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# 6_homework6 - the k th transform coefficient is given by...

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1 EE 211A Digital Image Processing I Fall Quarter, 2011 Handout 17 Instructor: John Villasenor Homework 6 Due: Thursday, November 10, 2011 1. For this problem we use a 2x2 matrix U defined as follows: a) Perform the SVD decomposition on the matrix U . Calculate the matrices such that b) Give the basis images of this SVD transform. Verify the result of a) using these basis images. 2. We have seen that for 1D Markov inputs, the KL transform becomes the DCT as the correlation ρ becomes 1. A critical question, however, concerns how fast this convergence occurs. In other words, for ρ values less than 1, how different are the transform domain energies between the KL and DCT? A zero mean 1 st order stationary Markov sequence can be generated using where is a zero-mean and white random variable. Derive, in terms of ρ , the variance of so that . For as described above, the autocorrelation matrix R is given as: Let be the KL transform of . We have seen that , the variance of

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Unformatted text preview: the k th transform coefficient , is given by the eigenvalue of the R matrix. Assume that the KL transform matrix is constructed according to decreasing eigenvalue magnitude. In other words, . 2 Denote the (i,j) element of a unitary matrix A as . Let denote the transform of obtained when transform matrix A is used. For an autocorrelation matrix as given above, the variance of is: The above summation can be used to compute the transform coefficient variances for any transform by using appropriate values for . Let , m = 1, … , N . where have been arranged in decreasing order. Note that gives the fraction of energy contained in the first m coefficients. • For N = 16, plot versus m for KLT, DCT, and DFT with ρ =0.95. Comment on the resulting plot. • For N = 16, plot log( of KLT – of DCT ) versus m with ρ =0.5, 0.9, 0.95, and 0.99. Comment on the resulting plot. (If you use MATLAB, you can use the command semilogy instead of taking logarithm of .)...
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6_homework6 - the k th transform coefficient is given by...

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