211A_1_Midterm2007

211A_1_Midterm2007 - 3 2D Unitary DFT(25 points Let the 2D...

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EE211A Digital Image Processing I Fall Quarter, 2007 Page 1 Midterm Note: For all problems please circle or otherwise clearly indicate your answers! 1. 2D Convolution and Autocorrelation (25 points) Consider the continuous 2D function f(x,y) given below, which has value 1 where the dark regions are located and 0 elsewhere. (a) (14 points) Make a sketch showing the boundaries of non-zero regions of the function g(x,y) , where g(x,y) = f(x,y) * f(x,y) . Be sure to place your sketch on a set of marked axis such that relevant dimensions and positions of features of g(x,y) are clearly indicated. (b) (11 points) Make a sketch showing the boundaries of non-zero regions of the autocorrelation of f(x,y) . Again, your answer must be clearly indicated for you to receive credit. 2. Integration (25 points) Evaluate the integral below. 2 sin 2 2 jt e t dt t π π −∞
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Unformatted text preview: 3. 2D Unitary DFT (25 points) Let the 2D unitary DFT of the 2D matrix Φ be Γ as indicated below. As is customary, the origin is the upper left element. 2 D DFT a b c d A B C D e f g h E F G H i j k l I J K L m n o p M N O P Φ = →Γ = Find the 2D unitary IDFT (not the forward DFT) of the matrix Ψ below in terms of the elements of Φ . Ψ = * * * * * * * * * * * * * * * * A B C D E F G H I J K L M N O P 4. DCT (25 points) Let the 1D unitary DCT of the 1D vector ( ) [ ] u n a b c d e f g h = be given by [ ] ( ) v k A B C D E F G H = . The underbars under the a and A indicate the origin locations; e.g. the n=0 and k=0 positions, respectively. Find the DCT of ( ) [ ] y n h g f e d c b a = in terms of the elements of ( ) v k ....
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