Transcript Digital Image Processing

Digital Image Processing
Filtering in the Frequency Domain
(Fundamentals)
Christophoros Nikou
[email protected]
University of Ioannina - Department of Computer Science
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Filtering in the Frequency Domain
Filter: A device or material for suppressing or
minimizing waves or oscillations of certain
frequencies.
Frequency: The number of times that a periodic
function repeats the same sequence of values
during a unit variation of the independent variable.
Webster’s New Collegiate Dictionary
C. Nikou – Digital Image Processing (E12)
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Jean Baptiste Joseph Fourier
Fourier was born in Auxerre,
France in 1768.
– Most famous for his work “La
Théorie Analitique de la
Chaleur” published in 1822.
– Translated into English in 1878:
“The Analytic Theory of Heat”.
Nobody paid much attention when the work
was first published.
One of the most important mathematical
theories in modern engineering.
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The Big Idea
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=
Any function that periodically repeats itself can
be expressed as a sum of sines and cosines of
different frequencies each multiplied by a
different coefficient – a Fourier series
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1D continuous signals
• It may be
considered both as
continuous and
discrete.
• Useful for the
representation of
discrete signals
through sampling
of continuous
signals.
x  x0
,
 ( x  x0 )  
 0 otherwise


f ( x) ( x  x0 )dx  f ( x0 )

C. Nikou – Digital Image Processing (E12)
1D continuous signals (cont.)
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Impulse train function
ST (t ) 

  (t  nT )
n 
x[n]  x(t )ST (t ) 


n 
n 
 x(t ) (t  nT )   x(nT ) (t  nT )
C. Nikou – Digital Image Processing (E12)
1D continuous signals (cont.)
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x[n]  x(t )ST (t )


 x(t ) (t  nT )
n 


 x(nT ) (t  nT )
n 
C. Nikou – Digital Image Processing (E12)
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1D continuous signals (cont.)
• The Fourier series expansion of a periodic
signal f (t).
f (t ) 

ce
n 
T /2
j
2
nt
T
n
1
cn 
f
(
t
)
e
T T/ 2
j
2
nt
T
dt
C. Nikou – Digital Image Processing (E12)
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1D continuous signals (cont.)
• The Fourier transform of a continuous
signal f (t).
F ( ) 


f (t )e
 j 2t
dt


f (t ) 

F (  )e j 2t d 

• Attention: the variable is the frequency (Hz) and
not the radial frequency (Ω=2πμ) as in the Signals
and Systems course.
C. Nikou – Digital Image Processing (E12)
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1D continuous signals (cont.)
sin(W )
f (t )  PW / 2 (t )  F (  )  W
(W )
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1D continuous signals (cont.)
• Convolution property of the FT.

f (t )* h(t ) 

f ( )h(t   )d

f (t )* h(t )  F (  ) H (  )
f (t )h(t )  F (  )* H (  )
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1D continuous signals (cont.)
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• Intermediate result
− The Fourier transform of the impulse train.

1
 (t  nT ) 

T
n 

n 

  


T 
n  
• It is also an impulse train in the frequency
domain.
• Impulses are equally spaced every 1/ΔΤ.
C. Nikou – Digital Image Processing (E12)
1D continuous signals (cont.)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
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Sampling
x[n]  x(t )ST (t )


 x(t ) (t  nT )
n 


 x(nT ) (t  nT )
n 
C. Nikou – Digital Image Processing (E12)
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1D continuous signals (cont.)
• Sampling
− The spectrum of the discrete signal consists of
repetitions of the spectrum of the continuous
signal every 1/ΔΤ.
− The Nyquist criterion should be satisfied.
f (t )  F (  )
1
f (nT )  f [n]  F (  ) 
T

n 

F  


T 

n 
C. Nikou – Digital Image Processing (E12)
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1D continuous signals (cont.)
Nyquist theorem
1
 2  max
T
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1D continuous signals (cont.)
FT of a continuous signal
Oversampling
Critical sampling with
the Nyquist frequency
Undersampling
Aliasing appears
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1D continuous signals (cont.)
• Reconstruction (under correct sampling).
F ( )  F ( ) H ( )
 t 
f (t )  f (t )* T sin c 


T


C. Nikou – Digital Image Processing (E12)
1D continuous signals (cont.)
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• Reconstruction
− Provided a correct sampling, the continuous
signal may be perfectly reconstructed by its
samples.
f (t ) 


n 
 (t  nT ) 
f (nT ) sinc 

n

T


C. Nikou – Digital Image Processing (E12)
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1D continuous signals (cont.)
• Under aliasing, the
reconstruction of
the continuous
signal not correct.
C. Nikou – Digital Image Processing (E12)
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1D continuous signals (cont.)
Aliased signal
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The Discrete Fourier Transform
• The Fourier transform of a sampled (discrete)
signal is a continuous function of the frequency.
1
F ( ) 
T

n 

F 



T


n 
• For a N-length discrete signal, taking N samples of
its Fourier transform at frequencies:
k
k 
, k  0,1,.., N  1
N T
provides the discrete Fourier transform (DFT) of
the signal.
C. Nikou – Digital Image Processing (E12)
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The Discrete Fourier Transform
(cont.)
• DFT pair of signal f [n] of length N.
N 1
F [k ]   f [n]e
j
2 nk
N
0  k  N 1
,
n 0
N 1
1
f [n]   F [k ]e
N n 0
j
2 nk
N
,
0  n  N 1
C. Nikou – Digital Image Processing (E12)
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The Discrete Fourier Transform
(cont.)
• Property
– The DFT of a N-length f [n] signal is periodic
with period N.
F [k  N ]  F[k ]
– This is due to the periodicity of the complex
exponential:
 wN 
n

 e

2
j
N
n

n
  wN  e

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j
2 n
N
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The Discrete Fourier Transform
(cont.)
• Property: sum of complex exponentials
1 N kn 1, k  rN , r 
wN  

otherwise
N n 0
0,
The proof is left as an exercise.
wN  e
j
2
N
C. Nikou – Digital Image Processing (E12)
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The Discrete Fourier Transform
(cont.)
• DFT pair of signal f [n] of length N may be
expressed in matrix-vector form.
N 1
F [k ]   f [n]w ,
nk
N
n 0
1
f [ n] 
N
wN  e
j
N 1
0  k  N 1
 F[k ]w
n 0
 nk
N
,
0  n  N 1
2
N
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The Discrete Fourier Transform
(cont.)
F = Af
  w0 0
 N
 1 0
wN 


A


 wN 1 0
 N
w 
w 
w 
w 
0 1
N
0 2
N
1 1
N
1 2
N
w  w 
N 1 1
N
N 1 2
N
w 
w 


1 N 1 
N



N 1 N 1 
 wN  
0
N
N 1
f =  f [0], f [1],..., f [ N  1] , F =  F [0], F [1],..., F [ N  1]
T
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T
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The Discrete Fourier Transform
(cont.)
Example for N=4
1 1 1 1 
1  j 1 j 

A
1 1 1 1


1 j 1  j 
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The Discrete Fourier Transform
(cont.)
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The inverse DFT is then expressed by:
-1
f =A F
 0
   wN 
 1 0
1  w 
  N
N 

  N 1 0
  wN 

0
A 1 
1
A

N

* T
w 
w 
0
N
1
1 1
N
w 
w 
0
N
2
1 2
N
w  w 
N 1 1
N
N 1 2
N
 w  
1 N 1 
 wN  


N

1
 wNN 1  
0
N
N 1
This is derived by the complex exponential sum
property.
C. Nikou – Digital Image Processing (E12)
* T









Linear convolution
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f [n]  {1, 2, 2}, h[n]  {1, 1}, N1  3, N2  2
g[n]  f [n]* h[n] 


f [m]h[n  m]
m 
is of length N=N1+N2-1=4
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Linear convolution (cont.)
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f [n]  {1, 2, 2}, h[n]  {1, 1}, N1  3, N2  2
g[n]  f [n]* h[n] 


f [m]h[n  m]
m 
f [ m]
n  0 h[0  m] 1
n  1 h [1  m]
n  2 h[2  m]
n  3 h [3  m]
1
2
2

1
1

1
1
1

1 1 
0
2
1
1
g[ n]
g[n]  {1, 1, 0,  2}
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Circular shift
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• Signal x[n] of length N.
• A circular shift ensures that the resulting
signal will keep its length N.
• It is a shift modulo N denoted by
x[(n  m) N ]  x[(n  m) mod N ]
• Example: x[n] is of length N=8.
x[(2) N ]  x[(2)8 ]  x[6]
x[(10)N ]  x[(10)8 ]  x[2]
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Circular convolution
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f [n]  {1, 2, 2}, h[n]  {1, 1}, N1  3, N2  2
g[n]=f [n]h[n] 


m 
f [m]h[(n  m) N ]
Circular shift modulo N
The result is of length N  max N1, N2  3
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Circular convolution (cont.)
f [n]  {1, 2, 2}, h[n]  {1, 1}, N1  3, N2  2
g[n]=f [n]h[n] 


m 
f [m]h[(n  m) N ]
f [ m]
1 2 2 g[ n]
n  0 h [(0  m) N ] 1 1
1
1
n  1 h [(1  m) N ]
1 1
1
n  2 h [(2  m) N ]
1 1
0
g[n]  { 1, 1, 0,}
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DFT and convolution
g[n]=f [n]h[n]  G[k ]  F[k ]H [k ]
• The property holds for the circular
convolution.
• In signal processing we are interested in
linear convolution.
• Is there a similar property for the linear
convolution?
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DFT and convolution (cont.)
g[n]=f [n]h[n]  G[k ]  F[k ]H [k ]
• Let f [n] be of length N1 and h[n] be of length N2.
• Then g[n]=f [n]*h[n] is of length N1+N2-1.
• If the signals are zero-padded to length N=N1+N2-1
then their circular convolution will be the same as
their linear convolution:
g[n]  f [n]* h[n]  G[k ]  F[k ]H[k ]
Zero-padded signals
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DFT and convolution (cont.)
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f [n]  {1, 2, 2}, h[n]  {1, 1}, N1  3, N2  2
Zero-padding to length N=N1+N2-1 =4
f [n]  {1, 2, 2, 0}, h[n]  {1, 1, 0, 0}
f [ m]
h[(n  0) 4 ] 0 0 1
h[(n  1) 4 ]
h[(n  2) 4 ]
h[(n  3) 4 ]
0
1
1
2
0
0 1 1
0 0 1
0
2 0 g[ n]
0 1
1
0
1
0
0
1
0
0 1
1
2
The result is the same as the linear convolution.
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DFT and convolution (cont.)
Verification using DFT
1 1 1 1  1   5 
1  j 1 j   2  1  j 2 
   

F = Af  
1 1 1 1  2   1 

  

1 j 1  j   0   1  j 2
1 1 1 1   1  0 
1  j 1 j   1 1  j 
   

H = Ah  
1 1 1 1  0   2 

  

1 j 1  j   0  1  j 
C. Nikou – Digital Image Processing (E12)
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DFT and convolution (cont.)
G[k ]  F[k ] H[k ]
Element-wise multiplication

(1 
G = FH  


(1 
5 0
  0 
j 2)  (1  j )  1  j3

  2 
1 2
 

j 2)  (1  j )  1  j3
C. Nikou – Digital Image Processing (E12)
DFT and convolution (cont.)
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Inverse DFT of the result
1 * T
g = A G = A 
4
1
1 1 1 1   0   1
1 j 1  j  1  j3  1


 
1 1 1 1  2   0 


  
1  j 1 j  1  j3  2
The same result as their linear convolution.
C. Nikou – Digital Image Processing (E12)
2D continuous signals
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, x  x0 , y  y0
 ( x  x0 , y  y0 )  
otherwise
 0
 

f ( x, y) ( x  x0 , y  y0 )dydx  f ( x0 , y0 )
 
Separable:  ( x  x0 , y  y0 )   ( x  x0 ) ( y  y0 )
C. Nikou – Digital Image Processing (E12)
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2D continuous signals (cont.)
The 2D impulse train is also separable:
SX Y ( x, y)  SX ( x) SY ( y) 


   ( x  nX , y  nY )
n  m 
C. Nikou – Digital Image Processing (E12)
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2D continuous signals (cont.)
• The Fourier transform of a continuous 2D
signal f (x,y).
F ( , ) 
 

f ( x, y)e
 j 2 (  x  vy )
dydx
 
 
f ( x, y) 
  F (, )e
j 2 (  x  vy )
 
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d d 
2D continuous signals (cont.)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
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• Example: FT of f (x,y)=δ(x)
F ( , ) 
y
f (x,y)=δ(x)
 

 j 2 (  x vy )

(
x
)
e
dydx

x
 





 j 2 x
 j 2 vy

(
x
)
e
dx
e
dy




e
 j 2 vy
dy   ( )

C. Nikou – Digital Image Processing (E12)
ν
F(μ,ν)=δ(ν)
μ
2D continuous signals (cont.)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
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• Example: FT of f (x,y)=δ(x-y)
F ( , ) 
f (x,y)=δ(x-y)
y
 

 j 2 (  x  vy )

(
x

y
)
e
dydx

x
 
 
  j 2 vy
 j 2 x
     ( x  y )e
dx e
dy
  




e
ν
F(μ,ν)=δ(μ+ν)

 j 2 y  j 2 y
e

dy 
 j 2 (   ) y
e
dy


  (   )
C. Nikou – Digital Image Processing (E12)
μ
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
45
2D continuous signals (cont.)
sin(W ) sin( W )
f ( x, y)   PW / 2,W / 2 ( x, y)  F (  , )  W
(W ) ( W )
2
C. Nikou – Digital Image Processing (E12)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
46
2D continuous signals (cont.)
• 2D continuous convolution
 
f ( x, y)* h( x, y) 

f ( x   , y   )h( ,  )d d 
 
• We will examine the discrete convolution
in more detail.
• Convolution property
f ( x, y)* h( x, y)  F ( , ) H (  , )
C. Nikou – Digital Image Processing (E12)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
47
2D continuous signals (cont.)
• 2D sampling is accomplished by
SX Y ( x, y ) 


   ( x  nX , y  nY )
n  m 
• The FT of the sampled 2D signal consists of
repetitions of the spectrum of the 1D
continuous signal.
1 1
F (  , ) 
X Y

m
n 

F  
, 


X
Y 

n 
C. Nikou – Digital Image Processing (E12)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
48
2D continuous signals (cont.)
• The Nyquist theorem involves both the
horizontal and vertical frequencies.
1
1
 2  max ,
 2vmax
X
Y
Over-sampled
Under-sampled
C. Nikou – Digital Image Processing (E12)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
49
Aliasing
C. Nikou – Digital Image Processing (E12)
50
Aliasing - Moiré Patterns
• Effect of sampling a scene with periodic or
nearly periodic components (e.g.
overlapping grids, TV raster lines and
stripped materials).
• In image processing the problem arises
when scanning media prints (e.g.
magazines, newspapers).
• The problem is more general than
sampling artifacts.
C. Nikou – Digital Image Processing (E12)
51
Aliasing - Moiré Patterns (cont.)
• Superimposed grid drawings (not digitized)
produce the effect of new frequencies not
existing in the original components.
C. Nikou – Digital Image Processing (E12)
52
Aliasing - Moiré Patterns (cont.)
• In printing industry the problem comes
when scanning photographs from the
superposition of:
• The sampling lattice (usually horizontal and
vertical).
• Dot patterns on the newspaper image.
C. Nikou – Digital Image Processing (E12)
53
Aliasing - Moiré Patterns (cont.)
C. Nikou – Digital Image Processing (E12)
54
Aliasing - Moiré Patterns (cont.)
• The printing
industry uses
halftoning to
cope with the
problem.
• The dot size is
inversely
proportional to
image intensity.
C. Nikou – Digital Image Processing (E12)
2D discrete convolution
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
55
m
m
f [m,n]
3 2
h [m,n]
1 1
1 -1
n
g[m, n]  f [m, n]* h[m, n] 

n

  f [k , l ]h[m  k , n  l ]
k  l 
• Take the symmetric of one of the signals with
respect to the origin.
• Shift it and compute the sum at every position
[m,n].
C. Nikou – Digital Image Processing (E12)
2D discrete convolution (cont.)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
56
n
n
f [m,n]
3 2
h [m,n]
1 1
1 -1
m
l
h [-k,-l]
l
g [0,0]=0
3 2
-1 1
1 1
m
k
h [1-k,1-l]
g [1,1]=0
3 2
-1 1
1 1
C. Nikou – Digital Image Processing (E12)
k
2D discrete convolution (cont.)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
57
n
n
f [m,n]
3 2
h [m,n]
1 1
1 -1
m
m
l
l
g [2,2]=3
h [2-k,2-l]
-1 13 2
1 1
h [3-k,2-l]
k
C. Nikou – Digital Image Processing (E12)
g [3,2]=-1
3 21
-1
1 1
k
2D discrete convolution (cont.)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
58
n
n
f [m,n]
3 2
h [m,n]
1 1
1 -1
m
n
3 5 2
3 -1 -3
g[m,n]
m
M1+M2-1=3
N1+N2-1=2
C. Nikou – Digital Image Processing (E12)
m
The 2D DFT
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
59
• 2D DFT pair of image f [m,n] of size MxN.
M 1 N 1
F [k , l ]    f [m, n]e
 km ln 
 j 2 
 
M N
m0 n 0
1
f [ m, n ] 
MN
M 1 N 1
  F [k , l ]e
 km ln 
j 2 
 
M
N

m 0 n 0
0  k  M  1
,

 0  l  N 1
0  m  M  1

 0  n  N 1
C. Nikou – Digital Image Processing (E12)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
60
The 2D DFT (cont.)
• All of the properties of 1D DFT hold.
• Particularly:
– Let f [m,n] be of size M1xN1 and h[m,n] of size
M2xN2 .
– If the signals are zero-padded to size (M1+M21)x(N1+N2-1) then their circular convolution will
be the same as their linear convolution and:
g[m, n]  f [m, n]* h[m, n]  G[k, l ]  F[k, l ]H[k, l ]
C. Nikou – Digital Image Processing (E12)