General Orthonormal MRA

Ref: Rao & Bopardikar, Ch. 3

Outline

• MRA characteristics– Nestedness, translation, dilation, …

• Properties of scaling functions

• Properties of wavelets

• Digital filter implementations

Recall Formal Definition of an MRA

An MRA consists of the nested linear vector space such that

• There exists a function (t) (called scaling function) such that is a basis for V0

• If and vice versa•

• Remarks:– Does not require the set of (t) and its integer translates

to be orthogonal (in general)– No mention of wavelet

2101 VVVV

integer:)( kkt

1)2( then )( kk VtfVtf

)(lim 2 RLV jj

}0{jV

Properties of Scaling Functions

1)( dtt

1)(

2dtt

0for basis

tindependenlinearly :)(

)()(),(

V

Zkkt

nntt

)(t

)1( t

Explained using Haar basis

Dilation of Scaling Functions

)(2)2

1(),

2

1(

)(2

1)2(),2(

nntt

nntt

)(2)2(),2( nntt kkk

)(t

)2( t

)12( t

)2/(t

1

2

1t

k

k

V

Zllt

for basistindependenlinearly

:)2(

Nested Spaces

• Every vector in V0 belongs to V1 as well

– In particular (t)

• Possible to express (t) as a linear combination of the basis for V1

ZkktV

ZkktV

VV

:)2(: of basis

:)(: of basis

1

0

10

Haar may be misleading …

• One can translate an arbitrary function by integers and compress it by 2; BUT there is no reason to think that the spaces Vj created by the function and its translates and dilates will necessarily be nested in each other

V0

V1

10 VV 10 VV

Remark

Two-Scale Relations(Scaling Fns)

n

ntnct )2()()(

2

1)(

)2()()(1

:sidesboth gintegratin

n

n

nc

dtntncdtt

n

nc 2)(

n

ntnct )2()()(

nn

lntncnltnclt )22()())(2()()(

)()2()(2

1)(),( llmcmcltt

m

Constraints on c(n)

n

kk ntnkatf )2(),()(

)(),()(),(

0)(),()(

)()()( :fn detail

01

000

010

nttfnttf

nttgVtg

tftftg

),0()(),(0 nanttf )2(),(),1()(),(1 mtntmanttfm

Orthogonal Projection in Subspaces

n

ntnatf )(),0()(0

n

ntnatf )2(),1()(1 Finer approx

Coarser approx

See next page

)(2

1

)2(),2()()2(),(

mc

mtntncmttn

)2(2

1)2(),(

)2(2

1

)2(),22()(

)2(),)1(2()()2(),1(

nmcmtnt

mc

mtntnc

mtntncmtt

n

n

m

nmcmana

2

)2(),1(),0(

From previous pageFiner coefficients and coarser ones are related by c(n)

0)( dtt

1)(

2dtt

0for basist independenlinearly

)()(),(

W

nntt

0)(),( ntt

Properties of Wavelets

Orthogonality

Two-Scale Relations(wavelet)

0)(

)(2

1)2()()(0

:sidesboth gintegratin

)2()()()( 1

n

nn

n

nd

nddtntnddtt

ntndtVt

)()2()(2

1)(),( llmcmcltt

m

We showed :

Similarly :

)()2()(2

1)(),( llmdmdltt

m

0)2()(2

1)(),(

m

lmcmdltt

Constraints on c(n) and d(n)

),0()(),(

)(),0()(),0(

)()()(

1

001

nbnttf

ntnbntna

tgtftf

nn

Function Reconstruction

m

m

mtntmanttf

mtmatf

)2(),(),1()(),(

)2(),1()(

1

1

See next page

m

nmdmanb

2

)2(),1(),0(

)(2

1

)2(),2()()2(),(

md

mtntndmttn

)2(2

1

)2(),22()()2(),1(

md

mtntndmttn

)2(2

1)2(),( nmdmtnt

Detail coefficients and finer representation are related by d(n)

jj

kj

kjk WVWV

WWWV

WWVWVV

WVV

WVV

1

1011

100112

110

001

Nested Space

VN

VN-1 WN-1

VN-2 WN-2

VN-3 WN-3

Digital Filter Implementation

Use existing methodology in signal processing for discrete wavelet comput

ation

Digital Filter Implementation

)()(~2

)()(

)()(~

2

)()(

Define

ngngnd

ng

nhnhnc

nh

m

nmcmana

2

)2(),1(),0(

m

nmdmanb

2

)2(),1(),0(

Recall

m

mnhmana )2(~

),1(),0(

m

mngmanb )2(~),1(),0(

Then

n

nh 1)(

n

n

n

kngnh

kkngng

kknhnh

0)2()(

)(2

1)2()(

)(2

1)2()(

n

nc 2)(

)()2()(2

1llmdmd

m

0)2()(2

1

m

lmcmd

)()2()(2

1llmcmc

m

n

ng 0)(

n

nd 0)(

)0(

~)2,1()1(

~)1,1()2(

~)0,1()3(

~)1,1(

)2(~

),1()1,0(

hahahaha

mhmaam

)0(

~)0,1()1(

~)1,1()2(

~)2,1()3(

~)3,1(

)(~

),1()0,0(

hahahaha

mhmaam

)0,0(a

)0(~h)1(

~h)2(

~h)3(

~h

)2,1( a )1,1( a )0,1(a )1,1(a)3,1( a )2,1(a )3,1(a

)1,0(a

)0(~h)1(

~h)2(

~h)3(

~h

)2,1( a )1,1( a )0,1(a )1,1(a)3,1( a )2,1(a )3,1(a

two)offactor aby n (decimatio

samples indexed retain and ~

with ),1( Convolve

:),0( getting ...

evenhma

naCoarsening

Similarly, …

two)offactor aby n (decimatio

samples indexed retain and ~ with ),1( Convolve

:),0( getting ...

evengma

nbencethe differComputing

)0,0(b

)0(~g)1(~g)2(~g)3(~g

)2,1( a )1,1( a )0,1(a )1,1(a)3,1( a )2,1(a )3,1(a

)1,0(b

)0(~g)1(~g)2(~g)3(~g

)2,1( a )1,1( a )0,1(a )1,1(a)3,1( a )2,1(a )3,1(a

Signal Reconstruction

m lm l

ll

mltmdlbmltmcla

ltlbltla

tgtftf

)22()(),0()22()(),0(

)(),0()(),0(

)()()( 001

m

mtmct )2()()(

m

mtmdt )2()()(

n ln l

ntlndlbntlnclatf

mln

)2()2(),0()2()2(),0( )(

2 ngSubstituti

1

)2()1,0()1(0)0()0,0()1(0)2()1,0(

)2()1,0()0()0,0()2()1,0()2(),0(

caccacca

cacacalclal

)1(c)0(c)1(c)2(c

0 )0,0(a 0 )1,0(a)1,0( a

)2(c

Subdivision … getting a(1,n):Zero insertion (upsampling) and convolve with 2H

n=0

)2(),1( )(1 ntnatfn

l l

l l

lnglblnhla

lndlblnclana

)2(),0(2)2(),0(2

)2(),0()2(),0(),1(

Hence

)2()1,0()1(0)0()0,0()1(0)2()1,0(

)2()1,0()0()0,0()2()1,0()2(),0(

dbddbddb

dbdbdbldlbl

)1(d)0(d)1(d)2(d

0 )0,0(b 0 )1,0(b)1,0( b

)2(d

Detail part: … getting a(1,n):upsampling and convolve with 2Gn=0

Similarly, …

Notations of Digital Filters

Interpolator and Decimator

nnMxny for )()(

otherwise0

,for )('

kkMnM

ny

nx

H~ H

G~ G

analysis filter bankperfect reconstruction pair:

Whatever goes into analysis bank isrecovered perfectly by the synthesisbank

synthesis filter bank

H~

H

G~

G

Haar Revisited

3,2,1,05379)( nnx

Analysis Filters

0

9

7

3

5 0

8

5

4

2.5

2

0

1

2

-1

2.5

2

-1 0

h(-n)

0.5 0.5

0

-1g(-n)0.5

-0.5

2

1)1()0( hh

2

1)1(,

2

1)0( gg

Haar:

Haar Revisited

Synthesis Filters

0 1

2 h(n)

1

0

2 g(n)

0

9

7

3

50

8

4

2

0

1

-1

2

-1

1

11