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Fuzzy set theory has developed a prolific armamentarium of mathematical tools for each of the topics that has fallen within its scope. One of such topics is data comparison, for which a range of operators has been presented in the past. These operators can be used within the fuzzy set theory, but can also be ported to other scenarios in which data are provided in various representations. In this work, we elaborate on notions for type-2 fuzzy sets, specifically for the comparison of type-2 fuzzy membership degrees, to create function comparison operators. We further apply these operators to hyperspectral imaging, in which pixelwise data are provided as functions over a certain energy spectra. The performance of the functional comparison operators is put to the test in the context of in-laboratory hyperspectral image segmentation. [ABSTRACT FROM AUTHOR]

In this work we introduce a method for constructing linear orders between pairs of intervals by using aggregation functions. We adapt this method to the case of interval-valued Atanassov intuitionistic fuzzy sets and we apply these sets and the considered orders to a decision making problem. [ABSTRACT FROM AUTHOR]

We study several functional equations, in two variables, that are closely related to the functional equation of permutability. We analyze the hierarchy among these equations, paying a particular attention to their use in Fuzzy Set Theory. [ABSTRACT FROM AUTHOR]

Abstract: The problem of choosing an appropriate total order is crucial for many applications that make use of extensions of fuzzy sets. In this work we introduce the concept of an admissible order as a total order that extends the usual partial order between intervals. We propose a method to build these admissible orders in terms of two aggregation functions and we prove that some of the most used examples of total orders that appear in the literature are specific cases of our construction. [Copyright &y& Elsevier]

The choice of membership functions plays an essential role in the success of fuzzy systems. This is a complex problem due to the possible lack of knowledge when assigning punctual values as membership degrees. To face this handicap, we propose a methodology called Ignorance functions based Interval-Valued Fuzzy Decision Tree with genetic tuning, IIVFDT for short, which allows to improve the performance of fuzzy decision trees by taking into account the ignorance degree. This ignorance degree is the result of a weak ignorance function applied to the punctual value set as membership degree. Our IIVFDT proposal is composed of four steps: (1) the base fuzzy decision tree is generated using the fuzzy ID3 algorithm; (2) the linguistic labels are modeled with Interval-Valued Fuzzy Sets. To do so, a new parametrized construction method of Interval-Valued Fuzzy Sets is defined, whose length represents such ignorance degree; (3) the fuzzy reasoning method is extended to work with this representation of the linguistic terms; (4) an evolutionary tuning step is applied for computing the optimal ignorance degree for each Interval-Valued Fuzzy Set. The experimental study shows that the IIVFDT method allows the results provided by the initial fuzzy ID3 with and without Interval-Valued Fuzzy Sets to be outperformed. The suitability of the proposed methodology is shown with respect to both several state-of-the-art fuzzy decision trees and C4.5. Furthermore, we analyze the quality of our approach versus two methods that learn the fuzzy decision tree using genetic algorithms. Finally, we show that a superior performance can be achieved by means of the positive synergy obtained when applying the well known genetic tuning of the lateral position after the application of the IIVFDT method. [ABSTRACT FROM AUTHOR]

Abstract: The representation and processing of edges in images based on notions from fuzzy set theory has become popular in recent years. There are several reasons for this direction, from the vague definition of edges to the inherent uncertainty of digital images. Here, we study the transition from a gradient image, a popular intermediate representation, to a fuzzy edge image. We consider different parametric membership functions to transform the gradients into membership degrees. A histogram-based strategy is then introduced for automatically determining the value of those parameters, adapting the membership functions to the characteristics of each image. The functions are applied on the Canny method for edge detection, resulting in an improvement compared to the classical normalizing approach. [Copyright &y& Elsevier]

Abstract: In this paper, we present the definition of restricted dissimilarity function. This definition arises from the concepts of dissimilarity and equivalence function. We analyze the relation there is between restricted dissimilarity functions, restricted equivalence functions (see [Bustince, H., Barrenechea, E., Pagola, M., 2006. Restricted equivalence functions. Fuzzy Sets Syst. 157, 2333–2346]) and normal E N -functions. We present characterization theorems from implication operators and automorphisms. Next, by aggregating restricted dissimilarity functions in a special way, we construct distance measures of Liu, proximity measures of Fan et al. and fuzzy entropies. We also study diverse interrelations between the above-mentioned concepts. These interrelations enable us to prove that under certain conditions, the threshold of an image calculated with the algorithm of Huang and Wang [Huang, L.K., Wang, M.J., 1995. Image thresholding by minimizing the measure of fuzziness. Pattern Recognit. 28 (1), 41–51], with the methods of Forero [Forero, M.G., 2003. Fuzzy thresholding and histogram analysis. In: Nachtegael, M., Van der Weken, D., Van de Ville, D., Kerre, E.E. (Eds.), Fuzzy Filters for Image Processing. Springer, pp. 129–152] or with the algorithms developed in [Bustince, H., Barrenechea, E., Pagola, M., 2007. Image thresholding using restricted equivalence functions and maximizing the measures of similarity. Fuzzy Sets Syst. 158, 496–516] is always the same, that is, it remains invariant. [Copyright &y& Elsevier]

In this paper we stress the relevance of a particular family of fuzzy sets, where each element can be viewed as the result of a classification problem. In particular, we assume that fuzzy sets are defined from a well-defined universe of objects into a valuation space where a particular graph is being defined, in such a way that each element of the considered universe has a degree of membership with respect to each state in the valuation space. The associated graph defines the structure of such a valuation space, where an ignorance state represents the beginning of a necessary learning procedure. Hence, every single state needs a positive definition, and possible queries are limited by such an associated graph. We then allocate this family of fuzzy sets with respect to other relevant families of fuzzy sets, and in particular with respect to Atanassov''s intuitionistic fuzzy sets. We postulate that introducing this graph allows a natural explanation of the different visions underlying Atanassov''s model and interval valued fuzzy sets, despite both models have been proven equivalent when such a structure in the valuation space is not assumed. [Copyright &y& Elsevier]