Etienne E. Kerre

A comparative study of fuzzy rough sets

The notion of a rough set was originally proposed by Pawlak (1982). Later on, Dubois and Prade (1990) introduced fuzzy rough sets as a fuzzy generalization of rough sets. In this paper, we present a more general approach to the fuzzification of rough sets. Specifically, we define a broad family of fuzzy rough sets, each one of which, called an (I, J)-fuzzy rough set, is determined by an implicator I and a triangular norm J. Basic properties of fuzzy rough sets are investigated. In particular, we define three classes of fuzzy rough sets, relatively to three main classes of implicators well known in the literature, and analyse their properties in the context of basic rough equalities. Finally, we refer to an operator-oriented characterization of rough sets as proposed by Lin and Liu (1994) and show soundness of this axiomatization for the Lukasiewicz fuzzy rough sets.

On the relationship between some extensions of fuzzy set theory

Since Zadeh introduced fuzzy sets in 1965, a lot of new theories treating imprecision and uncertainty have been introduced. Some of these theories are extensions of fuzzy set theory, others try to handle imprecision and uncertainty in a different (better?) way. Kerre (Computational Intelligence in Theory and Practice, Physica-Verlag, Heidelberg, 2001, pp. 55-72) has given a summary of the links that exist between fuzzy sets and other mathematical models such as flou sets (Gentilhomme), two-fold fuzzy sets (Dubois and Prade) and L-fuzzy sets (Goguen). In this paper, we establish the relationships between intuitionistic fuzzy sets (Atanassov, VII ITKRs Session, Sofia, June 1983 (Deposed in Central Sci.-Techn. Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)), L-fuzzy sets (J. Math. Anal. Appl. 18 (1967) 145), interval-valued fuzzy sets (Sambuc, Ph.D. Thesis, University of Marseille, France, 1975), interval-valued intuitionistic fuzzy sets (Intuitionistic fuzzy set, Physica-Verlag, Heidelberg, New York, 1999).

Reasonable properties for the ordering of fuzzy quantities (II)

As a continuation of the first part related to the first and second class of ordering approaches this paper deals with the fulfilment of reasonable properties in the third class of ordering approaches. To do so we briefly introduce fuzzy relations on which the third class of approaches is based. Then we recall some transitivity-related concepts and an ordering procedure based on a acyclic fuzzy relation. Acyclicity is a very weak restriction on a fuzzy relation. We prove that many fuzzy relations used for the comparison of fuzzy quantities satisfy some conditions stronger than acyclicity. So we give a widely applicable formulation to derive a total ranking order from a fuzzy relation. With our formulation we examine all the ordering indices in the third class with respect to the proposed axioms in part I.

Defuzzification: criteria and classification

In this paper, we contribute to the theory and development of defuzzification techniques. First we define the core of a fuzzy set. Then we formulate a set of criteria for defuzzification in arbitrary universes, ordered universes, and the set of the real numbers. Finally, we classify the most widely used defuzzification techniques into different groups and we examine the prototypes of each group with respect to the defuzzification criteria. We show that the maxima methods behave well with respect to the more basic defuzzification criteria, and hence are good candidates for fuzzy reasoning systems. On the other hand, the distribution methods and the area methods do not fulfill the basic criteria, but they exhibit the property of continuity that makes them suitable for fuzzy controllers.

Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application

Abstract With the demand for knowledge-handling systems capable of dealing with and distinguishing between various facets of imprecision ever increasing, a clear and formal characterization of the mathematical models implementing such services is quintessential. In this paper, this task is undertaken simultaneously for the definition of implication within two settings: first, within intuitionistic fuzzy set theory and secondly, within interval-valued fuzzy set theory. By tracing these models back to the underlying lattice that they are defined on, on one hand we keep up with an important tradition of using algebraic structures for developing logical calculi (e.g. residuated lattices and MV algebras), and on the other hand we are able to expose in a clear manner the two models’ formal equivalence. This equivalence, all too often neglected in literature, we exploit to construct operators extending the notions of classical and fuzzy implication on these structures; to initiate a meaningful classification framework for the resulting operators, based on logical and extra-logical criteria imposed on them; and finally, to re(de)fine the intuititive ideas giving rise to both approaches as models of imprecision and apply them in a practical context.

Noise reduction by fuzzy image filtering

A new fuzzy filter is presented for the noise reduction of images corrupted with additive noise. The filter consists of two stages. The first stage computes a fuzzy derivative for eight different directions. The second stage uses these fuzzy derivatives to perform fuzzy smoothing by weighting the contributions of neighboring pixel values. Both stages are based on fuzzy rules which make use of membership functions. The filter can be applied iteratively to effectively reduce heavy noise. In particular, the shape of the membership functions is adapted according to the remaining noise level after each iteration, making use of the distribution of the homogeneity in the image. A statistical model for the noise distribution can be incorporated to relate the homogeneity to the adaptation scheme of the membership functions. Experimental results are obtained to show the feasibility of the proposed approach. These results are also compared to other filters by numerical measures and visual inspection.

Fuzzy Interval Analysis

This chapter is an overview of past and present works dealing with fuzzy intervals and their operations. A fuzzy interval is a fuzzy set in the real line whose level-cuts are intervals. Particular cases include usual real numbers and intervals. Usual operations on the real line canonically extend to operations between fuzzy quantities, thus extending the usual interval (or error) analysis to membership functions. What is obtained is a counterpart of random variable calculus, but where, contrary to the latter case, there is no compensation between variables. Many results pertaining to basic properties of fuzzy interval analysis are summed up in the chapter. Computational methods are presented, exact or approximate ones, based on parametric representations, or level-cut approximations. The generalized fuzzy variable calculus involving interactive variables is also discussed with emphasis on triangular-norm based fuzzy additions. Dual ‘optimistic’ operations on fuzzy intervals, i.e., with maximal error compensation are also presented; its interest lies in providing tools for solving fuzzy interval equations. This chapter also contains a reasoned survey of methods for comparing and ranking fuzzy intervals. The chapter includes some historical background, as well as pointers to applications in mathematics and engineering.

Fuzzy techniques in image processing

Vision in general and images in particular have always played an important and essential role in human life. Today, image processing is a very active research area with many applications. In order to cope with the wide variety of image processing problems, several techniques have been introduced and developed, quite often with great success. Among the different techniques that are currently in use, we also encounter fuzzy techniques. The use of fuzzy techniques in image processing is one of the main topics of the Fuzziness and Uncertainty Modelling Research Group of Prof. Kerre. In this paper, we briefly summarize some achievements of the past years.

A fuzzy impulse noise detection and reduction method

Removing or reducing impulse noise is a very active research area in image processing. In this paper we describe a new algorithm that is especially developed for reducing all kinds of impulse noise: fuzzy impulse noise detection and reduction method (FIDRM). It can also be applied to images having a mixture of impulse noise and other types of noise. The result is an image quasi without (or with very little) impulse noise so that other filters can be used afterwards. This nonlinear filtering technique contains two separated steps: an impulse noise detection step and a reduction step that preserves edge sharpness. Based on the concept of fuzzy gradient values, our detection method constructs a fuzzy set impulse noise. This fuzzy set is represented by a membership function that will be used by the filtering method, which is a fuzzy averaging of neighboring pixels. Experimental results show that FIDRM provides a significant improvement on other existing filters. FIDRM is not only very fast, but also very effective for reducing little as well as very high impulse noise.

On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision

Intuitionistic fuzzy sets [K.T. Atanassov, Intuitionistic fuzzy sets, VII ITKRs Session, Sofia (deposed in Central Science-Technical Library of Bulgarian Academy of Science, 1697/84), 1983 (in Bulgarian)] are an extension of fuzzy set theory in which not only a membership degree is given, but also a non-membership degree, which is more or less independent. Considering the increasing interest in intuitionistic fuzzy sets, it is useful to determine the position of intuitionistic fuzzy set theory in the framework of the different theories modelling imprecision. In this paper we discuss the mathematical relationship between intuitionistic fuzzy sets and other models of imprecision.