Application of 2DPCA Based Techniques in DCT Domain for Face Recognition

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1 Application of 2DPCA Based Techniques in DCT Domain for Face Recognition essaoud Bengherabi, Lamia ezai, Farid Harizi, Abderraza Guessoum 2, and ohamed Cheriet 3 Centre de Développement des Technologies Avancées- Algeria Division Architecture des Systèmes et ultiédia Cité 20 Aout, BP, Baba Hassen, Algiers-Algeriabengherabi@yahoo.com, l_mezai@yahoo.fr, harizihourizi@yahoo.fr 2 Université Saad Dahlab de Blida Algeria Laboratoire Traitement de signal et d imagerie Route De Soumaa BP 270 Blida guessouma@hotmail.com 3 École des Technologies Supérieur Québec- Canada- Laboratoire d Imagerie, de Vision et d Intelligence Artificielle 00, Rue Notre-Dame Ouest, ontréal (Québec) H3C K3 Canada mohamed.cheriet@gpa.etsmtl.ca Abstract. In this paper, we introduce 2DPCA, DiaPCA and DiaPCA+2DPCA in DCT domain for the aim of face recognition. The 2D DCT transform has been used as a preprocessing step, then 2DPCA, DiaPCA and DiaPCA+2DPCA are applied on the upper left corner bloc of the global 2D DCT transform matrix of the original images. The ORL face database is used to compare the proposed approach with the conventional ones without DCT under Four matrix similarity measures: Frobenuis, Yang, Assembled atrix Distance (AD) and Volume easure (V). The experiments show that in addition to the significant gain in both the training and testing times, the recognition rate using 2DPCA, DiaPCA and DiaPCA+2DPCA in DCT domain is generally better or at least competitive with the recognition rates obtained by applying these three 2D appearance based statistical techniques directly on the raw pixel images; especially under the V similarity measure. Keywords: Two-Dimensional PCA (2DPCA), Diagonal PCA (DiaPCA), DiaPCA+2DPCA, face recognition, 2D Discrete Cosine Transform (2D DCT). Introduction Different appearance based statistical methods for face recognition have been proposed in literature. But the most popular ones are Principal Component Analysis (PCA) [] and Linear Discriminate Analysis (LDA) [2], which process images as 2D holistic patterns. However, a limitation of PCA and LDA is that both involve eigendecomposition, which is extremely time-consuming for high dimensional data. Recently, a new technique called two-dimensional principal component analysis 2DPCA was proposed by J. Yang et al. [3] for face recognition. Its idea is to estimate the covariance matrix based on the 2D original training image matrices, resulting in a E. Corchado et al. (Eds.): CISIS 2008, ASC 53, pp , springerlin.com Springer-Verlag Berlin Heidelberg 2009

2 244. Bengherabi et al. covariance matrix whose size is equal to the width of images, which is quite small compared with the one used in PCA. However, the projection vectors of 2DPCA reflect only the variations between the rows of images, while discarding the variations of columns. A method called Diagonal Principal Component Analysis (DiaPCA) is proposed by D. Zhang et al. [4] to resolve this problem. DiaPCA sees the projection vectors from diagonal face images [4] obtained from the original ones to ensure that the correlation between rows and those of columns is taen into account. An efficient 2D techniques that results from the combination of DiaPCA and 2DPCA (DiaPCA+2DPCA) is proposed also in [4]. Discrete cosine transform (DCT) has been used as a feature extraction step in various studies on face recognition. This results in a significant reduction of computational complexity and better recognition rates [5, 6]. DCT provides excellent energy compaction and a number of fast algorithms exist for calculating it. In this paper, we introduce 2DPCA, DiaPCA and DiaPCA+2DPCA in DCT domain for face recognition. The DCT transform has been used as a feature extraction step, then 2DPCA, DiaPCA and DiaPCA+2DPCA are applied only on the upper left corner bloc of the global DCT transform matrix of the original images. Our proposed approach is tested against conventional approaches without DCT under Four matrix similarity measures: Frobenuis, Yang, Assembled atrix Distance (AD) and Volume easure (V). The rest of this paper is organized as follows. In Section 2 we give a review of 2DPCA, DiaPCA and DiaPCA+2DPCA approaches and also we review different matrix similarity measures. In section 3, we present our contribution. In section 4 we report the experimental results and highlight a possible perspective of this wor. Finally, in section 5 we conclude this paper. 2 Overview of 2DPCA, DiaPCA, DiaPCA+2DPCA and atrix Similarity easures 2. Overview of 2D PCA, DiaPCA and DiaPCA+2DPCA 2.. Two-Dimensional PCA Given training face images, denoted by m n matrices A ( =, 2 ), twodimensional PCA (2DPCA) first uses all the training images to construct the image covariance matrix G given by [3] G = () T ( A A) ( A A) = Where A is the mean image of all training images. Then, the projection axes of 2DPCA, X opt =[x x d ] can be obtained by solving the algebraic eigenvalue problem Gx i =λ i x i, where x i is the eigenvector corresponding to the ith largest eigenvalue of G [3]. The low dimensional feature matrix C of a test image matrix A is extracted by C = AX opt (2) In Eq.(2) the dimension of 2DPCA projector X opt is n d, and the dimension of 2DPCA feature matrix C is m d.

3 Application of 2DPCA Based Techniques in DCT Domain Diagonal Principal Component Analysis Suppose that there are training face images, denoted by m n matrices A ( =, 2,, ). For each training face image A, we calculate the corresponding diagonal face image B as it is defined in [4]. Based on these diagonal faces, diagonal covariance matrix is defined as [4]: GDIAG = (3) = Where B = B is the mean diagonal face. According to Eq. (3), the projection = vectors X opt =[x,, x d ] can be obtained by computing the d eigenvectors corresponding to the d biggest eigenvalues of G DIAG. The training faces A s are projected onto X opt, yielding m d feature matrices. T ( B B) ( B B) C = A X (4) Given a test face image A, first use Eq. (4) to get the feature matrixc = matrix similarity metric can be used for classification. opt AX opt, then a 2..3 DiaPCA+2DPCA Suppose the n by d matrix X=[x,, x d ] is the projection matrix of DiaPCA. Let Y=[y,, y d ] the projection matrix of 2DPCA is computed as follows: When the height m is equal to the width n, Y is obtained by computing the q eigenvectors corresponding to the q biggest eigenvalues of the image covarinace matrix T A A A A. On the other hand, when the height m is not equal to the width ( ) ( ) = A A A A. = Projecting training faces A s onto X and Y together, yielding the q d feature matrices n, Y is obtained by computing the q eigenvectors corresponding to the q biggest ei- genvalues of the alternative image covariance matrix ( )( ) T D = T Y A X (5) T Given a test face image A, first use Eq. (5) to get the feature matrix D = Y AX, then a matrix similarity metric can be used for classification. 2.2 Overview of atrix Similarity easures An important aspect of 2D appearance based face recognition approaches is the similarity measure between matrix features used at the decision level. In our wor, we have used four matrix similarity measures Frobenius Distance Given two feature matrices A = (a ij ) m d and B = (b ij ) m d, the Frobenius distance [7] measure is given by: 2 2 m d d (, ) ( ) F A B = aij b (6) ij i= j=

4 246. Bengherabi et al Yang Distance easure Given two feature matrices A = (a ij ) m d and B = (b ij ) m d, the Yang distance [7] is given by: d m ( A B) = ( aij bij ) 2, j= i= d (7) Y Assembled atrix Distance (AD) A new distance called assembled matrix distance (AD) metric to calculate the distance between two feature matrices is proposed recently by Zuo et al [7]. Given two feature matrices A = (a ij ) m d and B = (b ij ) m d, the assembled matrix distance d AD (A,B) is defined as follows : d AD d m (, ) A B = ( aij bij ) j= i= ( 2) p ( p > 0) It was experimentally verified in [7] that best recognition rate can be obtained when p 0.25 while it decrease as p increases. In our wor the parameter p is set equal to Volume easure (V) The V similarity measure is based on the theory of high-dimensional geometry space. The volume of an m n matrix of ran p is given by [8] (8) Vol A 2 = det A (9) IJ ( I,J ) N where A IJ denotes the submatrix of A with rows I and columns J, N is the index set of p p nonsingular submatrix of A, and if p=0, then Vol A = 0 by definition. 3 The Proposed Approach In this section, we introduce 2DPCA, DiaPCA and DiaPCA+2DPCA in DCT domain for the aim of face recognition. The DCT is a popular technique in imaging and video compression, which was first applied in image compression in 974 by Ahmed et al [9]. Applying the DCT to an input sequence decomposes it into a weighted sum of basis cosine sequences. our methodology is based on the use of the 2D DCT as a feature extraction or preprocessing step, then 2DPCA, DiaPCA and DiaPCA+2DPCA are applied to w w upper left bloc of the global 2D DCT transform matrix of the original images. In this approach, we eep only a sub-bloc containing the first coefficients of the 2D DCT matrix as shown in Fig., from the fact that, the most significant information is contained in these coefficients. 2D DCT c c 2 c w. ċ w c w2 c ww Fig.. Feature extraction in our approach

5 Application of 2DPCA Based Techniques in DCT Domain 247 With this approach and inversely to what is presented in literature of DCT-based face recognition approaches, the 2D structure is ept and the dimensionality reduction is carried out. Then, the 2DPCA, DiaPCA and DiaPCA+2DPCA are applied to w w bloc of 2D DCT coefficients. The training and testing bloc diagrams describing the proposed approach is illustrated in Fig.2. 2D DCT Bloc w*w of 2D DCT coefficients Training algorithm based on 2DPCA DiaPCA DiaPCA+2DPCA Training data 2D DCT image 2D DCT Bloc Features Trained odel Test image 2D DCT 2D DCT image Bloc w*w of 2D DCT coefficients 2D DCT Bloc Features Projection of the DCT bloc of the test image using the eigenvectors of 2DPCA DiaPCA DiaPCA+2DPCA Comparison using Frobenius Ya ng AD V Decision Fig. 2. Bloc diagram of 2DPCA, DiaPCA and DiaPCA+2DPCA in DCT domain 4 Experimental Results and Discussion In this part, we evaluate the performance of 2DPCA, DiaPCA and DiaPCA+2DPCA in DCT domain and we compare it to the original 2DPCA, DiaPCA and DiaPCA+2DPCA methods. All the experiments are carried out on a PENTUI 4 PC with 3.2GHz CPU and Gbyte memory. atlab [0] is used to carry out these experiments. The database used in this research is the ORL [] (Olivetti Research Laboratory) face database. This database contains 400 images for 40 individuals, for each person we have 0 different images of size 2 92 pixels. For some subjects, the images captured at different times. The facial expressions and facial appearance also vary. Ten images of one person from the ORL database are shown in Fig.3. In our experiment, we have used the first five image samples per class for training and the remaining images for test. So, the total number of training samples and test samples were both 200. Herein and without DCT the size of diagonal covariance matrix is 92 92, and each feature matrix with a size of 2 p where p varies from to 92. However with DCT preprocessing the dimension of these matrices depends on the w w DCT bloc where w varies from 8 to 64. We have calculated the recognition rate of 2DPCA, DiaPCA, DiaPCA+2DPCA with and without DCT. In this experiment, we have investigated the effect of the matrix metric on the performance of the 2D face recognition approaches presented in section 2. We see from table, that the V provides the best results whereas the Frobenius gives the worst ones, this is justified by the fact that the Frobenius metric is just the sum of the

6 248. Bengherabi et al. (a) Fig. 3. Ten images of one subject in the ORL face database, (a) Training, (b) Testing (b) Euclidean distance between two feature vectors in a feature matrix. So, this measure is not compatible with the high-dimensional geometry theory [8]. Table. Best recognition rates of 2DPCA, DiaPCA and DiaPCA+2DPCA without DCT ethods Frobenius Yang AD p=0,25 Volume Distance 2DPCA 9.50 (2 8) (2 7) (2 4) (2 3) DiaPCA 9.50 (2 8) (2 0) 9.50 (2 8) (2 9) DiaPCA+2DPCA (6 0) (3 ) (2 6) (2 8) Tables 2, and Table 3 summarize the best performances under different 2D DCT bloc sizes and different matrix similarity measures. Table 2. 2DPCA, DiaPCA and DiaPCA+2DPCA under different DCT bloc sizes using the Frobenius and Yang matrix distance 2D DCT Best Recognition rate (feature matrix dimension) bloc 2DPCA DiaPCA DiaPCA+2DPCA 2DPCA DiaPCA DiaPCA+2DPCA size Frobenius Yang (8 8) 9.50 (8 6) 9.50 (6 6) (8 6) (8 5) (8 5) (9 9) (9 5) (9 5) (9 6) (9 9) (9 9) (0 5) (0 5) (0 5) (0 6) (0 9) (0 9) ( 8) 9.50 ( 5) (9 5) ( 6) ( 5) ( 5) (2 8) 9.50 (2 0) 9.50 (9 5) (2 6) (2 5) (2 5) (3 7) (3 ) (2 ) (3 6) (3 5) ( 5) (4 7) 9.50 (4 7) (2 7) (4 6) (4 5) (2 5) (5 5) 9.50 (5 5) (3 5) (5 9) (5 5) (2 5) (6 0) 9.50 (6 ) (4 0) (6 7) (6 5) (2 5) (32 6) 9.50 (32 6) ( 7) (32 6) (32 5) (2 5) (64 6) 9.00 (32 6) (4 2) (64 7) (64 5) (2 5) From these four tables, we notice that in addition to the importance of matrix similarity measures, by the use of DCT we have always better performance in terms of recognition rate and this is valid for all matrix measures, we have only to choose the DCT bloc size and appropriate feature matrix dimension. An important remar is that a bloc size of 6 6 or less is sufficient to have the optimal performance. So, this results in a significant reduction in training and testing time. This significant gain

7 Application of 2DPCA Based Techniques in DCT Domain 249 Table 3. 2DPCA, DiaPCA and DiaPCA+2DPCA under different DCT bloc sizes using the AD distance and V similarity measure on the ORL database 2D DCT Best Recognition rate (feature matrix dimension) bloc size 2DPCA DiaPCA DiaPCA+2DPCA 2DPCA DiaPCA DiaPCA+2DPCA AD V (8 4) (8 6) (7 5) (8 3) (8 4) (8 4) (9 4) (9 5) (9 5) (9 4) (9 5) (9 5) (0 4) (0 5) (9 7) (0 3) (0 4) (0 4) ( 5) ( 5) (9 6) ( 3) ( 3) ( 3) (2 5) (2 7) (9 7) (2 5) (2 5) ( 5) (3 4) (3 5) (2 5) (3 9) (3 5) (0 5) (4 4) (4 5) (0 5) (4 3) (4 5) (0 5) (5 4) (5 5) (9 7) (5 8) (5 5) (0 5) (6 4) (6 5) (2 5) (6 8) (6 5) (0 5) (32 4) (32 9) ( 5) (32 3) (32 5) (9 5) (64 4) (64 9) (2 5) (64 3) (64 5) (2 5) in computation is better illustrated in table 4 and table 5, which illustrate the total training and total testing time of 200 persons -in seconds - of the ORL database under 2DPCA, DiaPCA and DiaPCA+2DPCA without and with DCT, respectively. We should mention that the computation of DCT was not taen into consideration when computing the training and testing time of DCT based approaches. Table 4. Training and testing time without DCT using Frobenius matrix distance ethods 2DPCA DiaPCA DiaPCA+2DPCA Training time in sec (2 8) (2 8) 0.99 (6 0) Testing time in sec.294 (2 8) (2 8) 0.78 (6 0) Table 5. Training and testing time with DCT using the Frobenius distance and the same matrixfeature dimensions as in Table2 2D DCT Training time in sec Testing time in sec bloc size 2DPCA DiaPCA DiaPCA+2DPCA 2DPCA DiaPCA DiaPCA+2DPCA We can conclude from this experiment, that the proposed approach is very efficient in wealy constrained environments, which is the case of the ORL database. 5 Conclusion In this paper, 2DPCA, DiaPCA and DiaPCA+2PCA are introduced in DCT domain. The main advantage of the DCT transform is that it discards redundant information and it can be used as a feature extraction step. So, computational complexity is significantly reduced. The experimental results show that in addition to the significant gain in both the training and testing times, the recognition rate using 2DPCA, DiaPCA and DiaPCA+2DPCA in DCT domain is generally better or at least competitive with the

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