Pentland, Face Recognition using Eigenfaces, cvpr 1991 56 PCA General dimensionality reduction technique Preserves most of variance with a much more compact representation Lower storage requirements (eigenvectors a few numbers per face) Faster matching What are the proofing tools greek office 2010 problems for face recognition?
Fisherfaces, Belheumer., pami 1997 72 (No Transcript).
Find closest labeled face in database nearest-neighbor in K-dimensional space 49 Key Property of Eigenspace Representation Given 2 images that are used to construct the Eigenspace is the eigenspace projection of image Then, That is, distance in Eigenspace is approximately equal to the correlation between.Given m points in a n dimensional space, for large n, how does one project on to a low dimensional space while preserving broad trends in the data and allowing it to be visualized?14 2nd Principal Component, y2 1st Principal Component, y1 15, pCA Scores xi2 yi,1 yi,2 xi1 16, pCA Eigenvalues 2.As a result, we start picking the new basis vectors (new directions to project the data from the eigenvectors of the cov.The variance in each eigenvalue direction is lambda_i, so we sum the variance in the k direction and we require that it surpasses say 90 of the original variation.11 Change of basis x2 z1 Note that the vector 1 1 is longer than the vectors 1 0 or 0 1 hence the coefficient is still.Too many possible appearances!32 Principal Component Analysis (PCA) Linear transformation implied by PCA The linear transformation RN?Avv where eigenvalue, v eigenvector, pCA Method (6 step 4: Calculate the eigenvectors and eigenvalues of the covariance matrix.8, trick: Rotate Coordinate Axes, suppose we have a population measured on p random variables X1,Xp.Fisherfaces Reference Eigenfaces.How to predict/synthesis/match with novel views?Testing Steps (1) Testing Image: Preprocessing Transformed into Eigenface Components The weight describe the contribution of each eigenface in representing the testing image.
55 Recognition with eigenfaces Process labeled training images Find mean and covariance matrix S Find k principal components (eigenvectors of S) u1,uk Project each training image xi onto subspace spanned by principal components(wi1,wik) (u1T(xi, ukT(xi ) Given novel image x Project onto subspace(w1,wk) (u1T(x, ukT(x.
13, pCA: General From k original variables: x1,x2,.,xk: Produce k new variables: y1,y2,.,yk: y1 a11x1 a12x a1kxk y2 a21x1 a22x a2kxk.
57 Limitations Global appearance method not robust at all to misalignment not very robust to background variation, scale 58 Principal Component Analysis (PCA) Problems Background (de-emphasize the outside of the face.g., by multiplying the input image by a 2D Gaussian window centered on the.Presented by Miss Chayanut Petpairote.41 Problem: Size of Covariance Matrix A Suppose each data point is N-dimensional (N pixels) The size of covariance matrix A is N x N The number of eigenfaces is N Example: For N 256 x 256 pixels, Size of A will be x!42 Eigenfaces summary in words Eigenfaces are the eigenvectors of the covariance matrix of the probability distribution of the vector space of human faces Eigenfaces are the standardized face ingredients derived from the statistical analysis of many pictures of human faces A human face may.If scale is changed, the performance of recognition is very bad Face images are quite clear face and not occlusion References.23 PCA Example step 1 24 PCA Example step 1 zero mean data: data: x y.220.127.116.11 zero mean data: x y data: x y 25 PCA Example step 2 Calculate the covariance matrix since the non-diagonal elements in this covariance matrix.The eigenvectors with the largest eigenvalue results in the largest variance.54 Reconstruction P 4 P 200 P 400 Eigenfaces are computed using the 400 face images from ORL face database.Where K ltlt.Yk ak1x1 ak2x akkxk such that: yk's are uncorrelated (orthogonal) y1 explains as much as possible of original variance in data set y2 explains as much as possible of remaining variance etc.7, applications Uses: Examples: Data Visualization Data Reduction.