A New Method for Face Recognition with Fewer Features under Illumination and Expression Variations

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1 A New Method for Face Recognition with Fewer Features under Illumination and Expression Variations Chandan Tripathi Dept. of Computer Science Engg. Sharda University Greater Noida, India t chandan@yahoo.com Dr. K.P.Singh Dept. of Information Technology IIIT Allahabad Allahabad, India Abstract In this study, a new adaptive feature extraction method has been presented based on multi-dimensional discriminant analysis MLDA) over multi-dimensional principal components. Proposed work has been aimed to design a method that can predict required number of features for a particular dataset. This method use only effective features which have better discriminant power in different dimensions of an image. In order to ease the pre-processing we controlled the variance in each mode to make the feature selection adaptive in different datasets with facial variance present in the image. The Experiments with different datasets has been performed in order to check suitability for larger dataset, with lesser computational cost and higher efficiency. Moreover, when support vector machine operated as classifier, proposed algorithm shows its superiority of recognition over previous known methods like PCA,PCA-LDA,MPCA. Keywords-Principal Component AnalysisPCA);Linear Discriminant AnalysisLDA); Multilinear Principle Component AnalysisMPCA); K-Nearest Neighborhood ClassifierKNN); Support Vector Machine SVM); I. INTRODUCTION Face Recognition is one the field that extensively attracted the concentration of the researchers as in case of its methodology or its application in computer based vision and pattern recognition [1]. Several algorithms have been introduced and renovated to achieve higher recognition rate based on their own points of strength and restrictions. Still there are several challenges remains to improve the correctness of their systems under different illuminations, poses and image resolutions. Many of the researchers designed their algorithms that focuses over such issues but all of them required to adjust their features dimensions manually or based on some thresholds. In three decade research history of appearance based first face recognition algorithm was introduced by Turk and Pentlands [2] with use of Eigenfaces based principal component analysis PCA). Principal idea behind PCA was reduction of those dimensions which are non-repetitive in the image dataset i.e., with lower Eigen values. Comon Pierre [3] introduced a kernel Hilbert space based ICA which was further used to work with non-gaussian nature of the faces [4]. K. Kim, K. Jung and H. K. Kim introduced a kernel based PCA[5] to achieve optimal recognition speed. All these methods require reshaping of the series of k number images with dimension m n into matrix having higher dimensional size of mn k. It causes high computational and processing cost in terms of both memory and time. Yeng et al.[6] proposed a 2D-PCA approach which successfully reduces the computation complexity with an increase in recognition rate[7][8]. 2D-PCA approach was further modified by Nguyen et al. [9] with the help of random subspace technique in feature selection. This was the first time when some Eigenvectors with non-zero Eigenvalues were also considered with Eigenvectors with highest Eigenvalues. It enhanced the recognition rate when compared with traditional 2D-PCA. A Gabor Features based 2D) 2PCA was proposed by X. Pan and Q. Ruan [10]. W. Yu et al. [11] proposed an approach using LDA for feature reduction over both Upright vertical) and Level horizontal) directional information subspace of 2D-PCA. Three dimensional PCA was explored by Yu and Bennamoun [12] as nd-pca algorithm which used to test its efficiency on FRGC-3D face dataset as well as gray images. H. Lu et al. [13, 14] modified the MPCA framework for Tensor based object feature extraction with an application for gait recognition. D. Xu, S. Yan [15] proposed a solution of MPCA based algorithm but was only concerned with reconstruction not the recognition where the data is not centred. They were also not concerning with initialization issues, termination problems, convergence, and subspace dimensionality determination problems. After several experimentation it has been found that 2D-PCA and MPCA based methods designed to achieve computational efficiency but with an increase in feature dimension in order to retain recognition accuracy, when compared with traditional PCA. In process of classification these higher feature dimension of those methods were leading towards higher computational time. Keeping above views in this study, a new adaptive method has been proposed with optimal reduction in feature dimension without compromising with both, recognition accuracy as well as computational time. In order to justify the quality of proposed method we compared the results with different feature selection processes different PCA and LDA /12/$ IEEE

2 based methods and with their combinations) and classification techniques classifiers based on nearest neighbourhood and support vectors) examined in our experimentations. II. PRINCIPAL COMPONENT ANALYSIS PCA was the first algorithm used as feature selection based on Eigenfaces [2]. Eigenfaces is projection of input images into that subspace where principal components appear orthogonal to each other. PCA was designed under unsupervised learning model to perform its feature selection. Suppose the training images have P number of images F with size M N. Algorithm grounded on PCA is explained below: 1) Reshaping of F i into a column matrix with size of MN 1 in order to form a 2D matrix of size MN P. As for an i th image. F i M N to F i MN 1 1) 2) Obtain the mean image of reshaped image matrix as F MEAN = 1 P P FMN 1 i 2) i=1 3) Center the images by making difference between reshaped input image FMN i i and F MEAN as F i Diff = F i MN 1 F MEAN 3) 4) Construct the co-variance matrix C = 1 P P FDiff i )FDiff i ) T 4) i=1 5) Find out the Eigen Decomposition on covariance matrix C. Let X a matrix that contains the selected eigenvectors based on the largest n Eigenvalues, where the X can be imagined as X = [x 1,...,x n ] 6) Project the testing sample into the acquired subspace to find the features which will forwarded for classification as Y i = X T F i Diff ) T est 5) 7) Test the classifier with features obtained in Y i III. MULTILINEAR PRINCIPAL COMPONENT ANALYSIS This method works on tensor representation [13] of the faces in order to preserve the relationship among neighboring pixels as absent in PCA. A tensor object is a higher dimensional matrix of order N stand as T NS R T 1 T 2... T N ), where each L N represents each mode of that tensor. For M number of images where image is symbolized as F i, can be projected into a third dimensional tensor as T NS R T1 T2 T3). In TNS T 1 stands as row size height) of image as mode-1 of tensor TNS, T 2 is column size width) of image as mode-2 of tensor TNS and T 3 is number of images P used in training as mode-3 of tensor TNS. Multilinear PCA algorithm is explained below: 1) Obtain the mean Image value of all training images F M ean as F Mean = 1 T 3 FM N t where T 3 = P 6) T 3 t=1 2) Center the tensor object representation of training images {}}{ T NS = [F 1 M N F Mean, F 2 M N F Mean,..., F P M N F Mean ] 7) 3) Unfold the tensor for different n-modes into matrix for different modes. Unfolding of tensor TNS along the n-mode can be defined as T N Sn) R Tn T1... T N )) i.e.; for { T NS }}{ n ) indexval = f t1t 2t 3), where f is value present at t 1, t 2, t 3 ) th index value where index value selection can be done as n+1 n+1 ) 1 ) index val = [t n, t m 1) T q T q m=3 + 1 m=n 1 q=m t m 1) q=n 1 1 q=m 1 T q )] 8) 4) Evaluate the covariance matrix mentioned as C n and C n = T {}} NS { T ranspose {}}{ T NS of each n-mode and find the n) n) set of eigenvectors X n = [x 1,..., x kn) ]founded on the selected eigenvectors according to the largest k n) Eigenvalues. One can obtain optimal features set using any of the subsequent feature selection process: i) Using Y t = Fm n) t Input F Mean ) n different value for n for different modes), which will give k 1) T 2 + k 2) T 1 features. X T ranspose n) ii) Using Y t = X T ranspose 1) AB M N) t ) Input AB Mean )X 2) different value for n for different modes), which will givek 1) k 2) features. 5) Test the classifier with obtained feature set Y t IV. LINEAR DISCRIMINANT ANALYSIS LDA [16] [17] [18] aims to search the projection P a, a set of basis vectors which are normal vectors to discriminant hyper-plane of feature vector y m that can be obtained using any of the feature selection method. LDA intend to use Gaussian information present in image data using class information. For an m th training sample image having class C m LDA intended to maximize the ratio of the S between class between-class scattering matrix) and S within class withinclass scattering matrix), where S within class = M y m y cm )y m y cm ) T 9) m=1

3 where and S between class = where y c = 1 N c C Class=1 y = 1 M m,c m =C y m N Class y Class y)y Class y) T m y m 10) The purpose of LDA is find maximized set of basis vector P as P LDA = argmaxp P T S between class P ) P T S within class P ) 11) as above equation results P LDA = [p 1 p 2...p HZ ] Here p hz, where h Z = 1, 2,...H Z, is the set of generalized eigenvectors of S between class and Swithin class for H Z C 1 largest generalized Eigenvalues. The discriminant feature vector for m th training image can be obtained as yd m = P T LDA y m. V. PROPOSED ALGORITHM Based on above observations we are introducing our proposed algorithm based on n-mod Tensor unfolding features as input to Discriminant Analysis to find optimal features in each mod. The algorithm has been presented as below. 1) Obtain the mean Image value of all training images F M ean as F Mean = 1 T 3 FM N t where T 3 = P 12) T 3 t=1 2) Center the tensor object representation of training images {}}{ T NS = [F 1 M N F Mean, F 2 M N F Mean,..., F P M N F Mean ] 13) 3) Unfold the tensor for different n-modes into matrix for different modes. Unfolding of tensor TNS along the n-mode can be defined as T N Sn) R T n T 1... T N )) i.e.; for T { NS }}{ n ) indexval = f t1 t 2 t 3 ), where f is value present at t 1, t 2, t 3 ) th index value where index value selection can be done as n+1 n+1 ) 1 ) index val = [t n, t m 1) T q T q m=3 + 1 m=n 1 q=m t m 1) q=n 1 1 q=m 1 T q )] 4) Evaluate the covariance matrix mentioned as C n and C n = { T }} NS { T ranspose {}}{ T NS of each n-mode and find the set n) n) of eigenvectors X n = [x 1,..., x kn) ] according to their k n) Eigenvalues. 5) Search LDA subspace projection of each mod-n: i) A new projection P an) set of basis vectors for each mod-n which are normal vectors to discriminant hyper-plane of feature vector x kn) of m th training sample image in mod-n having class c m. ii) ii) Calculate within-class scattering matrix S within class using S within class = P class=1 x kn)class x kn)class ) x kn)class x kn)class ) T 14) here x kn) is within-class average image, where x kn) = 1 N C n,c n=c x kn) iii) Calculate between-class scattering matrix S between class using S between class = C class=1 N Class x kn)class x kn) ) x kn)class x kn) ) T 15) Here x kn) is between-class average image, where x n) = 1 P n x kn) iv) Calculate Eigenvaluesλ ldan) and Eigenvectors V ldan) corresponding each n-mod within-class scattering matrix S within class and between-class scattering matrix S between class S between class λ ldan) = λ ldan) S within class V ldan) 16) v) Find set of eigenvectors X ldan) = [x lda1,, x ldakn) ] from V ldan),as return by step iv), for each mod n according to their discrimination power over generalized k n) Eigenvalues obtained in step 5. 6) One can obtain optimal features set using Y t = X T ranspose lda 1) F t M N ) Input F Mean )X lda2) 17) Experimental outcomes of the proposed method have been compared with different previous known methods as Eigen face based PCA [2], fisher faces based combined approach of PCA-LDA [16], Eigen tensor based MPCA. In order to

4 TABLE I: Results along with their respective dimensions based on E-1 on AT&T Dataset based on SVM classifiers Sample Ratio training/testing) PCA PCA-LDA MPCA MLDA 1/ %40) 73.05%39) 75.28%69) 79.16% 39) 2/ %86) 86.87%39) 87.50%65) 89.38%39) 3/ %64) 91.43%39) 92.50%55) 92.34%39) 4/ %78) 94.17%39) 93.75%59) 95.42%39) 5/ %43) 97.00%39) 97.00%50) 98.50%39) 6/ %69) 96.25%39) 98.13%60) 99.32%39) 7/ %76) 98.33%39) 97.50%79) 98.97%39) 8/ %63) 97.50%39) 98.75%71) 100%39) 9/ %21) 97.50%39) 100%55) 100%39) TABLE II: Results along with their respective dimensions based on E-2 on AT&T Dataset based on SVM classifiers Sample Ratio training/testing) PCA PCA-LDA MPCA MLDA 1/ %40) 58.89%39) 61.94%69) 56.67% 39) 2/ %86) 71.88%39) 83.13%65) 77.50%39) 3/ %64) 78.21%39) 83.21%55) 88.57%39) 4/ %78) 79.58%39) 85.42%45) 91.67%39) 5/ %43) 79.50%39) 85.00%50) 93.50%39) 6/ %69) 80.63%39) 88.75%60) 96.88%39) 7/ %76) 83.33%39) 89.17%79) 95.00%39) 8/ %63) 81.25%39) 87.50%71) 96.25%39) 9/ %21) 85.00%39) 90.00%55) 95.00%39) Fig. 1: Example images of cropped AT&T dataset with size as used in E-1 Fig. 2: Example images of original AT&T data-set with as used in E-2 understand the roll of classifiers on the features based on the proposed method we tested it with the neighborhood classification approach and redial basis kernel function based support vector machine classifier. VI. EXPERIMENTAL ANALYSIS Proposed method has been tested over windows-7 platform of 32-bit based machine with Intel core-2 duo 2.5 GHz and 2GB RAM. The code has been simulated using Matlab Two data-sets AT&T and YALE has been involved to test the performance of proposed algorithm. The data-sets were partitioned into two random sets as one for training and other for testing. In order to ease of representation the data-set sample ratio has been represented as training/ testing. Pre processing methods have been used for face finding based on coordinate extraction with eye localization for alignment and normalization of the same. Proposed algorithm has been tested under two different experimental conditions of the original data-sets. The first experiment E-1) was tested over cropped and normalized facial area in both the data-sets. The motive behind E-1 was to test the strength of features selected through proposed algorithm. The second experiment E-2) tested over original data-set without any preprocessing. The idea behind E-2 was to test the selection strength of the proposed algorithm in presence of hair and white glass spectacles as occlusion followed by some shadow part in background. A. Experiment on AT&T data-set AT&T dataset is consists of 400 images of size 92x112 pixels of 40 individuals with 10 faces for each individual. Example images for E-1 with size 44x44 and for E-2 with 92x112 have been given in Figure-1 and Figure-2 respectively. The experiments have been performed by changing training and testing ratio from 1/9 to 9/1. The proposed algorithm has been comes up with percentage score 79.32% in E-1 and 56.67% in E-2 with ratio 1/9 when combined with SVM. Based on several trials it has been found that E-2 gives good result then the algorithm operated with 80% of variance in each mode MPCA feature dimension. This variance control was maintained in E-2 was just to give a guess of facial area to the algorithm. In E-1 there was no need of any variance

5 a) b) c) Fig. 3: Recognition rate of all four methods over ORL dataset: A) with two images taken for training B) with three images taken for training C) with four images taken for training D) with five images taken for training of each training sample d) control as we have taken the facial part only. Further, a change in sample ratio shows that recognition accuracy of proposed method gradually increases and found its superiority when it comes with its optimal feature size. Initially when fewer features have been used with small training size proposed method shows lower recognition accuracy. The reason behind this lies in way of feature selection of these methods. As PCA based methods seeks to find the features based on higher Eigen-values, whereas LDA based methods search those features which may not have higher Eigen-values but the higher discrimination power. The results based on Experiment- 1 and E-2 has been given in table-i and in table-ii. One comparison has been done to in order to understand the effect of feature dimension over recognition rate with different sample ratio in experiment-1 as shown in Figure-III. It shows that after reaching an optimal features dimension and a gradual increment upto the number of required features supposed by the system the proposed method shows higher recognition rate against rest of the methods. When the sample ratio of 5/5 has been operated then the proposed method shows its superiority at every feature dimension. A comparative study has been done over different combinations of classifiers and previous known algorithms along with the proposed algorithm. This comparison is to understand the effect of classifiers recognition rate as well as total processing time. Here total processing time contains sum of following time informations: i) Time in feature generation form raw facial images. ii) Time taken in training the SVM/ Time taken in finding K nearest neighbourhoods iii) Time taken in classification of testing images This test includes KNN and SVM classifiers in combination with PCA, PCA-LDA, MPCA, and proposed algorithm as shown in table-iii. Based on above comparisons we observed that time taken in proposed algorithm is quiet promising than the other methods. In both cases proposed algorithm recognize faster than other methods. B. Experiment on YALE data-set YALE database is consists of 165 images of 15 individuals with 11-type images per person in GIF format. Example images for E-1 with size and for E-2 with as original size, have been given in Figure-4 and Figure-5 respectively. The motive behind use of this database is to

6 TABLE III: Processing time comparisons of the methods based on KNN classifier and SVM classifier in E-1 on AT&T Dataset SVM KNN Sample Ratio Training/Testing) 1/9 2/8 3/7 4/6 5/5 6/4 Features MPCA Accuracy%) Total Processing Times) Features MLDA Accuracy%) Total Processing Times) Features MPCA Accuracy%) Total Processing Times) Features MLDA Accuracy%) Total Processing Times) Fig. 4: Example images of cropped YALE dataset with size as used in E-1 Fig. 5: Example images of original YALE dataset with size as used in E-2 TABLE IV: Results along with their respective dimensions based on E-1 on YALE Dataset based on SVM classifiers Sample Ratio training/testing) PCA PCA-LDA MPCA MLDA 2/ %30) 48.15%14) 49.63%25) 54.07%14) 3/ %25) 55.83%14) 76.30%20) 83.70%14) 4/ %28) 64.76%14) 79.17%15) 89.17%14) 5/ %28) 68.89%14) 83.81%23) 91.43%14) 6/ %20) 81.33%14) 87.78%25) 97.33%14) 7/ %25) 88.33%14) 89.33%20) 96.66%14) 8/ %21) 93.33%14) 93.33%18) 97.77%14) 9/ %29) 90.00%14) 90.00%21) 93.33%14) TABLE V: Results along with their respective dimensions based on E-2 on YALE Dataset based on SVM classifiers Sample Ratio training/testing) PCA PCA-LDA MPCA MLDA 2/ %30) 36.30%14) 46.67%25) 52.59%14) 3/ %25) 39.17%14) 74.17%20) 81.67%14) 4/ %28) 49.52%14) 76.19%15) 84.76%14) 5/ %28) 58.89%14) 80.00%23) 78.89%14) 6/ %20) 74.67%14) 77.33%25) 86.67%14) 7/ %25) 85.00%14) 76.67%20) 88.33%14) 8/ %21) 88.89%14) 82.22%18) 91.11%14) 9/ %29) 90.00%14) 83.33%21) 90.00%14)

7 TABLE VI: Processing time comparisons of the methods based on KNN classifier and SVM classifier in E-1 on YALE Dataset SVM KNN Sample Ratio Training/Testing) 2/9 3/8 4/7 5/6 6/5 Features MPCA Accuracy%) Total Processing Times) Features MLDA Accuracy%) Total Processing Times) Features MPCA Accuracy%) Total Processing Times) Features MLDA Accuracy%) Total Processing Times) a) b) c) Fig. 6: Recognition rate of all four methods over YALE dataset: A) with two images taken for training B) with three images taken for training C) with four images taken for training D) with five images taken for training of each training sample d)

8 understand the feature selection strength with change in illumination variation with different directional focus of light. This database is more challenging due to presence of occlusion as glasses and different facial appearance normal, sleepy, wink, surprised, happy and sad). The experiment has been done with different sample ratio varying from 2/9 to 9/2. Proposed algorithm scored 89.17% in E-1 and 84.76% in E-2 when operated with sample ratio of 4/7. The highest recognition rate of 97.77% in E-1and 91.11% in E-2 has been shown by proposed method. On further increment in sample ratio we found slight decrement in recognition accuracy because of some variations in hair style of last images of each person class, still higher than other methods. The results analysis has been given in table-iv. After several experimentations it has been found that in E-2 keeping 60% of variance in each mode with MPCA feature dimension gives best result as shown in table-v. As in E-2 when shadow part was higher in image part the system failed to understand true face variations and causes a decrement in recognition accuracy as compared with results based on E-1. One test has been done in order to understand the time complexity issue of proposed method over YALE dataset. When the total processing time has been compared then the proposed method proved its superiority than previous methods. Another test has been prepared with KNN and SVM classifiers in order to understand effect of classifiers on processing time with higher recognition rate. In both cases proposed MLDA works faster than other methods as shown in table-vi. We compared the recognition accuracy based on feature dimension variations in different methods with proposed methods with sample ratio starting from 2/9 to 5/6. Based on the results we found better accuracy of proposed method than previously known methods. All the comparisons have been performed with E-1. VII. CONCLUSION We have proposed a new algorithm for face recognition using fewer features. Proposed algorithm has been tested on ORL and YALE datasets with and without pre-processing techniques. Based on observations we concluded that proposed MLDA algorithm have more discrimination power under different illumination and expression conditions than previous existing methods such as PCA, PCA-LDA and MPCA. When data size has been increases the recognition accuracy also increases as information in the tenor space is also increases. In each case with increase in training size, the recognition accuracy of the proposed system shows its superiority when compared with other existing methods. It ensures that proposed method is suitable with larger datasets. Although it shows better results than existing methods, under pose and illumination variations but a slight degrade in accuracy when there is a larger variation in illumination is present. Another experiment that contains background with some shadow effect has been performed. We found encouraging results when compared with the other methods. SVM and KNN classifiers have been used to understand the effect of classifiers on the recognition rate. In both datasets proposed method gives better accuracy than other combinations when operated with SVM classifier. Finally it can be conclude that proposed MLDA method work well under illumination variation, pose variations and under shadow effect with promising accuracy and efficiency even with larger dataset size. REFERENCES [1] W. Jhao, R. Chellappa, P.J. Phillips, A. Rosenfeld, Face Recognition: A Literature Survey,ACM Computing Surveys CSUR, vol. 35,no. 4, pp , [2] M. Turk, A. Pentland, Eigen Faces for Recognition, Journal of cognitive neuroscience, vol. 3, no. 1, pp , [3] Comon Pierre Independent component analysis, A new concept?,signal Processing, vol. 36, no. 3, pp , [4] M.S. Bartlett, JavireR, Movellan, T.J. Sejnowski, Face Recognition by Independent Component Analysis,IEEE Transactions on Neural Networks, vol. 13, no. 6, pp , November [5] K. Kim, K. Jung, H. K. Kim, Face recognition using Kernel Principal Component Analysis,IEEE Signal Processing Letters, vol. 9, no. 2, pp , August [6] J. Yung, D. Jhang, et al., Two-dimensional PCA: A new approach to appearance based face representation and recognition,ieee Transaction on pattern analysis and Machine Intelligence, vol. 26, no. 1, pp , Jan [7] J. Ye., Generalized Low Rank Approximation of matrices, Machine Learning,vol. 61, no. 1-3, pp , [8] Liu, J. and Chen, S. and Zhou, Z.H. and Tan, X.,Generalized lowrank approximations of matrices revisited,ieee Transactions on Neural Networks, vol. 21, no.- 4, pp ,2010. [9] N. Nguyan, W. Liu, S. Venkates, Random subspace two-dimensional PCA for Face Recognition,Lecture Notes in Computer Science, Advances in Multimedia Information Processing-PCM 2007, vol. 4810, p.p , December [10] Pan, Q Ruan, Palmprint Recognition using Gabor Features-Based 2D)2PCA,Neurocomputing, vol. 71, no , pp , August [11] W. Yu., Z. Wang, W. Chen, A new Framework to Combine Vertical and Horizontal Information for Face Recognition, Neurocomputing, vol. 72, no. 4-6,pp , January [12] H. Yu, M. Bennamoun, 1D-PCA 2D-PCA to nd-pca, 18th International Conference on Pattern Recognition, ICPR 2006, vol. 4, pp ,August [13] H. Lu, K. N. Plataniotis, A. N. Venetsanopoulos, Multilinear Principal Component Analysis of Tensor Object for Recognition, 18th International Conference on Pattern Recognition, ICPR 200, vol. 2, pp , [14] D. Xu, S. Yan, L. Zhang, H. Zhang, Z. Liu, and H. Shum, Concurrent Subspace Analysis,IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR 2005, vol. -2, pp , [15] H. Lu, K. N. Plataniotis, A. N. Venetsanopoulos, MPCA: Multilinear Principal Component Analysis of Tensor Object,IEEE Transaction on Neural Networks, vol. 19, no. 1, pp , [16] S. Yan, D. Xu, Q. Yang, L. Jhang, X. Tang, and H.J. Jhang, Discriminant Analysis with Tensor Representation,IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR 2005, vol. 1, pp , June [17] P. N. Belhumeur, J. P. Hespanha, and D. J. Kriegman, Eigenfaces vs. Fisherfaces: Recognition using Class specific linear Projection, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 19, no. 7, pp , July [18] W. Zhao, R. Chellappa, A. Krishnaswamy, Discriminant Analysis of Principal Components for Face Recognition, Third IEEE International Conference on Automatic Face and Gesture Recognition, Proceedings, pp , April [19] J. Wang, K. N. Plataniotis, and A. N. Venetsanopoulos, Selecting Discriminant Eigenfaces for Face Recognition, Pattern Recognition Letters, vol. 26, no. 10, pp , 2005.

9 [20] K. Delac, M. Grgic, S. Grgic, Statistics in Face Recognition: Analyzing Probability Distributions of PCA, ICA and LDA Performance Results,Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, ISPA 2005, Zagreb, Croatia, pp , September [21] Vapnik. V.,An overview of statistical learning theory, IEEE Transactions on Neural Networks, vol. 10, no. 5, pp , [22] Gates, K. E., Fast and Accurate Face Recognition Using Support Vector Machines, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 3, pp. 163, 25 June [23] Daoqiang Zhang, Zhi-Hua Zhou, 2D)2PCA: Two-directional twodimensional PCA for efficient face representation and recognition, Neurocomputing,vol. 69, pp , December [24] Wangxin Yu, Zhizhong Wang, Weiting Chen, A new framework to combine vertical and horizontal information for face recognition, presented at Neurocomputing, Vol. 72, no. 4-6, pp , January [25] Jin Wang, Armando Barreto, Lu Wang, Yu Chen, Naphtali Rishe, Jean Andrian and Malek Adjouadi, Multilinear principal component analysis for face recognition with fewer features, Presented at Neurocomputing, vol. 73, no , pp , June [26] C. Gold, P. Sollich, Model Selection for Support Vector Machine Classification, Presented at Neurocomputing, vol. 55, no. 1-2, pp , September [27] F. Lauer, G. Bloch, Incorporating Prior Knowledge in Support Vector Machine for Classification: A Review, Presented at Neurocomputing, vol. 71, no. 7-9, pp , March [28] A. J. Goldstein, L. D. Harmon, and A. B. Lesk,Identification of human faces, Proceedings of the IEEE,vol. 59, no. 5, pp , [29] R. Brunelli and T. Poggio, Face recognition: Features versus templates, IEEE Transactions on Pattern Analysis and Machine Intelligence,vol. 15, no. 10, pp , [30] T. Kanade, Picture Processing by Computer Complex and Recognition of Human Faces, Ph.D. thesis, Kyoto University, 1973 [31] C. Tripathi,K.P. Singh,Face Recognition using Eigen Tensor based Linear Discriminant Analysis with SVM Classifier, M.Tech Thesis, Indian Institute of Information Technology Allahabad, [32] A. Samal and P. A.Iyengar,Automatic recognition and analysis of human faces and facial expressions: A survey, Pattern Recognition, vol. 25, no. 1, pp. 6577, [33] B. Scholkopf, A. Smola, and K. R. Muller, Nonlinear component analysis as a kernel eigenvalue problem, Neural Computation, MIT Press, vol. 10,no. 5, pp , [34] S. Mika, G. Ratsch, J. Weston, B. Scholkopf, and K.-R. Mller, Fisher discriminant analysis with kernels, Neural Networks for Signal Processing IX, Proceedings of the 1999 IEEE Signal Processing Society Workshop, pp. 4148, 1999.