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Introduction to The Lifting Scheme

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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Introduction to The Lifting Scheme

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  1. Introduction to The Lifting Scheme

  2. 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

  3. 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

  4. Understanding The Lifting Scheme signal Splitting Predicting Updating … Transmitting Inverse Updating Inverse Predicting Merge signal

  5. Lifting Scheme in the Z-Transform Domain Update stage Low band signal High band signal Prediction stage

  6. Lifting Scheme in the Z-Transform Domain Inverse update stage Inverse prediction stage

  7. 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)

  8. Two Main Stages • Prediction and Update

  9. 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

  10. 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

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

  12. 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.

  13. 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.

  14. 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

  15. 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.

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