Application of an iterative method and an evolutionary algorithm in fuzzy optimization

This work develops two approaches based on the fuzzy set theory to solve a class of fuzzy mathematical optimization problems with uncertainties in the objective function and in the set of constraints. The first approach is an adaptation of an iterative method that obtains cut levels and later maximizes the membership function of fuzzy decision making using the bound search method. The second one is a metaheuristic approach that adapts a standard genetic algorithm to use fuzzy numbers. Both approaches use a decision criterion called satisfaction level that reaches the best solution in the uncertain environment. Selected examples from the literature are presented to compare and to validate the efficiency of the methods addressed, emphasizing the fuzzy optimization problem in some import-export companies in the south of Spain. © 2012 Brazilian Operations Research Society.

Reducing the conservatism of LMI-based stabilisation conditions for TS fuzzy systems using fuzzy Lyapunov functions

In this article, the fuzzy Lyapunov function approach is considered for stabilising continuous-time Takagi-Sugeno fuzzy systems. Previous linear matrix inequality (LMI) stability conditions are relaxed by exploring further the properties of the time derivatives of premise membership functions and by introducing slack LMI variables into the problem formulation. The relaxation conditions given can also be used with a class of fuzzy Lyapunov functions which also depends on the membership function first-order time-derivative. The stability results are thus extended to systems with large number of rules under membership function order relations and used to design parallel-distributed compensation (PDC) fuzzy controllers which are also solved in terms of LMIs. Numerical examples illustrate the efficiency of the new stabilising conditions presented. © 2013 Copyright Taylor and Francis Group, LLC.

Projeto e implementação com chaveamento de reguladores fuzzy Takagi-Sugeno para um conjunto de pontos de operação

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Classificador Fisherface Fuzzy para o reconhecimento de faces

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Robust Switched Control Design for Nonlinear Systems Using Fuzzy Models

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

On Switched Regulator Design of Uncertain Nonlinear Systems Using Takagi-Sugeno Fuzzy Models

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Tecniche di elaborazione di dati macrosismici per la definizione di sorgenti sismogenetiche

La valutazione dell’intensità secondo una procedura formale trasparente, obiettiva e che permetta di ottenere valori numerici attraverso scelte e criteri rigorosi, rappresenta un passo ed un obiettivo per la trattazione e l’impiego delle informazioni macrosismiche. I dati macrosismici possono infatti avere importanti applicazioni per analisi sismotettoniche e per la stima della pericolosità sismica. Questa tesi ha affrontato il problema del formalismo della stima dell’intensità migliorando aspetti sia teorici che pratici attraverso tre passaggi fondamentali sviluppati in ambiente MS-Excel e Matlab: i) la raccolta e l’archiviazione del dataset macrosismico; ii), l’associazione (funzione di appartenenza o membership function) tra effetti e gradi di intensità della scala macrosismica attraverso i principi della logica dei fuzzy sets; iii) l’applicazione di algoritmi decisionali rigorosi ed obiettivi per la stima dell’intensità finale. L’intera procedura è stata applicata a sette terremoti italiani sfruttando varie possibilità, anche metodologiche, come la costruzione di funzioni di appartenenza combinando le informazioni macrosismiche di più terremoti: Monte Baldo (1876), Valle d’Illasi (1891), Marsica (1915), Santa Sofia (1918), Mugello (1919), Garfagnana (1920) e Irpinia (1930). I risultati ottenuti hanno fornito un buon accordo statistico con le intensità di un catalogo macrosismico di riferimento confermando la validità dell’intera metodologia. Le intensità ricavate sono state poi utilizzate per analisi sismotettoniche nelle aree dei terremoti studiati. I metodi di analisi statistica sui piani quotati (distribuzione geografica delle intensità assegnate) si sono rivelate in passato uno strumento potente per analisi e caratterizzazione sismotettonica, determinando i principali parametri (localizzazione epicentrale, lunghezza, larghezza, orientazione) della possibile sorgente sismogenica. Questa tesi ha implementato alcuni aspetti delle metodologie di analisi grazie a specifiche applicazioni sviluppate in Matlab che hanno permesso anche di stimare le incertezze associate ai parametri di sorgente, grazie a tecniche di ricampionamento statistico. Un’analisi sistematica per i terremoti studiati è stata portata avanti combinando i vari metodi per la stima dei parametri di sorgente con i piani quotati originali e ricalcolati attraverso le procedure decisionali fuzzy. I risultati ottenuti hanno consentito di valutare le caratteristiche delle possibili sorgenti e formulare ipotesi di natura sismotettonica che hanno avuto alcuni riscontri indiziali con dati di tipo geologico e geologico-strutturale. Alcuni eventi (1915, 1918, 1920) presentano una forte stabilità dei parametri calcolati (localizzazione epicentrale e geometria della possibile sorgente) con piccole incertezze associate. Altri eventi (1891, 1919 e 1930) hanno invece mostrato una maggiore variabilità sia nella localizzazione dell’epicentro che nella geometria delle box: per il primo evento ciò è probabilmente da mettere in relazione con la ridotta consistenza del dataset di intensità mentre per gli altri con la possibile molteplicità delle sorgenti sismogenetiche. Anche l’analisi bootstrap ha messo in evidenza, in alcuni casi, le possibili asimmetrie nelle distribuzioni di alcuni parametri (ad es. l’azimut della possibile struttura), che potrebbero suggerire meccanismi di rottura su più faglie distinte.

New methods for the estimation of Takagi-Sugeno model based extended Kalman filter and its applications to optimal control for nonlinear systems

This paper describes new approaches to improve the local and global approximation (matching) and modeling capability of Takagi–Sugeno (T-S) fuzzy model. The main aim is obtaining high function approximation accuracy and fast convergence. The main problem encountered is that T-S identification method cannot be applied when the membership functions are overlapped by pairs. This restricts the application of the T-S method because this type of membership function has been widely used during the last 2 decades in the stability, controller design of fuzzy systems and is popular in industrial control applications. The approach developed here can be considered as a generalized version of T-S identification method with optimized performance in approximating nonlinear functions. We propose a noniterative method through weighting of parameters approach and an iterative algorithm by applying the extended Kalman filter, based on the same idea of parameters’ weighting. We show that the Kalman filter is an effective tool in the identification of T-S fuzzy model. A fuzzy controller based linear quadratic regulator is proposed in order to show the effectiveness of the estimation method developed here in control applications. An illustrative example of an inverted pendulum is chosen to evaluate the robustness and remarkable performance of the proposed method locally and globally in comparison with the original T-S model. Simulation results indicate the potential, simplicity, and generality of the algorithm. An illustrative example is chosen to evaluate the robustness. In this paper, we prove that these algorithms converge very fast, thereby making them very practical to use.

Contribución al estudio de dos conceptos básicos de la lógica fuzzy

La tesis doctoral CONTRIBUCIÓN AL ESTUDIO DE DOS CONCEPTOS BÁSICOS DE LA LÓGICA FUZZY constituye un conjunto de nuevas aportaciones al análisis de dos elementos básicos de la lógica fuzzy: los mecanismos de inferencia y la representación de predicados vagos. La memoria se encuentra dividida en dos partes que corresponden a los dos aspectos señalados. En la Parte I se estudia el concepto básico de «estado lógico borroso». Un estado lógico borroso es un punto fijo de la aplicación generada a partir de la regla de inferencia conocida como modus ponens generalizado. Además, un preorden borroso puede ser representado mediante los preórdenes elementales generados por el conjunto de sus estados lógicos borrosos. El Capítulo 1 está dedicado a caracterizar cuándo dos estados lógicos dan lugar al mismo preorden elemental, obteniéndose también un representante de la clase de todos los estados lógicos que generan el mismo preorden elemental. El Capítulo finaliza con la caracterización del conjunto de estados lógicos borrosos de un preorden elemental. En el Capítulo 2 se obtiene un subconjunto borroso trapezoidal como una clase de una relación de indistinguibilidad. Finalmente, el Capítulo 3 se dedica a estudiar dos tipos de estados lógicos clásicos: los irreducibles y los minimales. En el Capítulo 4, que inicia la Parte II de la memoria, se aborda el problema de obtener la función de compatibilidad de un predicado vago. Se propone un método, basado en el conocimiento del uso del predicado mediante un conjunto de reglas y de ciertos elementos distinguidos, que permite obtener una expresión general de la función de pertenencia generalizada de un subconjunto borroso que realice la función de extensión del predicado borroso. Dicho método permite, en ciertos casos, definir un conjunto de conectivas multivaluadas asociadas al predicado. En el último capítulo se estudia la representación de antónimos y sinónimos en lógica fuzzy a través de auto-morfismos. Se caracterizan los automorfismos sobre el intervalo unidad cuando sobre él se consideran dos operaciones: una t-norma y una t-conorma ambas arquimedianas. The PhD Thesis CONTRIBUCIÓN AL ESTUDIO DE DOS CONCEPTOS BÁSICOS DE LA LÓGICA FUZZY is a contribution to two basic concepts of the Fuzzy Logic. It is divided in two parts, the first is devoted to a mechanism of inference in Fuzzy Logic, and the second to the representation of vague predicates. «Fuzzy Logic State» is the basic concept in Part I. A Fuzzy Logic State is a fixed-point for the mapping giving the Generalized Modus Ponens Rule of inference. Moreover, a fuzzy preordering can be represented by the elementary preorderings generated by its Fuzzy Logic States. Chapter 1 contemplates the identity of elementary preorderings and the selection of representatives for the classes modulo this identity. This chapter finishes with the characterization of the set of Fuzzy Logic States of an elementary preordering. In Chapter 2 a Trapezoidal Fuzzy Set as a class of a relation of Indistinguishability is obtained. Finally, Chapter 3 is devoted to study two types of Classical Logic States: irreducible and minimal. Part II begins with Chapter 4 dealing with the problem of obtaining a Compa¬tibility Function for a vague predicate. When the use of a predicate is known by means of a set of rules and some distinguished elements, a method to obtain the general expression of the Membership Function is presented. This method allows, in some cases, to reach a set of multivalued connectives associated to the predicate. Last Chapter is devoted to the representation of antonyms and synonyms in Fuzzy Logic. When the unit interval [0,1] is endowed with both an archimedean t-norm and a an archi-medean t-conorm, it is showed that the automorphisms' group is just reduced to the identity function.

A new approach to fuzzy estimation of Takagi-Sugeno model and its applications to optimal control for nonlinear systems

An efficient approach is presented to improve the local and global approximation and modelling capability of Takagi-Sugeno (T-S) fuzzy model. The main aim is obtaining high function approximation accuracy. The main problem is that T-S identification method cannot be applied when the membership functions are overlapped by pairs. This restricts the use of the T-S method because this type of membership function has been widely used during the last two decades in the stability, controller design and are popular in industrial control applications. The approach developed here can be considered as a generalized version of T-S method with optimized performance in approximating nonlinear functions. A simple approach with few computational effort, based on the well known parameters' weighting method is suggested for tuning T-S parameters to improve the choice of the performance index and minimize it. A global fuzzy controller (FC) based Linear Quadratic Regulator (LQR) is proposed in order to show the effectiveness of the estimation method developed here in control applications. Illustrative examples of an inverted pendulum and Van der Pol system are chosen to evaluate the robustness and remarkable performance of the proposed method and the high accuracy obtained in approximating nonlinear and unstable systems locally and globally in comparison with the original T-S model. Simulation results indicate the potential, simplicity and generality of the algorithm.

Variable structure control with chattering reduction of a generalized T-S model

In this paper, a fuzzy logic controller (FLC) based variable structure control (VSC) is presented. The main objective is to obtain an improved performance of highly non-linear unstable systems. New functions for chattering reduction and error convergence without sacrificing invariant properties are proposed. The main feature of the proposed method is that the switching function is added as an additional fuzzy variable and will be introduced in the premise part of the fuzzy rules; together with the state variables. In this work, a tuning of the well known weighting parameters approach is proposed to optimize local and global approximation and modelling capability of the Takagi-Sugeno (T-S) fuzzy model to improve the choice of the performance index and minimize it. The main problem encountered is that the T-S identification method can not be applied when the membership functions are overlapped by pairs. This in turn restricts the application of the T-S method because this type of membership function has been widely used in control applications. The approach developed here can be considered as a generalized version of the T-S method. An inverted pendulum mounted on a cart is chosen to evaluate the robustness, effectiveness, accuracy and remarkable performance of the proposed estimation approach in comparison with the original T-S model. Simulation results indicate the potential, simplicity and generality of the estimation method and the robustness of the chattering reduction algorithm. In this paper, we prove that the proposed estimation algorithm converge the very fast, thereby making it very practical to use. The application of the proposed FLC-VSC shows that both alleviation of chattering and robust performance are achieved.

Contribución al estudio de las negaciones, autocontradicción, t-normas y t-conormas en los conjuntos borrosos de tipo 2

Los conjuntos borrosos de tipo 2 (T2FSs) fueron introducidos por L.A. Zadeh en 1975 [65], como una extensión de los conjuntos borrosos de tipo 1 (FSs). Mientras que en estos últimos el grado de pertenencia de un elemento al conjunto viene determinado por un valor en el intervalo [0, 1], en el caso de los T2FSs el grado de pertenencia de un elemento es un conjunto borroso en [0,1], es decir, un T2FS queda determinado por una función de pertenencia μ : X → M, donde M = [0, 1][0,1] = Map([0, 1], [0, 1]), es el conjunto de las funciones de [0,1] en [0,1] (ver [39], [42], [43], [61]). Desde que los T2FSs fueron introducidos, se han generalizado a dicho conjunto (ver [39], [42], [43], [61], por ejemplo), a partir del “Principio de Extensión” de Zadeh [65] (ver Teorema 1.1), muchas de las definiciones, operaciones, propiedades y resultados obtenidos en los FSs. Sin embargo, como sucede en cualquier área de investigación, quedan muchas lagunas y problemas abiertos que suponen un reto para cualquiera que quiera hacer un estudio profundo en este campo. A este reto se ha dedicado el presente trabajo, logrando avances importantes en este sentido de “rellenar huecos” existentes en la teoría de los conjuntos borrosos de tipo 2, especialmente en las propiedades de autocontradicción y N-autocontradicción, y en las operaciones de negación, t-norma y t-conorma sobre los T2FSs. Cabe destacar que en [61] se justifica que las operaciones sobre los T2FSs (Map(X,M)) se pueden definir de forma natural a partir de las operaciones sobre M, verificando las mismas propiedades. Por tanto, por ser más fácil, en el presente trabajo se toma como objeto de estudio a M, y algunos de sus subconjuntos, en vez de Map(X,M). En cuanto a la operación de negación, en el marco de los conjuntos borrosos de tipo 2 (T2FSs), usualmente se emplea para representar la negación en M, una operación asociada a la negación estándar en [0,1]. Sin embargo, dicha operación no verifica los axiomas que, intuitivamente, debe verificar cualquier operación para ser considerada negación en el conjunto M. En este trabajo se presentan los axiomas de negación y negación fuerte en los T2FSs. También se define una operación asociada a cualquier negación suprayectiva en [0,1], incluyendo la negación estándar, y se estudia, junto con otras propiedades, si es negación y negación fuerte en L (conjunto de las funciones de M normales y convexas). Además, se comprueba en qué condiciones se cumplen las leyes de De Morgan para un extenso conjunto de pares de operaciones binarias en M. Por otra parte, las propiedades de N-autocontradicción y autocontradicción, han sido suficientemente estudiadas en los conjuntos borrosos de tipo 1 (FSs) y en los conjuntos borrosos intuicionistas de Atanassov (AIFSs). En el presente trabajo se inicia el estudio de las mencionadas propiedades, dentro del marco de los T2FSs cuyos grados de pertenencia están en L. En este sentido, aquí se extienden los conceptos de N-autocontradicción y autocontradicción al conjunto L, y se determinan algunos criterios para verificar tales propiedades. En cuanto a otras operaciones, Walker et al. ([61], [63]) definieron dos familias de operaciones binarias sobre M, y determinaron que, bajo ciertas condiciones, estas operaciones son t-normas (normas triangulares) o t-conormas sobre L. En este trabajo se introducen operaciones binarias sobre M, unas más generales y otras diferentes a las dadas por Walker et al., y se estudian varias propiedades de las mismas, con el objeto de deducir nuevas t-normas y t-conormas sobre L. ABSTRACT Type-2 fuzzy sets (T2FSs) were introduced by L.A. Zadeh in 1975 [65] as an extension of type-1 fuzzy sets (FSs). Whereas for FSs the degree of membership of an element of a set is determined by a value in the interval [0, 1] , the degree of membership of an element for T2FSs is a fuzzy set in [0,1], that is, a T2FS is determined by a membership function μ : X → M, where M = [0, 1][0,1] is the set of functions from [0,1] to [0,1] (see [39], [42], [43], [61]). Later, many definitions, operations, properties and results known on FSs, have been generalized to T2FSs (e.g. see [39], [42], [43], [61]) by employing Zadeh’s Extension Principle [65] (see Theorem 1.1). However, as in any area of research, there are still many open problems which represent a challenge for anyone who wants to make a deep study in this field. Then, we have been dedicated to such challenge, making significant progress in this direction to “fill gaps” (close open problems) in the theory of T2FSs, especially on the properties of self-contradiction and N-self-contradiction, and on the operations of negations, t-norms (triangular norms) and t-conorms on T2FSs. Walker and Walker justify in [61] that the operations on Map(X,M) can be defined naturally from the operations onMand have the same properties. Therefore, we will work onM(study subject), and some subsets of M, as all the results are easily and directly extensible to Map(X,M). About the operation of negation, usually has been employed in the framework of T2FSs, a operation associated to standard negation on [0,1], but such operation does not satisfy the negation axioms on M. In this work, we introduce the axioms that a function inMshould satisfy to qualify as a type-2 negation and strong type-2 negation. Also, we define a operation on M associated to any suprajective negation on [0,1], and analyse, among others properties, if such operation is negation or strong negation on L (all normal and convex functions of M). Besides, we study the De Morgan’s laws, with respect to some binary operations on M. On the other hand, The properties of self-contradiction and N-self-contradiction have been extensively studied on FSs and on the Atanassov’s intuitionistic fuzzy sets (AIFSs). Thereon, in this research we begin the study of the mentioned properties on the framework of T2FSs. In this sense, we give the definitions about self-contradiction and N-self-contradiction on L, and establish the criteria to verify these properties on L. Respect to the t-norms and t-conorms, Walker et al. ([61], [63]) defined two families of binary operations on M and found that, under some conditions, these operations are t-norms or t-conorms on L. In this work we introduce more general binary operations on M than those given by Walker et al. and study which are the minimum conditions necessary for these operations satisfy each of the axioms of the t-norm and t-conorm.

Vision-Based Tracking, Odometry and Control for UAV Autonomy

El principal objetivo de este trabajo es proporcionar una solución en tiempo real basada en visión estéreo o monocular precisa y robusta para que un vehículo aéreo no tripulado (UAV) sea autónomo en varios tipos de aplicaciones UAV, especialmente en entornos abarrotados sin señal GPS. Este trabajo principalmente consiste en tres temas de investigación de UAV basados en técnicas de visión por computador: (I) visual tracking, proporciona soluciones efectivas para localizar visualmente objetos de interés estáticos o en movimiento durante el tiempo que dura el vuelo del UAV mediante una aproximación adaptativa online y una estrategia de múltiple resolución, de este modo superamos los problemas generados por las diferentes situaciones desafiantes, tales como cambios significativos de aspecto, iluminación del entorno variante, fondo del tracking embarullado, oclusión parcial o total de objetos, variaciones rápidas de posición y vibraciones mecánicas a bordo. La solución ha sido utilizada en aterrizajes autónomos, inspección de plataformas mar adentro o tracking de aviones en pleno vuelo para su detección y evasión; (II) odometría visual: proporciona una solución eficiente al UAV para estimar la posición con 6 grados de libertad (6D) usando únicamente la entrada de una cámara estéreo a bordo del UAV. Un método Semi-Global Blocking Matching (SGBM) eficiente basado en una estrategia grueso-a-fino ha sido implementada para una rápida y profunda estimación del plano. Además, la solución toma provecho eficazmente de la información 2D y 3D para estimar la posición 6D, resolviendo de esta manera la limitación de un punto de referencia fijo en la cámara estéreo. Una robusta aproximación volumétrica de mapping basada en el framework Octomap ha sido utilizada para reconstruir entornos cerrados y al aire libre bastante abarrotados en 3D con memoria y errores correlacionados espacialmente o temporalmente; (III) visual control, ofrece soluciones de control prácticas para la navegación de un UAV usando Fuzzy Logic Controller (FLC) con la estimación visual. Y el framework de Cross-Entropy Optimization (CEO) ha sido usado para optimizar el factor de escala y la función de pertenencia en FLC. Todas las soluciones basadas en visión en este trabajo han sido probadas en test reales. Y los conjuntos de datos de imágenes reales grabados en estos test o disponibles para la comunidad pública han sido utilizados para evaluar el rendimiento de estas soluciones basadas en visión con ground truth. Además, las soluciones de visión presentadas han sido comparadas con algoritmos de visión del estado del arte. Los test reales y los resultados de evaluación muestran que las soluciones basadas en visión proporcionadas han obtenido rendimientos en tiempo real precisos y robustos, o han alcanzado un mejor rendimiento que aquellos algoritmos del estado del arte. La estimación basada en visión ha ganado un rol muy importante en controlar un UAV típico para alcanzar autonomía en aplicaciones UAV. ABSTRACT The main objective of this dissertation is providing real-time accurate robust monocular or stereo vision-based solution for Unmanned Aerial Vehicle (UAV) to achieve the autonomy in various types of UAV applications, especially in GPS-denied dynamic cluttered environments. This dissertation mainly consists of three UAV research topics based on computer vision technique: (I) visual tracking, it supplys effective solutions to visually locate interesting static or moving object over time during UAV flight with on-line adaptivity approach and multiple-resolution strategy, thereby overcoming the problems generated by the different challenging situations, such as significant appearance change, variant surrounding illumination, cluttered tracking background, partial or full object occlusion, rapid pose variation and onboard mechanical vibration. The solutions have been utilized in autonomous landing, offshore floating platform inspection and midair aircraft tracking for sense-and-avoid; (II) visual odometry: it provides the efficient solution for UAV to estimate the 6 Degree-of-freedom (6D) pose using only the input of stereo camera onboard UAV. An efficient Semi-Global Blocking Matching (SGBM) method based on a coarse-to-fine strategy has been implemented for fast depth map estimation. In addition, the solution effectively takes advantage of both 2D and 3D information to estimate the 6D pose, thereby solving the limitation of a fixed small baseline in the stereo camera. A robust volumetric occupancy mapping approach based on the Octomap framework has been utilized to reconstruct indoor and outdoor large-scale cluttered environments in 3D with less temporally or spatially correlated measurement errors and memory; (III) visual control, it offers practical control solutions to navigate UAV using Fuzzy Logic Controller (FLC) with the visual estimation. And the Cross-Entropy Optimization (CEO) framework has been used to optimize the scaling factor and the membership function in FLC. All the vision-based solutions in this dissertation have been tested in real tests. And the real image datasets recorded from these tests or available from public community have been utilized to evaluate the performance of these vision-based solutions with ground truth. Additionally, the presented vision solutions have compared with the state-of-art visual algorithms. Real tests and evaluation results show that the provided vision-based solutions have obtained real-time accurate robust performances, or gained better performance than those state-of-art visual algorithms. The vision-based estimation has played a critically important role for controlling a typical UAV to achieve autonomy in the UAV application.

Off-line signature verification and forgery detection using fuzzy modeling

Automatic signature verification is a well-established and an active area of research with numerous applications such as bank check verification, ATM access, etc. This paper proposes a novel approach to the problem of automatic off-line signature verification and forgery detection. The proposed approach is based on fuzzy modeling that employs the Takagi-Sugeno (TS) model. Signature verification and forgery detection are carried out using angle features extracted from box approach. Each feature corresponds to a fuzzy set. The features are fuzzified by an exponential membership function involved in the TS model, which is modified to include structural parameters. The structural parameters are devised to take account of possible variations due to handwriting styles and to reflect moods. The membership functions constitute weights in the TS model. The optimization of the output of the TS model with respect to the structural parameters yields the solution for the parameters. We have also derived two TS models by considering a rule for each input feature in the first formulation (Multiple rules) and by considering a single rule for all input features in the second formulation. In this work, we have found that TS model with multiple rules is better than TS model with single rule for detecting three types of forgeries; random, skilled and unskilled from a large database of sample signatures in addition to verifying genuine signatures. We have also devised three approaches, viz., an innovative approach and two intuitive approaches using the TS model with multiple rules for improved performance. (C) 2004 Pattern Recognition Society. Published by Elsevier Ltd. All rights reserved.

A fuzzy approach to texture segmentation

The texture segmentation techniques are diversified by the existence of several approaches. In this paper, we propose fuzzy features for the segmentation of texture image. For this purpose, a membership function is constructed to represent the effect of the neighboring pixels on the current pixel in a window. Using these membership function values, we find a feature by weighted average method for the current pixel. This is repeated for all pixels in the window treating each time one pixel as the current pixel. Using these fuzzy based features, we derive three descriptors such as maximum, entropy, and energy for each window. To segment the texture image, the modified mountain clustering that is unsupervised and fuzzy c-means clustering have been used. The performance of the proposed features is compared with that of fractal features.