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# l10 - CMPEN/EE455 – Lect 10 4 2-D Convolution h x,y = f...

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1 CMPEN/EE455 Lecture 10 Topics today: 1. 2-D Fourier transform properties Properties of 2-D Fourier Transform Shifting: f ( x - x o , y - y o ) ←→ F ( u, v ) e - j 2 π ( ux o + vy o ) Modulation: f ( x, y ) e j 2 π ( u o x + v o y ) ←→ F ( u - u o , v - v o ) Scaling: f ( ax, by ) ←→ 1 | ab | F ( u a , v b ) Convolution: f ( x, y ) * g ( x, y ) ←→ F ( u, v ) G ( u, v ) Multiplication: f ( x, y ) g ( x, y ) ←→ F ( u, v ) * G ( u, v ) (Modulation) Conjugation f * ( x, y ) ←→ F * ( - u, - v ) Symmetry f ( x, y ) real ←→ F ( - u, - v ) = F * ( u, v ) ←→ | F ( - u, - v ) | = | F ( u, v ) | ←→ φ ( - u, - v ) = - φ ( u, v )

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CMPEN/EE455 – Lect 10 2 DC Frequency { ( u, v ) = (0 , 0) } “High” frequencies ( | u | and | v | ) “large”): edges, region borders, details Rotation: If f ( x, y ) rotated an angle θ , then F ( u, v ) also rotated θ . Let x 0 , y 0 be the rotated spatial coordinates: x 0 = x cos θ + y sin θ y 0 = - x sin θ + y cos θ Thus, if f ( x, y ) rotated θ to f ( x 0 , y 0 ), Then, F ( u, v ) rotated θ to F ( u 0 , v 0 ). where u 0 = u cos θ + v sin θ v 0 = - u sin θ + v cos θ are the rotated frequency variables.
CMPEN/EE455 – Lect 10 3

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Unformatted text preview: CMPEN/EE455 – Lect 10 4 2-D Convolution h ( x,y ) = f ( x,y ) * g ( x,y ) = Z ∞-∞ Z ∞-∞ f ( α,β ) g ( x-α,y-β ) dαdβ Also, h ( x,y ) = g ( x,y ) * f ( x,y ) Image f ( x,y ) is put through a linear space-invariant ﬁlter g ( x,y ) → h ( x,y ) is ﬁltered image. Comments: 1. 2-D convolution often denoted as f ( x,y ) * * g ( x,y ) 2. Mask operations are often 2-D convolutions. 3. Hard to draw examples! CMPEN/EE455 – Lect 10 5 Convolution Property: f ( x,y ) * g ( x,y ) ↔ F ( u,v ) G ( u,v ) Convolution in one domain ↔ Multiplication in other domain....
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