Human Face Recognition 
Linear discriminant Analysis  

Linear Discriminant Analysis Linear Discriminant analysis or Fisherfaces method overcomes the limitations of eigenfaces method by applying the Fisher’s linear discriminant criterion. This criterion tries to maximize the ratio of the determinant of the betweenclass scatter matrix of the projected samples to the determinant of the withinclass scatter matrix of the projected samples. Fisher discriminants group images of the same class and separates images of different classes. Images are projected from N^{2}dimensional space to C dimensional space (where C is the number of classes of images). For example, consider two sets of points in 2dimensional space that are projected onto a single line. Depending on the direction of the line, the points can either be mixed together (Figure 4a) or separated (Figure 4b). Fisher discriminants find the line that best separates the points. To identify an input test image, the projected test image is compared to each projected training image, and the test image is identified as the closest training image. As with eigenspace projection, training images are projected into a subspace. The test images are projected into the same subspace and identified using a similarity measure. What differs is how the subspace is calculated. Unlike the PCA method that extracts features to best represent face images; the LDA method tries to find the subspace that best discriminates different face classes as shown in Figure 4. The withinclass scatter matrix, also called intrapersonal, represents variations in appearance of the same individual due to different lighting and face expression, while the betweenclass scatter matrix, also called the extrapersonal, represents variations in appearance due to a difference in identity. By applying this method, we find the projection directions that on one hand maximize the distance between the face images of different classes on the other hand minimize the distance between the face images of the same class. In another words, maximizing the betweenclass scatter matrix S_{b}, while minimizing the withinclass scatter matrix Sw in the projective subspace. Figure 5 shows good and bad class separation.
Figure 4. (a) Points mixed when projected onto a line. (b) Points separated when projected onto another line
Figure 5 Figure 5. (a) Good class separation. (b) Bad class separation The withinclass scatter matrix Sw and the betweenclass scatter matrix Sb are defined as
Where Γ_{i}^{ j} is the i^{th} sample of class j, μ_{j } is the mean of class j, C is the number of classes, Nj is the number of samples in class j.
where μ represents the mean of all classes. The subspace for LDA is spanned by a set of vectors W = [W1, W2, … , Wd], satisfying
Figure 6. LDA approach for face recognition The within class scatter matrix represents how face images are distributed closely within classes and between class scatter matrix describes how classes are separated from each other. When face images are projected into the discriminant vectors W, face images should be distributed closely within classes and should be separated between classes, as much as possible. In other words, these discriminant vectors minimize the denominator and maximize the numerator in Equation (14). W can therefore be constructed by the eigenvectors of S_{w}^{1} Sb. Figure 7 shows the first 16 eigenvectors with highest associated eigenvalues of S_{w}^{1} Sb. These eigenvectors are also referred to as the fisherfaces. There are various methods to solve the problem of LDA such as the pseudo inverse method, the subspace method, or the null space method . The LDA approach is similar to the eigenface method, which makes use of projection of training images into a subspace. The test images are projected into the same subspace and identified using a similarity measure. The only difference is the method of calculating the subspace characterizing the face space. The face which has the minimum distance with the test face image is labelled with the identity of that image. The minimum distance can be calculated using the Euclidian distance method as given earlier in Equation (11). Figure 6 shows the testing phase of the LDA approach..
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