Continuous Fuzzy Transform as Integral Operator

The fuzzy transform (F-transform) is ubiquitous in different research fields and applications, such as image and data compression, data mining, knowledge discovery, and the analysis of linguistic expressions. As a generalization of the F-transform, in this article, we introduce the continuous F-tran...

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Vydané v:IEEE transactions on fuzzy systems Ročník 29; číslo 10; s. 3093 - 3102
Hlavný autor: Patane, Giuseppe
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York IEEE 01.10.2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Predmet:
Computational efficiency
Data analysis
Data compression
Data mining
Data models
data-driven membership functions
Fuzzy transform (F-transform)
Image coding
Image compression
inverse F-transform
Kernel
Kernel functions
Laplace–Beltrami operator
Linear algebra
Operators (mathematics)
Splines (mathematics)
Transformations (mathematics)
Transforms
ISSN:1063-6706, 1941-0034
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Shrnutí:The fuzzy transform (F-transform) is ubiquitous in different research fields and applications, such as image and data compression, data mining, knowledge discovery, and the analysis of linguistic expressions. As a generalization of the F-transform, in this article, we introduce the continuous F-transform and its inverse, as an integral operator induced by a kernel function. Through the relation between membership functions and integral kernels, we show that the main properties (e.g., continuity and symmetry) of the membership functions are inherited by the continuous F-transform. Then, the relation between the continuous F-transform and integral operators is used to introduce a data-driven F-transform, which encodes intrinsic information (e.g., structure, geometry, and sampling density) about the input data. In this way, we avoid coarse fuzzy partitions, which group data into large clusters that do not adapt to their local behavior, or a too dense fuzzy partition, which generally has cells that are not covered by the data, thus being redundant and resulting in a higher computational cost. To this end, the data-driven membership functions are defined by properly filtering the spectrum of the Laplace-Beltrami operator associated with the input data. Finally, we introduce the space of continuous F-transforms, which is useful for the comparison of different continuous F-transforms and for their efficient computation.
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ISSN:1063-6706
1941-0034
DOI:10.1109/TFUZZ.2020.3012764