Label-based Type-2 Fuzzy Predicate Classification applied to the design of morphological W-operators for image processing Diego S. Comas #*1 , Gustavo J. Meschino #†2 , Marcel Brun #2 , Virginia L. Ballarin #4 # Digital Image Processing Group, Facultad de Ingeniería, Universidad Nacional de Mar del Plata Juan B. Justo 4302, Mar del Plata, Argentina 1 diego.comas@fi.mdp.edu.ar 2 gmeschin@fi.mdp.edu.ar 3 mbrun@fi.mdp.edu.ar 4 vballari@fi.mdp.edu.ar * Consejo Nacional de Investigaciones Científicas y Técnicas, CONICET Av. Rivadavia 1917, Ciudad Autónoma de Buenos Aires, Argentina † Bioengineering Lab, Facultad de Ingeniería, Universidad Nacional de Mar del Plata Juan B. Justo 4302, Mar del Plata, Argentina Abstract— Fuzzy logic is widely used in data classification. Due to its additional degrees of freedom compared to Type-1, Type-2 fuzzy logic has shown being more appropriate to classify data affected by noise. Algorithms for automatic design of classifiers using fuzzy logic have been developed using different approaches, with and without labeled data. W- operators are a set of nonlinear morphological operators widely used and their design can be approached as classifiers using training examples. This work aims to design W- operators through a new automatic method using labeled data, called Type-2 Label-based Fuzzy Predicate Classification. Results of the proposed approach show a great potential in achieving a lower classification error than other methods commonly used. Given the kind of analysis that the proposed method performs on the pattern space, this can contribute to both data classification and design of W-operators. I. INTRODUCTION Fuzzy Logic (FL) [1-3] is widely used in data classification, providing a tool to associate data (multidimensional vectors, also called patterns) with classes using logical operators and membership functions [1, 4]. One approach used in this context is based on fuzzy predicates [4, 5]. In this approach, the predicates describe, linguistically, properties that the components of a data vector presents, related to its membership to one class [1, 2]. The predicate that “explain” each class is formed by a logical combination of simple fuzzy predicates, whose truth value is obtained from membership functions. Depending on the type of the FL used, the truth values belong to the interval   0,1 in Type-1 FL, or to an interval   , ab , with   , 0,1 ab  , in interval Type-2 FL [6]. By defining truth values using intervals, Type-2 FL provides additional degrees of freedom in classification systems, compared to Type-1 FL. It has also been shown to be suitable in cases where it is not possible to define a unique truth value or in cases where the variables are affected by noise, obtaining better classification results compared to similar models of Type-1 FL [6-8]. In general, FL-based systems have been successfully used in data classification using both types of FL [1, 5]. As an essential feature, FL systems need expert knowledge [1] to define fuzzy predicates that “explain” each class, and to determine the membership functions which define the truth values of the simple predicates. We have developed, previously, algorithms for automatic design of classification systems using different approaches of FL, even using self-organized maps on data with and without labeling [1, 2]. Others authors have used clustering techniques in general approaches that combine supervised and unsupervised techniques [7, 9-11]. Within the field of digital image processing [12], window operators, also called W-operators, are a set of nonlinear morphological operators with application in image filtering and segmentation [13, 14]. The statistical design of W- operators can be approached as the design of a classifier from training examples, where the training set consists of pairs of images: the observed-ideal images. On each pixel of the observed image, we define a window W , which determines a subset of pixels associated. Each possible set of values of the image that are contained in W is called “configuration,” or “pattern,” in pattern recognition terminology. The design of a W-operator consists of determining the optimal classifier (with smallest error) that maps each pattern (or configuration) from the window W to one of the classes in the ideal image. This mapping, called characteristic function, is generally obtained via a statistical optimization process, based on the joint and conditional probabilities [14]. Like in other classifiers designed from data, designing a W-operator requires a careful consideration of the generalization; that is, which class to assign for a configuration that has not been presented in the training step [14]. This paper proposes the implementation of W-operators for binary images using fuzzy predicates based on interval Type-2 FL for the characteristic function, designing such systems through a new automatic method