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Throughout this paper, our main idea is to analyze the concepts of a strong negation and a restricted equivalence function, that appear in a natural way when dealing with theory and applications of fuzzy sets and fuzzy logic. Here we will use an analytical and topological approach, showing how to construct them in an easy way. In particular, we will also analyze some classical functional equation related to those key concepts. [ABSTRACT FROM AUTHOR]

In this work we study a new class of similarity measures between interval-valued fuzzy sets. The novelty of our approach lays, firstly, on the fact that we develop all the notions with respect to total orders of intervals; and secondly, on that we consider the width of intervals so that the uncertainty of the output is strongly related to the uncertainty of the input. For constructing the new interval-valued similarity, interval valued aggregation functions and interval-valued restricted equivalence functions which take into account the width of the intervals are needed, so we firstly study these functions, both in line with the two above stated features. Finally, we provide an illustrative example which makes use of an interval-valued similarity measure in stereo image matching and we show that the results obtained with the proposed interval-valued similarity measures improve numerically (according to the most widely used measures in the literature) the results obtained with interval valued similarity measures which do not consider the width of the intervals. [ABSTRACT FROM AUTHOR]

Ordered Weighted Averaging (OWA) operators are a profusely applied class of averaging aggregation functions, i.e. operators that always yield a value between the minimum and the maximum of the inputs. The orness measure was introduced to classify the behavior of the OWA operators depending on the weight vectors. Defining a suitable orness measure is an arduous task when we deal with OWA operators defined over more intricate spaces, such us intervals or lattices. In this work we propose a suitable definition for the orness measure to classify OWA operators defined on the set of m-dimensional intervals taking real values in [ 0 , 1 ]. The orness measure is applied to decide which is the best partition of a continuous range that should be divided into four linguistic labels. This example shows the good behavior of the proposed orness measure. [ABSTRACT FROM AUTHOR]

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]

In this work, we axiomatically describe the concepts of total homogeneity and local homogeneity of a fuzzy set. We also present different construction methods by means of restricted equivalence functions, restricted dissimilarity functions and implication operators. We show that the most common homogeneity expressions can be obtained as particular cases of the constructions developed here. The usefulness of these functions is shown in segmentation of fingerprint images. [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]

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: We study the method of Sun et al. for edge detection based on the Law of Universal Gravity. We analyze the effect of the substitution of the product operation by other triangular norms in the calculation of the gravitational forces. We treat edges as fuzzy sets for which membership degrees are extracted from the resulting gravitational force on each pixel. We consider several prototypical triangular norms and experimentally show that their features determine the kind of edges detected. The new method is tested on the Berkeley Segmentation Dataset, showing to be competitive compared to the Canny method. [ABSTRACT FROM AUTHOR]

Abstract: In this work we develop some reflections on the thresholding algorithm proposed by Tizhoosh in . The purpose of these reflections is to complete the considerations published recently in on said algorithm. We also prove that under certain constructions, Tizhoosh''s algorithm makes it possible to obtain additional information from commonly used fuzzy algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, we define the concept of an ignorance function and use it to determine the best threshold with which to binarize an image. We introduce a method to construct such functions from t-norms and automorphisms. By means of these new measures, we represent the degree of ignorance of the expert when given one fuzzy set to represent the background and another to represent the object. From this ignorance degree, we assign interval-valued fuzzy sets to the image in such a way that the best threshold is given by the interval-valued fuzzy set with the lowest associated ignorance. We prove that the proposed method provides better thresholds than the fuzzy classical methods when applied to transrectal prostate ultrasound images. The experimental results on ultrasound and natural images also allow us to determine the best choice of the function to represent the ignorance. [Copyright &y& Elsevier]

Abstract: In this paper we present a method to construct interval-valued fuzzy sets (or interval type 2 fuzzy sets) from a matrix (or image), in such a way that we obtain the length of the interval representing the membership of any element to the new set from the differences between the values assigned to that element and its neighbors in the starting matrix. Using the concepts of interval-valued fuzzy t-norm, interval-valued fuzzy t-conorm and interval-valued fuzzy entropy, we are able to detect big enough jumps (edges) between the values of an element and its neighbors in the starting matrix. We also prove that the unique t-representable interval-valued fuzzy t-norms and the unique s-representable interval-valued fuzzy t-conorms that preserve the length zero of the intervals are the ones generated by means of the t-norm minimum and the t-conorm maximum. [Copyright &y& Elsevier]

Starting from the definitions of the weak fuzzy subsethood measure (V.R. Young, Fuzzy subsethood. Fuzzy Sets and Systems, 77 (1996) 371–384) and of strong fuzzy inclusion, (D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, New York, Academic Press, 1980) we define weak fuzzy S-subsethood measures. We analyze their axiomatization and present various constructions obtained by aggregating implication operators with special properties. We conclude by presenting the conditions under which we can construct fuzzy entropies and overlap indices from these measures. [ABSTRACT FROM AUTHOR]

In this paper we present the definition and properties of intuitionistic fuzzy implication operators. We study the expression obtained from said operators when fuzzy implication and coimplication operators are applied to different aggregations of degrees of truth and non-truth of the propositions. [ABSTRACT FROM AUTHOR]

The necessity of dealing with uncertainty in real world problems has been a long-term research challenge which has originated different methodologies and theories. Recently, the concept of Hesitant Fuzzy Sets (HFSs) has been introduced to model the uncertainty that often appears when it is necessary to establish the membership degree of an element and there are some possible values that make to hesitate about which one would be the right one. Many researchers have paid attention on this concept who have proposed diverse extensions, relationships with other types of fuzzy sets, different types of operators to compute with this type of information, applications on information fusion and decision-making, etc. Nevertheless, some of these proposals are questionable, because they are straightforward extensions of previous works or they do not use the concept of HFSs in a suitable way. Therefore, this position paper studies the necessity of HFSs and provides a discussion about current proposals including a guideline that the proposals should follow and some challenges of HFSs. [ABSTRACT FROM AUTHOR]

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]

In this paper we propose an algorithm to solve group decision making problems using n -dimensional fuzzy sets, namely, sets in which the membership degree of each element to the set is given by an increasing tuple of n elements. The use of these sets has naturally led us to define admissible orders for n -dimensional fuzzy sets, to present a construction method for those orders and to study OWA operators for aggregating the tuples used to represent the membership degrees of the elements. In these conditions, we present an algorithm and apply it to a case study, in which we show that the exploitation phase which appears in many decision making methods can be omitted by just considering linear orders between tuples. [ABSTRACT FROM AUTHOR]

This paper deals with OWA (ordered weighted average) operators defined on any complete lattice endowed with a t-norm and a t-conorm and satisfying a certain finiteness local condition. A parametrization of these operators is suggested by introducing a quantitative orness measure for each OWA operator, based on its proximity to the OR operator. The meaning of this measure is analyzed for some concrete OWA operators used in color image reduction, as well as for some OWA operators used in a medical decision making process. [ABSTRACT FROM AUTHOR]

Abstract: In this work we present a construction method for Atanassov’s intuitionistic fuzzy preference relations starting from fuzzy preference relations and taking into account the ignorance of the expert in the construction of the latter. Moreover, we propose two generalizations of the weighted voting strategy to work with Atanassov’s intuitionistic fuzzy preference relations. An advantage of these algorithms is that they start from fuzzy preference relations and their results can be compared with those of any other decision-making algorithm based on fuzzy sets theory. We verify that our proposal is able to provide a unique solution in some cases in which the voting strategy is not able to order the alternatives. [Copyright &y& Elsevier]