scholarly journals Topological structuring of the digital plane

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Josef Šlapal
Keyword(s):  
Jordan Curve ◽  
Jordan Curve Theorem ◽  
Quotient Topology ◽  
Digital Plane ◽  
International Audience

Combinatorics International audience We discuss an Alexandroff topology on ℤ2 having the property that its quotient topologies include the Khalimsky and Marcus-Wyse topologies. We introduce a further quotient topology and prove a Jordan curve theorem for it.

Digital Jordan Curves and Surfaces with Respect to a Closure Operator

2021 ◽  
Vol 179 (1) ◽  
pp. 59-74
Author(s):  
Josef Šlapal
Keyword(s):  
Direct Product ◽  
Jordan Curve ◽  
Closure Operator ◽  
Jordan Curve Theorem ◽  
Closure Operators ◽  
Ternary Relation ◽  
Digital Space ◽  
Jordan Curves ◽  
Curves And Surfaces ◽  
Digital Plane

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

Walk-set induced connectedness in digital spaces

2017 ◽  
Vol 33 (2) ◽  
pp. 247-256
Author(s):  
JOSEF SLAPAL ◽  
Keyword(s):  
Jordan Curve ◽  
Digital Images ◽  
Simple Graph ◽  
Jordan Curve Theorem ◽  
Strong Product ◽  
Digital Plane ◽  
Vertex Set ◽  
Product Of Graphs ◽  
Digital Spaces

In an undirected simple graph, we define connectedness induced by a set of walks of the same lengths. We show that the connectedness is preserved by the strong product of graphs with walk sets. This result is used to introduce a graph on the vertex set Z2 with sets of walks that is obtained as the strong product of a pair of copies of a graph on the vertex set Z with certain walk sets. It is proved that each of the walk sets in the graph introduced induces connectedness on Z2 that satisfies a digital analogue of the Jordan curve theorem. It follows that the graph with any of the walk sets provides a convenient structure on the digital plane Z2 for the study of digital images.

scholarly journals Boundaries in digital planes

1990 ◽  
Vol 3 (1) ◽  
pp. 27-55 ◽  
Author(s):  
Efim Khalimsky ◽  
Ralph Kopperman ◽  
Paul R. Meyer
Keyword(s):  
Jordan Curve ◽  
Ordered Set ◽  
Continuous Functions ◽  
Jordan Curve Theorem ◽  
Connected Sets ◽  
Digital Plane ◽  
Parameter Interval ◽  
Totally Ordered Set ◽  
Definition Of ◽  
Natural Way

The importance of topological connectedness properties in processing digital pictures is well known. A natural way to begin a theory for this is to give a definition of connectedness for subsets of a digital plane which allows one to prove a Jordan curve theorem. The generally accepted approach to this has been a non-topological Jordan curve theorem which requires two different definitions, 4-connectedness, and 8-connectedness, one for the curve and the other for its complement.In [KKM] we introduced a purely topological context for a digital plane and proved a Jordan curve theorem. The present paper gives a topological proof of the non-topological Jordan curve theorem mentioned above and extends our previous work by considering some questions associated with image processing:How do more complicated curves separate the digital plane into connected sets? Conversely given a partition of the digital plane into connected sets, what are the boundaries like and how can we recover them? Our construction gives a unified answer to these questions.The crucial step in making our approach topological is to utilize a natural connected topology on a finite, totally ordered set; the topologies on the digital spaces are then just the associated product topologies. Furthermore, this permits us to define path, arc, and curve as certain continuous functions on such a parameter interval.

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.

scholarly journals A Proof of the Jordan Curve Theorem

2019 ◽  
Vol 11 (12) ◽  
pp. 345-360
Author(s):  
Xing Zhang
Keyword(s):  
Jordan Curve ◽  
Jordan Curve Theorem

The Jordan Curve Theorem

2021 ◽  
pp. 57-76
Author(s):  
James K. Peterson
Keyword(s):  
Jordan Curve ◽  
Jordan Curve Theorem
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