1. Lifting
Part 1: Introduction
Ref: SIGGRAPH96

2. Outline
• Introduction to wavelets and lifting scheme
• Basic Ideas
– Split, Predict, Update
– In-place computation
• Simple Examples
– Lifting version of Haar
– Linear interpolating wavelet

3. General Concepts
• Wavelets are building blocks that can quickly decorrelate data
• Most signals in life have correlation in time and
frequency
– temporal coherence and banded frequency
• Build wavelets that:
– Are compactly support (good time resolution; able to
more
localize spatial features)
later
– Have banded spectrum (good frequency resolution)
• smoothness (decay towards high freq)
• Have vanishing moments (decay towards low freq)

4. How lifting scheme differs from
classical wavelets
• Developed in 1994 by Wim Sweldens
• All constructions are derived in the spatial domain
• Faster implementation
Recall how PR
– In some cases, the number of operations halved
of orthogonal
• In-place computation
– No auxiliary memory required
wavelets are
verified …
• Easy to invert
– In classical derivations, perfect reconstruction must be
verified via Fourier transforms

5. Lifting in
Second Generation Wavelets
• True power of lifting is to construct wavelets in
settings where classical (translation and dilation)
and Fourier transform cannot be used:
– Bounded domain
• Avoid ad-hoc solutions: periodicity, zero-padding, reflection
around edges …
– Wavelets on curves and surfaces
– Irregular sampling
– …

6. Basic Ideas
• Forward transform: three stages
– Split the data into two smaller subsets: s/detail
• e.g., interlace sampling (lazy wavelet)
– Predict the subset based on the local correlation in the
original data
• Replace the detail as the difference between data and
prediction. (If prediction is reasonable, difference will be
small)
– Update and maintain some global properties of data
with original data
• (e.g., overall signal average)
• Inverse transform:
– Simply reverse order of operations and signs (+|-, *|/)

7. That is, … (Forward Transform)
Original signal
( s j −1 , d j −1 ) := Split( s j )
difference
signal
− = P(s j −1 )
d j −1
s j −1
s: even indices
d: odd indices
+ = U(d j −1 )
coarsened
signal
sj
s j ,0
s j ,1
...
...
s j ,2n
s j , 2 n +1
sj-1, dj-1
s j −1, 0
d j −1, 0
...
...
s j −1,n
d j −1,n

8. Inverse Transform
s j −1 − = U(d j −1 )
d j −1
+ = P(s j −1 )
s j := Merge(s j −1 , d j −1 )
• Observe the similarity with forward
transform !

9. Schematically, …
Hi-wire:
coarsened
signal
forward
Convention:
( 水平 ) – ( 垂
直)
Lo-wire:
difference
signal
inverse

10. Simple Examples
Haar (lifting version)
Linear Interpolating Wavelet

11. Revisit Haar
a slightly different version
Forward Transform
a+b
s=
2
d =b−a
[9
s
d
[ 8 4] [ − 2 2]
s
Preserve
“average”;
not
“energy”
7 3 5]
[ 6] [ − 4]
d

12. Haar (cont)
Inverse Transform
a = s−d /2
[9
7 3 5]
b = s+d /2
s
d
8 = 6 − (−4)
1
2
4 = 6 + 1 (−4)
2
9  8 1  − 2
  = −  
 3  4 2  2 
   
 
[ 8 4] [ − 2 2]
s
[ 6] [ − 4]
7 8 1  − 2
  = +  
 5  4 2  2 
   
 
d
a b

13. Haar and Lifting
• Rewrite expressions (forward)
a+b
s=
2
d =b−a
no need for
var d and s
b−=a
d =b−a
 →
a+ = b / 2
s = a+d /2
b replaces d
a replaces s

14. Haar and Lifting (cont)
• Inverse Transform:
– Reverse order of operations
– exchange plus/minus
a− = b / 2
b+=a
• Facilitate in-place computation

15. Ex: Haar (Lifting)
[9
7 3 5]
s2,0 s2,1 s2, 2 s2,3
b−=a
a+ = b / 2
[8
− 2 4 2]
s1,0 d1,0 s1,1 d1,1
b−=a
a+ = b / 2
[6
− 2 − 4 2]
s0,0 d1,0 d 0,0 d1,1
Note this order is different from Mallat’s order!
[6
− 4 − 2 2] = [ s0,0 d 0,0 d1,0 d1,1 ]

16. In-place Computation
• Only one set of
array is used
• Data are
overwritten during
the computation
• Saves overhead for
allocating multiple
arrays
s2,0 s2,1 s2, 2 s2,3
s1,0 d1,0 s1,1 d1,1
s0,0 d1,0 d 0,0 d1,1
Operate on the same piece
Operate on the same piece
of memory
of memory

17. Lifted Haar (inverse transform)
[9
7 3 5]
s2,0 s2,1 s2, 2 s2,3
a− = b / 2
b+=a
[8
− 2 4 2]
s1,0 d1,0 s1,1 d1,1
a− = b / 2
b+=a
[6
− 2 − 4 2]
s0,0 d1,0 d 0,0 d1,1

18. Give exact
prediction if
function were
constant
Predict: how the data fail to be
constant
• eliminate zeroth order
correlation
• Order of predictor = 1
Order: has to do
with polynomial
reproduction
(more later)
Update: preserve average
• zeroth order moment
• Order of Update operator = 1

19. Haar & Lifting
PHaar ( x) = x
U Haar ( x) = x / 2
( s j −1 , d j −1 ) := Split( s j )
d j −1
− = P( s j −1 )
s j −1
+ = U(d j −1 )
s j −1 − = U(d j −1 )
d j −1
+ = P( s j −1 )
s j := Merge(s j −1 , d j −1 )

20. Cascading
10
1100
10
00
Scaling Functions
Scaling Functions
00
-½ 0
-½ ½ 0 0
10
½0
Wavelets
Wavelets

21. Double Check
[9
7 3 5]
8×
1
+
[8
− 2 4 2]
s1,0 d1,0 s1,1 d1,1
4×
+
½
-2 ×
-½
Note the wavelet
definition is
different
1
2×
+
½
-½

22. Lifting Framework
93
84
93
9735
9735
75
-2 2
75
PHaar ( s j −1,k ) = s j −1,k
U: to ensure
coarsened signal
preserves
average
U Haar (d j −1,k ) = d j −1,k / 2

23. Pseudo Codes
Forward
Inverse

24. Lifting Ordering
f
s2
s1
s0
d2
d1
d0
final result
f0
f1
f2
f3
f4
f5
f6
f7
s2 , 0
(n=8)
d 2,0
s2,1
d 2,1
s2 , 2
d 2, 2
s2, 3
d 2,3
s1, 0
d1, 0
s1,1
s0 , 0
s0 , 0
d1,1
d 0, 0
d 2,0
d1, 0
d 2,1
d 0, 0
d 2, 2
d1,1
d 2,3

25. About Demo Implementation
#define S(j,l) ss[(l)*INCR[JMAX-(j)]]
// increment
#define D(j,l) ss[INCR[JMAX-((j)+1)]+(l)*INCR[JMAX-(j)]] // offset + increment
ndata = 16
JMAX = 4
INCR
2
3
4
2
4
8
16
s4,1
s4, 2
s4 , 3
s4 , 4
s3,0 d 3, 0
s3,1
d 3,1
s3, 2 d 3, 2
s3,3 d 3,3 s3, 4 d 3, 4
s3,5 d 3,5
s3,6 d 3,6
s3, 7 d 3,7
s2 , 0
d 2, 0
s2,1
d 2,1
d 2, 2
s2 , 3
d 2,3
s1, 0
s0 , 0
d1,0
s4 , 5
1
1
s4 , 0
0
s4 , 6
s4 , 7
s4 ,8
s2 , 2
s1,1
d 0,0
s4,9 s4,10 s4,11 s4,12 s4,13 s4,14 s4,15
d1,1

26. Linear Interpolating Wavelet
• more powerful lifting
• Predictor (Order = 2)
– Exact for linear data
• Update (Order = 2)
Predictor
d j −1,l = d j −1,l
1
− ( s j −1,l + s j −1,l +1 )
2
– Preserve the average and first moment

27. Linear Interpolating (Predictor)
d j −1,l = d j −1,l −
1
( s j −1,l + s j −1,l +1 )
2

28. Linear Interpolating Wavelet
(Update)
∑ s j −1,l =
Preserve average
l
1
∑ s j ,l
2 l
Propose update of s
j −1,l = s j , 2 l + A( d j −1,l −1 + d j −1,l )
the form :
∑s
l
j −1,l
use results already
computed
= ∑ s j , 2l + 2 A∑ d j −1,l
l
l
[
= ∑ s j , 2l + 2 A∑ s j , 2l +1 − 1 ( s j , 2l + s j , 2l + 2 )
2
l
l
= (1 − 2 A)∑ s j , 2l + 2 A∑ s j , 2l +1
l
Therefore, A = 1 / 4
l
]

29. Linear Wavelet (Update)
original signal
s j −1,l = s j −1,l
coarsened signal
1
+ ( d j −1,l −1 + d j −1,l )
4
Preserve average is equivalent
Preserve average is equivalent
to having zero mean difference
to having zero mean difference

30. Preservation of 1st Moment
1
∑ l ⋅ s j −1,l = 2 ∑ l ⋅ s j ,l satisfied due to symmetry
l
l
⇒ Order of Update = 2
We will refer this as the
dual order of MRA

31. In-place computation

32. assume data periodicity
assume data periodicity
Numeric Example (linear
wavelet)
2
4
4
6
10 12
8
6
2
4
4
6
2
1
4
−1 10
3
8
1
2
1
4
6
2
1
4
− 1 10.5
3
9
1
2.5
1
4
Forward
6
Average: 26/4
2
1
4
−1 10
2
4
4
6
3
8
1
2
1
4
6
10 12
8
6
2
4
4
6
Average: 52/8
Inverse

33. Remarks
• By substituting the predictor into update one gets
1
1
3
1
1
s j −1,l = − s j , 2l − 2 + s j , 2l −1 + s j , 2l + s j , 2l +1 − s j , 2l + 2
8
4
4
4
8
• This is biorthogonal (2,2) of CDF
– CDF: Cohen-Daubechies-Feauveau
– More computations in this form (and cannot be done inplace)
– Inverse transform harder to get (rely on Fourier-based
techniques)

34. Homeworks
• Review the derivation of PR for orthogonal
wavelets
• Verify that reversing order of operations
indeed inverses the transform
• Write a program that does general lifting.
Implement Haar and linear interpolation.
Compare.
• Verify the CDF (2,2) formula

35. Homework: lifting version of D4
Speed up ratio!?
Wiring diagram!?

36. undecided

37. Q
• In lifting, it seems that forward and inverse
use the same P and U boxes. Then, are
H_tilda (G_tilda) and H (G) are related? …
unlike what we mentioned in biorthogonal
wavelets?