Martina Daňková

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.

Relational compositions in Fuzzy Class Theory

We present a method for mass proofs of theorems of certain forms in a formal theory of fuzzy relations and classes. The method is based on formal identification of fuzzy classes and inner truth values with certain fuzzy relations, which allows transferring basic properties of sup-T and inf-R compositions to a family of more than 30 composition-related operations, including sup-T and inf-R images, pre-images, Cartesian products, domains, ranges, resizes, inclusion, height, plinth, etc. Besides yielding a large number of theorems on fuzzy relations as simple corollaries of a few basic principles, the method provides a systematization of the family of relational notions and generates a simple equational calculus for proving elementary identities between them, thus trivializing a large part of the theory of fuzzy relations.

Fuzzy transform as an additive normal form

In this paper, we recall a class of approximating formulas. Since they arose as a generalization of classical normal forms we denote them by the same term. Moreover, we will show that the fuzzy transform (F-transform) introduced in [Perfilieva, I., Fuzzy approach to solution of differential equations with imprecise data: application to reef growth problem, in: R.V. Demicco, G.J. Klir (Eds.), Fuzzy Logic in Geology, Academic Press, Amsterdam, 2003, pp. 275-300 (Chapter 9), Perfilieva, I., Fuzzy transforms, in: J.F. Peters, A. Skowron (Eds.), Transactions on Rough Sets II. Rough Sets and Fuzzy Sets, Lecture Notes in Computer Science, vol. 3135, 2004, pp. 63-81] as an approximation method for continuous functions can be viewed as a particular case of normal form. Therefore, the results valid for normal forms may be applied to F-transforms as well.

Image Fusion on the Basis of Fuzzy Transforms

We show that on the basis of fuzzy transform, the problem of reconstruction of corrupted images can be solved. The proposed technique is called image fusion. An algorithm of image fusion, based on fuzzy transform, is proposed and justified. A measure of fuzziness of an image is proposed as well.

On approximate reasoning with graded rules

This contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy IF-THEN rules formalized by approximating formulas of a special type. Two alternatives that follow from the dual character of approximating formulas are developed in parallel. The link to the theory of fuzzy control systems is also explained.

Approximation by pseudo-linear operators

The approximation operators provided by classical approximation theory use exclusively as underlying algebraic structure the linear structure of the reals. Also they are all linear operators. We address in the present paper the following problems: Need all the approximation operators be linear? Is the linear structure the only one which allows us to construct particular approximation operators? As an answer to this problem we propose new, particular, pseudo-linear approximation operators, which are defined in some ordered semirings. We study these approximations from a theoretical point of view and we obtain that these operators have very similar properties to those provided by classical approximation theory. In this sense we obtain uniform approximation theorems of Weierstrass type, and Jackson-type error estimates in approximation by these operators.

Continuity issues of the implicational interpretation of fuzzy rules

The implicational interpretation of fuzzy rules has received little attention in real-world applications so far. This is largely due to the fact that ensuring continuity of the resulting function is not a straightforward task. This paper targets this subject. Departing from consistent linguistic descriptions/rule bases, we introduce sufficient conditions for the continuity of the implicational interpretation with mean of maximum defuzzification. We demonstrate that continuity can be achieved under practically feasible conditions, regardless of the dimensionality of the input.

Generalized extensionality of fuzzy relations

In this work, we investigate a relationship between extensionality and continuity of fuzzy relations. We found conditions when extensionality implies continuity and vice versa.

Pseudo-Riemann-Stieltjes integral

An extension of the classical Riemann-Stieltjes integral to the field of pseudo-analysis is being investigated through this paper. Core of the construction presented here consists of generalized pseudo-operations given by monotone generating function.

Approximation of extensional fuzzy relations over a residuated lattice

We investigate a relationship between extensionality of fuzzy relations and their Lipschitz continuity on generalized metric spaces. The duality of these notions is shown, and moreover, two particular applications of the extensionality property in the field of approximation are given.