On the relationship among F-transform, fuzzy rough set and fuzzy topology

Perfilieva, Irina; Singh, Anand Pratap; Tiwari, S. P.

doi:10.1007/s00500-017-2559-x

Cited by 29 publications

(6 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The relational variant of Zadeh's extension was for the first time defined by Goguen [15], when he introduced the notion of the image of a fuzzy set under a fuzzy relation. Many examples using explicitly or implicitly powerset functors can be found in rough fuzzy sets theory, F-transform theory and many others (see, e..g., [16][17][18][19]). Powerset functors that convert morphisms f : X → Y between two fuzzy objects from a category to mappings between some sets of corresponding fuzzy objects T(X) and T(Y) defined above these objects sometimes represent also special types of transformation operators.…”

Section: Introductionmentioning

confidence: 99%

Fuzzy Type Relations and Transformation Operators Defined by Monads

Močkoř¹

2020

IJCIS

View full text Add to dashboard Cite

Using the theory of monads in categories and the theory of monadic relations, the concept of general transformation operator defined by a monadic relation is introduced. It is proven that a number of standard relations used in categories of fuzzy structures are monadic relations for monads defined in these categories. It is also proven that a number of standardly used transformation operators in fuzzy sets, fuzzy rough sets, or fuzzy soft sets are in the form of general operators defined by monadic relations.

show abstract

Section: Introductionmentioning

confidence: 99%

Fuzzy Type Relations and Transformation Operators Defined by Monads

Močkoř¹

2020

IJCIS

View full text Add to dashboard Cite

show abstract

“…This approximation was for the first time defined by Goguen [2], when he introduced the notion of the image of a fuzzy set under a fuzzy relation. Many examples using explicitly or implicitly approximation functors defined by various types of fuzzy relations can be found in rough fuzzy sets theory and many others (see, e.g., [16][17][18]).…”

Section: Introductionmentioning

confidence: 99%

“…These applications include in particular signal and image processing [19][20][21], data analysis [22][23][24], signal compression [25,26], and numerical solutions of differential equations [27][28][29]. This new method of a transformation of fuzzy sets is based on reduction of basic space of given fuzzy sets and it has been introduced and elaborated by Perfilieva in papers [16,24,25,[30][31][32][33][34]. The original form of a lattice-valued fuzzy transform is a map F : L X → L Y , where X is a "large" set (the original universe of fuzzy sets), Y is a "smaller" set, and L is an appropriate complete lattice.…”

Section: Introductionmentioning

confidence: 99%

Relational Variants of Lattice-Valued F-Transforms

Močkoř

2019

Axioms

View full text Add to dashboard Cite

Two categories of lower and upper lattice-valued F-transforms with fuzzy relations as morphisms are introduced, as generalisations of standard categories of F-transforms with maps as morphisms. Although F-transforms are defined using special structures called spaces with fuzzy partitions, it is shown that these categories are identical to the relational variants of the two categories of semimodule homomorphisms where these fuzzy partitions do not occur. This a priori independence of the F-transform on spaces with fuzzy partitions makes it possible, for example, to use a simple matrix calculus to calculate F-transforms, or to determine the image of F-transforms in relational morphisms of the two categories.

show abstract

“…The extension of the F-transform to functions in two variables shows powerful applications in signal and image processing, particularly, image compression (cf., [10,13,19]), edge detection (cf., [3,22]), image reconstruction [21], image fusion [32], coding/decoding of images [11] and solutions of wave equation [29]. After the study of lattice F-transform of one variable [16], recently, the relationship of lattice F-transform with fuzzy rough set and fuzzy topology is studied in [23], while mathematical morphology is studied in [27]. In continuation to such studies, in this paper we focus on the extension of the theory of lattice F-transform for functions in two variables based on the generalized fuzzy partition (not necessarily finite) of universe.…”

Section: Introductionmentioning

confidence: 99%