The Lifting Scheme - Topics

Reminder:approximations,details Haar wavelet transform Lifting scheme Update Higher order extensions

One Stage Filtering

Approximations and details: The low-frequency content is the most

important part in many applications, and gives the signal its identity.

This part is called “Approximations”

The high-frequency gives the ‘flavor’, and is called “Details”

Approximations and Details:

Approximations:low-frequency components of the signal

Details: high-frequency components

Input Signal

LPF

HPF

A

D

Decimation

The former process produces twice the data To correct this, we Down sample (or: Decimate) the

filter output by two.

A complete one stage block :

Input Signal

LPF

HPF

A*

D*

Example*:

* Wavelet used: db2

Multi-level Decomposition

Iterating the decomposition process, breaks the input signal into many lower-resolution components: Wavelet decomposition tree:

Low pass filter high pass filter

A Simple Example: The Haar Wavelet

Consider two neighboring samples a and b of a sequence.

- a and b have some correlation.

A simple linear transform:

High correlation

- small |d|, fewer bits representation.

(i.e. a=b,d=0)

No loss of any information Reconstruction formula of a and b:

The Haar Wavelet Con’t

The key behind Haar Wavelet Transform:these reconstruction formulas can be found by inverting a 2x2 matrix.

The Haar Wavelet Con’t Signal Sn of 2n sample values Sn,l:

Sn = {Sn,l | 0=< l <= 2n}

Apply average (Sn-1) and difference (dn-1) transform onto each pair:

a = S2l, b = S2l+1 Sn-1,l = Sn,2l+Sn,2l

- 2n-1 pairs (l=0…2n-1) 2 dn-1,l = Sn,2l+1-Sn,2l Recover the original signal Sn from Sn-1 and dn-1

The Haar Wavelet Con’t

Sn-1 as Approximations

dn-1 as Details

Signal with local coherence

- approximations closely resembles the

original signal

- detail is very small (efficient representation)

The Haar Wavelet Con’t

Applying the same transform (averages and differences) to Sn-1 itself.

Split Sn-1 to (yet) coarser signal Sn-2 and another difference signal dn-2,each of them contain 2n-2 samples.

We can repeat this transform n times till S0 contains only one sample S0,0.

This is the Haar transform

The Haar transform

Structure of the wavelet transform: recursively split into averages and differences

Sn

Sn-1 dn-1

Sn-2 dn-2

S0 d0

S1 d1

The Haar transform Con’t

We end up with:

- n detail signals dj (0<=j<=n-1),

each with 2j coefficients

- one signal s0 containing only one sample

s0,0 - the average of all the samples

of the original signal.

Inverse Haar transform

Start from S0 and dj for 0<=j<n and obtain Sn again.

* note that the total number of coefficients after transform is 1 for S0 plus 2j for each dj.

(the exact number of samples of the original signal). n-1

1+∑ 2j = 2n

j=0

The Haar transform

d0 d1 dn-2,l dn-1,l

S0 S1 S2 … Sn-1,l Sn,l

Structure of the inverse wavelet transform: recursively merge averages and differences.

The Haar transform Con’t

The Haar transform can be thought of as applying a NxN matrix (N = 2n) to the signal Sn.

- The cost of computing the transform is

O(N).

(FFT cost is O(NlogN) and linear

transformation of

an N vector is O(N2)).

Haar Transform in-place We want to overwrite old values with new values.

First step:

compute the difference: d=b-a

and store it in b location.

Second step:

We use a and d to compute the average: S=a+d/2

and store it in a location.

a+ = b/2 ; b -= a

Haar Transform in-place Con’t

Inverse: run code backwards!

a- = b/2 ; b+ = a

This particular scheme is a simple instance of the lifting scheme

The Lifting Scheme

Consider a signal Sn with 2n samples .

Goal: transform the signal into: - a coarser signal Sn-1 - a detail signal dn-1

3 steps: - Split - Predict - Update

The Lifting Scheme - Split divide the input data into: - even indexed samples Sn. - odd indexed samples Sn+1. Lazy wavelet transform

The Lifting Scheme - Predict

predict the odd elements from the even elements-output detail.

Sn-1

detail

Example – Predict using Haar

The even sample is the prediction for the odd sample.

The detail dn-1 is the difference between the

odd sample and its predictor:

odd sample left neighbouring even sample - predictor dn-1= Sn,2l+1- Sn,2l Predict

- - -

detail

Note: perfect if function is constant, detail coefficients zero

General Case - Predict Detail = difference between odd element and predict function calculated from the even elements

dn-1=oddn - P(evenn)

The Lifting Scheme - Update

follows the predict phase. The approximations Sn-1 (the signal for next

step) should maintain the average of the original signal Sn.

smoother input for the next step of the wavelet transform.

Sn-1 = evenj,i + U( dn-1)

Example – Update Haar

Sets Sn-1 to be the average of the even/odd

pair (e.g., the even element si and its odd successor, si+1):

Sn-1 = even j,i + odd j, i

2

Haar transform update step:

recover the original value (from predict step) :

dn-1= oddj,i - evenj,i => oddj,i = evenj,i + dn-1 Substituting this into the average (prev),

we get :

Sn-1 = even j,i + even j,i + dn-1

2

Sn-1 = even j,I + dn-1 = even + U(dn-1)

2

The Lifting Scheme

The averages (even elements) become the input for the next recursive step of the forward transform.

Sn

Sn-1

dn-1

dn-2

Sn-2

The Lifting Scheme

In place computation:

- (oddj-1,evenj-1): = Split(Sj)

- oddj-1- = P(evenj-1)

- evenj-1+ = U(oddj-1)

Inverse Lifting Scheme

Inverse Lifting Scheme-3 steps Undo Update

dn,S0 given

recover even samples by subtructing the update info:

Sn-1 = evenj,i + U( dn-1)

evenj,i = Sn-1 -U(dn-1)

Haar: Sn,2l = Sn-1,l - dn-1,l/2

Inverse Lifting Scheme - Undo Predict

evenn-1,dn-1 given recover odd samples by adding prediction info:

dn-1 =oddn-1 - P(evenn-1)

oddn-1= dn-1 + (evenn-1) Haar: Sn,2l+1=dn-1,l+Sn,2l

Inverse Lifting Scheme- Merge

zipper odd and even samples

recover original signal-inverse Lazy wavelet: Sn=Merge(evenn-1,oddn-1)

Inverse in place Evenj-1 - = U(oddj-1)

Oddj-1 + = P(evenj-1)

Sj := Merge(oddj-1,evenj-1) Inverse transform: reversing the order of the

operations and flipping the signs

Inverse Lifting Scheme

Undo Update

-

Undo Predict Merge

+

Even values

Odd values

The Lifting Scheme-Example

Sn = X = [ 1 2 3 4 5 6 7 8 ] 2n=8 , n=3

Split:Xe=[2 4 6 8] Xo=[1 3 5 7]

Pred: averaging neighboures (edges:simple subtruction of

one neighbour-can fix by zero padding or wrap

around or reflection)

Pred{Xe} = [2 3 5 7]

dn-1 = d2 = Xo- Pred{Xe} = [-1 0 0 0]

The Lifting Scheme-ExampleUpdate: Sn-1= even j,I + dn-1,l

2Xe=[2 4 6 8] dn-1

= [-1 0 0 0]S2 = even+d2/2 = [1.5 4 6 8]We repeat recursively: Split: S2 is splitted simillarly: Xe=[4 8] Xo=[1.5 6]Pred{Xe} = [4 6] dn-2

= d1 = Xo- Pred{Xe} = [-2.5 0]S1 = even+d1/2 = [2.75 8]Xe=[8] Xo=[2.75]Pred{Xe} = [8] dn-3

= d0 = Xo- Pred{Xe} = [-5.25]S0 = even+d0/2 = [5.375]

The Lifting Scheme-Example The pyramid will be: d2

= [-1 0 0 0] d1 = [-2.5 0] d0 = [-5.25] S0 = [5.375]

Good result(most of elements are 0).

The Lifting Scheme- Inverse Example

We will recover the original signal from the pyramid:

The pyramid : d2 = [-1 0 0 0] , d1 = [-2.5 0],

d0 = [-5.25] , S0 = [5.375] Inverse transform:

Xe =S0 – d0/2 = [8] Pred{Xe} = [8]

Xo=d0+Pred{Xe} = [2.75]

S1 = [2.75 8]

The Lifting Scheme- Inverse Example

Xe=S1 – d1/2 = [4 8] Pred{Xe} = 4 6

Xo=d1+ Pred{Xe} = [1.5 6]

S2 = [1.5 4 6 8]

Xe=S2– d2/2 = [2 4 6 8] Pred{Xe} = 2 3 5 7

Xo=d0+ Pred{Xe} = [1 3 5 7]

S3 = [1 2 3 4 5 6 7 8]

The Lifting Scheme

Suppose Predict and Update are linear.

description of their operation as matrices P and U:

Xonew=Xo-PXe

Xenew =Xe+UXo

new=Xe+UXo-UPXe

or: Xo

new I -P Xo

Xenew U I-UP Xe

The Lifting Scheme - Example 2

Assume: P=I and U=0.5I

Xonew

I -p Xo I -I Xo

Xenew

U I-UP Xe 0.5I 0.5I Xe

Odd elements Even elements

differences average

of

neighbouring pairs

An example:linear wavelet transform Haar - simple and fast wavelet transform Limitations - not smooth enough: blocky Erasing Haar Coefficients:

Fourier analysis not always applicable

The Lifting Scheme

Lifting

Build more powerful versions

An example

linear wavelet transform

linear wavelet transform – Prediction

Linear Prediction

- Use even on either side

- Predictor for odd sample Sn,2l+1: average of

neighboring samples:on left Sn,2l, on right Sn,2l+2

linear wavelet transform - Prediction

linear wavelet transform

Even values are subsampled

linear wavelet transform - Update

Sets Sn-1 to be the average of even and odd elements

2n elements => 2n-1 even elements/averages Example: 2n=8

linear wavelet transform - Update Ex.

Example: 2n=8

-

linear wavelet transform - Update Ex.

sum of the sj-1 elements is equal to the sum of the sj elements, divided by two:

linear wavelet transform - Update

Signal average is not preserved Update Sn-1,l using previously compute detail

signals dn-1,l. Using Neighboring wavelet coefficients:

Sn-1,l=Sn,2l+A(dn-1,l-1+dn-1,l)

Update

Sn-1,l = Sn,2l+A(dn-1,l-1+dn-1,l) A=1/4 to maintain the average

Inverse-easy to compute:

Sn,2l = Sn-1,l - 1/4(dn-1,l-1+dn-1,l) to recover even

Sn,2l+1=dn,l+1/2(Sn,2l+Sn,2l+2) to recover odd

samples

linear wavelet transform

Original signal

linear wavelet transform

linear wavelet transform

Extend

- build higher polynomial order predictors

- Use more (D) neighbors on left and right

Example:

- Cubic polynom interpolating 4 values

- Bilinear Interpolation: the assigned value is

an intermediate value between the 4

nearest pixels :

aweighted sum of the 4 nearest pixels

- Each weight is proportional to the distance

from each existing pixel.

effective weights: -1/16 9/16 9/16 -1/16

Higher Order Prediction

Lifting Scheme - Predict

Linearaverage of neighboring samples:on

dn-1= Sn,2l+1-1/2(S2n+S2n+2)

Bilinearintermediate value between the 4 nearest pixels

Haar The even sample is the

prediction for the odd sample.

dn-1= Sn,2l+1- Sn,2l

Summary

Lifting Scheme - construction of transforms - Haar example - rewriting Haar in place

Three steps - split

- Predict- Update

Summary Predict - detail coefficient is failure of prediction

Update - smooth coefficient to preserve average - B spline C2-ensure smoothness Higher order extensions - increase order of prediction and update

linear wavelet transform - Update (why A=1/4)

=1/2∑Sj[n]