Fourier Transform
Examples
1
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Example – Fourier transform properties
• Read in an
image
• Look at Fourier
transform
• Verify some
properties in
Table 4.1
clear all
close all
% Clear workspace of any old variables
% Close all open figures
% Read in a normal (real) image
I = imread('cameraman.tif');
F = fft2(I);
% Shift F so that zero frequency component is in the middle
Fshift = fftshift(F);
imshow(log(abs(Fshift)), []), impixelinfo
2
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Example – linearity of Fourier transform
• Read in two images
• Verify that Fourier transform is a linear operation
af1 ( x, y)  bf 2 ( x, y)  aF1 (u, v)  bF2 (u, v)
clear all
close all
% Clear workspace of any old variables
% Close all open figures
f1 = double(imread('cameraman.tif'));
F1 = fft2(f1);
f2 = double(imread('rice.png'));
F2 = fft2(f2);
a = 2;
b = 3;
Fsum1 = a*F1 + b*F2;
Fsum2 = fft2(a*f1 + b*f2);
figure, imshow(log(abs(fftshift(Fsum1))), []), impixelinfo
figure, imshow(log(abs(fftshift(Fsum2))), []), impixelinfo
3
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Example - differentiation property
• The differentiation property of the continuous 2D Fourier
transform is:
n
m
n
     
f
(
x
,
y
)

j
2

u
j
2

v
F (u, v)




   
 x   y 
m
• Show this for the case of the first derivative in the x
direction (i.e., m=1, n=0); i.e., show
f ( x, y)
  j 2 u  F (u, v)
x
4
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Differentiation property (continued)
• The inverse Fourier transform of F(u,v) is
f ( x, y)  F
1
 
F (u, v)    F (u, v) e j 2 uxvydu dv
 
• Taking the derivative with respect to x
5
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Differentiation property (continued)
• The inverse Fourier transform of F(u,v) is
f ( x, y)  F
1
 
F (u, v)    F (u, v) e j 2 uxvydu dv
 
• Taking the derivative with respect to x
 
f ( x, y ) 
j 2 ux  vy 

F
(
u
,
v
)
e
du dv


x
x  
 


F (u, v)
 
 


 j 2 ux  vy 
e
du dv
x
F (u, v)  j 2 u  e
j 2 ux  vy 
du dv
 
 


 
 j 2 u  F (u , v)  e j 2 ux  vy du dv
• So
f ( x, y)
  j 2 u  F (u, v)
x
 F 1  j 2 u  F (u , v) 
Colorado School of Mines
Department of Electrical Engineering and Computer Science
6
Differentiation property (continued)
• Think of the derivative as a filter that we can convolve
the image with.
• The Fourier pair of this is
f ( x, y )
 H (u , v) F (u , v)
x
• So in the frequency domain, we can perform the
derivative by multiplying by the H(u,v).
• From before, we saw that H(u,v) = j2u.
7
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Differentiation property (continued)
• Plot the spectrum (magnitude) of the filter |H(u,v)|
clear all
close all
400
M = 100;
N = 100;
300
[u,v] = meshgrid(-M/2:M/2, -N/2:N/2);
200
Hx = j*2*pi*u;
Habs = abs(Hx);
figure, imshow(Habs, []), impixelinfo
100
0
150
150
100
figure, surf(Habs), colormap jet;
100
50
50
0
0
8
Colorado School of Mines
Department of Electrical Engineering and Computer Science