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l12 - 1 CMPEN/EE455 Lecture 12 Topics today 1 2-D Discrete...

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1 CMPEN/EE455 Lecture 12 Topics today: 1. 2-D Discrete Fourier Transform (DFT) Pair 2. 2-D DFT Properties, periodicity 2-D Discrete Fourier Transform (DFT) Pair — G&W Ch. 4.5.5, 4.6; 1992 G&W Ch. 3.3 Discrete Fourier Transform (DFT): F ( u, v ) = M - 1 X x =0 N - 1 X y =0 f ( x, y ) e - j 2 π ( ux M + vy N ) , u = 0 , 1 , 2 , . . . M - 1 , and v = 0 , 1 , 2 , . . . N - 1 Inverse Discrete Fourier Transform (IDFT): f ( x, y ) = 1 MN M - 1 X u =0 N - 1 X v =0 F ( u, v ) e + j 2 π ( ux M + vy N ) , x = 0 , 1 , 2 , . . . M - 1 , and y = 0 , 1 , 2 , . . . N - 1
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CMPEN/EE455 – Lect 12 2 Properties of 2-D DFT: Periodicity f ( x, y ) = f ( x + M, y + N ) F ( u, v ) = F ( u + M, v + N ) Shifting: f ( x - x o , y - y o ) ←→ F ( u, v ) e - j 2 π ( ux o ) M + ( vy o ) N Modulation: f ( x, y ) e j 2 π ( u o x ) M + ( v o y ) N ←→ F ( u - u o , v - v o ) Convolution: f ( x, y ) * g ( x, y ) ←→ F ( u, v ) G ( u, v ) A few more properties to follow .... DC Frequency { ( u, v ) = (0 , 0) } “High” frequencies ( | u | and | v | ) “large”): edges, region borders, details
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CMPEN/EE455 – Lect 12 3 Special case of modulation property: f ( x, y )( - 1) x + y ←--→ F ( u - M 2 , v - N 2 )
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CMPEN/EE455 – Lect 12 4 (Assume M = N below.)
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CMPEN/EE455 – Lect 12 5 In “Discrete Fourier Transform” world: M × N image f ( x, y ) a) has 2-D period M × N b) periodically extends to -∞ ≤ x, y ≤ ∞ In picture below f ( x, y ) = f ( x, y + N ) =
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