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Axioms
Special Issues
Fuzzy Transforms and Their Applications
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Fuzzy Transforms and Their Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Logic".

Deadline for manuscript submissions: closed (30 October 2019) | Viewed by 21930

Special Issue Editors

Prof. Dr. Ferdinando Di Martino

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Guest Editor
Dipartimento di Architettura, Università degli Studi di Napoli Federico II, Via Toledo 402, 80134 Napoli, Italy
Interests: fuzzy sets and fuzzy relations; soft computing; fuzzy transform image processing theory; machine learning; data mining
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Irina Perfilieva

E-Mail Website
Guest Editor
Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 701 03 Ostrava, Czech Republic
Interests: fuzzy logic; fuzzy modeling; image processing; computer graphics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Salvatore Sessa

E-Mail Website
Guest Editor
Department of Architecture, Federico II Naples University, Via Toledo 402, 80134 Naples, Italy
Interests: fixed point theory in metric spaces; fuzzy clustering algorithms; fuzzy relations and their fuzzy calculus; fuzzy relation equations; fuzzy relational systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We propose to launch a new Special Issue of Axioms. The main topic is focused on “Fuzzy Transforms”. With this Special Issue, we aim to provide contributing authors an opportunity to present their recent results in the mathematical theory of Fuzzy Transforms with applications to various fields, such as signal processing, image processing, machine learning, and data analysis. Among the topics that this Special Issue will address, we consider the following non-exhaustive list:

Multidimensional Fuzzy Transform, higher order Fuzzy Transform, Fuzzy transforms applied in coding/decoding signals, images and videos, Fuzzy Transforms methods in image reduction, image fusion, image segmentation, image tamper detection, Fuzzy Transforms-based models for data classification,  forecasting, data mining, and Fuzzy Transforms in massive data knowledge extraction.

In addition, this Special Issue is open to discussing new ideas, apart from the aforementioned topics.

If this initiative meets your interests, we solicit you to send your contributions to be included in this Special Issue.

Prof. Dr. Ferdinando Di Martino
Prof. Dr. Irina Perfilieva
Prof. Dr. Salvatore Sessa
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Direct and inverse Fuzzy Transform
  • Discrete Fuzzy Transform
  • Multidimensional Fuzzy Transform
  • High order Fuzzy Transform
  • Fuzzy Transform applications

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Published Papers (7 papers)

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Research

32 pages, 4138 KiB  
Article
Bitcoin Analysis and Forecasting through Fuzzy Transform
by Maria Letizia Guerra, Laerte Sorini and Luciano Stefanini
Axioms 2020, 9(4), 139; https://doi.org/10.3390/axioms9040139 - 28 Nov 2020
Cited by 6 | Viewed by 3328
Abstract
Sentiment analysis to characterize the properties of Bitcoin prices and their forecasting is here developed thanks to the capability of the Fuzzy Transform (F-transform for short) to capture stylized facts and mutual connections between time series with different natures. The recently proposed L [...] Read more.
Sentiment analysis to characterize the properties of Bitcoin prices and their forecasting is here developed thanks to the capability of the Fuzzy Transform (F-transform for short) to capture stylized facts and mutual connections between time series with different natures. The recently proposed Lp-norm F-transform is a powerful and flexible methodology for data analysis, non-parametric smoothing and for fitting and forecasting. Its capabilities are illustrated by empirical analyses concerning Bitcoin prices and Google Trend scores (six years of daily data): we apply the (inverse) F-transform to both time series and, using clustering techniques, we identify stylized facts for Bitcoin prices, based on (local) smoothing and fitting F-transform, and we study their time evolution in terms of a transition matrix. Finally, we examine the dependence of Bitcoin prices on Google Trend scores and we estimate short-term forecasting models; the Diebold–Mariano (DM) test statistics, applied for their significance, shows that sentiment analysis is useful in short-term forecasting of Bitcoin cryptocurrency. Full article
(This article belongs to the Special Issue Fuzzy Transforms and Their Applications)
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37 pages, 1930 KiB  
Article
On the Numerical Solution of Ordinary, Interval and Fuzzy Differential Equations by Use of F-Transform
by Davide Radi, Laerte Sorini and Luciano Stefanini
Axioms 2020, 9(1), 15; https://doi.org/10.3390/axioms9010015 - 5 Feb 2020
Cited by 5 | Viewed by 3273
Abstract
An interesting property of the inverse F-transform f ^ of a continuous function f on a given interval [ a , b ] says that the integrals of f ^ and f on [ a , b ] coincide. Furthermore, the same property [...] Read more.
An interesting property of the inverse F-transform f ^ of a continuous function f on a given interval [ a , b ] says that the integrals of f ^ and f on [ a , b ] coincide. Furthermore, the same property can be established for the restrictions of the functions to all subintervals [ a , p k ] of the fuzzy partition of [ a , b ] used to define the F-transform. Based on this fact, we propose a new method for the numerical solution of ordinary differential equations (initial-value ordinary differential equation (ODE)) obtained by approximating the derivative x · ( t ) via F-transform, then computing (an approximation of) the solution x ( t ) by exact integration. For an ODE, a global second-order approximation is obtained. A similar construction is then applied to interval-valued and (level-wise) fuzzy differential equations in the setting of generalized differentiability (gH-derivative). Properties of the new method are analyzed and a computational section illustrates the performance of the obtained procedures, in comparison with well-known efficient algorithms. Full article
(This article belongs to the Special Issue Fuzzy Transforms and Their Applications)
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16 pages, 324 KiB  
Article
F-Transform Inspired Weak Solution to a Boundary Value Problem
by Linh Nguyen, Irina Perfilieva and Michal Holčapek
Axioms 2020, 9(1), 5; https://doi.org/10.3390/axioms9010005 - 31 Dec 2019
Cited by 2 | Viewed by 2718
Abstract
We propose and show efficiency of a new fuzzy-transform-based numerical method of solving ordinary differential equations with boundary conditions. The focus is on weak solutions and a special construction of a two-parameterized family of test functions. On theoretical and computational levels, we show [...] Read more.
We propose and show efficiency of a new fuzzy-transform-based numerical method of solving ordinary differential equations with boundary conditions. The focus is on weak solutions and a special construction of a two-parameterized family of test functions. On theoretical and computational levels, we show how the proposed technique relates to and outperforms the Ritz–Galerkin method. We emphasize the importance of the proposed technique by considering its application to a real-life problem—the option pricing policy. Full article
(This article belongs to the Special Issue Fuzzy Transforms and Their Applications)
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17 pages, 282 KiB  
Article
Relational Variants of Lattice-Valued F-Transforms
by Jiří Močkoř
Axioms 2020, 9(1), 1; https://doi.org/10.3390/axioms9010001 - 19 Dec 2019
Cited by 1 | Viewed by 2277
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fuzzy Transforms and Their Applications)
13 pages, 3053 KiB  
Article
A Fast Multilevel Fuzzy Transform Image Compression Method
by Ferdinando Di Martino, Irina Perfilieva and Salvatore Sessa
Axioms 2019, 8(4), 135; https://doi.org/10.3390/axioms8040135 - 3 Dec 2019
Cited by 4 | Viewed by 2666
Abstract
We present a fast algorithm that improves on the performance of the multilevel fuzzy transform image compression method. The multilevel F-transform (for short, MF-tr) algorithm is an image compression method based on fuzzy transforms that, compared to the classic fuzzy transform (F-transform) image [...] Read more.
We present a fast algorithm that improves on the performance of the multilevel fuzzy transform image compression method. The multilevel F-transform (for short, MF-tr) algorithm is an image compression method based on fuzzy transforms that, compared to the classic fuzzy transform (F-transform) image compression method, has the advantage of being able to reconstruct an image with the required quality. However, this method can be computationally expensive in terms of execution time since, based on the compression ratio used, different iterations may be necessary in order to reconstruct the image with the required quality. To solve this problem, we propose a fast variation of the multilevel F-transform algorithm in which the optimal compression ratio is found in order to reconstruct the image in as few iterations as possible. Comparison tests show that our method reconstructs the image in at most half of the CPU time used by the MF-tr algorithm. Full article
(This article belongs to the Special Issue Fuzzy Transforms and Their Applications)
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11 pages, 251 KiB  
Article
Why Triangular Membership Functions Are Successfully Used in F-Transform Applications: A Global Explanation to Supplement the Existing Local Ones
by Olga Kosheleva, Vladik Kreinovich and Thach Ngoc Nguyen
Axioms 2019, 8(3), 95; https://doi.org/10.3390/axioms8030095 - 5 Aug 2019
Cited by 4 | Viewed by 3663
Abstract
The main ideas of F-transform came from representing expert rules. It would be therefore reasonable to expect that the more accurately the membership functions describe human reasoning, the more successful will be the corresponding F-transform formulas. We know that an adequate description of [...] Read more.
The main ideas of F-transform came from representing expert rules. It would be therefore reasonable to expect that the more accurately the membership functions describe human reasoning, the more successful will be the corresponding F-transform formulas. We know that an adequate description of our reasoning corresponds to complicated membership functions—however, somewhat surprisingly, many successful applications of F-transform use the simplest possible triangular membership functions. There exist some explanations for this phenomenon, which are based on local behavior of the signal. In this paper, we supplement these local explanations by a global one: namely, we prove that triangular membership functions are the only one that provide the exact reconstruction of the appropriate global characteristic of the signal. Full article
(This article belongs to the Special Issue Fuzzy Transforms and Their Applications)
13 pages, 252 KiB  
Article
Why Use a Fuzzy Partition in F-Transform?
by Vladik Kreinovich, Olga Kosheleva and Songsak Sriboonchitta
Axioms 2019, 8(3), 94; https://doi.org/10.3390/axioms8030094 - 2 Aug 2019
Cited by 1 | Viewed by 2672
Abstract
In many application problems, F-transform algorithms are very efficient. In F-transform techniques, we replace the original signal or image with a finite number of weighted averages. The use of a weighted average can be naturally explained, e.g., by the fact that this is [...] Read more.
In many application problems, F-transform algorithms are very efficient. In F-transform techniques, we replace the original signal or image with a finite number of weighted averages. The use of a weighted average can be naturally explained, e.g., by the fact that this is what we get anyway when we measure the signal. However, most successful applications of F-transform have an additional not-so-easy-to-explain feature: the fuzzy partition requirement that the sum of all the related weighting functions is a constant. In this paper, we show that this seemingly difficult-to-explain requirement can also be naturally explained in signal-measurement terms: namely, this requirement can be derived from the natural desire to have all the signal values at different moments of time estimated with the same accuracy. This explanation is the main contribution of this paper. Full article
(This article belongs to the Special Issue Fuzzy Transforms and Their Applications)
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