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