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1_HW1Solutions

1_HW1Solutions - EE 211A Fall Quarter 2011 Instructor John...

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Unformatted text preview: EE 211A Fall Quarter, 2011 Instructor: John Villasenor Digital Image Processing I Handout 5 Homework 1 Solutions 1. (a) f ( x, y ) = rect (ax + b) → f ( x, y) = f1 ( x) f 2 ( y) where f1 = rect (ax + b) and f 2 ( y) = 1. F (u, v) = F1 (u) F2 (v), b 1 ⎛ྎ u ⎞ྏ j 2π a u rect (ax + b) ← → F1 (u ) = sinc ⎜ྎ ⎟ྏe , ⎯ྎ |a| ⎝ྎ a ⎠ྏ 1 ← → δ (v) = F2 (v) ⎯ྎ b 1 ⎛ྎ u ⎞ྏ j 2π u sinc ⎜ྎ ⎟ྏe a δ (v) ⇒ F (u, v) = |a| ⎝ྎ a ⎠ྏ (b) Use FT property that f (ax ) ↔ 2 e −bx = e ⇒e ⎛ྎ b ⎞ྏ ⎟ྏ −π ⎜ྎ ⎜ྎ π x ⎟ྏ ⎝ྎ ⎠ྏ ⎛ྎ b ⎞ྏ −π ⎜ྎ ⎜ྎ π x ⎟ྏ ⎟ྏ ⎝ྎ ⎠ྏ F (u, v) = 2 ←⎯ྎ → b e b = f (ax ) where a = 2 π 1 ⎛ྎ u ⎞ྏ F ⎜ྎ ⎟ྏ. a ⎝ྎ a ⎠ྏ −π 2 u 2 b π b π 22 e −π u b . rect (v ) 1 2. The region of nonzero autocorrelation is: y x 3. The function g ( x, y ) has a value given by the area of overlap of two unit squares whose centers are offset by ( x, y ). When both x and y are displaced simultaneously, as occurs along the line y = x, the x dimension and y dimension of the area of overlap will decrease linearly, leading to a quadratic overall decrease in g ( x, y ). The maximum value of g ( x, y ) will occur at (0,0), where there is complete overlap between the two unit squares. At this location, g ( x, y ) will have value 1. 2 ...
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1_HW1Solutions - EE 211A Fall Quarter 2011 Instructor John...

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