Feature selection, as one of the essential dimensionality reduction techniques, hasbecome one popular yet challenging area in the field, such as data mining andmachine learning. Unlike feature extraction (such as principle component analysis andnon-negative matrix factorization), preserving the entire information but losing thefeature relevance. Data processing in feature selection will lose information, leading tothe need to develop lightweight, efficient, and practical methods that preserve the datainformation as much as possible while performing dimensional reduction. In this paper,we propose a rankability-based feature selection method. The rankability concept wasproposed in 2019, similar to the entropy concept, and has not been studied widely yet.The proposed method is lightweight in terms of complexity, which requires no iterativeoptimization or auxiliary estimator tools.We experimented with sixteen datasets and compared our results with four otheralgorithms. The results show that our rankability-based feature selection methodoutperforms the fuzzy entropy-based method on five datasets in eight, and the averageaccuracy increased by 0.1482, 0.1078, and 0.1157, respectively. Then, in the varieddimension-reducing experiments, the proposed method shows superiority on fourdatasets out of eight and is competitive with others on two datasets out of eight.

The F-transform has proven to be effective in various applications, such as time series analysis, numerical solutions of differential equations, and signal or image processing. The most important parameter of the F-transform is a fuzzy partition initially introduced for 1-D spaces, with a few generalizations to higher dimensional spaces. However, these generalizations have a limited application to signals defined on domains with arbitrary geometry. To overcome this limitation, we propose using nonseparable membership functions induced by kernels, allowing the application of the F-transform to more general domains. We introduce a universal concept of a fuzzy partition that includes a kernel representation, a fuzzy partition, and the corresponding F-transform. In addition, we discuss the main properties of this generalized F-transform and characterize the nonlocal Laplace operator in terms of the F-transform. We also discuss image denoising as the main application and compare our results with state-of-the-art methods and different noise types and intensities.

In this article, a new iterative numerical method for solving bivariate nonlinear fuzzy Fredholm integral equations is proposed. The method combines two well-proven approaches-successive approximations and mixed trapezoidal and midpoint rules for the numerical integration. Both approaches are elaborated for fuzzy-valued functions. The main advantage of the proposed approach is that, by targeting the particular equation, another subordinate problem was solved under the common constraints. By this, we mean the development of a numerical method for fuzzy integrals. As a result, the proposed method is more accurate in comparison with any other mechanical combination of two separate and independent methods. We give conditions for the existence and uniqueness of a solution and estimate the error of the obtained approximation. We prove the stability and the method is performed on test problems to verify our theoretical results; numerical results are compared with those from existing methods in the literature to confirm the accuracy and efficiency of the proposed method.

We propose to assign the F-transform kernels to the CNN weights and compare them with commonly used initialization. By this, we develop a new initialization mechanism where the F-transform convolution kernels are used in the convolutional layers. Based on a series of experiments, we demonstrate the suitability of the F-transform-based deep neural network in the domain of image processing with the focus on classification. Moreover, we support our insight by revealing the similarity between the F-transform and first-layer kernels in certain deep neural networks.

This paper introduces a novel construction of test spaces in the Ritz-Galerkin method. These spaces are established on the basis of linear spaces of functions that are linear combinations of polynomials and fuzzy sets of a generalized uniform fuzzy partition.

In image processing, the inpainting (image restoration) problem is often considered with respect to the interpolation. We involve into the solution of this problem the F-transform-based nonlocal operators that define a new type of functionals, extending the ability of classical PDE-based algorithms in handling textures and repetitive structures. We showed that in the particular space with a fuzzy partition, the nonlocal Laplacian and partial derivatives can be represented by the F0- and F1-transforms. For the inpainting problem specified by relatively large damaged areas, we propose a new total variation model with the F-transform-based nonlocal operators. We show that the proposed model together with the corresponding algorithm increase the quality of a (usually considered) patch-based searching algorithm.

In this paper, we investigate a mechanism for fusing a set of inputs (values) in such a way that the procedure does not create new information during the process. In order to do so, we introduce internal fusion functions, a family of fusion functions in which the output always corresponds to some of the given inputs. We perform an in-depth theoretical study of internal fusion functions and, furthermore, we propose three different construction methods, which are based on 1) an arbitrary fusion function and a partition of the domain; 2) a linear order; and 3) a minimization mechanism using penalty functions. Finally, we illustrate this paper with the application of internal fusion functions in two image processing algorithms where a set of images must be fused, namely multifocus image and denoised image fusion, as well as in an example of multiclass problem, where we fuse a set of score matrices obtained by several classification algorithms.

We propose a new hybrid image compression algorithm which combines the F-transform and the JPEG. At first, we apply the direct F-transform and then, the JPEG compression. Conversly, the JPEG decompression is followed by the inverse F-transform to obtain the decompressed image. This scheme brings three benefits: (i) the direct F-transform filters out high frequencies so that the JPEG can reach a higher compression ratio; (ii) the JPEG color quantization can be omitted in order to achieve greater decompressed image quality; (iii) the JPEG-decompressed image is processed by by the inverse F-transform w.r.t. the adjoint partition almost lossless. The paper justifies the proposed hybrid algorithm by benchmarks which show that the hybrid algorithm achieves significantly higher decompressed image quality than the JPEG.

In this work we investigate a new family of fusion functions called internal fusion functions. The main characteristic of these functions is the fact that the output always corresponds to some of the given inputs. We propose a construction method and we study whether internal functions constructed in this way also satisfy properties of aggregation functions Finally, we apply internal functions in an example of a multi-class problem, where a set of matrices must be combine into a single representative collective matrix in order to obtain better classification rates.

A relationship between real-valued and lattice-valued F-transforms and aggregation functions is analyzed. For each direct F-transform, we find an axiomatic characterization of the corresponding mapping. We show that the real-valued F-transform is a set of images of linear aggregation functions and that the lattice-valued F-transforms are images of linear-like maps. In all cases, the involved maps respect certain partitions of the universe.

We solve the problem of network attack detection and classification. We discuss the way of generation and simulation of an artificial network traffic data. We propose an efficient algorithm for data classification that is based on the F-transform technique. The algorithm successfully passed all tests and moreover, it showed ability to perform classification in an on-line regime.

This contribution is focused on the complexity issue of implicative fuzzy associative memory (IFAM) applied to image processing. We propose to apply image reduction prior to IFAM and then reconstruction after it. Both reduction and reconstruction are based on the F-Transform.

We propose the technique of the semi-automatic image creation. By this we mean an automatic completion of an image that is partially defined on the given domain. The essential feature of this technique is that the complementary area is much larger than that where the image is defined. Moreover, the proposed technique can be used in image upsampling, image inpainting, etc. In this contribution, we propose the technique of F-transform as a universal tool for the image completion that is independent on the initial distribution of the image pixels.

Many interpolation techniques are available for image reconstruction, with differences in time complexity, memory complexity and quality. In this article, we compare the application of bilinear interpolation, nearest neighbor interpolation and the F-transform approximation technique to the problem of image reconstruction. Based on our results, F-transform achieves the best results in terms of quality.

The contribution is focused on a new method of grouping time series according to their local tendency indicator that is expressed by a linear coefficient of the F1-transform. The useful consequence of grouping is an effective procedure of forecasting such that only one time series from a group is forecasted.

The contribution describes newly developed technique used to improve image quality by fusion of information from the multiple images into one resulting image containing better information than each of the input ones. The presented approach is based on the F-Transform, integral transform used to detect gradients, similarity and image fusion, and noise reduction.

In this paper, we will focus on the application of fuzzy transform (F-transform) in the analysis of time series. We assume that the time series is decomposed into two constituents: the trend-cycle and random noise. We will demonstrate that using the F-transform we can reduce the variability of random noise which consequence is an extraction of the trend-cycle.

In many practical situations, e.g., in signal processing, image processing, analysis of temporal data, it is very useful to use fuzzy (F-) transforms. In an F-transform, we first replace a function x(t) by a few local averages (this is called forward F-transform), and then reconstruct the original function from these averages (this is called inverse F-transform). While the formula for the forward F-transform makes perfect intuitive sense, the formula for the inverse F-transform seems, at first glance, somewhat counter-intuitive. On the other hand, its empirical success shows that this formula must have a good justification. In this paper, we provide such a justification - a justification which is based on formulating a reasonable compression-based criterion.

This contribution shows how the technique of F-transform can be used for handling the problem of edge detection. A justification of the proposed approach is given and the based on it algorithm is presented. Various examples demonstrate effectiveness of the proposed algorithm and compare it with some established techniques.

The aim of this contribution is to show that the theory of F-transform can be successfully used in analysis and forecasting of time series. We propose relaxed constraints on a fuzzy partition as well as a matrix form of the F-transform for fast computation. For short time series, we propose to use the direct F-transform and make forecast on the basis of F-transform components.

We investigate the Cauchy problem for ordinary differential equation (ODE) with fuzzy initial condition. We define a solution to this problem and propose a new method of how an approximate solution can be constructed. The proposed method is based on the technique of fuzzy transforms and ends up with a solution of a system of fuzzy relation equations. The approximate solution is expressed symbolically and the quality of approximation is estimated.

A new methodology for forecasting of time series is proposed. It is based on combination of two techniques: fuzzy transform and perception-based logical deduction on the basis of learned linguistic description.

The problem of solvability of a system of fuzzy relation equations with sup-*composition is considered in semilinear vector spaces. Based on the fact that a complete set of solutions is determined by minimal solutions, we focused on characterization of them. At first, sets of all minimal solutions of a single equation have been described under different assumptions on an underlying algebra. Dependently on the ordering of the support set, either necessary or sufficient conditions, or criteria of being a minimal solution have been obtained. Then minimal solutions of a system are build from minimal solutions of single equations.

The technique of the direct and inverse fuzzy (F-) transforms of three different types is introduced and approximating properties of the inverse F-transforms are described. A number of theorems establishing best approximation properties have been proved. A method of lossy image compression and reconstruction on the basis of the F-transform is presented

In this article, we return to the problem of the derivation of a conclusion on the basis of fuzzy IF–THEN rules. The so-called Mamdani method is well elaborated and widely applied. In this article, we present an alternative to it. The fuzzy IF–THEN rules are here interpreted as genuine linguistic sentences consisting of the so-called evaluating linguistic expressions. Sets of fuzzy IF–THEN rules are called linguistic descriptions. Linguistic expressions derived on the basis of an observation in a concrete context are called perceptions. Together with the linguistic description, they can be used in logical deduction, which we will call a perception-based logical deduction. We focus on semantics only and confine ourselves to one specific model. If the perception-based deduction is repeated and the result interpreted in an appropriate model, we obtain a piecewise continuous and monotonous function. Though the method has already proved to work well in many applications, the nonsmoothness of the output may sometimes lead to problems. We propose in this article a method for how the resulting function can be made smooth so that the output preserves its good properties. The idea consists of postprocessing the output using a special fuzzy approximation method called F-transform. © 2004 Wiley Periodicals, Inc. Int J Int Syst 19: 1007–1031, 2004.

There are two main fuzzy system methodologies for translating expert rules into a logical formula: In Mamdani's methodology, we get a DNF formula (disjunction of conjunctions), and in a methodology which uses logical implications, we get, in effect, a CNF formula (conjunction of disjunctions). For both methodologies, universal approximation results have been proven which produce, for each approximated function f(x), two different approximating relations RDNF(x, y) and RCNF(x, y). Since, in fuzzy logic, there is a known relation FCNF(x) ≤ FDNF(x) between CNF and DNF forms of a propositional formula F, it is reasonable to expect that we would be able to prove the existence of approximations for which a similar relation RCNF(x, y) ≤ RDNF(x, y) holds. Such existence is proved in our paper. © 2002 Wiley Periodicals, Inc.

In this paper we report an application of fuzzy arithmetic to the reduction of geographical data sets. The proposed technique builds a “summary” of the data within a given subregion in the form of a suitable fuzzy real number. The membership function of this number is obtained with a procedure that resembles the mountain function method introduced by Yager and Filev [IEEE Trans Syst. Man, Cybern Aug. 1994, 24(8), 1279–1284]. The proposed approach is computationally efficient, theoretically sound, and quite robust in terms of experimental noise. In addition, it does not require any statistical assumption about the distribution of the data. The summarization technique reported has been successfully used in modeling a real terrain from collections of sparse elevation data on a terrain. Comparisons with similar approaches are also reported. ©1999 John Wiley & Sons, Inc.