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Digital Image Processing Unitary Transforms 21/11/15 Duong Anh Duc - Digital Image Processing 1 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 1 T H Hermitianconjugate If A is real-valued, i.e., A­1=A*, transform is „orthonormal“ 21/11/15 Duong Anh Duc - Digital Image Processing 2 Energy conservation with unitary transforms For any unitary transform c Af c 2 cHc f H AH Af we obtain f 2 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 3 Energy distribution for unitary transforms Energy is conserved, but often will be unevenly distributed among coefficients. Autocorrelation matrix Rcc E ccH E Af f H AH ARff AH Mean squared values („average energies“) of the coefficients ci are on the diagonal of Rcc E c2 Rcc i,i ARff AH i,i 21/11/15 Duong Anh Duc - Digital Image Processing 4 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 0 1 0 MN 1 is a diagonal matrix of eigenvalues. R is symmetric nonnegative definite, hence 0 for all i R is normal matrix, i.e., Rff Rff Rff Rff , hence unitary eigenmatrix exists 21/11/15 Duong Anh Duc - Digital Image Processing 5 ... - tailieumienphi.vn
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