Lifting

Part 1: Introduction

Ref: SIGGRAPH96

Outline

• Introduction to wavelets and lifting scheme

• Basic Ideas– Split, Predict, Update– In-place computation

• Simple Examples– Lifting version of Haar– Linear interpolating wavelet

General Concepts

• Wavelets are building blocks that can quickly de-correlate 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

localize spatial features)– Have banded spectrum (good frequency resolution)

• smoothness (decay towards high freq)• Have vanishing moments (decay towards low freq)

more later

How lifting scheme differs from classical wavelets

• Developed in 1994 by Wim Sweldens• All constructions are derived in the spatial domain• Faster implementation

– In some cases, the number of operations halved

• In-place computation– No auxiliary memory required

• Easy to invert – In classical derivations, perfect reconstruction must be

verified via Fourier transforms

Recall how PR of orthogonal wavelets are verified …

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

– …

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 (+|-, *|/)

That is, … (Forward Transform)

) U(d s

)P(s d

)Split( : ),(

11

11

11

jj

jj

jjj sds

Original signal

coarsened signal

difference signal

12,2,1,0, ...... njnjjj ssss

njnjjj dsds ,1,10,10,1 ......

sj

sj-1, dj-1

s: even indicesd: odd indices

Inverse Transform

• Observe the similarity with forward transform !

)d,Merge(s :

)P(s d

) U(d s

11

11

11

jjj

jj

jj

s

Schematically, …

forward

inverse

Convention:(水平 ) – (垂

直 )

Hi-wire:coarsened

signal

Lo-wire:difference

signal

Simple Examples

Haar (lifting version)

Linear Interpolating Wavelet

Revisit Haara slightly different version

abd

bas

2 5379

22 48

4 6

s d

s d

Forward Transform

Preserve “average”;

not “energy”

Haar (cont)

2/

2/

dsb

dsa

5379

22 48

4 6

s d

s d

ba

Inverse Transform

2

2

2

1

4

8

3

9

2

2

2

1

4

8

5

7

)4(64

)4(68

21

21

Haar and Lifting

• Rewrite expressions (forward)

2/das

abd

2/

ba

ab

sd and var for needno

b replaces da replaces s

abd

bas

2

Haar and Lifting (cont)

• Inverse Transform:– Reverse order of operations– exchange plus/minus

• Facilitate in-place computation

ab

ba

2/

Ex: Haar (Lifting)

2/

ba

ab

2/

ba

ab

Note this order is different from Mallat’s order!

1,10,10,00,02246 ddds

5379 3,22,21,20,2 ssss

2428 1,11,10,10,1 dsds

2426 1,10,00,10,0 ddds

In-place Computation

• Only one set of array is used

• Data are overwritten during the computation

• Saves overhead for allocating multiple arrays

3,22,21,20,2 ssss

1,11,10,10,1 dsds

1,10,00,10,0 ddds

Operate on the same piece of memory

Operate on the same piece of memory

Lifted Haar (inverse transform)

5379

2428

2426

ab

ba

2/

ab

ba

2/

3,22,21,20,2 ssss

1,11,10,10,1 dsds

1,10,00,10,0 ddds

Predict: how the data fail to be constant• eliminate zeroth order correlation• Order of predictor = 1

Update: preserve average• zeroth order moment• Order of Update operator = 1

Give exact prediction if

function were constant

Order: has to do with polynomial

reproduction (more later)

Haar & Lifting

2/)(

)(

xxU

xxP

Haar

Haar

2/)(

)(

xxU

xxP

Haar

Haar

),Merge( :

)P(

) U(

11

11

11

jjj

jj

jj

dss

sd

ds

) U(

)P(

)Split( : ),(

11

11

11

jj

jj

jjj

ds

sd

sds

Cascading1 0

0 0 1 0

1 1 0 0

1 0

-½ 0

½ 0

-½ ½ 0 0

0 0

Scaling FunctionsScaling Functions

WaveletsWavelets

Double Check

5379

2428

1,11,10,10,1 dsds

8 ×

4 ×

-2 ×

2 ×

+

½

1

1

-½

½

-½

+

+Note the wavelet definition is

different

Lifting Framework

9 7 3 5

9 3

7 5 -2 2

8 4 9 3

7 5

9 7 3 5

2/)(

)(

,1,1

,1,1

kjkjHaar

kjkjHaar

ddU

ssP

2/)(

)(

,1,1

,1,1

kjkjHaar

kjkjHaar

ddU

ssP

U: to ensure

coarsened signal preserves average

Pseudo CodesForward

Inverse

Lifting Ordering

(n=8)

3,23,22,22,21,21,20,20,2 dsdsdsds

1,11,10,10,1 dsds

0,00,0 ds

final result

76543210 ffffffff

2d

2sf 1s0s

1d 0d

3,21,12,20,01,20,10,20,0 ddddddds

About Demo Implementation

ndata = 16JMAX = 4

INCR

0 1 2 3 4

1 2 4 8 16

#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

15,414,413,412,411,410,49,48,47,46,45,44,43,42,41,40,4 ssssssssssssssss

7,37,36,36,35,35,34,34,33,33,32,32,31,31,30,30,3 dsdsdsdsdsdsdsds

3,23,22,22,21,21,20,20,2 dsdsdsds

1,11,10,10,1 dsds

0,00,0 ds

Linear Interpolating Wavelet

• more powerful lifting• Predictor (Order = 2)

– Exact for linear data

• Update (Order = 2)– Preserve the average and first moment

1,1,1,1,1 2

1

Predictor

ljljljlj ssdd 1,1,1,1,1 2

1

Predictor

ljljljlj ssdd

Linear Interpolating (Predictor)

1,1,1,1,1 2

1 ljljljlj ssdd 1,1,1,1,1 2

1 ljljljlj ssdd

Linear Interpolating Wavelet (Update)

)( ,11,12,,1 ljljljlj ddAss )( ,11,12,,1 ljljljlj ddAss

llj

llj

lljljlj

llj

llj

llj

llj

sAsA

sssAs

dAss

12,2,

22,2,21

12,2,

,12,,1

2)21(

2

2

4/1 Therefore, A

l

ljl

lj ss ,,1 2

1Preserve average

Propose update of the form :

use results already computed

Linear Wavelet (Update)

Preserve average is equivalent to having zero mean difference

Preserve average is equivalent to having zero mean difference

coarsened signal

original signal

ljljljlj ddss ,11,1,1,1 4

1 ljljljlj ddss ,11,1,1,1 4

1

Preservation of 1st Moment

2 UpdateofOrder

symmetry todue satisfied 2

1,,1

l

ljl

lj slsl

We will refer this as the dual order of MRA

In-place computation

Numeric Example (linear wavelet)

64426812106442

Average: 26/4

Average: 52/8

Forward

Inverse

6412183101412

6415.21935.101412

6412183101412

64426812106442

assume data periodicityassume data periodicity

Remarks

• By substituting the predictor into update one gets

• This is biorthogonal (2,2) of CDF– CDF: Cohen-Daubechies-Feauveau

– More computations in this form (and cannot be done in-place)

– Inverse transform harder to get (rely on Fourier-based techniques)

22,12,2,12,22,,1 8

1

4

1

4

3

4

1

8

1 ljljljljljlj ssssss

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

Homework: lifting version of D4

Speed up ratio!?Wiring diagram!?

undecided

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?