Transcript CS589-04 Digital Image Processing Lecture 3. Filtering in the Frequency Domain Spring 2008 New Mexico Tech.

CS589-04 Digital Image Processing
Lecture 3. Filtering in the
Frequency Domain
Spring 2008
New Mexico Tech
Outline
►
Fourier Transform
►
Filtering in Fourier Transform Domain
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2
Fourier Series and Fourier Transform: History
►
Jean Baptiste Joseph Fourier, French mathematician and physicist
(03/21/1768-05/16/1830)
Orphaned: at nine
Egyptian expedition
with Napoleon I:
1798
Governor of Lower
Egypt
http://en.wikipedia.org/wiki/Joseph_Fourier
Permanent
Secretary of the
French Academy of
Sciences: 1822
Théorie analytique
de la chaleur :
1822
(The Analytic
Theory of Heat)
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Fourier Series and Fourier Transform: History
►
Fourier Series
Any periodic function can be expressed as the sum of sines
and /or cosines of different frequencies, each multiplied by
a different coefficients
►
Fourier Transform
Any function that is not periodic can be expressed as the
integral of sines and /or cosines multiplied by a weighing
function
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Fourier Series: Example
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5
Preliminary Concepts
j  1, a complex number
C  R  jI
the conjugate
C*  R - jI
| C | R 2  I 2 and   arctan( I / R)
C | C | (cos   j sin  )
Using Euler's formula,
C | C | e j
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Fourier Series
A function f (t ) of a continuous variable t that is periodic
with period, T , can be expressed as the sum of sines and
cosines multiplied by appropriate coefficients
f (t ) 

ce
n 
j
2 n
t
T
n
where
2 n
j
t
1 T /2
T
cn  
f (t )e
dt
T T /2
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for n  0, 1, 2,...
7
Impulses and the Sifting Property (1)
A unit impulse of a continuous variable t located
at t =0, denoted  (t ), defined as
if t  0

 (t )  
if t  0
0
and is constrained also to satisfy the identity



 (t )dt  1
The sifting property



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


f (t ) (t  t0 )dt  f (t0 )
f (t ) (t ) dt  f (0)
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Impulses and the Sifting Property (2)
A unit impulse of a discrete variable x located
at x =0, denoted  ( x), defined as
if x  0
1
 ( x)  
if x  0
0
and is constrained also to satisfy the identity

  ( x)  1
x 
The sifting property


f ( x) ( x)  f (0)


x 
f ( x) ( x  x0 )  f ( x0 )
x 
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Impulses and the Sifting Property (3)
impulse train sT (t ),
sT (t ) 
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
  (t  nT )
n 
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Fourier Transform: One Continuous Variable
The Fourier Transform of a continous function f (t )
F (  )  { f (t )}  


f (t )e j 2t dt
The Inverse Fourier Transform of F (  )

f (t )   {F (  )}   F (  )e j 2t d 
1

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Fourier Transform: One Continuous Variable
F ( )  


f (t )e
 j 2 t
dt  
W /2
W /2
Ae  j 2t dt
A
A
 j 2 t W /2
e

e jW  e  jW 


W /2
j 2
j 2 W
sin(W )
 AW
(W )
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Fourier Transform: Impulses
The Fourier transform of a unit impulse located at the origin:

F (  )    (t )e j 2t dt

 e j 2 0
=1
The Fourier transform of a unit impulse located at t  t0 :

F (  )    (t  t0 )e  j 2t dt

 e  j 2t0
=cos(2 t0 )  j sin (2 t0 )
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Fourier Transform: Impulse Trains
sT (t ) 
Impulse train sT (t ),

  (t  nT )
n 
The Fourier series:
sT (t ) 

c
n 
n
e
j
2 n
t
T
where
1
cn 
T
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
T /2
T /2
sT (t )e
j
2 n
t
T
dt
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Fourier Transform: Impulse Trains
 j 21Tn t T /2  j 2Tjn2tTn t j 2t 1 T /2 j 2Tnjt2Tnt j 2t
s (et )e e dt = dt  e(t )e e dt dt
cn e T 
T /2 T
/2
T T 

 1  1
=

0
e

n

j
2

(


T
TT) t
n
 e
dt   (  
)

T


1
2 n n 
2 n n

  ( j T t ) 1   ( j  t )e j 2t du
 e T T
sT (t )  c n e T  
T n
n 2 nt
e
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j
T
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Fourier Transform: Impulse Trains
Let S (  ) denote the Fourier transform of the
periodic impulse train ST (t )
 1
S (  )  ST (t )   
 T

e
n 
j
2 n
t
T



2 n

j
t

1
T

  e

T n 

1 
n
=
 ( 
)

T n 
T
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Fourier Transform and Convolution
The convolution of two functions is denoted
by the operator
f (t )
 f (t )
h(t )  


f ( )h(t   )d


h(t )    f ( ) h(t   ) d e  j 2t dt

 
 



=  f ( )  h(t   )e  j 2t dt d
 



=


f ( )  H (  )e  j 2 d
=H (  ) 


f ( )e  j 2 d
=H (  ) F (  )
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Fourier Transform and Convolution
Fourier Transform Pairs
f (t ) h(t )  H ( ) F (  )
f (t )h(t )  H (  ) F (  )
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Fourier Transform of Sampled Functions
f (t )  f (t ) sT (t )



f (t ) (t  nT )
n 
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Fourier Transform of Sampled Functions
 
F (  )   f (t )   f (t ) sT (t )  F (  ) ? S (  )

F (  )  F1(  ) S (  )  n F ( ) S (    )d

S ( ) 

(


)

 T
1T n 

n
=
F ( )   (    
)d


T
T
n 
1  
n
=
F ( ) (   
)d



T n 
T
1 
n
=
F ( 
)

T n 
T
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Question
The Fourier transform of the
sampled function (shown in the
following figure) is
1. Continuous
2. Discrete
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Fourier Transform of Sampled Functions
►
A bandlimited signal is a signal whose Fourier transform
is zero above a certain finite frequency. In other words, if
the Fourier transform has finite support then the signal is
said to be bandlimited.
An example of a simple bandlimited signal is a sinusoid of
the form,
x(t )  sin(2 ft   )
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Fourier Transform of Sampled Functions
max
F ( ) 
1
T

n
F ( 
)

T
n 
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max
Over-sampling
1
 2  max
T
Critically-sampling
1
 2  max
T
under-sampling
1
 2  max
T
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Nyquist–Shannon sampling theorem
►
►
We can recover f (t ) from its sampled version if we can
isolate a copy of F (  ) from the periodic sequence of copies
of this function contained in F (  ), the transform of the
sampled function f (t )
Sufficient separation is guaranteed if
1
 2  max
T
Sampling theorem: A continuous, band-limited function
can be recovered completely from a set of its samples if
the samples are acquired at a rate exceeding twice the
highest frequency content of the function
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Nyquist–Shannon sampling theorem
?

f (t )   F (  )e j 2t d 

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Aliasing
If a band-limited function is sampled at a rate that is less
than twice its highest frequency?
The inverse transform will yield a corrupted function. This
effect is known as frequency aliasing or simply as
aliasing.
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Aliasing
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Aliasing
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Function Reconstruction from Sampled Data
F ( )  H ( ) F ( )
1
f (t )  
1

F ( )
H ( )F ( )
 h(t ) f (t )
f (t ) 


n 
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f (nT )sinc  (t  nT ) / nT 
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The Discrete Fourier Transform (DFT) of One
Variable
M 1
F (  )   f ( x)e j 2 x / M ,
 0,1,..., M  1
x 0
1
f ( x) 
M
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M 1
j 2 x / M
F
(

)
e
,

x  0,1, 2,..., M  1
 0
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2-D Impulse and Sifting Property: Continuous
The impulse  (t , z ),

 
and

 

 (t , z )  
0
if t  z  0
otherwise
 (t , z )dtdz  1
The sifting property

 



 
f (t , z ) (t , z )dtdz  f (0, 0)
and
 
 
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f (t , z ) (t  t0 , z  z0 )dtdz  f (t0 , z0 )
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2-D Impulse and Sifting Property: Discrete
The impulse  ( x, y),
1
 ( x, y)  
0
if x  y  0
otherwise
The sifting property



f ( x, y ) ( x, y )  f (0, 0)
x  y 
and



x  y 
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f ( x, y ) ( x  x0 , y  y0 )  f ( x0 , y0 )
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2-D Fourier Transform: Continuous
F (  , )  



 
f (t , z )e
 j 2 ( t  z )
dtdz
and
f (t , z )  



 
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f (  , )e j 2 ( t  z ) d  d
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2-D Fourier Transform: Continuous

F (  , )  


 

T /2

f (t , z )e  j 2 ( t  z ) dtdz
Z /2
T /2  Z /2
Ae  j 2 ( t  z ) dtdz
 sin(T )   sin( T ) 
 ATZ 
   T 

T



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2-D Sampling and 2-D Sampling Theorem
2  D impulse train:
sT Z (t , z ) 
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

   (t  mT , z  nZ )
m  n 
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2-D Sampling and 2-D Sampling Theorem
Function f (t , z ) is said to be band-limited if its Fourier transform
is 0 outside a rectangle established by the intervals [-max ,max ]
and [- max , max ], that is
F (  , )  0 for |  | max and |  |  max
Two-dimensional sampling theorem:
A continuous, band-limited function f (t , z ) can be recovered with
no error from a set of its samples if the sampling intervals are
T<
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1
2max
and Z<
1
2 max
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2-D Sampling and 2-D Sampling Theorem
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Aliasing in Images: Example
In an image system, the
number of samples is fixed at
96x96 pixels. If we use this
system to digitize checkerboard
patterns …
Under-sampling
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Aliasing in Images: Example
Re-sampling
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Aliasing in Images: Example
Re-sampling
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Moiré patterns
►
Moiré patterns are often an undesired artifact of images
produced by various digital imaging and computer graphics
techniques
e. g., when scanning a halftone picture or ray tracing a
checkered plane. This cause of moiré is a special case of
aliasing, due to under-sampling a fine regular pattern
http://en.wikipedia.org/wiki/Moiré_pattern
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Moiré patterns
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Moiré patterns
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Moiré patterns
A moiré pattern
formed by
incorrectly downsampling the
former image
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2-D Discrete Fourier Transform and Its
Inverse
DFT:
M 1 N 1
F (  , )    f ( x, y )e
 j 2 (  x / M  y / N )
x 0 y 0
  0,1, 2,..., M  1;  0,1, 2,..., N  1;
f ( x, y ) is a digital image of size M  N.
IDFT:
1
f ( x, y) 
MN
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M 1 N 1
 F (, )e
j 2 (  x / M  y / N )
x 0 y 0
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Properties of the 2-D DFT
relationships between spatial and frequency intervals
Let T and Z denote the separations between samples,
then the seperations between the corresponding discrete,
frequency domain variables are given by
1
 
M T
1
and
 
N Z
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Properties of the 2-D DFT
translation and rotation
f ( x, y )e j 2 ( 0 x / M  0 y / N )  F (   0 ,  0 )
and
f ( x - x0 , y - y0 )  F (  , )e
 j 2 (  x0 / M  y0 / N )
Using the polar coordinates
x  r cos  y=rsin  =cos
 = sin 
results in the following transform pair:
f (r ,   0 )  F (,   0 )
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Properties of the 2-D DFT
periodicity
2  D Fourier transform and its inverse are infinitely periodic
F (  , )  F (   k1M , )  F ( ,  k2 N )  F (  k1M ,  k2 N )
f ( x, y)  f ( x  k1M , y)  f ( x, y  k2 N )  f ( x  k1M , y  k2 N )
f ( x)e
j 2 ( 0 x / M )
0  M / 2,
f ( x, y)(1)
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 F (  0 )
f ( x)(1)  F (  M / 2)
x
x y
 F (  M / 2,  N / 2)
48
Properties of the 2-D DFT
periodicity
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Properties of the 2-D DFT
Symmetry
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Properties of the 2-D DFT
Fourier Spectrum and Phase Angle
2-D DFT in polar form
F (u, v) | F (u, v) | e j (u ,v )
Fourier spectrum
1/2
| F (u, v) |  R (u, v)  I (u, v) 
2
2
Power spectrum
P(u, v) | F (u, v) |2  R 2 (u, v)  I 2 (u, v)
Phase angle
 I (u, v) 
 (u,v)=arctan 

R
(
u
,
v
)


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Example: Phase Angles
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Example: Phase Angles and The Reconstructed
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2-D Convolution Theorem
1-D convolution
M 1
f ( x ) h ( x )   f ( m) h ( x  m)
m 0
2-D convolution
M 1 N 1
f ( x, y ) h( x, y )   f (m, n )h( x  m, y  n )
m 0 n 0
x  0,1,2,..., M  1; y  0,1,2,..., N  1.
f ( x, y ) h( x, y )  F (u, v) H (u, v)
f ( x, y )h( x, y )  F (u, v) H (u, v)
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An Example of Convolution
Mirroring h
about the
origin
Translating
the mirrored
function by x
Computing the
sum for each
x
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An Example of Convolution
It causes the
wraparoun
d error
It can be
solved by
appending
zeros
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Zero Padding
►
Consider two functions f(x) and h(x) composed of A and B
samples, respectively
►
Append zeros to both functions so that they have the same
length, denoted by P, then wraparound is avoided by
choosing
P ≥A+B-1
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Zero Padding
►
Let f(x,y) and h(x,y) be two image arrays of sizes A×B and
C×D pixels, respectively. Wraparound error in their
convolution can be avoided by padding these functions
with zeros
 f ( x, y )
f p ( x, y )  
 0
0  x  A -1 and 0  y  B -1
 h ( x, y )
h p ( x, y )  
 0
0  x  C -1 and 0  y  D -1
A  x  P or B  y  Q
C  x  P or D  y  Q
Here P  A  C  1; Q  B  D  1
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Summary
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Summary
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Summary
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Summary
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The Basic Filtering in the Frequency Domain
Why is the spectrum at
almost ±45 degree stronger
than the spectrum at other
directions?
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The Basic Filtering in the Frequency Domain
►
Modifying the Fourier transform of an image
►
Computing the inverse transform to obtain the processed
result
g ( x, y )  1{H (u, v) F (u, v)}
F (u, v) is the DFT of the input image
H (u, v) is a filter function.
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The Basic Filtering in the Frequency Domain
►
In a filter H(u,v) that is 0 at the center of the transform
and 1 elsewhere, what’s the output image?
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The Basic Filtering in the Frequency Domain
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The Basic Filtering in the Frequency Domain
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Zero-Phase-Shift Filters
1
g ( x, y)   {H (u, v) F (u, v)}
F (u, v)  R(u, v)  jI (u, v)
1
g ( x, y)  
 H (u, v)R(u, v)  jH (u, v)I (u, v)
Filters affect the real and imaginary parts equally,
and thus no effect on the phase.
These filters are called zero-phase-shift filters
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Examples: Nonzero-Phase-Shift Filters
Even small
in the phase angle
can ishave
Phasechanges
angle is
Phase angle
by undesirable) effects
dramaticmultiplied
(usually
on the
multiplied
by filtered
0.5
0.5
output
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Summary:
Steps for Filtering in the Frequency Domain
1. Given an input image f(x,y) of size MxN, obtain the
padding parameters P and Q. Typically, P = 2M and Q = 2N.
2. Form a padded image, fp(x,y) of size PxQ by
appending the necessary number of zeros to f(x,y)
3. Multiply fp(x,y) by (-1)x+y to center its transform
4. Compute the DFT, F(u,v) of the image from step 3
5. Generate a real, symmetric filter function*, H(u,v), of
size PxQ with center at coordinates (P/2, Q/2)
*generate from a given spatial filter, we pad the spatial filter, multiply the expadded
array by (-1)x+y, and compute the DFT of the result to obtain a centered H(u,v).
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Summary:
Steps for Filtering in the Frequency Domain
6. Form the product G(u,v) = H(u,v)F(u,v) using array
multiplication
7. Obtain the processed image


g p ( x, y )  real  1 G (u, v) (1) x  y
8. Obtain the final processed result, g(x,y), by extracting
the MxN region from the top, left quadrant of gp(x,y)
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An Example:
Steps for Filtering in the Frequency Domain
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Correspondence Between Filtering in the
Spatial and Frequency Domains (1)
Let H(u) denote the 1-D frequency domain Gaussian filter
H (u)  Ae
-u 2 /2 2
The corresponding filter in the spatial domain
h( x)  2 Ae
2 2 2 x2
1. Both components are Gaussian and real
2. The functions behave reciprocally
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Correspondence Between Filtering in the
Spatial and Frequency Domains (2)
Let H (u ) denote the difference of Gaussian filter
H (u )  Ae
- u 2 /212
 Be
- u 2 /2 22
with A  B and  1   2
The corresponding filter in the spatial domain
h( x)  2 1 Ae
2 212 x 2
 2 2 Ae
2 2 22 x2
High-pass filter or low-pass filter ?
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Correspondence Between Filtering in the
Spatial and Frequency Domains (3)
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Correspondence Between Filtering in the
Spatial and Frequency Domains: Example
600x600
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Correspondence Between Filtering in the
Spatial and Frequency Domains: Example
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Generate H(u,v)
0  x  599 and 0  y  599
 f ( x, y)
f p ( x, y)  
600  x  602 or 600  y  602
 0
h( x, y)
hp ( x, y)  
0
0  x  2 and 0  y  2
3  x  602 or 3  y  602
Here P  A(600)  C (3)  1  602;
Q  B(600)  D(3)  1  602.
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Generate H(u,v)
1. Multiply hp ( x, y) by (-1)x y to center the frequency domain filter
2. Compute the forward DFT of the result in (1)
3. Set the real part of the resulting DFT to 0 to account for
parasitic real parts
4. Multiply the result by (-1)u v , which is implicit when h( x, y)
was moved to the center of hp ( x, y).
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Image Smoothing Using Filter Domain Filters:
ILPF
Ideal Lowpass Filters (ILPF)
1 if D(u, v)  D0
H (u, v)  
0 if D(u, v)  D0
D0 is a positive constant and D(u, v) is the distance between a point (u, v)
in the frequency domain and the center of the frequency rectangle
2 1/2
D(u, v)  (u  P / 2)  (v  Q / 2) 
2
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Image Smoothing Using Filter Domain Filters:
ILPF
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ILPF Filtering Example
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ILPF
Filtering
Example
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The Spatial Representation of ILPF
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Image Smoothing Using Filter Domain Filters:
BLPF
Butterworth Lowpass Filters (BLPF) of order n and
with cutoff frequency D0
H (u , v ) 
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1   D(u, v) / D0 
2n
87
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The Spatial Representation of BLPF
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Image Smoothing Using Filter Domain Filters:
GLPF
Gaussian Lowpass Filters (GLPF) in two dimensions is given
H (u, v)  e
 D2 ( u ,v )/2 2
By letting   D0
H (u, v)  e
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 D 2 ( u ,v )/2 D02
90
Image Smoothing Using Filter Domain Filters:
GLPF
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Examples of smoothing by GLPF (1)
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Examples of smoothing by GLPF (2)
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Examples of smoothing by GLPF (3)
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Image Sharpening Using Frequency Domain
Filters
A highpass filter is obtained from a given lowpass filter
using
H HP (u, v)  1  H LP (u, v)
A 2-D ideal highpass filter (IHPL) is defined as
0 if D(u, v)  D0
H (u, v)  
1 if D(u, v)  D0
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Image Sharpening Using Frequency Domain
Filters
A 2-D Butterworth highpass filter (BHPL) is defined as
1
H (u, v) 
2n
1   D0 / D(u, v)
A 2-D Gaussian highpass filter (GHPL) is defined as
H (u , v )  1  e
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 D2 ( u ,v )/2 D02
97
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The Spatial Representation of Highpass
Filters
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Filtering Results by IHPF
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Filtering Results by BHPF
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Filtering Results by GHPF
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Using Highpass Filtering and Threshold for
Image Enhancement
BHPF
(order 4 with a cutoff
frequency 50)
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The Laplacian in the Frequency Domain
H (u, v)  4 2 (u 2  v2 )
H (u, v)  4 2 (u  P / 2) 2  (v  Q / 2) 2 ) 
 4 2 D 2 (u, v)
The Laplacian image
2 f ( x, y)  1 H (u, v) F (u, v)
Enhancement is obtained
g ( x, y)  f ( x, y)  c2 f ( x, y)
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c  -1
104
The Laplacian in the Frequency Domain
The enhanced image
F (u, v)  H (u, v) F (u, v)
 1 1  H (u, v)  F (u, v)
1
g ( x, y )  


 1 1  4 2 D 2 (u, v)  F (u, v)
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The Laplacian in the Frequency Domain
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Unsharp Masking, Highboost Filtering and
High-Frequency-Emphasis Fitering
gmask ( x, y)  f ( x, y)  f LP ( x, y)
f LP ( x, y)  1  HLP (u, v)F (u, v)
Unsharp masking and highboost filtering
g ( x, y)  f ( x, y)  k * g mask ( x, y)


g ( x, y)  1 1  k *1  H LP (u, v) F (u, v)
 1 1  k * H HP (u, v) F (u, v)
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Unsharp Masking, Highboost Filtering and
High-Frequency-Emphasis Fitering
g ( x, y )  1  k1  k2 * H HP (u, v)  F (u, v)
k1  0 and k2  0
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Gaussian Filter
D0=40
High-Frequency-Emphasis Filtering
Gaussian Filter
K1=0.5, k2=0.75
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Homomorphic Filtering
f ( x, y)  i( x, y)r ( x, y)
= i( x, y) r ( x, y)
 f ( x, y) 
?
z ( x, y)  ln f ( x, y)  ln i( x, y)  ln r ( x, y)
z( x, y)  ln f ( x, y)  ln i( x, y)  ln r( x, y)
Z (u, v)  Fi (u, v)  Fr (u, v)
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Homomorphic Filtering
S (u, v)  H (u, v) Z (u, v)
 H (u, v) Fi (u, v)  H (u, v) Fr (u, v)
s( x, y)  1 S (u, v)
 1 H (u, v) Fi (u, v)  H (u, v) Fr (u, v)
 1 H (u, v) Fi (u, v)  1 H (u, v) Fr (u, v)
 i '( x, y)  r '( x, y)
g ( x, y)  es ( x, y )  ei '( x, y )er '( x, y)  i0 ( x, y)r0 ( x, y)
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Homomorphic Filtering
The illumination component of an image generally is
characterized by slow spatial variations, while the
reflectance component tends to vary abruptly
These characteristics lead to associating the low
frequencies of the Fourier transform of the logarithm of an
image with illumination the high frequencies with
reflectance.
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Homomorphic Filtering
H (u, v)  ( H   L ) 1  e

 c  D 2 ( u , v )/ D02 


 
 L
Attenuate the contribution
made by illumination and
amplify the contribution made
by reflectance
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 L  0.25
Homomorphic
H  2
Filtering
c 1
D0  80
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Homomorphic Filtering
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Selective Filtering
Non-Selective Filters:
operate over the entire frequency rectangle
Selective Filters
operate over some part, not entire frequency rectangle
• bandreject or bandpass: process specific bands
• notch filters: process small regions of the frequency
rectangle
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Selective Filtering:
Bandreject and Bandpass Filters
H BP (u, v)  1  H BR (u, v)
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Selective Filtering:
Bandreject and Bandpass Filters
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Selective Filtering:
Notch Filters
Zero-phase-shift filters must be symmetric about the origin.
A notch with center at (u0, v0) must have a corresponding
notch at location (-u0,-v0).
Notch reject filters are constructed as products of highpass
filters whose centers have been translated to the centers of
the notches.
Q
H NR (u, v)   H k (u, v) H  k (u, v)
k 1
where H k (u, v) and H - k (u, v) are highpass filters whose centers are
at (uk , vk ) and (-uk , -vk ), respectively.
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Selective Filtering:
Notch Filters
Q
H NR (u, v)   H k (u, v) H  k (u, v)
k 1
where H k (u, v) and H - k (u, v) are highpass filters whose centers are
at (uk , vk ) and (-uk , -vk ), respectively.
A Butterworth notch reject filter of order n



1
1
H NR (u, v)   
2n  
2n 
k 1 1   D0 k / Dk (u , v)   1   D0 k / D k (u , v)  



3
2 1/2
Dk (u, v)  (u  M / 2  uk )  (v  N / 2  vk ) 
2
2 1/2
D k (u, v)  (u  M / 2  uk )  (v  N / 2  vk ) 
2
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Examples:
Notch
Filters (1)
A Butterworth notch
reject filter D0 =3
and n=4 for all
notch pairs
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Examples:
Notch Filters
(2)
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