Two-Dimensional Signals and Systems
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Two-Dimensional Signals and Systems Fundamental of Digital Image Processing ANIL K.JAIN Chap.2
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Two-Dimensional Signals and Systems. Fundamental of Digital Image Processing ANIL K.JAIN Chap.2. Notation and definitions. One-dimensional signal Continuous signal : Sampled signal : Two-dimensional signal Continuous signal : Sampled signal : - PowerPoint PPT Presentation
Transcript of Two-Dimensional Signals and Systems
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
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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
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Some special signals(or functions)
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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( )
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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)
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' '
)','()','(),(),(),(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
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Stability Definition : bounded input, bounded output
Stable LSI systems(necessary and sufficient condition)
)],([|,|),(| nmxHthennmxif
m n
nmh |),(|
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dxuxjxfuF )2exp()()(
duuxjuFxf )2exp()()(
dydxvyuxjyxfvuF
))(2exp(),(),(
dvduyvxujvuFyxf
))(2exp(),(),(
2-D Fourier transform
The Fourier transformThe Fourier transform
Definition 1-D Fourier transform
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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
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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
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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
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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
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original 256x256 lena
normalized spectrum(log-scale)
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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)
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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
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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
