211A_1_final2009sol

# 211A_1_final2009sol - EE211A Fall Quarter 2009 Digital...

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EE211A Digital Image Processing I Fall Quarter, 2009 1 Final Exam Solutions 1. 2D DFT (20 points) The above 256x256 image contains a periodic pattern of 64x64 squares of intensity 255 on a background of intensity 0. The center (not the upper left corner) of the above image is the (0, 0) location. The square centered at (0, 0) goes from . All the other squares are the same size and are centered at locations offset from the central square (both vertically and horizontally) by integer multiples of 128. Identify all the locations where , the 2D DFT of will be nonzero. Answer: The image in the space domain can be understood as a 2D rect of dimension 64x64 convolved with a 2D array of delta functions with spacing (128, 128) in each dimension. Convolution in the space domain corresponds to multiplication in the frequency domain. The nonzero locations of the 2D DFT of the image above can be answered by determining the nonzero locations of the 2D DFT of the array of delta functions multiplied by the 2D DFT of the rect. An array of delta functions with space domain spacing 128 transforms into an array of delta functions with a spacing of 256/128 = 2 pixels in each dimension in the frequency domain. A 64x64 rect is transformed into a 2D sinc function with nulls spaced at all and multiples of 256/64 = 4 (except for ).

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EE211A Digital Image Processing I Fall Quarter, 2009 2 When a function that is nonzero only at = multiple of 2 is multiplied by a function that is zero at all multiples of 4 except 0, the result is a function that is nonzero in the following places: All combinations where is one of 0, ± 2, ± 6, ± 10, ± 14, ± 18 . . . . ± 126 etc. AND is one of 0, ± 2, ± 6, ± 10, ± 14, ± 18 . . . . ± 126. Note: This answer can also be equivalently expressed using the range from 0 to 255 by taking the above values modulo 256.
EE211A Digital Image Processing I Fall Quarter, 2009 3 2. Transforms ( 20 points) Consider a 1D transform defined as follows: a) (5 points) In matrix notation, the transform can be described as

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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_final2009sol - EE211A Fall Quarter 2009 Digital...

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