STUDY OF FACE AUTHENTICATION USING EUCLIDEAN AND MAHALANOBIS DISTANCE CLASSIFICATION METHOD

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1 STUDY OF FACE AUTHENTICATION USING EUCLIDEAN AND MAHALANOBIS DISTANCE CLASSIFICATION METHOD M.Brindha 1, C.Raviraj 2, K.S.Srikanth 3 1 (Department of EIE, SNS College of Technology, Coimbatore, India, mbrindhaeie@gmail.com) 2 (Department of EIE, SNS College of Technology, Coimbatore, India, raviraj51095@gmail.com) 3 (Department of EIE, SNS College of Technology, Coimbatore, India, srikanth24zin@gmail.com) Abstract In face recognition feature extraction and classification are the two aspects to be focused. In principle component analysis (PCA) based face recognition technique, the 2D face image matrices must be previously transformed in to one dimensional image vectors. In this paper two dimensional principle component analysis(2dpca) is used to extract the features. Comparing to conventional principle component analysis, two dimensional principle component analysis is based on 2D matrices rather than 1D vectors. The image matrix is formed directly using original image matrices Recognition rate seems to be higher using two dimensional principle component analysis. The Mahalanobis distance is a metric which is better adapted than the usual Euclidean distance to settings involving non spherically symmetric distribution. Recall, precision, fmeasure, recognition rate are calculated and the results are analyzed for Oracle Research Laboratory (ORL) database and for the database taken using normal digital camera. This paper includes the comparison of Euclidean and Mahalanobis Distance classification methods and analyzes the results. Keywords Fmeasure; Precision; Recall; Recognition rate; Two dimensional principle component analysis (2DPCA) 1. INTRODUCTION Biometrics is becoming important in our securityheightened world [1].Automatic face recognition is an interesting computer vision problem with many commercial and law enforcement applications [2]. If an effective face recognition system can be implemented then Mugshot matching, user verification and user access control, crowd surveillance, enhanced human computer interaction all become possible [3]. However, still face recognition is an active area of research to solve the face recognition problem. In the PCA-based face recognition technique, the 2D face image matrices must be previously transformed into 1D image vectors. Usually, the resulting image vectors of faces lead to a high dimensional image vector space, where covariance matrix is difficult to evaluate accurately due to its large size and the relatively small number of training samples. Fortunately, the eigenvectors (eigenfaces) can be calculated efficiently using the Singular Value Decomposition (SVD) techniques [6], [7] and the process of generating the covariance matrix is actually avoided. In this paper, a straightforward image projection technique, called two-dimensional principal component analysis (2DPCA), is developed for image feature extraction. As opposed to conventional PCA, 2DPCA is based on 2D matrices rather than 1D vectors. That is, the image matrix does not need to be previously transformed into a vector. Instead, an image covariance matrix can be constructed directly using the original image matrices. In contrast to the covariance matrix Of PCA, the size of the image covariance matrix using 2DPCA is much smaller. As a result, 2DPCA has two important advantages over PCA. First, it is easier to evaluate the covariance matrix accurately. Second, less time is required to determine the corresponding eigen vectors[5]. This paper is organized as follows. Section II describes about the steps for feature extraction, Section III deals with mathematical formulations of 2DPCA,Section IV shows experimental results and observations. The paper concludes with section V 2. PRINCIPLE COMPONENT ANALYSIS Principle component analysis, also called as Karhunen Loeve expansion. It is a classical feature extraction and data representation technique and it is widely used in the areas of pattern recognition and computer vision. Let a face image I(x,y) n be a two-dimensional N by N array of intensity values or a vector of dimension N 2. A typical image of size 256 by 256 describes a vector of dimension 65,536, or equivalently, a point in 65,536-dimensional space [5]. Consider our training set of images of 85 by 95 pixels. The main objective of PCA is to find the vectors which best account for the distribution of the face images within the entire image space. Fig 1.Flow chart for face detection Volume: 04 Issue:

2 2.1 STEPS FOR FEATURE EXTRACTION 1. A set S is obtained with M face images in the first step. Each image is transformed into a vector of size N and placed into the set. S={ Γ, Γ₂, Γ₃,,Γmm} (1) 2. Second step is to obtain the mean image Ψ. Ψ = 1 mm Гnn nn=1 (2) 3.Find the difference F between the input image and the mean image φᵢ =Γᵢ- Ψ (3) 4. Next it seek a set of M orthonormal vectors,u n, which best describes the distribution of the data. The k th vector, u k,is chosen such that λ k= 1 ( NN=1 uᵀkɸ n ) 2 (4) where λk is the eigenvalues of the covariance matrix C 5. The covariance matrix C has been obtained in the following manner C= 1 ɸ nn=1 nɸᵀn (5) =AAᵀ A={ɸ1,ɸ2,ɸ3, ɸn} 6. The huge computational task is to find eigenvectors from the covariance matrix. Since M is far less than N 2 by N 2, we can construct the M by M matrix L= Aᵀ A. 7. Find the M eigenvector, v 1 of L. 8. These vectors (v 1 ) determine linear combinations of the M training set face images to form the eigenfaces u 1. u 1= VV lk ɸ k l=1,2,.m (6) 9. Project each of the original images into eigenspace. This gives a vector of weights representing the contribution of each eigenfaces to the reconstruction of the given image. ὡ kk = uᵀk(γ- Ψ ) Ω T =[ω 1,ω 2,ω 3, ω M ] (7) where uk is the k th eigenvector and ω k is the k th weight in the vector. 3. TWO DIMENSIONAL PRINCIPLE COMPONENT ANALYSIS Normally, in PCA-based face recognition methods, concatenation technique is used to transform the 2D face image samples into 1D image vectors. In 2DPCA model, the 2D features are obtained directly from original vector space of a face image. rather than from a vectorized 1D space. The 2DPCA overcomes the drawbacks of PCA. The usage of 2DPCA for face recognition is a novel idea and is discussed in this section. The steps of 2DPCA face recognition model are given below. To form a training set, acquire face images (X1, X2,..XN) 2DPCA is used to extract features for each training sample and each testing sample. The image is recognized after classification. Recognition result is obtained. Fig 2.Five sample images of one subject in the ORL face database. Fig 3.Five sample images of face database taken using normal camera. 3.1 MATHEMATICAL FORMULATION: Let X denotes an n-dimensional unitary column vector. By using linear transformation the image A, an m x n random matrix onto X is projected. Y=AX (8) To measure the discriminatory power of the projection vector X, the total scatter of the projected samples can be introduced. The total scatter of the projected samples can be characterized by the trace of the covariance matrix of the projected feature vectors. The covariance matrix of the projected feature vectors of the training samples can be denoted as Sx and tr(sx) denotes the trace of Sx. The physical significance of maximizing the criterion is to find a projection direction X, onto which all samples are projected, so that the total scatter of the resulting projected samples is maximized. The covariance matrix Sx can be denoted by Sx=E(Y-EY)](Y-EY) T =E(AX-EAX)(AX-EAX) T = E[(A-EA)X][(A-EA)X] T (9) So, tr(sx)= X T [E(A-EA) T (A-EA)]X. (10) Then the following matrix will be, G t =E[(A-EA) T (A-EA)] (11) The matrix G t is called the image covariance (scatter) matrix.to verify that G t is an nxn non negative definite matrix is easy. We can evaluate G t directly using the training image samples. Suppose that there are M training image samples in total, the jth training image is denoted by an m x n matrix A j (j=1,2, M), and the average image of all training samples is denoted by A. Then, Gt can be calculated by M 1 G = ( Ak A)( Ak A) M k = 1 Alternatively, the value of J(X) can be expressed by J(X)=X T G t X t (12) Where X is a unitary column vector. This criterion is called the generalized total scatter criterion. The unitary vector X that maximizes the criterion is called the optimal projection axis. Intuitively, this means that the total scatter of the T Volume: 04 Issue:

3 projected samples is maximized after the projection of an image matrix onto X.The optimal projection axis is denoted as Xopt and it is the unitary vector that maximizes J(X), i.e., the eigenvector of G t corresponding to the largest eigenvalue.in fact, the optimal projection axes, X1, Xd, are the orthonormal eigenvectors of G t corresponding to the first d largest eigenvalues. Feature Extraction The optimal projection vectors of 2DPCA, X1 Xd, are used for feature extraction. For a given image sample A, let Y K =AX K ; K=1,2,,d (13) Then, we obtain a family of projected feature vectors, Y1 Yd, which are called the principal component (vectors) of the sample image A. It should be noted that each principal component of 2DPCA is a vector, whereas the principal component of PCA is a scalar. The principal component vectors obtained are used to form an m x d matrix B =[Y1, Yd], which is called the feature matrix or feature image of the image sample A. 3.2 CLASSIFICATION METHOD USING MAHALANOBIS DISTANCE Transformation is done by 2DPCA and a feature matrix is obtained for each image. Then, a nearest neighbor classifier is used for classification. Here, the distance between two arbitrary feature matrices, B i =[Y 1 (i), Y 2 (i) Y d (i) ] and Bj= [Y 1 (j), Y 2 (j) Y d (j) ] is defined by d(b i, B j ) which denotes the Mahalanobis distance between the two principal component vectors Y k (i) and Y k (j). 3.3 CLASSIFICATION METHOD USING EUCLIDEAN DISTANCE We calculate the distance between the test image from each image from the training base. If the distance between them is minimum, then it seems to be that which image from the database matches the test image best. The Euclidean distance is measured or calculated using the formula given below. DD D(A,B)= (aaᵢ bbᵢ) 2 = A-B (14) ii=1 4. EXPERIMENT RESULTS AND OBSERVATIONS The figure 2 shows the sample ORL database and the figure 3 shows the sample database taken using normal digital camera contains images from 40 individuals, each providing 10 different images. The facial expressions (open or closed eyes, smiling or non smiling) and facial details (glasses or no glasses) also vary. The images were taken with a tolerance for some tilting and rotation of the face of up to 20 degrees. Moreover, there is also some variation in the scale of up to about 10 percent. The resolutions of all images are of 92x112 pixels. (a) (b) Fig 4.Face recognition for (a) ORL database (b)database taken using normal digital camera TABLE 1.TRAINING AND TESTING VALUES FOR ORL DATSABASE train=1 train=2 testing=5 train=6 testing=4 Recall Precision Fmeasure Recognition Rate Fig 4 (a) and (b) shows verification of images for ORL database and the database taken using normal digital camera. Face recognition is done by calculating the distance of test image from each image from the training base. Minimum distance shows us which image from the database matches the test image best. The Table 1 and 2 shows Training and testing values for ORL database and for the database taken using normal digital camera. The values of recall, precision, fmeasure and recognition rate are obtained and the results are analyzed. TABLE 2.TRAINING AND TESTING VALUES FOR THE DATABASE TAKEN USING NORMAL DIGITAL CAMERA train=1 train=2 Recall Precision Fmeasure Recognition Rate Volume: 04 Issue:

4 testing=5 train=6 testing= COMPARISION OF RECOGNITION RATE Recognition Rate ORL Database Database Taken Using Normal Digital Camera 75 Fig 5. Comparison of recognition rate for ORL database and for the database taken using normal digital camera The PCA method and the proposed 2DPCA method were used for feature extraction. In 2DPCA, Mahalanobis distance is used to calculate the distance between two feature matrices (formed by the principal component vectors) whereas in PCA (Eigenfaces), the common Euclidean distance measure is used. The 2DPCA algorithm was first used for feature extraction.here, the size of image covariance matrix Gt was 92 x 92, so it was very easy to calculate its eigenvectors. We choose the eigenvectors corresponding to 10 largest eigenvalues, X 1 X 10, as projection axes. After the projection of the image sample onto these axes, we obtained ten principal component vectors, Y 1 Y 10.Hence it can be concluded that the energy of an image is concentrated on its first small number of component vectors. Therefore, it is reasonable to use these component vectors to represent the image for recognition purposes. On the other hand, by adding up the first d sub images together, we obtain an approximate reconstruction of the original image. For comparison, the PCA (Eigenfaces) was also used to represent and reconstruct the same face image..the PCA did not perform as well in the reconstruction of this image The time taken by 2DPCA for feature extraction is less compared to conventional PCA. As the number of training samples per class is increased, the relative gain between 2DPCA & PCA becomes apparent.here, it should be pointed out that PCA used all components for achieving the maximal recognition accuracy when there are one or two samples per person for training. The computational efficiency of 2DPCA method is also superior to PCA in terms of computational efficiency for feature extraction. As the number of training samples per class is increased, the relative gain between 2DPCA and PCA becomes more apparent. However, one disadvantage of 2DPCA (compared to PCA) is that more coefficients are needed to represent an image. It is clear that dimension of the 2DPCA feature vector is always much higher than PCA at top recognition accuracy. For the ORL database, 2DPCA outperformed PCA significantly in the trials. A. Selected Test Image Fig 6.Test image The fig 6 shows the selected test image and the figure 7 shows the recognised output for the given test image. Fig 7. Recognition output of the given test image 5. CONCLUSION AND FUTURE WORK In this paper Two Dimensional Principle Component Analysis is used as a proposed feature extraction technique. Mahalanobis Distance is a metric better adapted than the usual Euclidean distance and it involves non spherical symmetric distributions. It increases the quality of image reconstruction. By using PCA the values of fmeasure, recall, precision are calculated. The recognition rate for 2DPCA is better compared to conventional PCA. When a small number of the principal components of PCA are used to represent an image, the mean square error (MSE) between the approximation and the original pattern is minimal. The future work includes the comparison of various classification methods with the proposed method and analyzes the results. REFERENCES [1] Zhao, W., Chellappa, R., Rosenfeld, A., Phillips, Face Recognition: A Literature Survey, ACM Computing Surveys, pp , [2] P. Sinha, B. Balas, Y. Ostrovsky, R. Russell, Face Recognition by Humans: 19 Results All Computer Vision Researchers Should Know About, Proceedings of the IEEE, Vol. 94, No. 11,, pp , Volume: 04 Issue:

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