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# l8 - CMPEN/EE455 – Lect 8 4 When f x,y is an image then...

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1 CMPEN/EE455 Lecture 8 Topics today: 1. 1-D Fourier Transform Properties 2. 2-D Fourier Transform 1-D Fourier-Transform Properties Shifting: f ( x - x o ) ←→ F ( u ) e - j 2 πux o Modulation: f ( x ) e j 2 πu o x ←→ F ( u - u o ) Scaling: f ( ax ) ←→ 1 | a | F ( u a ) Time/Space Convolution: f ( x ) * g ( x ) ←→ F ( u ) G ( u ) Correlation: f ( x ) g ( x ) ←→ F ( u ) G ( - u ) Multiplication(Modulation): f ( x ) g ( x ) ←→ F ( u ) * G ( u ) Conjugation f * ( x ) ←→ F * ( - u ) Diﬀerentiation d n dx n f ( x ) ←→ ( j 2 πu ) n F ( u ) Symmetry f ( x ) real ←→ F ( - u ) = F * ( u ) ←→ | F ( - u ) | = | F ( u ) | ←→ φ ( - u ) = - φ ( u )

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CMPEN/EE455 – Lect 8 2 In the table above, f * ( x ) = complex conjugate of f ( x ) F * ( u ) = complex conjugate of F ( u ) 3 Example: Proof of shifting property 3
CMPEN/EE455 – Lect 8 3 2-D Fourier Transform Pair Fourier transform easily extended to two dimensions: 2-D (forward) Fourier Transform: F ( u,v ) = F [ f ( x,y )] = Z -∞ Z -∞ f ( x,y ) e - j 2 π ( ux + vy ) dxdy 2-D Inverse Fourier Transform: f ( x,y ) = F - 1 [ F ( u,v )] = Z -∞ Z -∞ F ( u,v ) e + j 2 π ( ux + vy ) dudv Units of measure: x,y space (pixels) u,v (spatial) frequency (cycles/pixel)

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Unformatted text preview: CMPEN/EE455 – Lect 8 4 When f ( x,y ) is an image , then it is real and ≥ 0. F ( u,v ) = Re { F ( u,v ) } + j Im { F ( u,v ) } = R ( u,v ) + jI ( u,v ) ↑ ↑ real part imaginary part Also, in magnitude-phase form, F ( u,v ) = | F ( u,v ) | e j Φ( u,v ) where | F ( u,v ) | = r R 2 ( u,v ) + I 2 ( u,v ) Φ( u,v ) = tan-1 I ( u,v ) R ( u,v ) power spectrum or spectral density of f ( x,y ): P ( u,v ) = | F ( u,v ) | 2 = R 2 ( u,v ) + I 2 ( u,v ) CMPEN/EE455 – Lect 8 5 3 Example Let f ( x,y ) = δ ( x,y ) – 2-D impulse. F [ f ( x,y )] = Z ∞-∞ Z ∞-∞ δ ( x,y ) e-j 2 π ( ux + vy ) dxdy = ↑ impulse sift at ( x = 0 ,y = 0) 3 CMPEN/EE455 – Lect 8 6 Notes:...
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l8 - CMPEN/EE455 – Lect 8 4 When f x,y is an image then...

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