211A_1_final2010

# 211A_1_final2010 - EE211A Fall Quarter 2010 Digital Image...

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Digital Image Processing I Fall Quarter, 2010 1 Final Exam Solutions 1. KL Transform (25 points) Consider a three-dimensional random vector u that has the following three equiprobable realizations: 0 1 2 2 1 1 1 , 2 , 1 1 1 2 u u u = = = . Let the KL transform of this random vector u be denoted by v. (a) (20 points) Find the transform coefficient energies. In other words find ( 29 ( 29 ( 29 2 2 2 0 , 1 , and 2 . E v E v E v (b) (5 points) Find the KL transform basis vector corresponding to the transform coefficient with the largest energy. 2. Image Transform (25 points) Consider the image ( 29 , u m n of size 256 by 256 consisting of an array of regular spaced, equal-strength delta functions as shown in Figure 1. (a) (18 points) Suppose that a 2D DFT of Figure 1 is taken to get ( 29 , v k l . Now, suppose that a zonal low-pass filter is applied to ( 29 , v k l . If operating on a window 0 k,l 255 where the origin is at a corner of the window, the zonal filter can be specified by identifying four different regions of the ( 29 , k l plane when doing the low-pass filtering because of the Fourier domain mapping of low, negative frequencies to high, positive frequencies. More specifically, assume the filter is such that all frequency components satisfying 2 2 16 or k l + ( 29 2 2 256 16 or k l - + ( 29 ( 29 2 2 256 256 16 or k l - + - ( 29 2 2 256 16 k l + - are passed with an amplification of 1 and all others are set to zero. (In other words, all frequency components within a distance of 16 from the origin of the k,l plane are set to zero.) After the appropriate frequency domain coefficients are set to zero, the inverse transformation of the filtered ( 29 , v k l is taken to get ( 29 2 , u m n . What is the ratio of the total energy in

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## This note was uploaded on 12/27/2011 for the course EE211A 211A taught by Professor Villasenor during the Spring '11 term at UCLA.

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211A_1_final2010 - EE211A Fall Quarter 2010 Digital Image...

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