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INCORPORATING INVARIANTS IN MAHALANOBIS DISTANCE-BASED CLASSIFIERS: APPLICATIONS TO FACE 187 RECOGNITION are first histogram equalized and then shifted and scaled such that the mean value of all pixels in the face region is zero and the standard deviation is one.â Each recognition algorithm calculates subspaces and fits parameters using the preprocessed training images and knowledge of the identity of the individuals in the images. Then, using those parameters, each algorithm constructs a matrix consisting of the distances between each pair of images in the testing set of 640 images. Thus, in the training phase, one can calculate the mean image, µk, of an individual, but in the testing phase, the algorithm has no information about the identity of the individuals in the images. We developed three recognition algorithms: the first consists of the general techniques of Section II combined with minor modifications to fit the test task. We developed the second two algorithms after observing that the CSU algorithms based on angular distance perform best (see Fig. 2). In Section II we supposed that we would have several examples of each class, making an estimate of each class mean µk plausible, but for the task defined by the CSU evaluation procedure, we must simply provide 640Ã640 interimage distances. The most obvious method for fitting our classification approach within this distance-based framework is to define the distance between image Yk and Yl as the Mahalanobis distance Note, however, that this distance is not symmetric, since the augmented covariance is only relevant to one of the two images. Consequently, the symmetrized distance is used for the distance matrix. After observing that of the CSU algorithms, those based on angular distance perform best (see Fig. 2), we developed two additional algorithms. The âMahalanobis Angleâ distance is with symmetrized version Instead of symmetrizing d1(Yk, Yl), we also define the symmetric distance where Evaluating each of the first two distances on the test set of 640 images takes about 30 minutes on a 2.2 GHz Pentium III. We found that the second distance performed better than the first. Because we estimated that evaluating the third distance would take about 160 hours, we instead implemented a hybrid, constructed by computing and then computing only for those distance below some threshold (further detail may be found in [4]). Each of our algorithms operates in a subspace learned from the training data and uses an estimated covariance, associated with each image Yk. We list the key ideas here: ⢠Use the training data (which includes image identities) to calculate raw within-class sample covariances, . Regularize the raw covariances as follows: (1) Do an eigenvalue-eigenvector decomposition to find (2) Sum the eigenvalues, (3) Set Cw= which has no eigenvalues less than δS. ⢠Conceptually convolve the test image with a Gaussian kernel that has mean zero and variance where h is an adjustable parameter in the code that must be an odd integer. Change variables to transfer differentiation from the image to the kernel. Evaluate the matrices Vk and by convolving (using FFT methods) differentiated kernels with the image. Thus α, δ, and h are three adjustable parameters in the estimate of Ck. We investigated the dependence of the performance on these parameters [4], and chose the values α=100, h=11, and δ=0.0003. Our experiments indicated that the classification performance was not sensitive to small changes in these choices. Results are displayed in Fig. 2 and Fig. 3. Each of our algorithms performs better than all of the algorithms in the CSU package. IV. CONCLUSIONS We have presented techniques for constructing classifiers that combine statistical information from training data with tangent approximations to known transformations, and we demonstrated the techniques by applying them to a face recognition task. The techniques we created are a significant step forward from the work of Simard et al. due to the careful use of the curvature term for the control of the approximation errors implicit in the procedure. For the face recognition task we used a five parameter group of invariant transformations consisting of rotation, shifts, and scalings. On the face test case, a classifier based on our techniques has an error rate more than 20% lower than that of the best algorithm in a reference software distribution. The improvement we obtained is surprising because our techniques handle rotation, shifts, and scalings, but we also preprocessed the FERET data with a program from CSU that centers, rotates, and scales each image based on measured eye coordinates. While our techniques may compensate for errors in the measured eye coordinates or weaknesses in
INCORPORATING INVARIANTS IN MAHALANOBIS DISTANCE-BASED CLASSIFIERS: APPLICATIONS TO FACE 188 RECOGNITION Fig. 2. Approximate distributions for the rank one recognition performance of the algorithms. For each algorithm, a Gaussian is plotted with a mean and variance estimated by a Monte-Carlo study. Note that the key lists the algorithms in order of decreasing mean of the distributions; the first three are the algorithms described in Section III, and the remainder are those implemented in the CSU software distribution. Fig. 3. The mean recognition rate and 95% confidence intervals as a function of rank for the following algorithms: hybrid (the hybrid of and ), (the symmetrized Mahalanobis Angle with tangent augmentation), (the symmetrized Mahalanobis Angle with no tangent augmentation, illustrating the benefit obtained from the regularization of ), (the symmetrized Mahalanobis distance), and LDA Correlation (the best performing algorithm in the CSU distribution).
