Two-Dimensional Signals
and Systems
Two-Dimensional Signals
and Systems
Fundamental of Digital Image Processing
ANIL K.JAIN Chap.2
Fundamental of Digital Image Processing
ANIL K.JAIN Chap.2
2
Notation and definitionsNotation and definitions
One-dimensional signal Continuous signal : Sampled signal :
Two-dimensional signal Continuous signal : Sampled signal :
i, j, k, l, m, n, … are usually used to specify integer indices Separable form :
),...(),(),( tsxuxf
),....(, nuun
),...,(),,(),,( yxfyxvyxu
),...,(),,(,, jiunmvu nm
)()(),( yfxfyxf
3
2-D delta function Dirac :
Property
Scaling :
Kronecker delta : Property
)()(),( yxyx
,),('')','()','(
yxfdydxyyxxyxf
1),(lim 0 dxdyyx
)()(),( nmnm
,)','()','(),(' '
m n
nnmmnmxnmx
|,|/)()( axax
|,|/),(),( abyxbyax
1),(
m n
nm
Notation and definitionsNotation and definitions
4
Some special signals(or functions)
5
Linear and shift invariant systems Linear and shift invariant systems
Linearity
)(),(,,),,(),( 21212211 xxaafornmyanmya
)],([)],([)],(),([ 22112211 nmxHanmxHanmxanmxaH
Definition of impulse response )]','([)',';,( nnmmHnmnmh
Output of linear systems
' '
])','()','([)],([),(m n
nnmmnmxHnmxHnmy
' '
])','([)','(m n
nnmmHnmx
by superpositionimpulse response, unit sample response,
point spread function(PSF)
H[ ] y(m,n)=H x(m,n)[ ]x m,n( )
6
Shift invarianceandnmxHnmyIf )],([),(
Output of LSI(linear shift invariant) systems
' '
)','()','(),(m n
nnmmhnmxnmy
),(),;,(, 0000 nnmmhnmnmhthen 000000 ,)],,([),( nmfornnmmxHnnmmy
definition ofshift invariance
' '
])','()','([)],([),(m n
nnmmnmxHnmxHnmy
' '
])','([)','(m n
nnmmHnmx
' '
)',';,()','(m n
nmnmhnmx
' '
)','()','(m n
nnmmhnmx
by superpositionof linearity
by definition of impulse response
by shift invariance
(2-D convolution)
7
' '
)','()','(),(),(),(m n
nnmmhnmxnmxnmhnmy
)','( nmh
)','( nmx
A
B C'm
'n
)','( nnmmh
)','( nmx
A
BC
'm
'n
m
n
(a) impulse response(b) output at location (m,n) is the sum of product
of quantities in the area of overlap
rotate by 180 degree and shift by (m,n)
2-D convolution
(ex)352
141
11
11
m
n
m
n
m
n
11
1111
11
m
n
),( nmx ),( nmh ),( nmh ),1( nmh
2)0,0( y 352)0,1( y
m
n
3232
25103
1551
8
Stability Definition : bounded input, bounded output
Stable LSI systems(necessary and sufficient condition)
)],([|,|),(| nmxHthennmxif
m n
nmh |),(|
9
dxuxjxfuF )2exp()()(
duuxjuFxf )2exp()()(
dydxvyuxjyxfvuF
))(2exp(),(),(
dvduyvxujvuFyxf
))(2exp(),(),(
2-D Fourier transform
The Fourier transformThe Fourier transform
Definition 1-D Fourier transform
10
edgehigh spatial frequencies
Properties Spatial frequencies : u,v (reciprocals of x and y)
f(t) F(w) ; w = frequency f(x,y) F(u,v) ; u,v = spatial frequencies that represent
the luminance change with respect to spatial distance
representing luminance change with respect to spatial distance
11
Uniqueness and are unique with respect to one another
Separarability
Eigenfunction of a linear shift invariant system
))(2exp(),(),( vyuxjvuHyxg
))(2exp( yvxuj
',' yyYxxX
),( yxh H
''))''(2exp()','(),(),(),( dydxvyuxjyyxxhyxyxhyxg
dyvyjdxuxjyxfvuF
)2exp(])2exp(),([),(
property of eignfunctionfrequency response
Performing the change of variables
),( yxf ),( vuF
12
Convolution theorem
Inner product preservation
Hankel transform : polar coordinate form of FT
),(),(),( yxfyxhyxg ),(),(),( vuFvuHvuG
dudvvuHvuFdxdyyxhyxf ),(),(),(),( **
Setting h=f, Parseval energy conservation formula
dudvvuFdxdyyxf 22 |),(||),(|
)sin,cos(),( FFp
2
0 0)]cos(2exp[),( rdrdrjrf p
)sin,cos(),( rrfrf p where
13
1-D case
2-D case
is periodic : period = 1
Sufficient condition for existence of
m n
vunvmujnmxvuX 5.0,5.0),)(2exp(),(),(
5.0
5.0
5.0
5.0))(2exp(),(),( dudvnvmujvuXnmx
),( vuX
,2,1,0,),,(),( lklvkuXvuX
Fourier transform of sequences(Table2.4)
n
unujnxuX 5.05.0),2exp()()(
5.0
5.0)2exp()()( dunujuXnx
|))(2exp(),(||),(|
m n
nvmujnmxvuX
m nm n
nmxnvmujnmx |),(||))(2exp(||),(|
),( vuX
14
original 256x256 lena
normalized spectrum(log-scale)
15
16
Definition
ROC(region of convergence)
m n
nm zznmxzzX 2121 ),(),(
211
21
1212),(
)2(
1),( dzdzzzzzX
jnmx nm
m n
nm zznmxzzX |),(||),(| 2121
m n
nm zznmx |),(| 21
m n
nm zznmx |||||),(| 21
z-plane
ujez 2
u2
1
}|),(|),{(),( 212121 zzXzzzzXofROC
The Z-transform(or Laurent series)The Z-transform(or Laurent series)
17
otherwise
nmrrnmx
nm
,0
0,0,),( 21
0 0
111
122
0 0212121 )()(),(
n m
mn
n m
nmnm zrzrzzrrzzX
}1||,1||),{(),( 122
1112121 zrandzrzzzzXofROC
0 02121 ||
n m
nmnm zzrr
For convergence of ),( 21 zzX
By definition
,1
1
1
1),(
122
111
21
zrzrzzX
}|||||,|||,{),( 22112121 rzrzzzzzXofROC
Example
18
19
Optical and modulation transfer functions
Optical transfer function(OTF) Normalized frequency response
Modulation transfer function(MTF) Magnitude of the OTF
)0,0(
),(
H
vuHOTF
|)0,0(|
|),(|||
H
vuHOTFMTF