Irina Perfilieva

Mathematical principles of fuzzy logic

Preface. 1. Fuzzy Logic: What, Why, for Which? 2. Algebraic Structures for Logical Calculi. 3. Logical Calculi and Model Theory. 4. Fuzzy Logic in Narrow Sense. 5. Functional Systems in Fuzzy Logic Theories. 6. Fuzzy Logic in Broader Sense. 7. Topoi and Categories of Fuzzy Sets. 8. Few Historical and Concluding Remarks. References. Index.

Fuzzy transforms: Theory and applications

The technique of direct and inverse fuzzy (F-)transforms of three different types is introduced and approximating properties of the inverse F-transforms are described. All three types of the direct F-transform are transformations from a function space to a finite dimensional vector space. The first (ordinary) F-transform is constructed on the basis of the ordinary algebra of reals, while the other two types of the F-transform are constructed on the basis of residuated lattice. The core idea of the technique of F-transforms is a fuzzy partition of a universe into fuzzy subsets (factors, clusters, granules etc.). We claim that for a sufficient representation of a function defined on this universe, we may consider its average values over fuzzy subsets from the partition. Thus, a function can be associated with a mapping from a set of fuzzy subsets to the set of its thus obtained average values. A number of theorems establishing best approximation properties of the inverse F-transforms are proved. In fact, three types of the inverse F-transform are the best approximations in average, from below, and from above respectively. As one of many possible applications, we present a method of image compression and reconstruction on the basis of the F-transform.

An image coding/decoding method based on direct and inverse fuzzy transforms

With some modifications, we adopt the coding/decoding method of image processing based on the direct and inverse fuzzy transforms defined in previous papers. By normalizing the values of its pixels, any image can be considered as a fuzzy matrix (relation) which is subdivided in submatrices (possibly square) called blocks. Each block is compressed with the formula of the discrete fuzzy transform of a function in two variables and successively it is decompressed via the related inverse fuzzy transform. The decompressed blocks are recomposed for the reconstruction of the image, whose quality is evaluated by calculating the PSNR (Peak Signal to Noise Ratio) with respect to the original image. A comparison with the coding/decoding method of image processing based on the fuzzy relation equations with the Lukasiewicz triangular norm and the DCT method are also presented. By using the same compression rate in the three methods, the results show that the PSNR obtained with the usage of direct and inverse fuzzy transforms is higher than the PSNR determined either with fuzzy relation equations method or in the DCT one and it is close to the PSNR determined in JPEG method for small values of the compression rate.

Fuzzy transform in the analysis of data

Fuzzy transform is a novel, mathematically well founded soft computing method with many applications. In this paper, we present this technique with applications to data analysis. First, we show how it can be used for detection and characterization of dependencies among attributes. Second, we apply it to mining associations that have a functional character. Moreover, the mined associations are characterized linguistically which means that their antecedent consists of fuzzy numbers and the consequent is characterized using pure evaluative linguistic expressions (i.e. expressions such as small, very big, more or less medium, etc).

Fuzzy Transforms: A Challenge to Conventional Transforms

Publisher Summary This chapter focuses on the notion of a fuzzy transform (F-transform) that explains modeling with fuzzy IF–THEN rules as a specific transformation. This allows the comparison of the success of fuzzy modeling with the success of classical transforms. This is a fairly powerful technique with many useful properties and great potential for various applications, such as special numerical methods, the solving of ordinary and partial differential equations, mining associations from numerical data, applications to signal processing, compression and decompression of images, and the fusion of images. The chapter focuses on the lattice-based F-transforms, overviews the main properties of the ordinary F-transform, and discusses the F-transforms applications.

Towards a higher degree F-transform

The aim of this study is to show how the F-transform technique can be generalized from the case of constant components to the case of polynomial components. After a general presentation of an F^m- transform, m>=0, a detailed characterization of the F^1- transform is given. We apply a numeric integration technique in order to simplify the computation of F^1- transform components. The inverse F^m- transform, m>0, is defined similarly to the ordinary inverse F-transform. The quality of approximation using the inverse F^m- transform increases with an increase in m.

Fuzzy function as an approximate solution to a system of fuzzy relation equations

This paper is mostly focused on the problem of approximate solvability of a system of fuzzy relation equations. However, we put a new light on this problem connecting it with the interpolation of a fuzzy function. We introduce a notion of fuzzy function and its representation by fuzzy relation. We demonstrate how problems of interpolation and approximation of fuzzy functions are connected with solvability of systems of fuzzy relation equations. First, we explain the general framework and then we prove some results related to the problem of the best approximation. In particular, we have shown that fuzzy relations introduced by Sanchez and Mamdani are the best approximations in certain approximation spaces.

Fuzzy transforms of monotone functions with application to image compression

This paper is focused on the special properties of functions (monotonicity, Lipschitz continuity) that are invariant with fuzzy transform. On the basis of the monotonicity invariance, we propose an efficient algorithm for improved image compression and reconstruction based on fuzzy transform.

Analysis of seasonal time series using fuzzy approach

A new methodology for the analysis and forecasting of time series is proposed. It directly employs two soft computing techniques: the fuzzy transform and the perception-based logical deduction. Thanks to the use of both these methods, and to the innovative approach, consisting of the construction of several independent models, the methodology is successfully applicable to robust long-time predictions.

On the semantics of perception‐based fuzzy logic deduction

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.