114_1_image_proc_exam_solns_2009

# 114_1_image_proc_exam_solns_2009 - EE114D Winter Quarter...

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EE114D Introduction to Speech and Image Processing Winter Quarter, 2009 Page 1 of 7 Image Processing Exam Solutions Notes: - For all problems please circle or otherwise clearly indicate your answers! Note: 2 log 2 1 = , 2 5 log 3 3 , 2 log 4 2 = 2 7 log 5 3 , 2 17 log 7 6 The use of calculators or other electronic devices with calculator-like functionality is not permitted on this test. 1. 2D Convolution (20 points) Consider the continuous 2D function ) , f( y x given below, which has value 1 where the dark regions are located and 0 elsewhere. Make a sketch showing the boundaries of non-zero regions of the self-convolution of ) , f( y x . Be sure to place your sketch on a set of marked axis such that relevant dimensions and positions of features are clearly indicated. Solution:

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EE114D Introduction to Speech and Image Processing Winter Quarter, 2009 Page 2 of 7 2. Integration (20 points) Find the total area under the function ( 29 2 2 sin ( ) x f x x π π = Solution: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 { } ( 29 ( 29 ( 29 ( 29 { } ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 { } 2 2 2 2 2 2 sin sinc sin ' : . . sinc sin 1 1 . . rect * rect rect 2 2 1 . . rect 4 total area under f x f x dx x dx x x x dx using Parseval s Theorem g x dx F T g x du here g x x x so F T g x u u u u u j j F T g x u π π π π δ π δ π π π +∞ -∞ +∞ -∞ +∞ -∞ +∞ +∞ -∞ -∞ = = = = = = - - + = - - + = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2
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