The Lifting Scheme:
a custom-design construction of
biorthogonal wavelets
Sweldens95, Sweldens 98
(appeared in SIAM Journal on
Mathematical Analysis)
Relations of Biorthogonal Filters
~
 ( n)
h ( m  2 n ) h ( m) 

2
m
 ( n)
~
g (m  2n) g (m) 

2
m
h(m  2n) g~(m)  0

m
~
 h ( m  2n ) g ( m)  0
m
Biorthogonal Scaling Functions
and Wavelets
~
Dual
 (t ),  (t  k )   (k )
Dual
 (t ),~(t  k )   (k )
~
wavelet  dual scaling fns   (t ),  (t  n)  0
dual wavelet  scaling fns  ~(t ),  (t  n)  0
Wavelet Transform
(in operator notation)
transpose
~
 j  H j  j 1
~
 j  G j  j 1
Filter operators are matrices
encoded with filter coefficients
with proper dimensions
 j 1  H  j  G  j
*
j
*
j
Note that up/down-sampling is
absorbed into the filter operators
Operator Notation
Relations on Filter Operators
Biorthogonality
Write in matrix form:
~ * ~ *
H j H j  G jG j  1
~ * ~ *
H jG j  G j H j  0
~
H j  *
 ~  Hj
 G j 

H
Exact Reconstruction
* ~
*~
H j H j  G jG j  1
~

~
 j 1  H *j H j  j 1  G *j G j  j 1


~
~
 H *j H j  G *j G j  j 1

*
j

1 0 
G 

0
1


~


* Hj
Gj  ~  1
 G j 
*
j

Theorem 8 (Lifting)
• Take an initial set of biorthogonal filter operators
H
old
j
~ old old ~ old
, H j ,Gj ,Gj

• A new set of biorthogonal filter operators can be
found as H j , H~ j , G j , G~ j 
~
• Scaling functions and H and G untouched
Hj  H
~ old
~
~ old
H j  H j  S jG j
old
j
*
old
G j  G old

S
H
j
j
j
~
~ old
Gj  Gj
~
~ old
 H j  1 S   H j 
~ 
 ~ old 

 G j  0 1   G j 

H j   1
0  H old
j
 old 
G    *

 j   S 1  G j 
Proof of Biorthogonality
H
~


* Hj
*
old*
G j  ~   H old
G
j
j
G
 j 




*
j

 H
~
H j  *
1
*
 ~  H j Gj  
 G j 
0
1

0


old*
j
G
old*
j
1
0

1
0

~ old

 S  1 S  H j 
 ~ old 



1  0 1   G j 
~ old

0 H j 
 ~ old   1

1  G j 
~ old
S   H j  old*
 S
old* 1
Gj 
 ~ old  H j


1   G j 
0
1


S  1 0 1  S  1 0






1  0 1 0 1  0 1


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
Same thing expressed
in frequency domain
Corollary 6.
• Take an initial set of finite biorthogonal
filters h, h~ 0 , g 0 , g~
• Then a new set of finite biorthogonal filters
can be found as h, h~, g , g~
~
~
h (w )  h 0 (w )  g~ (w ) s (2w )
g (w )  g 0 (w )  h(w ) s(2w )
• where s(w) is a trigonometric polynomial
Details
Theorem 7 (Lifting scheme)
• Take an initial set of biorthogonal scaling
functions and wavelets  , ~ 0 , 0 ,~
• Then a new set ,  ,~, ,~ which is formally
biorthognal can be found as
Same thing expressed
in indexed notation
• where the coefficients sk can be freely chosen.
Dual Lifting
~
• Now leave dual scaling function and H and
G filters untouched
Hj  H
~
~ old
Hj  Hj
old
j
~ old
 S jG j
G j  G old
j
~
~ old ~ * ~ old
Gj  Gj  S j H j
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.
Before Lifting
~ old
 j  H j  j 1
~ old
 j  G j  j 1
Forward Transform


~ old
~
~ old
 j  H j  j 1  H j  S j G j  j 1
~ old
 H j  j 1  S j j
Inverse Transform
 j 1  H *j  j  G *j j

H
old *
j
j  G
H
old *
j

j
old *
j
 S jH
 S j j   G
old *
j
j
old *
j

j
Examples
Interpolating Wavelet Transform
Biorthogonal Haar Transform
The Lazy Wavelet
• Subsampling operators E (even) and D (odd)

E *
 D E
 
E
H
*
lazy
j

1 0
D 

0
1


* E
D   1
 D
*

~ lazy
~ lazy
lazy
 H j  E and G j  G j  D
Interpolating Scaling Functions
and Wavelets
• Interpolating filter: always pass through the
data points
• Can always take Dirac function as a formal
dual
~
H  E  S jD
~ int
Hj  E
int
j
G D
~ int
~*
Gj  D  S j E
int
j
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.
~
H j  H  E  S jD
~*
~
~ int ~ ~ int
H j  H j  S j G j  (1  S j S j ) E  S j D
int
*
int
*
*~
G j  G j  S j H j   S j E  (1  S j S j ) D
~
~ int
~*
Gj  Gj  D  S j E
int
j
Algorithm of Interpolating
Wavelet Transform
(indexed form)
Example: Improved Haar
• Increase vanishing moments of the wavelets
from 1 to 2
• We have
~0
h (w )  h(w )  12  12 e  iw
0
~
1
1  iw
g (w )  g (w )   e
2
2
After lifting : g (w )  g 0 (w )  h(w ) s(2w )
Details
Verify Biorthogonality
~
 ( n)
h
(
m

2
n
)
h
(
m
)


2
m
 ( n)
~
g (m  2n) g (m) 

2
m
 h(m  2n) g~(m)  0
m
~
 h ( m  2n ) g ( m)  0
m
~0
hn  hn  { 12 12 }n 0,1
g~n  g n0  { 21 12 }n 0,1
Improved Haar (cont)
0th moment van ishes : g (0)  0
g 0 (0)  h(0) s(0) 
1
2
 12  12 s(0)  0
s(0)  0
1st moment van ishes : g ' (0)  0
g ' (w )  g 0' (w )  h' (w ) s (2w )  2h(w ) s' (2w )
g 0' (w )   2i e iw
g ' (0)  g 0 ' (0)  h' (0) s (0)  2h(0) s ' (0)
  2i  0  2( 12  12 ) s ' (0)
  2i  2s' (0)  0
s' (0)   4i
Choose : s(w ) 
i
4
sin w 

1 e iw  e  iw
4
2

 e iw  e  iw
8
 ei 2w  e i 2w
ei 2w  e i 2w
s(2w ) 
and s(2w ) 
8
8
g (w )  g 0 (w )  h(w ) s(2w )
g(0) = g’(0)
=0
i 2w
i 2w

e

e
 21  12 e iw  12  12 e iw
8
 161 ei 2w  161 eiw  12  12 e iw  161 e i 2w  161 e i 3w


~
~0
h (w )  h (w )  g~ (w ) s (2w )
i 2w
i 2w
e

e
 12  12 e iw  -21  12 e iw
8
  161 e i 2w  161 e iw  12  12 e iw  161 e i 2w  161 ei 3w


Details
Verify Biorthogonality
~
 ( n)
h
(
m

2
n
)
h
(
m
)


2
m
~(m)  0
h
(
m

2
n
)
g

m
~
 h ( m  2n ) g ( m)  0
 ( n)
~
g (m  2n) g (m) 

2
m
hn  { 12 12 }n 0,1
~
1
1
1
hn  16
16
2
g n  161
g~n  { 21
1
16
1
2
1
2 n  0 ,1
}
m
1
2
1
16
1
2
1
16

1

16 n  2 , 1, 0 ,1, 2 , 3
1
16 n  2 , 1, 0 ,1, 2 , 3