Computer Vision –
2D Signal & System
Hanyang University
Jong-Il Park
Notation and definitions
 One-dimensional signal
 Continuous signal : f ( x), u ( x), s(t ),...
 Sampled signal : un , u (n),....
 Two-dimensional signal



Continuous signal : u ( x, y), v( x, y), f ( x, y),...
Sampled signal : um,n , v(m, n), u (i, j ),...
 i, j, k, l, m, n, … are usually used to specify integer
indices
Separable form : f ( x, y)  f ( x) f ( y)
Department of Computer Science and Engineering, Hanyang University
Delta function
 2-D delta function
 Dirac :  ( x, y )   ( x) ( y )
 Property

 

 
f ( x' , y' ) ( x  x' , y  y' )dx' dy'  f ( x, y),

lim  0 


 


 ( x, y)dxdy  1
Scaling :  (ax)   ( x) / | a |,
 (ax, by )   ( x, y ) / | ab |,
Kronecker delta :  (m, n)   (m) (n)
 Property
x(m, n) 




  x(m' , n' ) (m  m' , n  n' ),    (m, n)  1
m '  n ' 
m   n  
Department of Computer Science and Engineering, Hanyang University
Special signals
 Some special signals(or functions)
Department of Computer Science and Engineering, Hanyang University
Linear and shift invariant systems
H[ ]
x(m,n)
y(m,n)=H[x(m,n)]
 Linearity
H[a1x1 (m, n)  a2 x2 (m, n)]  a1H[ x1 (m, n)]  a2 H[ x2 (m, n)]
 a1 y1 (m, n)  a2 y2 (m, n), for a1 , a2 , x1 (), x2 ()
• Output of linear systems
y(m, n)  H [ x(m, n)]  H [ x(m' , n' ) (m  m' , n  n' )]
m'
n'
  x(m' , n' ) H [ (m  m' , n  n' )]
m'
n'
by superposition
impulse response, unit sample response,
point spread function(PSF)
• Definition of impulse response
h(m, n; m' , n' )  H [ (m  m' , n  n' )]
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Shift invariance
 Shift invariance
If
y (m, n)  H [ x(m, n)] and
y(m  m0 , n  n0 )  H [ x(m  m0 , n  n0 )], for m0 , n0
definition of
shift invariance
then, h(m, n; m0 , n0 )  h(m  m0 , n  n0 )
Output of LSI(linear shift invariant) systems
y(m, n)   x(m' , n' )h(m  m' , n  n' )
m'

(2-D convolution)
n'
y(m, n)  H [ x(m, n)]  H [ x(m' , n' ) (m  m' , n  n' )]
m'
n'
  x(m' , n' ) H [ (m  m' , n  n' )]
m'
n'
m'
n'
m'
n'
  x(m' , n' )h(m, n; m' , n' )
  x(m' , n' )h(m  m' , n  n' )
by superposition
of linearity
by definition of
impulse response
by shift invariance
Department of Computer Science and Engineering, Hanyang University
2D convolution
2-D convolution
y(m, n)  h(m, n)  x(m, n)   x(m' , n' )h(m  m' , n  n' )
m'
n'
n'
n'
x ( m' , n' )
B
C
rotate by 180 degree
and shift by (m,n)
x ( m' , n' )
n
A
h ( m  m' , n  n ' )
h( m' , n' )
A
B
n
n
1 4 1
2 5 3
1
1 1
m
x ( m, n )
n
1
n
h ( m, n )
1 1
n
1 1
1 1
m
m'
(b) output at location (m,n) is the sum of product
of quantities in the area of overlap
(a) impulse response
(ex)
m
m'
C
m
h (  m,  n )
1 1
1
m
h(1  m,n)
5
5
1
3 10
5
2
2
3 2 3
m
y (0,0)  2
y (1,0)  2  5  3
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Stability
 Stability

Definition : bounded input, bounded output
if | x(m, n) | , then | H [ x(m, n)]  

Stable LSI systems(necessary and sufficient condition)


  | h(m, n) |  
m   n  
Department of Computer Science and Engineering, Hanyang University
The Fourier transform
 Definition
 1-D Fourier transform

f ( x)   F (u) exp( j 2ux)du


F (u)   f ( x) exp(  j 2ux)dx


2-D Fourier transform
f ( x, y)  



 
F (u, v) exp( j 2 ( xu  yv))dudv
F (u, v)  



 
f ( x, y) exp(  j 2 (ux  vy))dxdy
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Frequency domain
 Properties
 f(t)  F() ;  = angular frequency
 f(x,y)  F(u,v) ; u,v = spatial frequencies that represent
the luminance change with respect
to spatial distance
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Properties of Fourier transform


Uniqueness
f ( x, y ) and

another
F (u, v)
are unique with respect to one
Separability




F (u, v)   [ f ( x, y) exp(  j 2ux)dx] exp(  j 2vy)dy

Convolution theorem
g ( x, y )  h ( x, y )  f ( x, y )
G (u, v)  H (u, v) F (u, v)
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
Inner product preservation

 

 
f ( x, y)h ( x, y)dxdy  
*



 
F (u, v) H * (u, v)dudv
Setting h=f, Parseval energy conservation formula

 

 

| f ( x, y) | dxdy  
2



 
| F (u, v) |2 dudv
Hankel transform : polar coordinate form of FT
Fp ( ,  )  F ( cos  ,  sin  )

2
0


0
f p (r , ) exp[  j 2r cos(   )]rdrd
where f p (r , )  f (r cos  , r sin  )
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Fourier series
 1-D case
x(n)  
0.5
0.5
X (u) exp(  j 2nu)du
X (u ) 

 x(n) exp(  j 2nu),
 0.5  u  0.5
n  
Department of Computer Science and Engineering, Hanyang University
2D Fourier series
 2-D case
x(m, n)  
0.5

0.5
0.5 0.5
X (u, v) exp( j 2 (mu1  nv))dudv


  x(m, n) exp(  j 2 (mu  nv)),
X (u, v) 
 0.5  u, v  0.5
m   n  

X (u , v ) is periodic : period = 1
X (u, v)  X (u  k , v  l ), k , l  0,1,2,

Sufficient condition for existence of X (u , v)
| X (u, v) |  |





  x(m, n) exp(  j 2 (mu  nv)) |
m   n  
  | x(m, n) | | exp(  j 2 (mu  nv)) |
m   n  



  | x(m, n) |

m   n  
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Eg. 2D Fourier transform
original 256x256 lena
normalized spectrum
(log-scale)
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Department of Computer Science and Engineering, Hanyang University
OTF & MTF
 Optical and modulation transfer functions

Optical transfer function(OTF)
 Normalized frequency response
OTF 


H (u, v)
H (0,0)
Modulation transfer function(MTF)
 Magnitude of the OTF
MTF | OTF |
| H (u, v) |
| H (0,0) |
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Readings
 Gonzalez and Woods, Digital Image Processing, 3rd
ed.


Sect. 2.4. Sampling & Quantization
Sect. 3.4. Fundamentals of Spatial Filtering
Department of Computer Science and Engineering, Hanyang University