tailieunhanh - Digital Image Processing: Unitary Transforms - Duong Anh Duc

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. | 5/14/2020 4:35:05 AM Duong Anh Duc - Digital Image Processing Digital Image Processing Unitary Transforms 5/14/2020 4:35:05 AM 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 where A is a matrix of size MNxMN The transform A is unitary, iff If A is real-valued, ., A-1=A*, transform is „orthonormal“ 5/14/2020 4:35:05 AM Duong Anh Duc - Digital Image Processing Energy conservation with unitary transforms For any unitary transform we obtain Interpretation: every unitary transform is simply a rotation of the coordinate system. Vector lengths („energies“) are conserved. 5/14/2020 4:35:05 AM Duong Anh Duc - Digital Image Processing Energy distribution for unitary transforms Energy is conserved, but often will be unevenly distributed among coefficients. Autocorrelation matrix Mean squared values („average energies“) of the . | 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing Digital Image Processing Unitary Transforms 5/14/2020 5:31:15 AM 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 where A is a matrix of size MNxMN The transform A is unitary, iff If A is real-valued, ., A-1=A*, transform is „orthonormal“ 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing Energy conservation with unitary transforms For any unitary transform we obtain Interpretation: every unitary transform is simply a rotation of the coordinate system. Vector lengths („energies“) are conserved. 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing Energy distribution for unitary transforms Energy is conserved, but often will be unevenly distributed among coefficients. Autocorrelation matrix Mean squared values („average energies“) of the coefficients ci are on the diagonal of Rcc 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing Eigenmatrix of the autocorrelation matrix Definition: eigenmatrix F of autocorrelation matrix Rff F is unitary The columns of F form an orthonormalized set of eigenvectors of Rff, ., Rff F = FL is a diagonal matrix of eigenvalues. Rff is symmetric nonnegative definite, hence i 0 for all i Rff is normal matrix, ., , hence unitary eigenmatrix exists 5/14/2020 5:31:15 AM Duong Anh Duc - Digital Image Processing Karhunen-Loeve transform Unitary transform with matrix A = FH where the columns of F are ordered according to decreasing eigenvalues. Transform coefficients are pairwise uncorrelated Rcc = ARffAH = FHRffF = FHFL = L 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. 5/14/2020 5:31:15 AM Duong Anh Duc -