Introduction to The Lifting Scheme

1 / 15

# Introduction to The Lifting Scheme - PowerPoint PPT Presentation

Introduction to The Lifting Scheme . Wavelet Transforms. Two approaches to make a wavelet transform: Scaling function and wavelets (dilation equation and wavelet equation) Filter banks (low-pass filter and high-pass filter) The two approaches produce same results, proved by Doubeches.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Introduction to The Lifting Scheme

Wavelet Transforms

• Two approaches to make a wavelet transform:
• Scaling function and wavelets (dilation equation and wavelet equation)
• Filter banks (low-pass filter and high-pass filter)
• The two approaches produce same results, proved by Doubeches.
• Filter bank approach is preferable in signal processing literatures

Wavelet Transforms

H

(

z

)

G

(

z

)

2

2

0

0

X

'

(

z

)

X

(

z

)

Y

'

(

z

)

Y

(

z

)

0

0

0

0

+

X

(

z

)

Y

(

z

)

H

(

z

)

G

(

z

)

2

2

1

1

X

'

(

z

)

X

(

z

)

Y

'

(

z

)

Y

(

z

)

1

1

1

1

Practical Filter

Understanding The Lifting Scheme

signal

Splitting

Predicting

Updating

Transmitting

Inverse Updating

Inverse Predicting

Merge

signal

Lifting Scheme in the Z-Transform Domain

Update stage

Low band signal

High band signal

Prediction stage

Lifting Scheme in the Z-Transform Domain

Inverse update stage

Inverse prediction stage

Four Basic Stages

• A spatial domain construction of bi-orthogonal wavelets, consists of the following four basic operations:
• Split : sk(0)=x2i(0), dk(0)=x2i+1(0)
• Predict : dk(r)= dk(r-1) –pj(r) sk+j(r-1)
• Update : sk(r)= sk(r-1) + uj(r) dk+j(r)
• Normalize : sk(R)=K0sk(R), dk(R)=K1dk(R)

Two Main Stages

• Prediction and Update

Prediction Stage

• A prediction rule : interpolation
• Linear interpolation coefficients: [1,1]/2
• used in the 5/3 filter
• Cubic interpolation coefficients: [-1,9,9,-1]/16
• used in the 13/7 CRF and the 13/7 SWE

Update Stage

• An update rule : preservation of average (moments) of the signal
• The update coefficients in the 5/3 are [1,1]/4
• The update coefficients in the 13/7 SWE are[-1,9,9,-1]/32
• The update coefficients in the 13/7 CRF are[-1,5,5,-1]/16

Example

• The 5/3 wavelet
• The (2,2) lifting scheme

Example

• We have p0 = 1/2 by linear interpolation and the detailed coefficient are given by
• In the update stage, we first assure that the average of the signal be preserved
• From an update of the form, we have
• From this, we get A=1/4 as the correct choice to maintain the average.

Example

• The coefficients of the corresponding high pass filter are {h1} = ½{-1,2,-1}
• The coefficients of the corresponding low pass filter are {h0} = ⅛{-1,2,6,2,-1}
• So, the (2,2) lifting scheme is equal to the 5/3 wavelet.

Example

• Complexity of the lifting version and the conventional version
• The conventional 5/3 filter
• X_low = ( 4*x[0]+2*x[0]+2*(x[-1]+x[1])-(x[2]+x[-2]) )/8
• X_high = x[0]-(x[1]+x[-1])/2
• Number of operations per pixel = 9+3 = 12
• The (2,2) lifting
• D[0] = x[0]- (x[1]+x[-1])/2
• S[0] = x[0] + (D[0]+D[1])/4
• Number of operations per pixel = 6
Conclusions
• The lifting scheme is an alternative method of computing the wavelet coefficients
• Advantages of the lifting scheme:
• Requires less computation and less memory.
• Easily produces integer-to-integer wavelet transforms for lossless compression.
• Linear, nonlinear, and adaptive wavelet transform is feasible, and the resulting transform is invertible and reversible.