Min Kang ◽  
Sik Lee ◽  
...  
Keyword(s):  
Crucial Role ◽  
Dimensional Space ◽  
Digital Topology ◽  
Topological Spaces ◽  
Khalimsky Topology

<abstract><p>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.</p></abstract>

On Khalimsky topology and applications on the digital image segmentation

2015 ◽  
Vol 9 ◽  
pp. 3687-3701 ◽  
Author(s):  
M. Al Hajri ◽  
K. Belaid ◽  
L. Jaafar Belaid
Keyword(s):  
Image Segmentation ◽  
Digital Image ◽  
Khalimsky Topology

scholarly journals Topological graphs based on a new topology on Zn and its applications

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6313-6328 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Extension Problem ◽  
Topological Spaces ◽  
Natural Numbers ◽  
Topological Graph ◽  
Topological Graphs ◽  
Graph Homomorphism ◽  
Continuous Map ◽  
Higher Dimensional ◽  
Khalimsky Topology ◽  
Digital Spaces

Up to now there is no homotopy for Marcus-Wyse (for short M-) topological spaces. In relation to the development of a homotopy for the category of Marcus-Wyse (for short M-) topological spaces on Z2, we need to generalize the M-topology on Z2 to higher dimensional spaces X ? Zn, n ? 3 [18]. Hence the present paper establishes a new topology on Zn; n 2 N, where N is the set of natural numbers. It is called the generalized Marcus-Wyse (for short H-) topology and is denoted by (Zn, n). Besides, we prove that (Z3, 3) induces only 6- or 18-adjacency relations. Namely, (Z3, 3) does not support a 26-adjacency, which is quite different from the Khalimsky topology for 3D digital spaces. After developing an H-adjacency induced by the connectedness of (Zn; n), the present paper establishes topological graphs based on the H-topology, which is called an HA-space, so that we can establish a category of HA-spaces. By using the H-adjacency, we propose an H-topological graph homomorphism (for short HA-map) and an HA-isomorphism. Besides, we prove that an HA-map (resp. an HA-isomorphism) is broader than an H-continuous map (resp. an Hhomeomorphism) and is an H-connectedness preserving map. Finally, after investigating some properties of an HA-isomorphism, we propose both an HA-retract and an extension problem of an HA-map for studying HA-spaces.

scholarly journals Digital straight lines in the Khalimsky plane

2005 ◽  
Vol 96 (1) ◽  
pp. 49 ◽  
Author(s):  
Erik Melin
Keyword(s):  
Line Segment ◽  
Straight Line Segment ◽  
Integer Points ◽  
Straight Line ◽  
Digital Plane ◽  
Straight Lines ◽  
Khalimsky Topology

We consider the digital plane of integer points equipped with the Khalimsky topology. We suggest a digitization of straight lines such that the digitized image is homeomorphic to the Khalimsky line and a digitized line segment is a Khalimsky arc. It is demonstrated that a Khalimsky arc is the digitization of a straight line segment if and only if it satisfies a generalized version of the chord property introduced by Rosenfeld.

scholarly journals A closure operator for the digital plane

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3229-3237
Author(s):  
Josef Slapal
Keyword(s):  
Jordan Curve ◽  
Closure Operator ◽  
Digital Images ◽  
Jordan Curve Theorem ◽  
Digital Plane ◽  
Khalimsky Topology

We introduce and study a closure operator on the digital plane Z2. The closure operator is shown to provide connectedness that allows for a digital analogue of the Jordan curve theorem. This enables using the closure operator for structuring the digital plane in order to study and process digital images. An advantage of the closure operator over the Khalimsky topology on Z2 is demonstrated, too.

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