Digitizations associated with several types of digital topological approaches

Kang, Jeong Min; Han, Sang-Eon; Min, Kyung Chan

doi:10.1007/s40314-015-0245-0

Cited by 20 publications

(43 citation statements)

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“…Then we obtain the product topology on R n , denoted by (R n , E n U ), induced by (R, E U ). Based on (R n , E n U ), we use a U -local rule [16] which is used to digitize (…”

Section: Some Properties Of a U And An L-local Rulementioning

confidence: 99%

“…Let us now recall the L-local rule in [16]. The L-topology on R, denoted by (R, E L ), is induced by the set of closed open intervals in R, {[a, b) | a, b ∈ R and a < b}, as a base [23].…”

Section: Definition 3 [16]mentioning

confidence: 99%

“…It is clear that the relation L X in the set (X, L X ) of Definition 7 is an equivalence relation [16].…”

Section: Establishments Of a U (K)-and An L(k)-digitizationmentioning

confidence: 99%

“…Hereafter, based on the U -topology and the L-topology, after proceeding with a digitization of (X, E n X ) [16], we obtain a U -(resp. an L-) digitized space denoted by D U (X) (resp.…”

Section: Introductionmentioning

confidence: 99%

“…an L-) digitized space denoted by D U (X) (resp. D L (X)) in Z n [16]. Further considering D U (X) (resp.…”

Section: Introductionmentioning

confidence: 99%

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U(k)- AND L(k)-HOMOTOPIC PROPERTIES OF DIGITIZATIONS OF nD HAUSDORFF SPACES

Han¹

2017

HJMS

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Abstract. For X ⊂ R n let (X, E n X ) be the usual topological space induced by the nD Euclidean topological space (R n , E n ). Based on the upper limit (U -, for short) topology (resp. the lower limit (L-, for brevity) topology), after proceeding with a digitization of (X,) with a digital k-connnectivity, we obtain a digital image from the viewpoint of digital topology in a graph-theoretical approach, i.e. Rosenfeld model [25], denoted by D U (k) (X) (resp. D L(k) (X)) in the present paper. Since a Euclidean topological homotopy has some limitations of studying a digitization of (X, E n X ), the present paper establishes the so called U (k)-homotopy (resp. L(k)-homotopy) which can be used to study homotopic properties of both (X, E n X ) and D U (k) (X) (resp. both (X, E n X ) and D L(k) (X)). The goal of the paper is to study some relationships among an ordinary homotopy equivalence, a U (k)-homotopy equivalence, an L(k)-homotopy equivalence and a khomotopy equivalence. Finally, we classify (X, E n X ) in terms of a U (k)-homotopy equivalence and an L(k)-homotopy equivalence. This approach can be used to study applied topology, approximation theory and digital geometry.

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“…Then we obtain the product topology on R n , denoted by (R n , E n U ), induced by (R, E U ). Based on (R n , E n U ), we use a U -local rule [16] which is used to digitize (…”

Section: Some Properties Of a U And An L-local Rulementioning

confidence: 99%

Section: Definition 3 [16]mentioning

confidence: 99%

“…It is clear that the relation L X in the set (X, L X ) of Definition 7 is an equivalence relation [16].…”

Section: Establishments Of a U (K)-and An L(k)-digitizationmentioning

confidence: 99%

“…Hereafter, based on the U -topology and the L-topology, after proceeding with a digitization of (X, E n X ) [16], we obtain a U -(resp. an L-) digitized space denoted by D U (X) (resp.…”

Section: Introductionmentioning

confidence: 99%

“…an L-) digitized space denoted by D U (X) (resp. D L (X)) in Z n [16]. Further considering D U (X) (resp.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations