FACE RECOGNITION USING ORTHO-DIFFUSION BASES Sravan Gudivada and Adrian G. Bors Dept. of Computer Science, University of York, York YO10 5GH, UK E-mail: adrian.bors@york.ac.uk ABSTRACT This paper proposes a new approach for face recognition by representing inter-face variation using orthogonal decomposi- tions with embedded diffusion. The modified Gram-Schmidt with pivoting the columns orthogonal decomposition, called also QR algorithm, is applied recursively to the covariance matrix of a set of images forming the training set. At each re- cursion a set of orthonormal bases functions are extracted for a specific scale. A diffusion step is embedded at each scale in the QR decomposition. The algorithm models the main vari- ations of face features from the training set by preserving only the most significant bases while eliminating noise and non- essential features. Each face is represented by a weighted sum of such representative bases functions, called ortho-diffusion faces. Index Terms— Diffusion wavelets, Face recognition, Gram-Schmidt orthogonal decomposition, Eigenfaces. 1. INTRODUCTION A challenge in complex data analysis is when we have to re- cover a low dimensional intrinsic manifold which manifests itself through a very complex representation in the observable space. The use of kernels on undirected graphs was shown to lead to good results in machine learning tasks. Various approaches have been adopted for modeling data represen- tations using diffusion processes on graphs [1, 2, 3]. Diffu- sion maps [2, 3, 4] achieve dimensionality reduction by re- organizing data according to the parametrization of its under- lying geometry on orthogonal sub-spaces. When the diffusion is propagated, it integrates the local data structure to reveal relational properties of the data set at different scales [3, 4]. Maggioni and Mahadevan proposed in [5] a new multi-scale orthogonal decomposition on graphs into sets of bases func- tions and diffusion wavelets. One of the classical face recognition methods was the eigenfaces method proposed by Turk and Pentland [6]. Each face is represented as a linear combination of the eigenvec- tors resulting from eigendecomposing the covariance matrix of a training set of faces. Extensions of this method include the Fisherfaces [7], the kernel PCA method [8] and the Lapla- cianfaces [9]. Orthogonal Laplacianfaces approach was pro- posed in [10] while a multilinear discriminant analysis was employed in [11] for face recognition. Nevertheless, human identification depends on incorporating multi-modal human features such as the 3-D appearance and speech characteris- tics as well [12]. In this paper we propose to use orthonormal diffusion wavelet methodology for face recognition. The modified Gram-Schmidt algorithm with pivoting the columns QR, is used recursively for extracting a set of orthogonal basis functions, each representing an ortho-diffusion face. At each recursion, we consider data diffusion on the graph leading to the application of the QR algorithm in the following step on a dilated scale of the data. Ortho-diffusion bases which do not represent significant information are removed from further processing at each level. Each human face is defined by its projections onto the ortho-diffusion face space. In the testing stage, faces are classified according to the nearest neigh- borhood to the training data defined in the ortho-diffusion face space. The proposed ortho-diffusion faces method is described in Section 2. The face recognition method using ortho-diffusion faces is detailed in Section 3. In Section 4 we provide the experimental results while the conclusions are given in Section 5. 2. ORTHONORMAL DECOMPOSITIONS USING QR ALGORITHM Let us consider a training set of M face images {I i |,i = 1,...,M }, of size m × n, and consider each of them as a vector of mn pixel entries. In the following we model the face variation within the given training set by calculating the deviation of each face from the mean face as in [6]: ¯ I = 1 M M  i=1 I i , (1) where I i represents the mean face S i = I i − ¯ I. (2) Each S i forms a column in a matrix A of size M × mn. The spread of the face variation within the training set is calcu- lated by means of the covariance matrix C: C = A τ A (3) where matrix C is the diagonal covariance matrix of the train- ing set, of size M × mn, with each column corresponding to 20th European Signal Processing Conference (EUSIPCO 2012) Bucharest, Romania, August 27 - 31, 2012 © EURASIP, 2012 - ISSN 2076-1465 1578