1. The Lifting Scheme: a custom-design construction of biorthogonal wavelets Sweldens95, Sweldens 98 (appeared in SIAM Journal on Mathematical Analysis)

2. Relations of Biorthogonal Filters

3. Biorthogonal Scaling Functions and Wavelets Dual

4. Wavelet Transform (in operator notation) Note that up/down-sampling is absorbed into the filter operators Filter operators are matrices encoded with filter coefficients with proper dimensions transpose

5. Operator Notation

6. Relations on Filter Operators Biorthogonality Exact Reconstruction Write in matrix form:

7. Theorem 8 (Lifting) Take an initial set of biorthogonal filter operators A new set of biorthogonal filter operators can be found as Scaling functions and H and untouched

8. Proof of Biorthogonality

9. Choice of S Choose S to increase the number of vanishing moments of the wavelets Or, choose S so that the wavelet resembles a particular shape –This has important applications in automated target recognition and medical imaging

10. Corollary 6. Take an initial set of finite biorthogonal filters Then a new set of finite biorthogonal filters can be found as where s(  ) is a trigonometric polynomial Same thing expressed in frequency domain

11. Details

12. Theorem 7 (Lifting scheme) Take an initial set of biorthogonal scaling functions and wavelets Then a new set, which is formally biorthognal can be found as where the coefficients s k can be freely chosen. Same thing expressed in indexed notation

13. Dual Lifting Now leave dual scaling function and and G filters untouched

14. Fast Lifted Wavelet Transform Basic Idea: never explicitly form the new filters, but only work with the old filter, which can be trivial, and the S filter.

15. Before Lifting Forward Transform Inverse Transform

16. Examples Interpolating Wavelet Transform Biorthogonal Haar Transform

17. The Lazy Wavelet Subsampling operators E (even) and D (odd)

18. Interpolating Scaling Functions and Wavelets Interpolating filter: always pass through the data points Can always take Dirac function as a formal dual

19. Theorem 15 The set of filters resulting from interpolating scaling functions, and Diracs as their formal dual, can be seen as a dual lifting of the Lazy wavelet.

21. Algorithm of Interpolating Wavelet Transform (indexed form)

22. Example: Improved Haar Increase vanishing moments of the wavelets from 1 to 2 We have

23. Verify Biorthogonality Details

24. Improved Haar (cont)

25. g(0) = g’(0) = 0

26. Verify Biorthogonality Details