IntroductionEigenfacesFacesAverage (mean) FaceDifferenceCovariance MatrixEigenvalues and EigenvectorsBasisFace RecognitionIntroductionEigenfacesFace RecognitionHome PageTitle PageJJ IIJ IPage 1 of 11Go BackFull ScreenCloseQuitEigenfacesCrystal Pepper and Chris WilsonOctober 02, 2009AbstractIn this paper we will explore the concept of facial recognition using eigenfaces.Eigenfaces are pro duced by transforming the pixels in an image to (x, y) co ordi-nates and forming a matrix with the coordinates. The eigenfaces or the principalcomp onents of the faces are the eigenvectors of the matrix and it is the eigenvectorsthat are used in the facial recognition pro cess.IntroductionFacial recognition is inherent in humans. Not only can we recognize thousands of facesin a lifetime, we also recognize people we have not seen for years with a passing glance.Since the computer revolution, we have attempted to recreate this phenomenon intoan algorithm that the computer will recognize. There are several uses for this interestincluding criminal identification, security and film development. However, creating acomputer based model for facial recognition is very difficult due to the fact that “ facesare complex, multidimensional, and meaningful visual stimuli.” Therefore, we will takethe simplistic approach and developing our model by avoiding the three-dimensionalworld.IntroductionEigenfacesFace RecognitionHome PageTitle PageJJ IIJ IPage 2 of 11Go BackFull ScreenCloseQuitFigure 1: Training SetThe idea is to create a program that can define each facial image into its character-istic elements called “eigenfaces”. The group of images are placed in a span of spacewe call “face space.” Each image’s location is known relative to the others. Therefore,to recognize any one image, the face image is projected into the subspace spanned bythe eigenfaces and then, by comparing its position in face space with the positions ofthe known images, a new image is created.IntroductionEigenfacesFace RecognitionHome PageTitle PageJJ IIJ IPage 3 of 11Go BackFull ScreenCloseQuitEigenfacesFacesWe first obtained a set of 20 digital images. Each face was captured by positioning thecamera and face at the same distance from each person. This is required so that theaverage face will look like a face. A single digital image is comprised of N by M pixels,which is essentially a matrix of N rows and M columns. We then resized each imageI to a standard size of 120 × 160 pixels which results in a 120 × 160 matrix. In orderto compare images, we converted each 120 × 160 matrix to a single column 19200 × 1vector. This is done by taking each face image of N × M and transforming into a singlecolumn vector Γi.I =i1,1i1,2. . . i1,160i2,1i2,2. . . i2,160............i120,1i120,2. . . i120,160120×160=⇒ Γ =γ1,1...γ120,1γ1,2...γ120,2γ1,160...γ120,16019200×1Next, we placed each image into a matrix to represent all 20 images as a set S. Thiswas done by placing each image vector side by side in the same matrix. Doing this,results in no data loss. Our matrix is now 19200 × 20. The training set of our imagescomputed by matlab is shown in Figure 1. The set of all images is defined byIntroductionEigenfacesFace RecognitionHome PageTitle PageJJ IIJ IPage 4 of 11Go BackFull ScreenCloseQuitFigure 2: Normal Training SetIntroductionEigenfacesFace RecognitionHome PageTitle PageJJ IIJ IPage 5 of 11Go BackFull ScreenCloseQuitFigure 3: Average Faceγ1,1γ1,2. . . γ1,160............γ120,160γ120,2. . . γ120,160=⇒ [Γ1Γ2· · · Γ20]. =⇒ SAverage (mean) FaceThe next step in our process was to compute the average face vector Ψ. This can bedone by taking the sum of all the images Γ1+ Γ2+ · · · + Γ20and dividing by the totalnumber of images:Ψ =12020Xi=1Γi(Average Face)The result is the mean value image Ψ, which produces the blurred image of a face. Theaverage image of our set of face images can be seen in Figure 3.IntroductionEigenfacesFace RecognitionHome PageTitle PageJJ IIJ IPage 6 of 11Go BackFull ScreenCloseQuitDifferenceSubtracting any one of the images Γifrom the average image Ψ results in a vector thatdescribes the difference between the subtracted image and the average image. This givesus the variance, or in other words, the distinctive features that seperate one image fromthe rest such as a wider nose, or eyes that are closer together then the average face.This process allows us to remove the redundent freatures from each face matrix andthereby make the recognition process easier.Φi= Γi− Ψ (Deviation from the mean)Once we determined the deviation from the mean of each image, the images were placedinto a matrix A:φ1,1φ1,2. . . φ1,160............φ120,160φ120,2. . . φ120,160=⇒ [Φ1Φ2· · · Φ20]. =⇒ ACovariance MatrixOnce the common features were removed, we use the new matrix A (with the greatestof distinguishing features) to compute the dispersion of each image around the mean.This is the covariance matrix C. In other words, we compare the first row of elementsto the second row of elements, then the second row of elements with the third, the thirdrow with the fourth, and so on. This describes the correlation of elements amonst all 20images. The diagonal elements of the covariance matrix are the variance, they are theunique structure of the image. The off-diagonal elements are the redundant features inthe set of images.IntroductionEigenfacesFace RecognitionHome PageTitle PageJJ IIJ IPage 7 of 11Go BackFull ScreenCloseQuitC =12020Xn=1ΦnΦTn=120AATThis is a matrix that is spanned by the row space of A. This creates a very large matrixwith 19200 rows and 19200 columns.Eigenvalues and EigenvectorsUnfortunatley, the resulting matrix C = (1/20)AATis 19200 × 19200. This is a largematrix that would require much computing and processing time. However, the matrixATA is a 20 × 20 matrix, which is much easier to work with. Therefore, we simplifiedour matrix to ATA by using the mathematical relation between the eigenvectors ofAATand ATA. The singular values of AATand ATA are the nonzero square roots ofthe eigenvalues. By retaining the nonzero eigenvalues, a singular value decompositioncan be constructed revealing the relation between AATand ATA. We first calculatethe singular value decomposition of AATto find the eigenvalues and