KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
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IMAGE ENHANCEMENT
IN THE FREQUENCY DOMAIN (1)
KOM3212 Image Processing in Industrial
Systems
Some of the contents are adopted from
R. C. Gonzalez, R. E. Woods, Digital Image Processing, 2nd edition, Prentice Hall, 2008
KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
Questions
In-depth understanding
• Why do we need to conduct image processing in the frequency
• domain?
• What does Fourier series do?
Properties
• Is FT a linear or nonlinear process?
• What would the FT of a rotated image look like?
• What is FFT?
• What is F(0,0)?
• Why is image padding necessary?
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
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Fourier series
The Fourier transform can separate the
frequencies which contribute to the signal
which is emitted from the image slice.
Crucially, it also tells us the amplitude of
those waves, which will correspond to
signal intensity levels in an image.
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
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Fourier series vs Fourier Series
• Any function that periodically repeats itself can be expressed as the sum of
sines and/or cosines of different frequencies, each multiplied by a different
coefficient (Fourier series).
• Even functions that are not periodic (but whose area under the curve is finite)
can be expressed as the integral of sines and/or cosines multiplied by a
weighting function (Fourier transform).
KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
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What is the frequency domain
• The frequency domain
refers to the plane of the
two dimensional discrete
Fourier transform of an
image.
• The purpose of the Fourier
transform is to represent a
signal as a linear
combination of sinusoidal
signals of various
frequencies.
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
Continuous FT
• The one-dimensional Fourier transform and its inverse
• Fourier transform (continuous case)
F (u )  


f ( x)e  j 2ux dx where j   1
• Inverse Fourier transform:

f ( x)   F (u )e j 2uxdu
e j  cos   j sin 

• The two-dimensional Fourier transform and its inverse
• Fourier transform (continuous case)
F (u, v)  



 
f ( x, y)e  j 2 (uxvy ) dxdy
• Inverse Fourier transform:
f ( x, y)  



 
F (u, v)e j 2 (uxvy ) dudv
KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
Discrete FT (DFT)
• The one-dimensional Fourier transform and its inverse
• Fourier transform (discrete case) DTC
1
F (u ) 
M
M 1
 f ( x )e
 j 2ux / M
for u  0,1,2,..., M  1
x 0
• Inverse Fourier transform:
M 1
f ( x)   F (u )e j 2ux / M
u 0
for x  0,1,2,..., M  1
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
Discrete FT (DFT)
• F(u) can be expressed in polar coordinates:
F (u )  F (u ) e j (u )

2
2
where F (u )  R (u )  I (u )

1 
1
2
(magnitude or spectrum)
I (u ) 
 (u )  tan 
(phase angle or phase spectrum)

 R(u ) 
• R(u): the real part of F(u)
• I(u): the imaginary part of F(u)
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
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Discrete FT (DFT)
e j  cos   j sin 
discrete Fourier transform can be redefined
1
F (u ) 
M
M 1
 f ( x)[cos 2ux / M  j sin 2ux / M ]
x 0
• Frequency domain: the domain (values of u) over which the values of F(u)
range; because u determines the frequency of the components of the
transform.
KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
Discrete FT (DFT)
• The two-dimensional Fourier transform and its inverse
• Fourier transform (discrete case) DTC
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F (u , v) 
MN
M 1N 1
  f ( x, y)e  j 2 (ux / M vy / N )
x 0 y 0
u  0,1,2,..., M  1, v  0,1,2,..., N  1
• Inverse Fourier transform:
f ( x, y ) 
M 1N 1
  F (u, v)e j 2 (ux / M vy / N )
u 0 v 0
x  0,1,2,..., M  1, y  0,1,2,..., N  1
• u, v : the transform or frequency variables
• x, y : the spatial or image variables
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
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Discrete FT (DFT)
• We define the Fourier spectrum, phase angle, and power spectrum as follows:

F (u, v)  R (u, v)  I (u, v)
2
2

1
2
( spectrum)
 I (u, v) 
 (u, v)  tan 
(phase angle)

 R(u, v) 
1
2
P(u,v)  F (u, v)  R 2 (u, v)  I 2 (u, v) (power spectrum)
• R(u,v): the real part of F(u,v)
• I(u,v): the imaginary part of F(u,v)
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
Discrete FT (DFT)
magnitude
phase
KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
Some properties of Fourier transform
ℱ  f ( x, y )(1)
x y
1
F (0,0) 
MN

M
N
 F (u  , v  ) (shift)
2
2
M 1N 1
  f ( x, y )
(average)
x 0 y 0
F (u , v)  F * (u ,v)
(conjugate symmetric)
F (u , v)  F (u ,v)
(symmetric)
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
Some properties of Fourier transform
Spatial Domain (, )
Linearity
c1 f x, y   c2 g x, y 
f ax, by 
Scaling
Shifting
f  x  x0 , y  y0 
Shifting for a period
f  x, y (1x  y )
c1F u, v   c2G u, v 
1 u v
F , 
ab  a b 
e  j 2 (ux0 vy0 ) F u , v 
F u  M / 2, u  N / 2
| F u, v  || F  u,v  |
Symmetry
Convolution
Frequency Domain (, )
f  x, y   g  x, y 
F u, v G u, v 
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
Convolution in Fourier transform
Spatial Domain (x)
Frequency Domain (u)
g  f h
g  fh
G  FH
G  F H
So, we can find g(x) by Fourier transform
g

IFT
G
f

FT

F
h
FT

H
KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
FT examples
rect(x) function
sinc(x)=sin(x)/x
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
FT examples
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
FT examples
• Typically, we visualize |F(u,v)|
Typically, we visualize |F(u,v)|
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
FT examples
DFT
DFT
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
FT examples
Sine wave
2D Gaussian
function
Its DFT
Its DFT
KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
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Image processing in the Frequency domain
f  x, y (1x  y )
g  x, y (1x  y )
KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
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Image processing in the Frequency domain
• Noise reduction in the frequency domain
KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
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Image processing in the Frequency domain
1. Take the FT of f(x):
2. Remove undesired frequencies:
3. Convert back to a signal
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
Image processing in the Frequency domain
• We want a smoothed function of f(x)
g x   f x  hx 
• Let us use a Gaussian kernel
h x 
 1 x2 
1
h x  
exp 
2
2

2 


• Then
 1
2 2




H u  exp  2u  
 2

Gu   F u H u 

x
H u 
1
2
u
H(u) attenuates high frequencies in F(u) (Low-pass Filter)!
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KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
Image vs. its frequency domain representation
Magnitude of the FT
Does not look anything like what we have seen
KOM3212 Image Processing in Industrial Systems | Dr Muharrem Mercimek
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Image vs. its frequency domain representation
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Image processing in the Frequency domain