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

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
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 )
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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)
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
This is the end of the preview. Sign up to access the rest of the document.

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 ( u,v ) where | F ( u,v ) | = r R 2 ( u,v ) + I 2 ( u,v ) ( 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 ) = ( x,y ) 2-D impulse. F [ f ( x,y )] = Z - Z - ( x,y ) e-j 2 ( ux + vy ) dxdy = impulse sift at ( x = 0 ,y = 0) 3 CMPEN/EE455 Lect 8 6 Notes:...
View Full Document

This note was uploaded on 09/24/2009 for the course EE 455 taught by Professor Staff during the Fall '08 term at Pennsylvania State University, University Park.

Page1 / 6

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online