1. Introduction to The Lifting Scheme

2. 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

3. Practical Filter

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

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

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

7. A spatial domain construction of bi-orthogonal wavelets, consists of the following four basic operations: Split : s k (0) = x 2i (0), d k (0) = x 2i+1 (0) Predict : d k (r) = d k (r-1) –  p j (r) s k+j (r-1) Update : s k (r) = s k (r-1) +  u j (r) d k+j (r) Normalize : s k (R) = K 0 s k (R), d k (R) = K 1 d k (R) Four Basic Stages

8. Prediction and Update Two Main Stages

9. 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 Prediction Stage

10. 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 Update Stage

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

12. We have p 0 = 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

13. The coefficients of the corresponding high pass filter are { h 1 } = ½{-1,2,-1} The coefficients of the corresponding low pass filter are { h 0 } = ⅛{-1,2,6,2,-1} So, the (2,2) lifting scheme is equal to the 5/3 wavelet. Example

14. 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 Example

15. 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. Conclusions