Michel Ménard

Possibilistic and probabilistic fuzzy clustering: unification within the framework of the non-extensive thermostatistics

Fuzzy clustering algorithms are becoming the major technique in cluster analysis. In this paper, we consider the fuzzy clustering based on objective functions. They can be divided into two categories: possibilistic and probabilistic approaches leading to two different function families depending on the conditions required to state that fuzzy clusters are a fuzzy c-partition of the input data. Recently, we have presented in Menard and Eboueya (Fuzzy Sets and Systems, 27, to be published) an axiomatic derivation of the Possibilistic and Maximum Entropy Inference (MEI) clustering approaches, based upon an unifying principle of physics, that of extreme physical information (EPI) defined by Frieden (Physics from Fisher information, A unification, Cambridge University Press, Cambridge, 1999). Here, using the same formalism, we explicitly give a new criterion in order to provide a theoretical justification of the objective functions, constraint terms, membership functions and weighting exponent m used in the probabilistic and possibilistic fuzzy clustering. Moreover, we propose an unified framework including the two procedures. This approach is inspired by the work of Frieden and Plastino and Plastino and Miller (Physics A 235, 577) extending the principle of extremal information in the framework of the non-extensive thermostatistics. Then, we show how, with the help of EPI, one can propose extensions of the FcM and Possibilistic algorithms.

Combination, cooperation and selection of classifiers: a state of the art

When several classifiers are brought to contribute to the same task of recognition, various strategies of decisions, implying these classifiers in different ways, are possible. A first strategy consists in deciding using different opinions: it corresponds to the combination of classifiers. A second strategy consists in using one or more opinions for better guiding other classifiers in their training stages, and/or to improve the decision-making of other classifiers in the classification stage: it corresponds to the cooperation of classifiers. The third and last strategy consists in giving more importance to one or more classifiers according to various criteria or situations: it corresponds to the selection of classifiers. The temporal aspect of Pattern Recognition (PR), i.e. the possible evolution of the classes to be recognized, can be treated by the strategy of selection.

The fuzzy c+2-means: solving the ambiguity rejection in clustering

Abstract In this paper we deal with the clustering problem whose goal consists of computing a partition of a family of patterns into disjoint classes. The method that we propose is formulated as a constrained minimization problem, whose solution depends on a fuzzy objective function in which reject options are introduced. Two types of rejection have been included: the ambiguity rejection which concerns patterns lying near the class boundaries and the distance rejection dealing with patterns that are far away from all the classes. To compute these rejections, we propose an extension of the fuzzy c-means (FcM) algorithm of Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, New York, 1981. This algorithm is called the fuzzy c+2-means (Fc+2M). These measures allow to manage uncertainty due both to imprecise and incomplete definition of the classes. The advantages of our method are (1) the degree of membership to the reject classes for a pattern xk are learned in the iterative clustering problem; (2) it is not necessary to compute other characteristics to determine the reject and ambiguity degrees; (3) the partial ambiguity rejections introduce a discounting process between the classical FcM membership functions in order to avoid the memberships to be spread across the classes; (4) the membership functions are more immune to noise and correspond more closely to the notion of compatibility. Preliminary computational experiences on the developed algorithm are encouraging and compared favorably with results from other methods as FcM, FPcM and F(c+1)M (fuzzy c+1-means: clustering with solely distance rejection) algorithms on the same data sets. The differences in the performance can be attributed to the fact that ambiguous patterns are less accounted in for the computing of the centers.

Fuzzy clustering and switching regression models using ambiguity and distance rejects

This paper examines how reject options can be used in performing fuzzy clustering and switching regression models. We define an objective function in which reject options are introduced to optimization of certain clustering models. This approach can be directly applied to any clustering model which can be represented as a functional dependent upon a set of cluster centers. The approach can be further generalized for models that require parameters other than the cluster centers. Two types of reject have been included: (1) the ambiguity reject which concerns patterns lying near the cluster boundaries or in the case of switching regression problems, the data points which fit several models equally well; (2) the distance or error reject dealing with patterns that are far away from all the clusters. Clustering and fuzzy c-regression algorithms such as FcM (fuzzy c-means) and FcRM (fuzzy c-regression models) which use calculus-based optimization methods suffer from several drawbacks. They are very sensitive to the presence of noise. Moreover, the memberships are relative numbers. The membership of a point in a cluster depends on the membership of the point in all other clusters. So, the cluster centers or estimates for the parameters are poor. This can be a serious problem in situations where one wishes to generate membership functions from training data. This paper provides answers to these problems: to avoid the memberships to be spread across the clusters and to allow the distinction between “equally likely” and “unknown”, we define partial ambiguity rejects which introduce a discounting process between the classical FcM or FcRM membership functions; to improve the performance of our algorithm in the presence of noise, we use an amorphous noise cluster defined in Demko et al. (Actes des sixiemes rencontres del la societe francophone de classification, Montpellier, France, September 1998). To compute these rejects, we propose an extension of FcRM algorithm (Hathaway and Bezdek, IEEE Trans. Fuzzy Systems 1 (3) (1993) 195–203). This algorithm is called the fuzzy (c+2)-regression model (Fc+2RM). Preliminary computational experiences on the developed algorithm are encouraging and compare favorably with results from other methods as FcRM and AFC algorithms on the same data sets.

Weighted and extended total variation for image restoration and decomposition

In various information processing tasks obtaining regularized versions of a noisy or corrupted image data is often a prerequisite for successful use of classical image analysis algorithms. Image restoration and decomposition methods need to be robust if they are to be useful in practice. In particular, this property has to be verified in engineering and scientific applications. By robustness, we mean that the performance of an algorithm should not be affected significantly by small deviations from the assumed model. In image processing, total variation (TV) is a powerful tool to increase robustness. In this paper, we define several concepts that are useful in robust restoration and robust decomposition. We propose two extended total variation models, weighted total variation (WTV) and extended total variation (ETV). We state generic approaches. The idea is to replace the TV penalty term with more general terms. The motivation is to increase the robustness of ROF (Rudin, Osher, Fatemi) model and to prevent the staircasing effect due to this method. Moreover, rewriting the non-convex sublinear regularizing terms as WTV, we provide a new approach to perform minimization via the well-known Chambolles algorithm. The implementation is then more straightforward than the half-quadratic algorithm. The behavior of image decomposition methods is also a challenging problem, which is closely related to anisotropic diffusion. ETV leads to an anisotropic decomposition close to edges improving the robustness. It allows to respect desired geometric properties during the restoration, and to control more precisely the regularization process. We also discuss why compression algorithms can be an objective method to evaluate the image decomposition quality.

Attention analysis in interactive software for children with autism

This work is a part of an ongoing project that focuses on potential applications of an interactive system that helps children with autism. Autism is classified as a neurodevelopmental disorder that manifests itself in markedly abnormal social interaction, communication ability, patterns of interests, and patterns of behavior [1]. Children with autism are socially impaired and usually do not attend to the people around them. An interesting point which characterized children with autism is that they are unable to choose which event is more or less important. As a consequence they are often saturated because of too many stimuli and thus they adopt an extremely repetitive, unusual, self-injurious, or aggressive behaviour. Recently, a new trend of using human computer interface (HCI) technology and computer science in the treatment of autism has emerged [2, 3]. The platform we developed helps children with autism to focus their attention on a specific task. In this article, we only present the attention analysis system which is a part of a more general system that used a multi-agent architecture [4]. Each task proposed on our system fit to each child, is reproducible and evolutive following a specific scenario defined by the expert. This scenario takes into account age, ability, and degree of autism of each child. In order to focus a childs attention onto the relevant object, our system displays or plays specific stimulus; once again the specific stimulus is defined for each child. Symbol or sound represents an emotional and satisfaction value for the child. The major problem is to define the correct moment when the system has to (dis)play this signal. We tackle this problem by defining a robust measure of attention. This measure is defined by analyzing the gaze direction and the face orientation, and incorporating the childs specific profile. Following expert directives, our system helps children to categorize elementary perception (strong, smooth, quick, slow, big, small...). Our objective is that children re-use these classifications in others situations.

A Comparison of Wavelet Based Spatio-temporal Decomposition Methods for Dynamic Texture Recognition

This paper presents four spatio-temporal wavelet decompositions for characterizing dynamic textures. The main goal of this work is to compare the influence of spatial and temporal variables in the wavelet decomposition scheme. Its novelty is to establish a comparison between the only existing method [11] and three other spatio-temporal decompositions. The four decomposition schemes are presented and successfully applied on a large dynamic texture database. Construction of feature descriptors are tackled as well their relevance, and performances of the methods are discussed. Finally, future prospects are exposed.

Characterization and recognition of dynamic textures based on the 2D+T curvelet transform

The research context of this article is the recognition and description of dynamic textures. In image processing, the wavelet transform has been successfully used for characterizing static textures. To our best knowledge, only two works are using spatio-temporal multiscale decomposition based on the tensor product for dynamic texture recognition. One contribution of this article is to analyze and compare the ability of the 2D+T curvelet transform, a geometric multiscale decomposition, for characterizing dynamic textures in image sequences. Two approaches using the 2D+T curvelet transform are presented and compared using three new large databases. A second contribution is the construction of these three publicly available benchmarks of increasing complexity. Existing benchmarks are either too small not available or not always constructed using a reference database. Feature vectors used for recognition are described as well as their relevance, and performances of the different methods are discussed. Finally, future prospects are exposed.

Decomposition of Dynamic Textures Using Morphological Component Analysis

The research context of this paper is dynamic texture analysis and characterization. Many dynamic textures can be modeled as large scale propagating wavefronts and local oscillating phenomena. After introducing a formal model for dynamic textures, the morphological component analysis (MCA) approach with a well-chosen dictionary is used to retrieve the components of dynamic textures. We define two new strategies for adaptive thresholding in the MCA framework, which greatly reduce the computation time when applied on videos. Tests on real image sequences illustrate the efficiency of the proposed method. An application to global motion estimation is proposed and future prospects are finally exposed.

Cooperation of fuzzy segmentation operators for correction aliasing phenomenon in 3D color Doppler imaging.

The study described in this paper concerns natural object modeling in the context of uncertain, imprecise and inconsistent representation. We propose a fuzzy system which offers a global modeling of object properties such as color, shape, velocity, etc. This modeling makes a transition from a low level reasoning (pixel level), which implies a local precise but uncertain representation, to a high level reasoning (region level), inducing a certain assignment. So, we use fuzzy structured partitions characterizing these properties. At this level. each property will have its own global modeling. Then, these different models are merged for decision making. Our approach was tested with several applications. In particular, we show here its performance in the area of blood flow analysis from 3D color Doppler images in order to quantify and study the development of this flow. We present methods that detect and correct aliasing phenomenon, i.e. inconsistent information. At first, the flow space is partitioned into fuzzy sectors where each sector is defined by a center, an angle and a direction. In parallel, the velocity information carried by the pixels is classified into fuzzy classes. Then, by combining these two partitions, we obtain the velocity distribution into sectors. Moreover, for each found path (from the first sector to the last one), we locate and correct inconsistent velocities by applying global rules. After extracting some meaningful sector features, the fuzzy modeling, applied to the aliasing correction, makes it possible to simplify and synthesize the blood flow direction.