Transcript The Fourier Transform Jean Baptiste Joseph Fourier Image Operations in Different Domains 1) Gray value (histogram) domain - Histogram stretching, equalization, specification, etc... 2)

The Fourier Transform
Jean Baptiste Joseph Fourier
Image Operations in Different Domains
1) Gray value (histogram) domain
- Histogram stretching, equalization, specification, etc...
2) Spatial (image) domain
- Average filter, median filter, gradient, laplacian, etc…
3) Frequency (Fourier) domain
+
Original histogram
3500
3000
3 X 3 Average
3000
2500
2500
Noisy image
(Salt & Pepper noise) Original image f
2000
2000
1500
1500
1000
1000
Blurry Image
500
0
5 X 5 Average
=
Equalized histogram
3500
0
50
100
Laplacian
150
200
250
Gradient magnitude
2
f Sharpened
f x 2  f yImage
500
0
0
7 X 7 Average
50
100
150
Median
200
250
A sum of sines and cosines
=
3 sin(x)
A
+ 1 sin(3x)
B
+ 0.8 sin(5x)
C
+ 0.4 sin(7x)
D
A+B
A+B+C
A+B+C+D
Higher frequencies due
to sharp image variations
(e.g., edges, noise, etc.)
The Continuous Fourier Transform

f ( x)   F(u)e
 2iux
du
u  
e
2iux
 cos( 2ux)  i sin( 2ux)
Complex Numbers
Imaginary
Z=(a,b)
b

Real
a
i  1
Z  Re( Z )  i Im( Z )
 a  ib
 Z e i
(i 2  1)
i
e  cos  i sin 
(a unit vector)
Z  a 2  b2
(Fourier spectrum)
  tg 1 (b / a )
(phase)
Z  Z *  a  ib  Z e i (conjugate )
The 1D Basis Functions e
Re( e
Im( e
2iux
)  cos( 2ux)
2iux
)  sin( 2ux)
2iux
cos2ux 
1
x
1/u
– The wavelength is 1/u .
– The frequency is u .
The Continuous Fourier Transform
1D Continuous Fourier Transform:

The Inverse
1
 ( F (u )) Fourier
Transform
f ( x)   F(u)e
 2iux
du 
F (u )   f(x)e
 2iux
dx  ( f ( x))
u  

The Fourier
Transform
x  
2D Continuous Fourier Transform:

f ( x, y )  

 F(u, v)e
u   v  


F (u, v)  
dudv
The Inverse Transform
 f(x, y)e
x   y  
 2i ( ux  vy)
 2i ( ux vy)
dxdy
The Transform
The 2D Basis Functions e
2i ( ux  vy )
V
u=-2, v=2
u=-1, v=2
u=0, v=2
u=1, v=2
u=2, v=2
u=-2, v=1
u=-1, v=1
u=0, v=1
u=1, v=1
u=2, v=1
U
u=-2, v=0
u=-1, v=0
u=0, v=0
u=1, v=0
u=2, v=0
u=-2, v=-1
u=-1, v=-1
u=0, v=-1
u=1, v=-1
u=2, v=-1
u=-2, v=-2
u=-1, v=-2
u=0, v=-2
u=1, v=-2
u=2, v=-2
The wavelength is 1/ u 2  v 2 .
The direction is u/v .
Discrete Functions
f(x)
f(n) = f(x0 + nDx)
f(x0+2Dx)
f(x0+3Dx)
f(x0+Dx)
f(x0)
x0
x0+Dx x0+2Dx x0+3Dx
0
1
2
3
...
N-1
The discrete function f:
{ f(0), f(1), f(2), … , f(N-1) }
The Discrete Fourier Transform
1D Discrete Fourier Transform:
1
F (u ) 
N
f ( x) 
 2iux
N 1

x  0
(u = 0,..., N-1)
 2iu x
N 1

u  0
N
f ( x )e
F (u )e
N
(x = 0,..., N-1)
2D Discrete Fourier Transform:
 2i (
N 1 M 1
1 1
F (u, v) 
  f ( x, y )e
N M x 0 y 0 (u = 0,..., N-1;
N 1 M 1
f ( x, y )    F (u, v)e
u 0 v 0
ux vy

)
N M
v = 0,…,M-1)
 2i (
ux vy

)
N M
(x = 0,..., N-1; y = 0,…,M-1)
The Fourier Image
Image f
Fourier spectrum |F(u,v)|
log(1 + |F(u,v)|)
Frequency Bands
Image
Fourier Spectrum
Percentage of image power enclosed in circles (small to large) :
90%, 95%, 98%, 99%, 99.5%, 99.9%
Low pass Filtering
90%
95%
98%
99%
99.5%
99.9%
Noise Removal
Noisy image
Fourier Spectrum
Noise-cleaned image
High Pass Filtering
Original
High Pass Filtered
High Frequency Emphasis
Original
+
High Pass Filtered
High Frequency Emphasis
Original
Original
High Frequency Emphasis
High Frequency
Emphasis
High Frequency Emphasis
Original
High Frequency
Emphasis
High pass Filter
High Frequency Emphasis
+
Histogram Equalization
Rotation
2D Image
Fourier Spectrum
2D Image - Rotated
Fourier Spectrum
Fourier Transform -- Examples
Image
Domain
Frequency
Domain
Fourier Transform -- Examples
Image
Fourier spectrum