Due to its reasonable properties, the Kulisch and Miranker binary relation on the family of all closed and bounded real intervals has attracted the attention of many researchers, especially in the field of Computation. However, it is not total, so there are intervals that are not comparable. To face this problem, Bustince et al. introduced the notion of admissible order , which is coherent to the Kulisch and Miranker binary relation. Due to its technical construction, most of the examples of admissible orders are defined by only employing the extremes of such intervals. In this paper we introduce a non-countable family of admissible orders in the set of all closed and bounded subintervals contained in a concrete closed and bounded real interval. The approach is novel in two senses: on the one hand, due to the mathematical objects that are involved (a dense sequence and a family of continuous functions); and, on the other hand, we do not handle the intervals through their extremes, but only by their interior points.

This paper introduces a unified framework of type-(k1, k2) fuzzy subsethood measures (FSMs), where k1, k2 ∈ {0, 1, 2} with k1 ≥ k2. The proposed formulation generalizes classical subsethood measures by defining inclusion mappings between fuzzy sets of different types while preserving fundamental properties such as monotonicity, boundary conditions, and normality with respect to strong negations. We provide construction methods showing that well-known measures arise as special cases. Illustrative examples and an application to digital image processing demonstrate the applicability of the proposed FSMs in uncertain environments.

The 2019 coronavirus disease outbreak, caused by the severe acute respiratory syndrome type-2 virus (SARS-CoV-2), was declared a pandemic in March 2020. Since its emergence to the present day, this disease has brought multiple countries to the brink of health care collapse during several waves of the disease. One of the most common tests performed on patients is chest x-ray imaging. These images show the severity of the patient’s illness and whether it is indeed covid or another type of pneumonia. Automated assessment of this type of imaging could alleviate the time required for physicians to treat and diagnose each patient. To this end, in this paper we propose the use of Convolutional Neural Networks (CNNs) to carry out this process. The aim of this paper is twofold. Firstly, we present a pipeline adapted to this problem, covering all steps from the preprocessing of the datasets to the generation of classification models based on CNNs. Secondly, we have focused our study on the modification of the information fusion processes of this type of architectures, in the pooling layers. We propose a number of aggregation theory functions that are suitable to replace classical processes and have shown their benefits in past applications, and study their performance in the context of the x-ray classification problem. We find that replacing the feature reduction processes of CNNs leads to drastically different behaviours of the final model, which can be benefitial when prioritizing certain metrics such as precision or recall.

In this paper we propose a new version of penalty-based aggregation functions, the Multi Cost Aggregation choosing functions (MCAs), in which the function to minimize is constructed using a convex combination of two relaxed versions of restricted equivalence and dissimilarity functions instead of a penalty function. We additionally suggest two different alternatives to train a MCA in a supervised classification task in order to adapt the aggregation to each vector of inputs. We apply the proposed MCA in a Motor Imagery-based Brain Computer Interface (MI-BCI) system to improve its decision making phase. We also evaluate the classical aggregation with our new aggregation procedure in two publicly available datasets. We obtain an accuracy of 82.31% for a left vs. right hand in the Clinical BCI challenge (CBCIC) dataset, and a performance of 62.43% for the four-class case in the BCI Competition IV 2a dataset compared to a 82.15% and 60.56% using the arithmetic mean. Finally, we have also tested the goodness of our proposal against other MI-BCI systems, obtaining better results than those using other decision making schemes and Deep Learning on the same datasets.

Choquet integral is a widely used aggregation operator on 1-D and interval-valued information, since it is able to take into account the possible interaction among data. However, there are many cases where the information taken into account is vectorial, such as long short-term memories (LSTM). LSTM units are a kind of recurrent neural networks that have become one of the most powerful tools to deal with sequential information since they have the power of controlling the information flow. In this article, we first generalize the standard Choquet integral to admit an input composed by$n$-dimensional vectors, which produces an$n$-dimensional vector output. We study several properties and construction methods of vector Choquet integrals (VCIs). Then, we use this integral in the place of the summation operator, introducing in this way the new VCI-LSTM architecture. Finally, we use the proposed VCI-LSTM to deal with two problems: 1) sequential image classification; 2) text classification.

Restricted equivalence functions are well-known functions to compare two numbers in the interval between 0 and 1. Despite the numerous works studying the properties of restricted equivalence functions and their multiple applications as support for different similarity measures, an extension of these functions to an n-dimensional space is absent from the literature. In this article, we present a novel contribution to the restricted equivalence function theory, allowing to compare multivalued elements. Specifically, we extend the notion of restricted equivalence functions from$L$to$L^{n}$and present a new similarity construction on$L^{n}$. proposal is tested in the context of color image anisotropic diffusion as an example of one of its many applications.

Automatic image detection is one of the most important areas in computing due to its potential application in numerous real-world scenarios. One important tool to deal with that is calledoverlap indices. They were introduced as a procedure to provide the maximum lack of knowledge when comparing two fuzzy objects. They have been successfully applied in the following fields: image processing, fuzzy rule-based systems, decision making, and computational brain interfaces. This notion ofoverlap indicesis also necessary for applications in which type-2 fuzzy sets are required. In this article, we introduce the notion oftype-$(2,k)$overlap index($k \in \lbrace 0,1,2\rbrace$) in the setting of type-2 fuzzy sets. We describe both the reasons that have led to this notion and the relationships that naturally arise among the algebraic underlying structures. Finally, we illustrate how type-$(2,k)$overlap indices can be employed in the setting of fuzzy rule-based systems when the involved objects are type-2 fuzzy sets.

The Choquet integral (CI) is an averaging aggregation function that has been used, e.g., in the fuzzy reasoning method (FRM) of fuzzy rule-based classification systems (FRBCSs) and in multicriteria decision making in order to take into account the interactions among data/criteria. Several generalizations of the CI have been proposed in the literature in order to improve the performance of FRBCSs and also to provide more flexibility in the different models by relaxing both the monotonicity requirement and averaging conditions of aggregation functions. An important generalization is the$C_{F}$-integrals, which are preaggregation functions that may present interesting nonaveraging behavior depending on the function$F$adopted in the construction and, in this case, offering competitive results in classification. Recently, the concept of d-Choquet integrals was introduced as a generalization of the CI by restricted dissimilarity functions (RDFs), improving the usability of CIs, as when comparing inputs by the usual difference may not be viable. The objective of this article is to introduce the concept of$dC_{F}$-integrals, which is a generalization of$C_{F}$-integrals by RDFs. The aim is to analyze whether the usage of$dC_{F}$-integrals in the FRM of FRBCSs represents a good alternative toward the standard$C_{F}$-integrals that just consider the difference as a dissimilarity measure. For that, we consider six RDFs combined with five fuzzy measures, applied with more than 20 functions$F$. The analysis of the results is based on statistical tests, demonstrating their efficiency. Additionally, comparing the applicability of$dC_{F}$-integrals versus$C_{F}$-integrals, the range of the good generalizations of the former is much larger than that of the latter.

Aggregation functions have been extensively studied and applied in several practical problems involving some sort of fuzzy modeling, by enacting the fusion process of data from the unit interval. T-norms and t-conorms, as well as overlap and grouping functions, are examples of pairs of aggregation functions that are related through the duality property, which is associated with some definition of fuzzy negation. By constructing pairs of dual aggregation functions and applying them in some practical problem, one can analyze which type of behaviour (conjunctive or disjunctive, for example) of the aggregation operator can benefit the whole system. However, when dealing with applications that do not involve fuzzy modeling, such as classification via convolutional neural networks, the data to be aggregated do not necessarily comes from the unit interval. Recently, a framework for defining classes aggregation functions on an arbitrary closed real interval (namely, (a, b)-aggregation functions) based on core known classes of aggregation functions have been introduced, but the study of negations and duality in this context is yet to be developed. Thus, in this paper we introduce and study the concept of negations defined on a arbitrary closed real intervals, called (a, b)-negations, presenting a construction method for them based core fuzzy negations. From that, we develop the concept of duality between (a, b)-aggregation functions, showing that the duality property is preserved when constructing (a, b)-aggregation functions from dual aggregation functions.

A function that takes n numbers as input and outputs one number is said to be homogeneous whenever the result of multiplying each input by a certain factor λ yields the original output multiplied by that same factor.This concept has been extended by the notion of abstract homogeneity, which generalizes the product in the expression of homogeneity by a general function g and the effect of the factor λ by an automorphism.However, the effect of parameter λ remains unchanged for all the input values.In this study, we generalize further the condition of abstract homogeneity by introducing ℱ-homogeneity, which is defined with respect to a family of functions, enabling a different behavior for each of the inputs.Next, we study the properties that are satisfied by this family of functions and, moreover, we link this concept with the condition of directional monotonicity, which is a trendy property in the framework of aggregation functions.To achieve that, we generalize directional monotonicity by ℱ directional monotonicity, which is defined with respect to a family of functions ℱ and a family of vectors  .Finally, we show how the introduced concepts could be applied

Brain-computer interface (BCI) technologies are popular methods of communication between the human brain and external devices. One of the most popular approaches to BCI is motor imagery (MI). In BCI applications, the electroencephalography (EEG) is a very popular measurement for brain dynamics because of its noninvasive nature. Although there is a high interest in the BCI topic, the performance of existing systems is still far from ideal, due to the difficulty of performing pattern recognition tasks in EEG signals. This difficulty lies in the selection of the correct EEG channels, the signal-to-noise ratio of these signals, and how to discern the redundant information among them. BCI systems are composed of a wide range of components that perform signal preprocessing, feature extraction, and decision making. In this article, we define a new BCI framework, called enhanced fusion framework, where we propose three different ideas to improve the existing MI-based BCI frameworks. First, we include an additional preprocessing step of the signal: a differentiation of the EEG signal that makes it time invariant. Second, we add an additional frequency band as a feature for the system: the sensorimotor rhythm band, and we show its effect on the performance of the system. Finally, we make a profound study of how to make the final decision in the system. We propose the usage of both up to six types of different classifiers and a wide range of aggregation functions (including classical aggregations, Choquet and Sugeno integrals, and their extensions and overlap functions) to fuse the information given by the considered classifiers. We have tested this new system on a dataset of 20 volunteers performing MI-based brain-computer interface experiments. On this dataset, the new system achieved 88.80% accuracy. We also propose an optimized version of our system that is able to obtain up to 90.76%. Furthermore, we find that the pair Choquet/Sugeno integrals and overlap functions are the ones providing the best results.

Overlap functions are a type of aggregation functions that are not required to be associative, generally used to indicate the overlapping degree between two values. They have been successfully used as a conjunction operator in several practical problems, such as fuzzy-rule-based classification systems (FRBCSs) and image processing. Some extensions of overlap functions were recently proposed, such as general overlap functions and, in the interval-valued context,n-dimensional interval-valued overlap functions. The latter allow them to be applied inn-dimensional problems with interval-valued inputs, such as interval-valued classification problems, where one can apply interval-valued FRBCSs (IV-FRBCSs). In this case, the choice of an appropriate total order for intervals, such as an admissible order, can play an important role. However, neither the relationship between the interval order and then-dimensional interval-valued overlap function (which may or may not be increasing for that order) nor the impact of this relationship in the classification process have been studied in the literature. Moreover, there is not a clear preferredn-dimensional interval-valued overlap function to be applied in an IV-FRBCS. Hence, in this article, we: first, present some new results on admissible orders, which allow us to introduce the concept ofn-dimensional admissibly ordered interval-valued overlap functions, that is,n-dimensional interval-valued overlap functions that are increasing with respect to an admissible order; second, develop a width-preserving construction method for this kind of function, derived from an admissible order and ann-dimensional overlap function, discussing some of its features; finally, analyze the behavior of several combinations of admissible orders andn-dimensional (admissibly ordered) interval-valued overlap functions when applied in IV-FRBCSs. All in all, the contribution of this article resides in pointing out the effect of admissible orders andn-dimensional admissibly ordered interval-valued overlap functions, both from a theoretical and applied points of view, the latter when considering classification problems.

Throughout this article, our main idea is to analyze from a theoretical and normative point of view different methods to aggregate individual rankings. To do so, first, we introduce the concept of a general mean on an abstract set. This new concept conciliates the social choice—where well-known impossibility results as the Arrovian ones are encountered—and the decision-making approaches—where the necessity of fusing rankings is unavoidable. Moreover, it gives rise to a reasonable definition of the concept of a ranking fusion function that does indeed satisfy the axioms of a general mean. Then, we will introduce some methods to build ranking fusion functions, paying a special attention to the use of score functions, and pointing out the equivalence between ranking and scoring. To conclude, we prove that any ranking fusion function introduces a partial order on rankings implemented on a finite set of alternatives. Therefore, this allows us to compare rankings and different methods of aggregation, so that in practice, one should look for the maximal elements with respect to such orders defined on rankings.

In this article, we develop moderate deviation functions to measure similarity and dissimilarity among a set of given interval-valued data to construct interval-valued aggregation functions, and we apply these functions in two motor-imagery brain–computer interface (MI-BCI) systems to classify electroencephalography signals. To do so, we introduce the notion of interval-valued moderate deviation function and, in particular, we study those interval-valued moderate deviation functions, which preserve the width of the input intervals. In order to apply them in an MI-BCI system, we first use fuzzy implication operators to measure the uncertainty linked to the output of each classifier in the ensemble of the system, and then we perform the decision making phase using the new interval-valued aggregation functions. We have tested the goodness of our proposal in two MI-BCI frameworks, obtaining better results than those obtained using other numerical aggregation and interval-valued ordered weighted averaging operators, and obtaining competitive results versus some nonaggregation-based frameworks.

In this article, we propose a new generalization for the notion of homogeneous functions. We show some properties and how it appears in some scenarios. Finally, we show how this generalization can be used in order to provide a new paradigm for decision-making theory calledconsistent influenced/disturbed decision making. In order to illustrate the applicability of this new paradigm, we provide a toy example.

A new concept of equivalence between intervals and the induced indistinguishability between interval-valued (IV) fuzzy sets are proposed and considered. A new notion of the degree of IV equivalence is presented where partial or linear orders and the width of intervals are involved reflecting uncertainty. Furthermore, construction methods of the considered equivalences are provided and the relation between them and other notions of IV equivalences are studied. Finally, a methodology to apply the proposed equivalences in the field of image processing is shown, along with an illustrative example.

The notion of an orness measure for aggregation functions has been a relevant study subject whose history can be traced back to the early works of Dujmović in 1973. Intuitively, an orness measure quantifies the similarity of an aggregation function to the “or” function and results in an essential tool for decision engineering, field in which the choice of aggregation function is sometimes restricted to a desired value of orness (orness-directed aggregation). In 1988, Yager presented a particular example of orness measure for ordered weighted averaging (OWA) functions and initiated a series of contributions aiming at proposing an axiomatic definition of orness measure for OWA functions. In this paper, we go much further and present an axiomatic definition of orness measure for the whole family of aggregation functions. We end by proposing two natural construction methods for an orness measure for aggregation functions. The particular examples of the (discrete) Choquet integral and uninorms are studied in detail.

In this article, some formal relationships between the different axiomatic definitions of inclusion measure are analyzed. In particular, the links between the different proposals about the null-space (the collection of pairs associated with a null degree of inclusion) are studied. Taking as starting point the well-known axiomatics of Kitainik and Sinha-Dougherty, we observe that other alternative proposals about the null-space are incompatible with both the null-space and the decomposition axioms of these authors. We also conclude that both the axiomatics of Kitainik and those of Sinha-Dougherty contain certain redundancies. Reduced equivalent lists of axioms are proposed.

Fuzzy relations are compared by membership values and as a consequence new types of local properties of fuzzy relations are introduced. In the new properties of fuzzy relations an arbitrary binary relation is involved. Particularly, a binary aggregation function may be used to define these properties. Connections between the new local properties of fuzzy relations are described. Furthermore, preservation of these properties in aggregation process is considered. Finally, notes on applications of the presented local properties in the context of fuzzy multisets and decision making are provided.

Two of the main research lines in the theory of aggregation functions is the extension to more general domains and the relaxation of the monotonicity conditions. In this work, we discuss the state-of-the-art of the main introduced relaxed forms of monotonicity that can be found in the literature, i.e., weak, directional, ordered directional and strengthened ordered directional monotonicity. We pay special attention to the extension of a relaxed form of monotonicity to the interval-valued setting and we propose the concepts of ordered and strengthened ordered directional monotonicity for this general setting. Moreover, we study the main properties of the functions that satisfy the introduced properties and present some construction methods.

In this article, we propose a generalization of the Choquet integral, starting from its definition in terms of the Möbius transform. We modify the product on R considered in the Lovász extension form of the Choquet integral into a function F, and we discuss the properties of this new functional. For a fixed n, a complete description of all F yielding an n-ary aggregation function with a fixed diagonal section, independent of the considered fuzzy measure, is given, and several particular examples are presented. Finally, all functions F yielding an aggregation function, independent of the number n of inputs and of the considered fuzzy measure, are characterized, and related aggregation functions are shown to be just the Choquet integrals over the distorted inputs.

It is common sense the relevance of overlap and grouping functions, specially in applications in which associativity is not required. In this work, based on previous investigations concerning classes of implication functions derived from overlap functions O, grouping functions G and fuzzy negations N (namely, (G, N)-implications, RO-implications and QL-implications constructed from (O, G, N)), a deep study on D-implications constructed from grouping functions G is carried out. Such fuzzy implications, also known as Dishkant implications, are obtained from D-operations derived from (O, G, N), which are used for the generalization of the implication p → q ≡ q(¬p∧¬q) of orthomodular lattices. We investigate under which conditions such D-operations are fuzzy implication functions, presenting a general form for obtaining D-implications. We also provide a comparative study of D-implications derived from grouping functions and other classes of fuzzy implications constructed from N, O and G, analysing the intersections among such classes.

In this paper, the issue of measuring the degree of inclusion and equivalence measure for Atanassov intuitionistic fuzzy setting is considered. We propose an application of the inclusion and equivalence measures created by using the partial or linear order on Atanassov intuitionistic fuzzy setting. Moreover, some properties of inclusion and equivalence measures and some correlation between them and aggregation operators are examined and their possible application in image processing is indicated.

Pre-aggregation functions have extended the framework of valid functions to fuse data with respect to aggregation functions, in the sense that the monotonicity requirements are relaxed. Moreover, this class of functions has proven to be useful in classification problems. In this work we present the concept of directional monotonicity for interval-valued functions and use it to define the concept of interval-valued pre-aggregation functions. Additionally, we present some relevant properties of directionally monotone interval-valued functions and a method to construct interval-valued pre-aggregation functions from standard pre-aggregation functions.

In this work we propose a new axiomatic definition of distance between interval-valued fuzzy sets which takes into account the width of the membership intervals and linear orders. We discuss some construction methods using aggregation functions which are defined in terms of admissible orders.

A key component of fuzzy rule-based classification systems (FRBCS) is the fuzzy reasoning method (FRM) since it infers the class predicted for new examples. A crucial stage in any FRM is the way in which the information given by the fired rules during the inference process is aggregated. A widely used FRM is the winning rule, which applies the maximum to accomplish this aggregation. The maximum is an averaging operator, which means that its result is within the range delimited by the minimum and the maximum of the aggregated values. Recently, new averaging operators based on generalizations of the Choquet integral have been proposed to perform this aggregation process. However, the most accurate FRBCSs use the FRM known as additive combination that considers the normalized sum as the aggregation operator, which is nonaveraging. For this reason, this paper is aimed at introducing a new nonaveraging operator named CF1F2-integral, which is a generalization of the Choquet-like Copula-based integral (CC-integral). CF1F2-integrals present the desired properties of an aggregation-like operator since they satisfy appropriate boundary conditions and have some kind of increasingness property. We show that CF1F2-integrals, when used to cope with classification problems, enhance the results of the previous averaging generalizations of the Choquet integral and provide competitive results (even better) when compared with state-of-the-art FRBCSs.

The aim of this paper is to introduce the concept of pre-t-conorms, based on the notion of pre-aggregation function, which was introduced by Lucca et al. as an “aggregation” concept that it is not monotonic in all its domain. We also study the concept of light pre-t-conorms, which are non necessarily associative commutative functions with neutral element e = 0. We present some properties of (light) pre-t-conorms and the classes of (light) pre-t-conorms, showing interesting examples. Finally, we present an application of pre-conorms and pre-negations for defining directional fuzzy implication functions. We notice that the introduction of pre-t-conorms allows applications where the full monotonicity is not required (as in classification problems) and the light pre-t-conorms can be used in applications that do not require the associativity property (as in image processing and decision making).

In a recent paper, it was introduced the concept of Choquet-like Copula-based integral (CC-integral for short). This kind of function extends the standard Choquet integral and generalizes it by copula functions. These functions were applied in the Fuzzy Reasoning Method (FRM) of a Fuzzy Rule-Based Classification System (FRBCS), presenting an example where the CC-integral based on the minimum t-norm had different behaviors according to the values being aggregated. Therefore, the resulting FRM is theoretically more flexible than those associated with classical aggregation functions like the maximum. In this work, we present a methodology to study the flexibility of the aggregation function used in the FRM. Specifically, we conduct an analysis of 3 different methods to aggregate values in the FRM, namely, the CC-integral based on the minimum t-norm, the standard Choquet integral and the maximum (classical FRM of the winning rule - WR). We prove that the CC-integral behaves in different ways according to the values to be aggregated, whereas the Choquet integral offers an averaging behavior and the WR presents an strict behavior, since it considers only the rule having the maximum compatibility with the example.

In this contribution, we discuss the role and potential of pre-aggregation functions. This class of functions has the same boundary conditions as aggregation functions, but differs in the constraints related to function increase. Specifically, monotonicity is replaced by the so-called directional monotonicity. With this work, we attempt to shed light on the relationship between aggregation and pre-aggregation functions, as well as to discuss some construction methods of pre-aggregation functions.

In this paper we study interval-valued fuzzy relations. We consider preference relations, i.e. a triplet consisting of strict preference, indifference and incomparability which are defined with the use of a fuzzy negation. We analyze the preservation of the fuzzy negation based reciprocity property of interval-valued fuzzy relations by aggregation functions and by some basic interval-valued fuzzy relations. We use diverse representa-tions of aggregation functions. We also consider the connection between N-reciprocal relations and transitivity properties. We provide a numerical example where the final alternative is chosen with the use of generalized voting method, where admissible linear orders for intervals are applied.

Differentiation of interval-valued functions is an intricate problem, since it cannot be defined as a direct generalization of differentiation of scalar ones. Literature on interval arithmetic contains proposals and definitions for differentiation, but their semantic is unclear for the cases in which intervals represent the ambiguity due to hesitancy or lack of knowledge. In this work we analyze the needs, tools and goals for interval-valued differentiation, focusing on the case of interval-valued images. This leads to the formulation of a differentiation schema inspired by bilateral filters, which allows for the accommodation of most of the methods for scalar image differentiation, but also takes support from interval-valued arithmetic. This schema can produce area-, segment- and vector-valued gradients, according to the needs of the image processing task it is applied to. Our developments are put to the test in the context of edge detection.

In this paper, we review the definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature. We also analyze the relationships between them and enumerate some of the applications in which they have been used.

In the paper properties of intuitionistic fuzzy relations are considered and preservation of some properties by operations, including lattice operations, composition and related operators are studied. Properties of intuitionistic fuzzy relations, namely reflexivity, irreflexivity, connectedness, symmetry, antisymmetry, perfect antisymmetry and transitivity are considered. Moreover, the authors study assumptions under which intuitionistic fuzzy preference relations and related operators fulfil these properties.

Most fuzzy associative memories (FAMs) in the literature correspond to neural networks with a single layer of weights that distributively contains the information on associations to be stored. The main applications of these types of associative memory can be found in fuzzy rule-based systems. In contrast, Θ-fuzzy associative memories ( Θ-FAMs) represent parametrized fuzzy neural networks with a hidden layer and these FAM models extend (dual) S-FAMs and SM-FAMs based on fuzzy subsethood and similarity measures. In this paper, we provide theoretical results concerning the storage capacity and error correction capability of Θ-FAMs. In addition, we introduce a training algorithm for Θ-FAMs and we compare the error rates produced by Θ-FAMs and some well-known classifiers in some benchmark classification problems that are available on the internet. Finally, we apply Θ-FAMs to a problem of vision-based self-localization in mobile robotics.

In this paper, we will present a wider view on the relationship between interval-valued fuzzy sets and interval type-2 fuzzy sets, where we will show that interval-valued fuzzy sets are a particular case of the interval type-2 fuzzy sets. For this reason, both concepts should be treated in a different way. In addition, the view presented in this paper will allow a more general perspective of interval type-2 fuzzy sets, which will allow representing concepts that could not be presented by interval-valued fuzzy sets.

An important problem in working with fuzzy sets is the correct construction of the membership functions that represent the objects of the system. Different experts construct different membership functions to represent the same object. In this paper, we construct an interval type-2 fuzzy set (IT2FS) with different fuzzy sets such that the length of the (membership) interval represents the uncertainty of the expert with respect to the choice of the membership function. We analyze this problem in the context of image segmentation. We propose a new version of the classical fuzzy thresholding algorithm, in which an expert can select multiple membership functions, to avoid the problem of selecting only one to represent the image. From these membership functions, we construct an IT2FS, and by minimizing its entropy, we find a threshold with which to binarize the image. We present experimental results that show that it is advisable to use this methodology when it is not known which membership function is the most suitable.

Atanassov's intuitionistic fuzzy sets (AIFS) and interval valued fuzzy sets (IVFS) are two generalizations of a fuzzy set, which are equivalent mathematically although different semantically. We analyze the median aggregation operator for AIFS and IVFS. Different mathematical theories have lead to different definitions of the median operator. We look at the median from various perspectives: as an instance of the intuitionistic ordered weighted averaging operator, as a Fermat point in a plane, as a minimizer of input disagreement, and as an operation on distributive lattices. We underline several connections between these approaches and summarize essential properties of the median in different representations.

In this work we present a new image thresholding algorithm for the segmentation of MRI brain images into two classes: gray matter and white matter. The proposed algorithm is based on the concept of incomparability proposed by Fodor and Roubens for fuzzy preference relations. We test our algorithm for local and global segmentation of brain images. We proof that global segmentation performs better results than local segmentation and improves the results obtained by other thresholding algorithm.

In this paper we introduce the concept of Generalized Atanassov's Intuitionistic Fuzzy Index. We characterize it in terms of fuzzy implication operators and propose a construction method with order automorphisms. Finally, we obtain, by means of special aggregation functions applied to the generalized Atanassov's intuitionistic fuzzy index, the Atanassov's intuitionistic fuzzy entropy given by Burillo and Bustince.

Stereo matching problem attempts to find corresponding locations between pairs of displaced images of the same scene. Correspondence estimation between pixels suffers from occlusions, noise, and bias. This paper introduces a novel approach to represent images by means of interval-valued fuzzy sets. These sets allow one to overcome the uncertainty due to the aforementioned problems. The aim is to take advantage of the new representation to develop a stereo matching algorithm. The interval-valued fuzzification process for images that is proposed here is based on image segmentation. Interval-valued fuzzy similarities are introduced to compare windows whose pixels are represented by intervals. To make use of color information, the similarities of the RGB channels were aggregated using the luminance formula. The experimental analysis makes a comparison with other methods. The new representation that is proposed together with the new similarity measure show a better overall behavior, providing more accurate correspondences, mainly near depth discontinuities and for images with a large amount of color.

In this work we consider the concept of contractive interval-valued fuzzy negation, as a negation such that it does not increase the length or amplitude of an interval. We relate this to the concept of Lipschitz function. In particular, we prove that the only strict (strong) contractive interval-valued fuzzy negation is the one generated from the standard (Zadeh's) negation.

The edge detection process can be broken down into four basic transformations, modifying the image from the original presentation to the final edges one. The adoption of this framework makes the process far more understandable, and offers an starting point for the combination and comparison of different edge detection methods. In this work we analyze the role of the third of the transformations, the blending, where the edge features are combined to obtain the edginess values. This work studies the use of quasi-aritmethic means for the combination of the edge features. Moreover, we show results obtained with different operators on real images, in order to illustrate the importance of the blending phase in the edge detection process. Results will show the impact of the function selection in the final results.

In this work we propose an image reduction algorithm based on weak local reduction operators. We use several averaging functions to build these operators and we analyze their properties. We present experimental results where we apply the algorithm and weak local reduction operators in procedures of reduction, and later, reconstruction of images. We analyze these results over natural images and noisy images.

This work explores the possibilities of extracting edges using a t-conorm based gravity approach and its relation with the t-norm based one.

Abstract Text Categorization is the process of assigning documents to a set of previously fixed categories. A lot of research is going on with the goal of automating this time‐consuming task. Several different algorithms have been applied, and Support Vector Machines (SVM) have shown very good results. In this report, we try to prove that a previous filtering of the words used by SVM in the classification can improve the overall performance. This hypothesis is systematically tested with three different measures of word relevance, on two different corpus (one of them considered in three different splits), and with both local and global vocabularies. The results show that filtering significantly improves the recall of the method, and that also has the effect of significantly improving the overall performance.