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211A_1_final2008 - EE211A Fall Quarter 2008 Digital Image...

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EE211A Digital Image Processing I Fall Quarter, 2008 1 Final Exam 1. 2D DFT (20 points) The above 128x128 image consists of two equal amplitude sinusoids added to a gray background. The intensity of the background before the addition of the sinusoids was 0.5. 0 represents black and 1 is white. Find the location of the non- zero elements in the two-dimensional DFT of the above image. 2. SVD Transform (20 points) For this problem consider the 4 × 4 matrix U given as follows: 1 0 0 2 0 1 0 0 0 0 2 0 1 0 0 2 U = Perform the SVD decomposition on the matrix U . For your answer, specify (as numbers, not as expressions or equations) all of the elements of each of the three matrices 1/ 2 , , and T Ψ Λ Φ such that 1/ 2 T U = ΨΛ Φ .
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EE211A Digital Image Processing I Fall Quarter, 2008 2 3. KL Transform (20 points) Consider a random vector u that has three possible realizations as below. 0 1 1 1 u     =       , 1 1 1 0 u = - , 2 1 1 2 u =
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