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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.

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Presentation Transcript
slide2

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
slide3

Wavelet Transforms

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Practical Filter

understanding the lifting scheme
Understanding The Lifting Scheme

signal

Splitting

Predicting

Updating

Transmitting

Inverse Updating

Inverse Predicting

Merge

signal

lifting scheme in the z transform domain
Lifting Scheme in the Z-Transform Domain

Update stage

Low band signal

High band signal

Prediction stage

lifting scheme in the z transform domain1
Lifting Scheme in the Z-Transform Domain

Inverse update stage

Inverse prediction stage

slide7

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)
slide8

Two Main Stages

  • Prediction and Update
slide9

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
slide10

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
slide11

Example

  • The 5/3 wavelet
    • The (2,2) lifting scheme
slide12

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.
slide13

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.
slide14

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
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.