A Comparative Study of Face Recognition System Using PCA and LDA

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1 12 A Comparative Study of Face Recognition System Using PCA and LDA Raj Kumar Sahu, Research Scholar, J.N.U., Jodhpur, India Dr. Yash Pal Singh, Director, (P.G.)I.T.M. Rewari, India Dr. Abhijit Kulshrestha, Professor, J.N.U., Jodhpur, India ABSTRACT This paper concern on Face recognition has a major impact in security measures which makes it one of the most appealing areas to explore. To perform face recognition, researchers adopt mathematical calculations to develop automatic recognition systems. As a face recognition system has to perform over wide range of database, dimension reduction techniques become a prime requirement to reduce time and increase accuracy. In this paper, face recognition is performed using Principal Component Analysis followed by Linear Discriminant Analysis based dimension reduction techniques. Sequencing of this paper is preprocessing, dimension reduction of training database set by PCA, extraction of features for class separability by LDA and finally testing by nearest mean classification techniques. The proposed method is tested over ORL face database. It is found that recognition rate on this database is 96.00% and hence showing efficiency of the proposed method than previously adopted methods of face recognition systems. Keywords Face Recognition, PCA, LDA, City Block Distance, Recognition Rate 1. INTRODUCTION Over the last few years, Face recognition researches have accelerated technological development in human identification for security and safety measures. A face recognition system that basically performs the functions of face detection, human authentication and finally person recognition has innumerable applications in defense, security and automatic access control, human-computer interface. Development of such systems are however very challenging, which includes variations in illumination conditions, poses, facial expression, aging, and disguises such as moustaches, beard, glasses or cosmetics. Among various methods developed by researchers, appearancebased approaches for face recognition yielded good results, which generally work straight on face and hence processed it as two dimensional patterns. The extracted face image features are then represented to belong with a class and separate faces to different classes. Features which are having high separability are continued for further processing and lower one is discarded [26] [14]. To reduce computational costs, face image data, generated after various linear or non-linear transformations, are represented in a lower dimensional space. Some successful appearance-based methods like Principal Component Analysis, Linear Discriminant Analysis and Independent Component Analysis, employ arithmetical techniques like eigenfaces, fischerfaces and independent components respectively, have been widely used to extract the abstract features of the face in facial analysis for face recognition. PCA gives class representations which are in an orthogonal linear space. However LDA generates class discriminatory information in a Linear separable space which is not necessarily orthogonal [1]. Good results are being obtained by these techniques under varying condition. Advantages of both of these techniques can be derived by first reducing dimension of face images by eigen faces and then applying fischer space for class separability. 2. LITERATURE STUDY AND RELATED WORK With the study of literature and journals for the PCA and LDA it is found that, feature based approach key information of facial features, like nose, eyes, chin and mouth, are collected with the help of deformable templates and extensive mathematics and converted into a feature vector. Yuille et. al. [23] proposed deformable templates techniques, where facial features are determined by interactions with the face images. In Image based approach information theoretical concepts like class information, class separability, independent components, face image energy contents are utilised. As whole face image is used so this method is also termed as holistic method. Turk et. al. [19] developed Principal Component Analysis based face recognition using eigenface techniques. As mathematical algorithms utilizing eigenvectors for representation of primary components of the face, it is termed as eigenface technique. It creates a unique weight for each face image, which are used to represent the eigenface features. Hence comparison of these weights permits identification of individual faces from a database. Zhao et. al. [25] uses linear discriminant analysis to maximize the scatter between different classes and

2 13 minimize the scatter between the input data in the same class. PCA tries to simplify the input data by extracting the features and LDA tries to distinguish the input data by dimension reduction. By combining both these techniques, subspace LDA based face recognition system is developed [7]. 3. IDENTIFICATION OF THE PROBLEM AND PRESENT RESEARCH WORK: To study the comparison between the PCA and LDA Also to find the integration with regard to Develop the Robust face recognition system. 4.PROPOSED FACE RECOGNITION ALGORITHM A. PRINCIPAL COMPONENT ANALYSIS A gray scale image is a two dimensional identity. In spatial domain its dimension is represented by number of pixels within the image.in gray scale images, each pixel value lies in the range of 0 to 255. Suppose a grey scale image of size (R x C), the dimension of the image space is Q, then Q will be R times C. Turk et. al. [19] used eigen values for principal component analysis, so it is also termed as Eigenface method. In this method, variance due to non-face images are eliminated and variation just coming out of the variation between the face images are collected for analysis. Hence features of images are obtained by finding maximum deviation of each image, in the spatial domain of image, from the mean image. Then eigenvectors of the covariance matrix of all the images are calculated. Then the training images are projected into the eigenface space. In the same way, test image is projected into this eigenface space and then distances of the projected test image with each of projected training images are calculated to identify the test image. Figure 1 is representing flowchart of Principal Component Analysis Based Face Recognition System. Steps are then explained with the help of mathematical equations after the flowchart. Face Acquisition Face Database formation Training Dataset Test Image Mean Image calculation Difference Matrix Difference Matrix Covariance Matrix Eigen Vector Calculation Weight Matrix Weight Matrix Eigen Face Space Training Projection Matrix Eigen Face Space Test Projection Matrix Simillarity Measurement {L 1 norm or L 2 norm or Covariance} Output Face Image Fig.1: Flowchart of PCA Based Face Recognition System 4.11 Training Phase Image: - The image matrix I of size (R x C) pixels, where R is number of rows and C is number of columns of a two dimensional grey scale image. It is then transformed into a single column vector, Γ, of size (Q x 1) where Q = (R x C) Training Set: - Image matrix is formed by appending column vector of each image one after another. Γ = [Γ 1 Γ 2... Γ M ] (1) and its size is (Q x M) where M is the number of the training images Mean Face: - Now column wise mean of this image matrix is found out. It is the arithmetic average of the training image vectors at each pixel point M Ψ = 1 M Γ 1 Its size is Q x 1. Figure 2 is showing mean image of the training database images used (2)

3 14 Mean Image Fig.2: Mean Face Image 4.14 Mean Subtracted Image: - Now difference of each training image vector is calculated from just found mean image vector. Φ = Γ Ψ (3) It size will be Q x Difference Matrix: - Each of mean subtracted training image vectors are then appended column wise to form difference matrix A= [Φ 1 Φ 2 Φ 3 (4) Φ 4 Φ M ] It size will be Q x M. Here also, M is the number of the training images 4.16 Covariance Matrix: - Covariance of difference matrix is then found, which its multiplication with its own transpose is. X = A. A T M = 1 M Φ T (5) i Φ i Hence its size will be Q x Q Eigen Vectors & Eigen Values: - In PCA, eigenvectors of the covariance matrix is calculated. For a face image having Q numbers pixels, the covariance matrix is of size (Q x Q). Recognition can be done with the help of this covariance matrix. But because of its very big size, calculation with take lots of time and hence cost of recognition will get increased tremendously. Hence PCA adopts eigenface method to calculate the eigenvectors of size (M x M) matrix, where M being the number of training face images. In eigen face method, instead of X, another matrix Y is calculated which is, Y = A T. A = 1 M M Γ i T (6) which is of size (M x M). Then, the eigenvectors v i and the eigenvalues μ i of Y are obtained, Y. v i = μ i.v i (7) i.e A T. A. v i = μ i.v i (8) On multiplying A both sides, A. A T. A. v i =A. μ i.v i (9) Since μ i is a scalar, it can also be written as A. A T. A. v i = μ i. A.v i (10) X. A. v i = μ i. A. v i (11) Γ i If Α. ν i is substituted by υ i = Α. ν i. Then υ i = A. v i (12) is one of the eigenvectors of Χ = Α.Α T and its size is (Q x 1). So it can be observed that eigenvectors of X can also be obtained from eigenvectors of Y. Instead of using a very big covariance matrix X of (Q X Q), another matrix Y of size (M x M) is used Selecting highest eigen vectors:- Now since most of the generalization power is contained in the first few eigenvectors. Around 40% of the total eigenvectors have % of the total generalization power [29]. So instead of using all the M numbers of the eigenfaces, M' M numbers of the eigenfaces is sufficient to show its effect. Larger number of eigenvalue signifies larger variance, which an eigenvector represents. Hence, the eigenvectors are arranged in descending order with respect to their corresponding eigenvalues. So, the most generalizing eigenvector comes first in the eigenvector matrix Testing Phase An image under test is also projected over eigenface space like that of training images. It is mean subtracted and projected onto the eigenface space and then classified; that is, the test image is assumed to belong to the nearest class by evaluating the shortest distance of the test image, with all other training images in eigen face space Test image vector: - Image under test, will be of (R x C) size. It is also converted into a single column vector, Γ T, to represent in image space. It size will be (Q x 1). Mean subtracted image: - It is the difference of the test image from the mean image. Φ T = Г T - Ψ (15) Its size will also be (Q x 1) Projected Test Image: - This mean subtracted test image is then projected over pre calculated eigenface space, ω, which was of size (Q x M ). Ω T = ω. Φ T (16) Its size will be (M x1) 4.22 Classification: - Classification is performed by similarity measures, which is defined as the distance between Projected Test Image and each image in Projection Matrix. δδ ii = Ω TT Ω = (Ω TT Ω kk ) 2 MM kk=1 (17)

4 15 It is the Euclidean distance (L2 norm) between projections. By finding location of minimum value of this Euclidean distance matrix, test image can be classified as, in which image class it lies Reconstructed image vector:- The training and test image vectors can be reconstructed by a back transformation from the eigenface space to the image vector space. Image can be reconstructed from in eigenface domain. Figure 4.7 is facial images, of first 16 individuals, reconstructed from eigenface space projection matrix. Г f = υ. Ω + Ψ = (18) Φ f + Ψ A representative image vector can be formed from the class of images for an individual. 4.24Average Class Projection: - Average class projection is the average of the projections of each class is the mean of all the class projected image vectors. c Ω Ψ = 1 c Ω i (19) This average class projection can be used as one of the vectors (representing an image class instead of an image vector) to compare with the test image vector Distance Threshold: - It is the maximum allowable distance, which is half of the distance between the two most distant classes. Θ = 1 2 max Ω Ψi (20) Ω Ψj 4.26Distance measure: - It is the distance between the mean subtracted image and the reconstructed image. ε 2 = Φ Φ f 2 (21) Criterion of inside database and out of face database image:- (a) If ε < Θ and for all i > Θ, δ i the image is an unknown face. (b) If ε < Θ and for one of i < Θ, δ i the image is a known face. Reconstructed Image no.1 Reconstructed Image no.2 Reconstructed Image no.3 Reconstructed Image no.4 Reconstructed Image no.5 Reconstructed Image no.6 Reconstructed Image no.7 Reconstructed Image no.8 Reconstructed Image no.9 Reconstructed Image no.10 Reconstructed Image no.11 Reconstructed Image no.12 Reconstructed Image no.13 Reconstructed Image no.14 Reconstructed Image no.15 Reconstructed Image no.16 Fig.4: Reconstructed Images of first 16 Individuals B. SUBSPACE LINEAR DISCRIMINANT ANALYSIS Linear discriminant analysis is used to overcome drawback of Principal Component Analysis of its application restricted to small image database. It is achieved by projecting the image onto the eigenface space by PCA and then implementing pure LDA over it, to classify the eigenface space projected data. Linear Discriminant Analysis is also termed as fischerface method. Belhumeour et. al. [6] uses linear discriminant analysis to maximize the scatter between different classes and minimize the scatter between the input data in the same class. Figure 5 is flowchart representation of LDA Based Face Recognition System, followed by their mathematical explanations. Here first part i.e. PCA tries to simplify the input data by extracting the features and then its next part, which is, LDA tries to distinguish the input data by dimension reduction. By combining both these techniques, subspace LDA based face recognition system is developed Training Phase 4.28Image: - The image matrix I of size (R x C) pixels, where R is number of rows and C is number of columns of a two dimensional grey scale image. It is then transformed

5 16 into a single column vector, Γ, of size (Q x 1) where Q = (R x C) Training Set: - Image matrix is formed by appending column vector of each image one after another. Γ = [Γ 1 Γ 2... Γ M ] (22) and its size is (Q x M) where M is the number of the training images. Its size is Q x Mean Face: - Now column wise mean of this image matrix is found out. It is the arithmetic average of the training image vectors at each pixel point. Face acquisition M Ψ = 1 M Γ i (23) Face Database formation Training Dataset Test Image Mean Image calculation Diffrence Matrix Diffrence Vector Covariance Matrix Eigen Vector Calculation Projection vector Weight Matrix Training Database Projection Matrix (Eigenface Space) Testing Image Projection Marix Eigenface Space & Class Mean Within Class & Between Class Scatter Matrix Eigenvector Calculation Eigenface Weight Matrix Classification Space Training Matrix Classification Space Testing Matrix Simillarity Measurement {L 1 norm or L 2 norm or Covariance} Fig.5: Flowchart of Subspace LDA based Face Recognition System 4.31 Mean subtracted image: - Now difference of each training image vector is calculated from just found mean image vector. Φ = Γ Ψ (24) It size will be Q x Difference Matrix: - Each of mean subtracted training image vectors are then appended column wise to form difference matrix. Output Face Image A= [Φ 1 Φ 2 Φ 3 Φ 4 Φ M ] (25) It size will be Q x M. Here also, M is the number of the training images 4.33 Covariance Matrix: - Covariance of difference matrix is then found, which its multiplication with its own transpose is.

6 17 X = A. A T M = 1 M Φ T (26) i Φ i Hence its size will be Q x Q Eigen Vectors & Eigen Values: - In PCA, eigenvectors of the covariance matrix is calculated. For a face image having Q numbers pixels, the covariance matrix is of size (Q x Q). Recognition can be done with the help of this covariance matrix. But because of its very big size, calculation with take lots of time and hence cost of recognition will get increased tremendously. Hence PCA adopts eigenface method to calculate the eigenvectors of size (M x M) matrix, where M being the number of training face images. Then, the eigenvectors v i and the eigenvalues μ i of Y are obtained, Y. v i = μ i.v i (27) i.e A T. A. v i = μ i.v i (28) Multiplying A both A. A T. A. v i (29) sides, =A. μ i.v i Since μ i is a scalar A. A T. A. v i = (30) quantity μ i. A.v i X. A. v i = μ i. (31) A. v i If υ i = Α. ν i X. υ i = μ i. υ i (32) So, υ i = A. v i is one of the eigenvectors of Χ = Α.Α T and its size is (Q x 1). So it can be observed that eigenvectors of X can also be obtained from eigenvectors of Y. Instead of using a very big covariance matrix X of (Q X Q), another matrix Y of size (M x M) is used Selecting highest eigen vectors: - Now since most of the generalization power is contained in the first few eigenvectors. Around 40% of the total eigenvectors have % of the total generalization power. So instead of using all the M numbers of the eigenfaces, M' M numbers of the eigenfaces is sufficient to show its effect. Eigen face Space: - It can be formed by calculating dot product of difference matrix with selected numbers of eigen vectors. ω=a. υ k (33) where k = 1, 2,, M. Hence its size will be (Q x M ). Projection Matrix: - Training images are then projected over eigenface space. Ω= ω. A (34) So its size is (M x M). This method transforms each image from image space to eigen face space. Earlier in image space, each image was of size (Q x 1), however eigenface transformation each image is of size (M x 1) in the eigenface space. Hence dimension reduction is achieved, as M <Q Linear Discriminant Analysis Instead of using the pixel values of the images, the eigenface projections of PCA transformation are used in the Subspace LDA method. Suppose in the training database, there are C individual person classes having q i number of training images in each of the class, such that total number of facial images M = C x q i. Eigenface Class Mean: - It is the arithmetic average of the eigenface projected training image vectors corresponding to the same individual class. q i m i = 1 q i Ω k k=1 (35) where i = 1, 2,, C and and size of each eigenface class mean is (M x 1). Eigenface Mean Face: - It is the arithmetic average of all the eigenface projected training image vectors. M m 0 = 1 M Ω k k=1 (36) Its size is (M x 1). Within Class Scatter Matrix: - This matrix represents the average scattering of the projection matrix Ω in the eigenface space of different individuals C i around their respective class means m i. Its size will be (M x M ), which is based on largest number of eigen vectors M, selected for eigenface space. Where average scattering is C S w = P(C i )Σ i Σ i = E[(Ω m i )(Ω m i ) T ] (37) (38) And Prior Class Probability is P(C i ) = 1 (39) C which is the selected with assumption that each class has equal prior probabilities. Between-Class Scatter Matrix: - This matrix represents the scatter of each projection classes mean mi around the overall mean vector m 0. C S b = P(C i )(m i m 0 )(m i m 0 ) T Its size is (M x M ). For LDA, a discriminatory power is defined as; Discriminatory Power J(T) = TT. S b. T (41) T T. S w. T where S b is the between-class and S w is the within-class scatter matrix. The objective is to maximize J(T); This can be achieved by obtaining projection W which maximizes between-class scatter and minimizes within-class scatter. Then, W can be obtained by solving the generalized eigenvalue problem; S b W = S w W λ w (42) Here also some highest numbers of fischer features, F, are selected such that F < M. Next, the eigenface projections of the training image vectors are projected to the fischeface space by the dot (40)

7 18 product of optimum projection, W and eigenface projection matrix, Ω. Classification space projection matrix: - This matrix is the projection of the training image vectors eigenface projections to the classification space. g(ω) = W T. Ω (43) It is of size (F x M). Testing Phase Test image vector: - Image under test, will be of (R x C) size. It is also converted into a single column vector, Γ T, to represent in image space. It size will be (Q x 1). Mean subtracted image: - It is the difference of the test image from the mean image. Φ T = Г T - Ψ (44) Its size will also be (Q x 1). Projected Test Image: - This mean subtracted test image is then projected on each of the eigenvectors. where k = 1, 2,, M Weight Matrix: - It is representation of the test image in the eigenface space. Ω T = [ω 1 ω 3.. ω M ] T ω 2 (46) Its size is (M x 1). Individual Class Classification test space projection: - This matrix is formed by eigenface projection of the test image vector (i.e. weight matrix) to the classification space in the same manner. g (Ω T ) = W T. Ω T (47) It is of size (F x 1). Distance measure: - Then distance between the classification test space projection and classification space projection matrix is determined by the Euclidean distance calculation. C i = g (Ω T ) g (Ω i ) = g k (Ω T ) g k (Ω i ) 2 images under test. T T ω k = υ k. Φ T = υ k. (Г T Ψ) (45) PCA (out of 59) LDA (out of 59) k=1 (48) Where for i = 1, 2 M. Location of shortest Euclidean distance represents class of RESULTS Table 1 shows, correctly recognized facial images out of 38 different test images. These results are of PCA and Subspace LDA algorithms. Further their recognition rates are also compared. In this comparison method, numbers of PCA features are taken as 80. Recognition Rate (%) By PCA Recognition Rate (%) By LDA

8 Table 1: Class Wise Result Variation of PCA and LDA Based Face Recognition System Feature wise variation in recognition rate of PCA & LDA System is shown in figure 6. With increase in features, in PCA recognition rate keep increasing upto 60 features and then it becomes constant. Moreover recognition rate of LDA is always greater than PCA PCA LDA Recognition Rate (%) Dimensions (Number of PCA Features) Fig.6: Feature Based Recognition Rate Comparison of PCA & LDA

9 20 Table II COMPARISON OF RECOGNITION RATE OF PROPOSED ALGORITHM WITH THAT OF OTHER METHODS Methods Recognition Rate ICA[28] 70.91% Linear Subspace[27] 79.40% PCA with L 2 Norm [19] 84.00% DCT Based Face Recognition [24] 84.50% K-means [10] 86.75% Kernel-Fisher[28] 93.94% PCA & LDA with L 2 Norm [23][25] 94.80% Fuzzy Ant with fuzzy C- means [10] 94.82% Proposed Algorithm 96.00% CONCLUSION In this paper a new face recognition system based on, dimension reduction technique by PCA followed by LDA and classification with City Block Distance Calculation, is presented. The Recognition Rates are compared with varying PCA features and LDA features. For experiment ORL Face Database is used, which provided face images under varying facial condition [12]. It is found that increasing PCA features and reducing LDA features increases recognition rate as shown graphically in figure 6. In our simulations, we considered 8 training images and 2 test images for each of 40 persons (totally 320 training and 80 test face images). Our experimental results show a recognition rate equal to %, which demonstrated an improvement in comparison with some other previous methods, as shown in table II. Hence from these experiments, it can be concluded that when the number of face images per subject is small PCA is better, but for large training face images and different training data set, its combination with LDA shows better result than pure PCA. Future research work: With the help and utilizing the comparative study of PCA and LDA we may able to develop the Robusr face recognition system. It may open varied areas of the Study. REFERENCES [1] Belhumeur P.N., Hespanha J.P., and Kriegman D. J., Eigen Faces vs. Fisher Faces: Recognition Using Class Specific Linear Projection, IEEE Transaction on Pattern Analysis and Machine Intelligence, vol. 19, PP , [2] Eleyan Alaa and Demirel Hasan, PCA and LDA based Neural Networks for Human Face Recognition, Face Recognitin Book, ISBN , pp.558, I-Tech, Vienna, Austria, June [3] Etemad K. and Chellappa R., Face Recognition Using Discriminant Eigenvectors, IEEE Transaction for Pattern Recognition, pp ,1996. [4] Hu H., Zhang P. and Torre F. De la, Face recognition using enhanced linear discriminant analysis, IET Computer Vision,, Vol. 4, Iss. 3, pp , [5] Laurens van der Maaten, Eric Postma, and Jaap van den Herik, Dimensionality Reduction: A Comparative Review, Tilburg centre for Creative Computing, Technical Report,2009. [6] Lee S. J., Yung S. B, Kwon J. W., and Hong S. H., Face Detection and Recognition Using PCA, IEEE TENCON, pp , [7] Lone Manzoor Ahmad, Zakariya S. M. And Ali Rashid, Automatic Face Recognition System by Combining Four Individual Algorithms, proceeding of International Conference on Computational Intelligence and Communication Systems, [8] Lu Juwei, Plataniotis K.N., and Venetsanopoulos A.N., Face Recognition Using LDA Based Algorithms, IEEE Transactions On Neural Networks, [9] Mahyabadi M. P., Soltanizadeh H., Shokouhi Sh. B., Facial Detection based on PCA and Adaptive Resonance Theory 2A Neural Network, Proceedings of IJME - INTERTECH Conference, [10] Makdee S., Kimpan C., Pansang S., Invariant range image multi pose face recognition using Fuzzy ant algorithm and membership matching score Proceedings of IEEE International Symposium on Signal Processing and Information Technology, pp , 2007.

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