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The Lifting Scheme: a custom-design construction of biorthogonal wavelets. Sweldens95, Sweldens 98 (appeared in SIAM Journal on Mathematical Analysis ). Relations of Biorthogonal Filters. Dual. Dual. Biorthogonal Scaling Functions and Wavelets. transpose. - PowerPoint PPT Presentation
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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