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211A_1_Midterm2009Solutions

# 211A_1_Midterm2009Solutions - EE211A Fall Quarter 2009...

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EE211A Digital Image Processing I Fall Quarter, 2009 Page 1 Midterm Solutions 1. 2D Convolution and Autocorrelation (20 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) (10 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. Answer:

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EE211A Digital Image Processing I Fall Quarter, 2009 Page 2 (b) (10 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. Answer:
EE211A Digital Image Processing I Fall Quarter, 2009 Page 3 2. 1D Fourier Transform (20 points) Find F ( u ) , the Fourier transform of the 1D function f ( x ) = δ ( x 2 - 1). You should express F ( u ) in its simplest form. Answer: ( ( ( ( 29 ( 29 ( 29 ( 29 2 ( 1) 1 1 2 1 2 1 x x x x x δ δ δ δ - = - + = - + - + Scaling property of Fourier Transform: 1 ( ( )) u FT f ax F a a = Using this property, δ 2 x - 1 ( 29 ( 29 + δ - 2 x + 1 ( 29 ( 29 1 2 e - 2 π iu + 1 2 e 2 π iu = cos2 π u Answer: f ( x ) = δ ( x 2 - 1) F ( u ) = cos2 π u

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EE211A Digital Image Processing I Fall Quarter, 2009 Page 4 3. 3D Linear, Shift-invariant Systems (20 points) Consider a linear, shift invariant system described by the following 3D (not 2D) difference equation: ( , , ) ( 1, , ) ( , 1, ) ( , 1, 1) ( , , 1) y k l m y k l m y k l m y k l m x k l m = - + - + - - + + a) (15 points) In the above 3D system, h R , the region of support of the impulse response, occupies three dimensions. The impulse response is the response of the system due to an input of the form x ( k , l , m ) = δ (0,0,0). Provide a 2D plot showing the intersection of h R with the 0 k = plane. In other words, your plot should show the
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