The Lifting Scheme: a custom-design construction of biorthogonal wavelets

The Lifting Scheme:a custom-design construction of biorthog

onal wavelets

Sweldens95, Sweldens 98

(appeared in SIAM Journal on Mathematical Analysis)

Relations of Biorthogonal Filters

0)(~)2( m

mgnmh

0)()2(~

m

mgnmh

2

)()(

~)2(

nmhnmh

m

2

)()(~)2(

nmgnmg

m

Biorthogonal Scaling Functions and Wavelets

0)(~

),(fns scaling dualwavelet ntt

0)(),(~fns scaling waveletdual ntt

)()(~),( kktt Dual

)()(~

),( kktt Dual

Wavelet Transform(in operator notation)

jjjjj

jjj

jjj

GH

G

H

**1

1

1

~

~

Note that up/down-sampling is absorbed into the filter operators

Note that up/down-sampling is absorbed into the filter operators

Filter operators are matrices encoded with filter coefficients with proper dimensions

Filter operators are matrices encoded with filter coefficients with proper dimensions

transpose

Operator Notation

Relations on Filter Operators

0~~

1~~

**

**

jjjj

jjjj

HGGH

GGHH

1~~ ** jjjj GGHH

Biorthogonality

Exact Reconstruction

1

**

1*

1*

1

~~

~~

jjjjj

jjjjjjj

GGHH

GGHH

1~

~

10

01~

~

**

**

j

jjj

jjj

j

G

HGH

GHG

H

Write in matrix form:

Theorem 8 (Lifting)• Take an initial set of biorthogonal filter operators

• A new set of biorthogonal filter operators can be found as

• Scaling functions and H and untouched

oldjj

oldjj

oldjj

oldjj

oldjj

oldjj

GG

HSGG

GSHH

HH

~~

~~~

*

oldj

oldj

oldj

oldj GGHH

~,,

~,

jjjj GGHH~

,,~

,

oldj

oldj

j

j

oldj

oldj

j

j

G

H

SG

H

G

HS

G

H

1

01

~

~

10

1~

~

*

G~

Proof of Biorthogonality

1~

~

10

01

~

~

10

1

10

1~

~

**

****

oldj

oldjold

joldj

oldj

oldjold

joldj

j

jjj

G

HGH

G

HSSGH

G

HGH

10

01

10

1

10

01

10

1

10

1~

~

10

1~

~****

SS

SGH

G

HSGH

G

H oldj

oldjold

j

oldj

jjj

j

Choice of S

• Choose S to increase the number of vanishing moments of the wavelets

• Or, choose S so that the wavelet resembles a particular shape– This has important applications in automated

target recognition and medical imaging

Corollary 6.

• Take an initial set of finite biorthogonal filters

• Then a new set of finite biorthogonal filters can be found as

• where s() is a trigonometric polynomial

gghh ~,,~

, 00

gghh ~,,~

,

)2()(~)(~

)(~ 0 sghh

)2()()()( 0 shgg

Same thing expressed in frequency domain

Details

Theorem 7 (Lifting scheme)

• Take an initial set of biorthogonal scaling functions and wavelets

• Then a new set , which is formally biorthognal can be found as

• where the coefficients sk can be freely chosen.

~,,~, 00

~,,~,

Same thing expressed in indexed notation

Dual Lifting

• Now leave dual scaling function and and G filters untouched

oldjj

oldjj

oldjj

oldjj

oldjj

oldjj

HSGG

GG

HH

GSHH

~~~~

~~

~

*

H~

Fast Lifted Wavelet Transform

• Basic Idea: never explicitly form the new filters, but only work with the old filter, which can be trivial, and the S filter.

1

1

~

~

joldjj

joldjj

G

H

Before Lifting

jjj

oldj

joldjj

oldjjjj

SH

GSHH

1

11

~

~~~Forward Transform

j

oldjjjj

oldj

joldjj

oldjj

oldj

jjjjj

GSH

HSGH

GH

**

***

**1

Inverse Transform

Examples

Interpolating Wavelet Transform

Biorthogonal Haar Transform

The Lazy Wavelet

• Subsampling operators E (even) and D (odd)

1

10

01

**

**

D

EDE

DED

E

DGGEHH lazyj

lazyj

lazyj

lazyj

~ and

~

Interpolating Scaling Functions and Wavelets

• Interpolating filter: always pass through the data points

• Can always take Dirac function as a formal dual

ESDG

DG

EH

DSEH

jj

j

j

jj

*int

int

int

int

~~

~

~

Theorem 15

• The set of filters resulting from interpolating scaling functions, and Diracs as their formal dual, can be seen as a dual lifting of the Lazy wavelet.

ESDGG

DSSESHSGG

DSESSGSHH

DSEHH

jjj

jjjjjjj

jjjjjjj

jjj

*int

**int*int

*intint

int

~~~)

~1(

)~

1(~~~~

~

Algorithm of Interpolating Wavelet Transform

(indexed form)

Example: Improved Haar

• Increase vanishing moments of the wavelets from 1 to 2

• We have

i

i

egg

ehh

21

210

21

210

)()(~)()(

~

)2()()()( :liftingAfter 0 shgg

Verify Biorthogonality

2

)()(

~)2(

nmhnmh

m

2

)()(~)2(

nmgnmg

m

0)(~)2( m

mgnmh

0)()2(~

m

mgnmh

1,021

210

1,021

210

} {~

} {~

nnn

nn

gg

hhn

Details

Improved Haar (cont)

0)0(

0)0()0()0()0(

0)0( :ishesmoment van0th

21

21

210

s

sshg

g

0)0(' :ishesmoment van1st g

4

2

21

21

2

'0

2'0

'0

)0('

0)0('2

)0(')(20

)0(')0(2)0()0(')0()0('

)(

)2(')(2)2()(')()('

i

i

i

ii

s

s

s

shshgg

eg

shshgg

8)2( and

8)2(

sin)( :Choose2222

8241

4

iiii

eeeei

ees

ees

siiii

31612

161

21

21

1612

161

22

21

21-

21

21

0

8

)2()(~)(~

)(~

iiiii

iiii

eeeee

eeee

sghh

31612

161

21

21

1612

161

22

21

21

21

21

0

8

)2()()()(

iiiii

iiii

eeeee

eeee

shgg

g(0) = g’(0) = 0

Verify Biorthogonality

2

)()(

~)2(

nmhnmh

m

2

)()(~)2(

nmgnmg

m

0)(~)2( m

mgnmh

0)()2(~

m

mgnmh

1,021

21

3,2,1,0,1,2161

161

21

21

161

161

3,2,1,0,1,2161

161

21

21

161

161

1,021

21

} {~

~} {

nn

nn

nn

nn

g

g

h

h

Details