Digital Image Processing: Unitary Transforms - Duong Anh Duc present about Unitary Transforms; Energy conservation with unitary transforms; Karhunen-Loeve transform; Optimum energy concentration by KL transform; Basis images and eigenimages; Sirovich and Kirby method; Gender recognition using eigenfaces.
Digital Image Processing Unitary Transforms 21/11/15 Duong Anh Duc - Digital Image Processing Unitary Transforms Sort samples f(x,y) in an MxN image (or a rectangular block in the image) into colunm vector of length MN Compute transform coefficients c Af where A is a matrix of size MNxMN The transform A is unitary, iff A *T A A H Hermitian conjugate If A is real-valued, i.e., A1=A*, transform is „orthonormal“ 21/11/15 Duong Anh Duc - Digital Image Processing Energy conservation with unitary transforms For any unitary transform c Af we obtain H H H c c c f A Af f Interpretation: every unitary transform is simply a rotation of the coordinate system Vector lengths („energies“) are conserved 21/11/15 Duong Anh Duc - Digital Image Processing Energy distribution for unitary transforms Energy is conserved, but often will be unevenly distributed among coefficients Autocorrelation matrix H E cc Rcc H E Af f A H AR ff A H Mean squared values („average energies“) of the coefficients ci are on the diagonal of Rcc i Ec 21/11/15 Rcc i ,i AR ff A H i ,i Duong Anh Duc - Digital Image Processing Eigenmatrix of the autocorrelation matrix Definition: eigenmatrix of autocorrelation matrix Rff is unitary The columns of form an orthonormalized set of eigenvectors of Rff, i.e., Rff 0 is a diagonal matrix of eigenvalues MN Rff is symmetric nonnegative definite,H hence i 0 for all i H R ff R ff R ff R ff Rff is normal matrix, i.e., , hence unitary eigenmatrix exists 21/11/15 Duong Anh Duc - Digital Image Processing Karhunen-Loeve transform Unitary transform with matrix A= H where the columns of are ordered according to decreasing eigenvalues Transform coefficients are pairwise uncorrelated Rcc = ARffAH = HRff = H = Energy concentration property: No other unitary transform packs as much energy into the first J coefficients, where J is arbitrary Mean squared approximation error by choosing only first J coefficients is minimized 21/11/15 Duong Anh Duc - Digital Image Processing Optimum energy concentration by KL transform To show optimum energy concentration property, consider the truncated coefficient vector b IJc where IJ contain ones on the first J diagonal positions, else zeros Energy in first J coefficients for arbitrary transform A J E Tr Rbb Tr I J Rcc I J Tr I J AR ff A H I J akT R ff ak* k where akT is the k th row of A Lagrangian cost function to enforce unit-length basis vectors J J J L E k k akT ak* akT R ff ak* k k akT ak* k Differentiating L with respect to aj yields neccessary condition R ff a*j 21/11/15 * a J j j for all j Duong Anh Duc - Digital Image Processing Basis images and eigenimages For any unitary transform, the inverse transform f A c H can be interpreted in terms of the superposition of „basis images“ (columns of AH) of size MN If the transform is a KL transform, the basis images, which are the eigenvectors of the autocorrelation matrix Rff , are called „eigenimages.“ If energy concentration works well, only a limited number of eigenimages is needed to approximate a set of images with small error These eigenimages form an optimal linear subspace of dimensionality J 21/11/15 Duong Anh Duc - Digital Image Processing Computing eigenimages from a training set How to measure MNxMN autocorrelation matrix? Use training set , , , L Define training set matrix S and calculate L H R L l l l , , , L H SS L Problem 1: Training set size should be L >> MN If L < MN, autocorrelation matrix Rff is rank - deficient Problem 2: Finding eigenvectors of an MNxMN matrix Can we find a small set of the most important eigenimages from a small training set L