In this paper, two supervised methods for enhancing the classification accuracy of the Nonnegative Matrix Factorization (NMF) algorithm are presented. The idea is to extend the NMF algorithm in order to extract features that enforce not only the spatial locality, but also the separability between classes in a discriminant manner. The first method employs discriminant analysis in the features derived from NMF. In this way, a two phase discriminant feature extraction procedure is implemented, namely NMF plus Linear Discriminant Analysis (LDA). The second method incorporates the discriminant constraints inside the NMF decomposition. Thus, a decomposition of a face to its discriminant parts is obtained and new update rules for both the weights and the basis images are derived. The introduced methods have been applied to the problem of frontal face verification using the well known XM2VTS database. Both methods greatly enhance the performance of NMF for frontal face verification. Index TermsSubspace techniques, non-negative matrix factorization, linear discriminant analysis, frontal face verification.

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In this paper two image representation approaches called non-negative matrix factorization (NMF) and local non-negative matrix factorization (LNMF) have been applied to two facial databases for recognizing six basic facial expressions. A principal component analysis (PCA) approach was performed as well for facial expression recognition for comparison purposes. We found that, for the first database, LNMF outperforms both PCA and NMF, while NMF produces the poorest recognition performance. Results are approximately the same for the second database, with slightly performance improvement on behalf of NMF.

Two hybrid systems for classifying seven categories of human facial expression are proposed. The first system combines independent component analysis (ICA) and support vector machines (SVMs). The original face image database is decomposed into linear combinations of several basis images, where the corresponding coefficients of these combinations are fed up into SVMs instead of an original feature vector comprised of grayscale image pixel values. The classification accuracy of this system is compared against that of baseline techniques that combine ICA with either two-class cosine similarity classifiers or twoclass maximum correlation classifiers, when we classify facial expressions into these seven classes. We found that, ICA decomposition combined with SVMs outperforms the aforementioned baseline classifiers. The second system proposed operates in two steps: first, a set of Gabor wavelets (GWs) is applied to the original face image database and, second, the new features obtained are classified by using either SVMs or cosine similarity classifiers or maximum correlation classifier. The best facial expression recognition rate is achieved when Gabor wavelets are combined with SVMs. 1NTRODUCTlONFrom the perspective of psychology and anthropology, the evolution of human brain is reflected by the human language that represents a major milestone in the human evolution. The language alone does not seem to be sufficient for successful social (human-to-human) interaction. Therefore. the evolution o f a nonverbal signaling system, such as the facial expression mechanism has captured an increased attention in psychology and anthropology for a better understanding of the social context [I]. Unlike the human-to-human interaction that takes into consideration the facial expression, human computer interaction systems that use facial expression analysis have been introduced only recently 121, [3], [4] and [51. Reliable facial expression modeling and, particularly, human emotions recognition, are challenging tasks since there is no pure emotion. A particular emotion rather is a complex combination of several facial expressions. Moreover, the emotions can vary in intensity, which makes emotion recognition even more difficult. Basically, there are two types of approaches to cope with facial expression recognition: appearance-based methods and geometric feature-based methods. For the first method the fiducial points ofthe face are selected either manually [b] or automaticallyThis work was supported by the European Union Research Training Network "Multi-modal Human-Computer Interartion (HPRN-CT-2000-001 I I).[7]. The face images are convolved with Gabor filters and the responses extracted from the face images at fiducial points form vectors that are futher used for classification. Alternatively, the Gabor filters can he applied to the entire face image instead to specific face regions. Regarding the geometric feature-based methods, the positions of a set of fiducial points in a face form a feature vector that represents the face geom...

Plenty of methods have been proposed in order to discover latent variables (features) in data sets. Such approaches include the principal component analysis (PCA), independent component analysis (ICA), factor analysis (FA), etc., to mention only a few. A recently investigated approach to decompose a data set with a given dimensionality into a lower dimensional space is the so-called nonnegative matrix factorization (NMF). Its only requirement is that both decomposition factors are nonnegative. To approximate the original data, the minimization of the NMF objective function is performed in the Euclidean space, where the difference between the original data and the factors can be minimized by employing L(2)-norm. In this paper, we propose a generalization of the NMF algorithm by translating the objective function into a Hilbert space (also called feature space) under nonnegativity constraints. With the help of kernel functions, we developed an approach that allows high-order dependencies between the basis images while keeping the nonnegativity constraints on both basis images and coefficients. Two practical applications, namely, facial expression and face recognition, show the potential of the proposed approach.