2D_FFT

2D_FFT - F u N v N Fee u v Feo u v W Foe u v W Foo u v W N...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
EEL-5820: Digital Image Processing Task: Fast Fourier Transform for 2D case. Input Function: f(x,y) sampled function in a square array, for x,y = 0,1,. ..N-1 The Fourier Transform for f(x,y) is ( 29 F u v N f x y W N ux vy y N x N ( , ) ( , ) = + = - = - 1 0 1 0 1 (I) where: W e N j N = - 2 π if N is in the form of N=2 n , then N can be expresed as: N=2M where M is a positive integer. Then Eq.(I). can be expressed as: [ ( 29 ( 29 ( 29 ( 29 ( 29 ] F u v M f x y W M f x y W W M f x y W W M f x y W W y M x M M ux vy y M x M M ux vy N v y M x M M ux vy N u y M x M M ux vy N u v ( , ) ( , ) ( , ) ( , ) ( , ) = + + + + + + + = - = - + = - = - + = - = - + = - = - + + 1 2 1 2 2 1 2 2 1 1 2 1 2 1 2 1 2 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 (II) October 3, 1994 Patricio Vidal
Background image of page 1

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

View Full DocumentRight Arrow Icon
Defining: ( 29 ( 29 ( 29 ( 29 Fee u v M f x y W Feo u v M f x y W Foe u v M f x y W Foo u v M f x y W y M x M M ux vy y M x M M ux vy y M x M M ux vy y M x M M ux vy ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) = = + = + = + + = - = - + = - = - + = - = - + = - = - + 1 2 2 1 2 2 1 1 2 1 2 1 2 1 2 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 (III) for u,v=0. .M-1, reduces Eq.(II) to: ( 29 F u v Fee u v Feo u v W Foe u v W Foo u v W N v N u N u v ( , ) ( , ) ( , ) ( , ) ( , ) = + + + + (IV) Since W M (a+N/2) =W M a and W N (a+N/2) =-W N a , from Eqs.(III) and (IV) we can obtain: ( 29 ( 29 ( 29 F u v N Fee u v Feo u v W Foe u v W Foo u v W F u N v Fee u v Feo u v W Foe u v W Foo u v W
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: F u N v N Fee u v Feo u v W Foe u v W Foo u v W N v N u N u v N v N u N u v N v N u N u v ( , / ) ( , ) ( , ) ( , ) ( , ) ( / , ) ( , ) ( , ) ( , ) ( , ) ( / , / ) ( , ) ( , ) ( , ) ( , ) + =-+-+ = +--+ + =--+ + + + 2 2 2 2 (V) Therefore, the Fourier Transform for a 2D funtion f(x,y) of NxN points can be obtained by simple operations (additions and multiplications) of four Fourier Transforms of N/2xN/2 points (Eq.(III)-(V)). Also, each one of this four Fourier Transforms can be obtained by simple operations of four Fourier Transforms of N/4xN/4 points. This procedure can be done until the number points of each Fourier Transform is 2x2 . October 3, 1994 Patricio Vidal...
View Full Document

Page1 / 2

2D_FFT - F u N v N Fee u v Feo u v W Foe u v W Foo u v W N...

This preview shows document pages 1 - 2. Sign up to view the full document.

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