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The Lifting Scheme: a custom-design construction of biorthogonal wavelets

The Lifting Scheme: a custom-design construction of biorthogonal wavelets. Sweldens95, Sweldens 98 (appeared in SIAM Journal on Mathematical Analysis ). Relations of Biorthogonal Filters. Dual. Dual. Biorthogonal Scaling Functions and Wavelets. transpose.

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The Lifting Scheme: a custom-design construction of biorthogonal wavelets

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  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. Dual Dual Biorthogonal Scaling Functions and Wavelets

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

  5. Operator Notation

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

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

  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 sk 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.

  20. Algorithm of Interpolating Wavelet Transform (indexed form)

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

  22. Details Verify Biorthogonality

  23. Improved Haar (cont)

  24. g(0) = g’(0) = 0

  25. Details Verify Biorthogonality

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