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Image processing represents an important challenge in different fields. Mathematical Morphology uses concepts from set theory, geometry, algebra and topology to analyze the geometrical structure of an image. Using them, powerful tools as dilation and erosion operators are introduced to solve problems related to edge detection or image segmentation among others. In addition, fuzzy relations are also a useful tool for image processing, primarily those methods based on interval-valued fuzzy relations. Actually, they can be understood as a gradient from a morphological point of view, although they are not dilations neither erosions. In this work a comparison between different approaches is performed, checking some configurations and evaluating in terms of least squares fitting the configuration with the best performance. [ABSTRACT FROM AUTHOR]

Hesitant fuzzy sets represent a useful tool in many areas such as decision making or image processing. Finite interval-valued hesitant fuzzy sets are a particular kind of hesitant fuzzy sets that generalize fuzzy sets, interval-valued fuzzy sets or Atanassov's intuitionistic fuzzy sets, among others. Partitioning is a long-standing open problem due to its remarkable importance in many areas such as clustering. Thus, many different partitioning approaches have been developed for crisp and fuzzy sets. This work presents a partitioning method for the so-called finite interval-valued hesitant fuzzy sets. The definition of this partitioning method involves a definition of an ordering relation for finite interval-valued fuzzy sets membership degrees, i.e, finitely generated sets, as well as the definitions of t-norm and t-conorm for these kinds of sets. [ABSTRACT FROM AUTHOR]

Interval-valued fuzzy sets are an extension of fuzzy sets and are helpful when there is not enough information to define a membership function. This paper studies the behavior of a construction method for an interval-valued fuzzy relation built from a fuzzy relation. The behavior of this construction method is analyzed depending on the used t-norms and t-conorms, showing that different combinations of them produce a big variation in the results. Furthermore, a hybrid construction method that considers weight functions and a smoothing procedure is also introduced. Among the different applications of this method, the detection of edges in images is one of the most challenging. Thus, the performance of the proposal in detecting image edges is tested, showing that the hybrid approach that combines weights and a smoothing procedure provides better results than the non-weighted methods. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this article, we analyze how classical techniques of data privacy, as k-anonymity, can be developed within a framework of fuzzy information. In particular, the main notion that has to be studied is the concept of cardinality, in order to obtain a proper way to count how many elements belong to a given equivalence class. [ABSTRACT FROM AUTHOR]

Multiple definitions have been put forward in the literature to measure the differences between two interval-valued fuzzy sets. However, in most cases, the outcome is just a real value, although an interval could be more appropriate in this environment. This is the starting point of this contribution. Thus, we revisit the axioms that a measure of the difference between two interval-valued fuzzy sets should satisfy, paying special attention to the condition of monotonicity in the sense that the closer the intervals are, the smaller the measure of difference between them is. Its formalisation leads to very different concepts: distances, divergences and dissimilarities. We have proven that distances and divergences lead to contradictory properties for this kind of sets. Therefore, we conclude that dissimilarities are the only appropriate measures to measure the difference between two interval-valued fuzzy sets when the outcome is an interval. [ABSTRACT FROM AUTHOR]

Interval-valued fuzzy sets are a generalization of classical fuzzy sets where the membership values are intervals. The epistemic interpretation of interval-valued fuzzy sets assumes that there is one real-valued membership degree of an element within the membership interval of possible membership degrees. Considering this epistemic interpretation, we propose a new measure, called IV-embedding, to compare the precision of two interval-valued fuzzy sets. An axiomatic definition for this concept as well as a construction method are provided. The construction method is based on aggregation operators and the concept of interval embedding, which is also introduced and deeply studied. [ABSTRACT FROM AUTHOR]

In this work, a definition of entropy is studied in an interval-valued hesitant fuzzy environment, instead of the classical fuzzy logic or the interval-valued one. As the properties of this kind of sets are more complex, the entropy is built by three different functions, where each one represents a different measure: fuzziness, lack of knowledge and hesitance. Using all, an entropy measure for interval-valued hesitant fuzzy sets is obtained, quantifying various types of uncertainty. From this definition, several results have been developed for each mapping that shapes the entropy measure in order to get such functions with ease, and as a consequence, allowing to obtain this new entropy in a simpler way. [ABSTRACT FROM AUTHOR]