the lifting scheme: a custom-design construction of biorthogonal wavelets sweldens95, sweldens 98...
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![Page 1: The Lifting Scheme: a custom-design construction of biorthogonal wavelets Sweldens95, Sweldens 98 (appeared in SIAM Journal on Mathematical Analysis)](https://reader036.vdocument.in/reader036/viewer/2022062804/5697bf801a28abf838c84e9e/html5/thumbnails/1.jpg)
The Lifting Scheme:a custom-design construction of biorthog
onal wavelets
Sweldens95, Sweldens 98
(appeared in SIAM Journal on Mathematical Analysis)
![Page 2: The Lifting Scheme: a custom-design construction of biorthogonal wavelets Sweldens95, Sweldens 98 (appeared in SIAM Journal on Mathematical Analysis)](https://reader036.vdocument.in/reader036/viewer/2022062804/5697bf801a28abf838c84e9e/html5/thumbnails/2.jpg)
Relations of Biorthogonal Filters
0)(~)2( m
mgnmh
0)()2(~
m
mgnmh
2
)()(
~)2(
nmhnmh
m
2
)()(~)2(
nmgnmg
m
![Page 3: The Lifting Scheme: a custom-design construction of biorthogonal wavelets Sweldens95, Sweldens 98 (appeared in SIAM Journal on Mathematical Analysis)](https://reader036.vdocument.in/reader036/viewer/2022062804/5697bf801a28abf838c84e9e/html5/thumbnails/3.jpg)
Biorthogonal Scaling Functions and Wavelets
0)(~
),(fns scaling dualwavelet ntt
0)(),(~fns scaling waveletdual ntt
)()(~),( kktt Dual
)()(~
),( kktt Dual
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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
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Operator Notation
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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:
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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~
![Page 8: The Lifting Scheme: a custom-design construction of biorthogonal wavelets Sweldens95, Sweldens 98 (appeared in SIAM Journal on Mathematical Analysis)](https://reader036.vdocument.in/reader036/viewer/2022062804/5697bf801a28abf838c84e9e/html5/thumbnails/8.jpg)
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
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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
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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
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Details
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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
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Dual Lifting
• Now leave dual scaling function and and G filters untouched
oldjj
oldjj
oldjj
oldjj
oldjj
oldjj
HSGG
GG
HH
GSHH
~~~~
~~
~
*
H~
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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.
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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
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Examples
Interpolating Wavelet Transform
Biorthogonal Haar Transform
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The Lazy Wavelet
• Subsampling operators E (even) and D (odd)
1
10
01
**
**
D
EDE
DED
E
DGGEHH lazyj
lazyj
lazyj
lazyj
~ and
~
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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
~~
~
~
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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.
![Page 20: The Lifting Scheme: a custom-design construction of biorthogonal wavelets Sweldens95, Sweldens 98 (appeared in SIAM Journal on Mathematical Analysis)](https://reader036.vdocument.in/reader036/viewer/2022062804/5697bf801a28abf838c84e9e/html5/thumbnails/20.jpg)
ESDGG
DSSESHSGG
DSESSGSHH
DSEHH
jjj
jjjjjjj
jjjjjjj
jjj
*int
**int*int
*intint
int
~~~)
~1(
)~
1(~~~~
~
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Algorithm of Interpolating Wavelet Transform
(indexed form)
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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
![Page 23: The Lifting Scheme: a custom-design construction of biorthogonal wavelets Sweldens95, Sweldens 98 (appeared in SIAM Journal on Mathematical Analysis)](https://reader036.vdocument.in/reader036/viewer/2022062804/5697bf801a28abf838c84e9e/html5/thumbnails/23.jpg)
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
![Page 24: The Lifting Scheme: a custom-design construction of biorthogonal wavelets Sweldens95, Sweldens 98 (appeared in SIAM Journal on Mathematical Analysis)](https://reader036.vdocument.in/reader036/viewer/2022062804/5697bf801a28abf838c84e9e/html5/thumbnails/24.jpg)
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
![Page 25: The Lifting Scheme: a custom-design construction of biorthogonal wavelets Sweldens95, Sweldens 98 (appeared in SIAM Journal on Mathematical Analysis)](https://reader036.vdocument.in/reader036/viewer/2022062804/5697bf801a28abf838c84e9e/html5/thumbnails/25.jpg)
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
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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