Miguel Pagola

Interval-valued fuzzy sets constructed from matrices: Application to edge detection

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.

Construction of fuzzy indices from fuzzy DI-subsethood measures: Application to the global comparison of images

Two measures are presented for comparing fuzzy sets. The method for constructing these measures is studied, starting from fuzzy DI-subsethood measures (see [H. Bustince, V. Mohedano, E. Barrenechea, M. Pagola, Definition and construction of fuzzy DI-subsethood measures, Information Sciences 176 (2006) 3190-3231]). We then analyze and compare the properties satisfied by the measures and those satisfied by other classical indices used in the literature on fuzzy sets. The minimal set of conditions are studied that, from our point of view, must be met by any given measure for comparing images. We also prove that only one of the measures identified fulfills such conditions. Finally, all the measures studied are applied to different images and the results are analyzed, indicating that measures constructed from fuzzy DI-subsethood measures provide the best results.

A Historical Account of Types of Fuzzy Sets and Their Relationships

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.

Restricted equivalence functions

In this paper we present the concept of a restricted equivalence function. This concept arises on the one hand, from the definition of equivalence given by J. Fodor and M. Roubens, and on the other, from the properties usually demanded from the measures used for comparing images. We also study different methods for the construction of restricted equivalence functions from automorphisms and implication operators. Finally we analyze the manner of generating similarity measures of X. Liu and of J. Fan et al. from our restricted equivalence functions.

Image thresholding using restricted equivalence functions and maximizing the measures of similarity

In this paper we apply restricted equivalence functions to the computation of the threshold of an image. In the first part we present an algorithm for obtaining the best threshold of a grayscale image with a single object. In the second part we study different algorithms for calculating the optimal threshold. Then we analyze two algorithms for obtaining a sequence of optimal thresholds in images with several objects. Lastly, we compare our results with those obtained with other methods and carry out a study of the time efficiency of the methods we propose.

Definition and construction of fuzzy DI-subsethood measures

In this paper we study a method of construction of the fuzzy subsethood measures of V.R. Young [V.R. Young, Fuzzy subsethood, Fuzzy Sets and Systems 77 (1996) 371-384] and the fuzzy subsethood measures of J. Fan, X. Xie, and J. Pei [J. Fan, X. Xie, J. Pei, Subsethood measures: new definitions, Fuzzy Sets and Systems 106 (1999) 201-209]. We establish the conditions under which our constructions satisfy the axioms of Sinha and Doughertys inclusion measures and we present different methods for obtaining fuzzy entropies from said measures. Next we present a particular case of Youngs subsethood measures, DI-subsethood measures. For these we also analyze their construction and the conditions under which they satisfy different axioms.

Grouping, Overlap, and Generalized Bientropic Functions for Fuzzy Modeling of Pairwise Comparisons

In this paper, we propose new aggregation functions for the pairwise comparison of alternatives in fuzzy preference modeling. More specifically, we introduce the concept of a grouping function, i.e., a specific type of aggregation function that combines two degrees of support (weak preference) into a degree of information or, say, a degree of comparability between two alternatives, and we relate this new concept to that of incomparability. Grouping functions of this type complement the existing concept of overlap functions in a natural way, since the latter can be used to turn two degrees of weak preference into a degree of indifference. We also define the so-called generalized bientropic functions that allow for a unified representation of overlap and grouping functions. Apart from analyzing mathematical properties of these types of functions and exploring relationships between them, we elaborate on their use in fuzzy preference modeling and decision making. We present an algorithm to elaborate on an alternative preference ranking that penalizes those alternatives for which the expert is not sure of his/her preference.

Ignorance functions. An application to the calculation of the threshold in prostate ultrasound images

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.

Medical diagnosis of cardiovascular diseases using an interval-valued fuzzy rule-based classification system

This work was partially supported by the Spanish Ministry of Science and Technology under project TIN2010-15055 and the Research Services of the Universidad Publica de Navarra.

Uncertainties with Atanassov's intuitionistic fuzzy sets: Fuzziness and lack of knowledge

We review the existing measures of uncertainty (entropy) for Atanassovs intuitionistic fuzzy sets (AIFSs). We demonstrate that the existing measures of uncertainty for AIFS cannot capture all facets of uncertainty associated with an AIFS. We point out and justify that there are at least two facets of uncertainty of an AIFS, one of which is related to fuzziness while the other is related to lack of knowledge or non-specificity. For each facet of uncertainty, we propose a separate set of axioms. Then for each of fuzziness and non-specificity we propose a generating family (class) of measures. Each family is illustrated with several examples. In this context we prove several interesting results about the measures of uncertainty. We prove some results that help us to construct new measures of uncertainty of both kinds.