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Lifting

Lifting. Part 3: Designing Update Operators Ref: SIGGRAPH 96. General Concepts. We discussed several ways of realizing predictors in the context of an inverse transform with zero wavelet coefficients Now we discuss how to design “Update” boxes Design objective:

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Lifting

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  1. Lifting Part 3: Designing Update Operators Ref: SIGGRAPH 96

  2. General Concepts • We discussed several ways of realizing predictors in the context of an inverse transform with zero wavelet coefficients • Now we discuss how to design “Update” boxes • Design objective: • Ensure coarsened signal has the same average (and/or other properties) as the original (higher resolution) signal

  3. Begin with m DOFs 2m DOFs after one step The extra m DOFs: the difference between Vj and Vj+1 in resolution Design update boxes is all about manipulating the inbetweeen DOFs So that the coarser signal (m DOFs) and finer signal (2m DOFs) are “similar” DOF Analysis of Inverse Transform

  4. Design Update for Interpolating Lifting

  5. Recall Cascade Algorithm • Scaling functions and wavelets

  6. Interpolating Lifting (linear) w/o U boxes !

  7. Merge PI Details (lo-wire)

  8. Merge Details

  9. Observation • These “wavelet” generated by the low wire do not have zero integral, a condition must be satisfied to ensure equality of the sample average • Manipulate the extra m DOF to achieve this goal • Design methodology: • Start from lazy wavelet and incorporate scaling functions of the same level to build a more performing wavelet Interlace splitting

  10. …0,0,1,0,0… Hi-Wire (after U) Lo-Wire (after P) Merged result should sum up to zero: A=1/4

  11. Summary: Linear Update Operator

  12. Graphically, …

  13. even odd Forward Linear Wavelet Transform

  14. even odd Inverse Linear Wavelet Transform

  15. Wavelet Basis • Cascading on the lower wire (with update) to get wavelet function

  16. sj upsampling Refinement Relations dj+1

  17. 0,-1/4, -1/4,0 0,0,0,0 0,-1/8,-1/4,3/4,-1/4,-1/8,0,0 0,1,0,0 -1/8,3/4,-1/8,0 Hi-wire (after U) Lo-wire (after P) Merged result Cascading of Linear Wavelet

  18. 0,0,0,0 0,1,0,0 Cascading to Reach Limit Function

  19. Wavelets from Interpolating Subdivision

  20. More Powerful Update Operators • Ensure not only the average but also the first moments of the sequences are preserved:

  21. N = 4, = 4 0th moment (average): 2nd moment: Additionally, 1st and 3rd moment vanishes due to symmetry

  22. Design Higher Order Update • Note that symmetry is the key of this form • Solve A and B by requiring the vanishing of 0th and 2nd moment • Due to symmetry, 1st and 3rd moments are zero • Dual order is 4

  23. N = 2, = 4

  24. Update for Average-Interpolating Lifting

  25. sj-1 sj PAI UHaar PHaar Split dj-1 Average-Interpolating (AI) Lifting • Note that there is a slight difference for AI Lifting: • The lazy wavelet (interlace splitting) is followed by a Haar transform. Therefore, the forward transform looks like:

  26. sj-1 sj PAI UHaar PHaar Split dj-1 Properties of AI Lifting • Use the update of Haar to preserve average • No need to further design new update operator (unless further vanishing moments requested) Same Average! (from Haar)

  27. sj-1 sj dj-1 AI Lifting (cont) • Inverse Transform

  28. Ex: AI Transform (forward) sj+1,k After PHaar After UHaar After PAI sj,k: coarsened signal dj,k: difference signal

  29. Ex: AI Transform (inverse) 3, 4, 1, 4 4, 3, 2, 3.5 3, 5, 4, 2, 1, 3, 4, 3 2.125, -1.5, 1.875, -1.5 2, -2, 2, -1 5, 2, 3, 3

  30. 0, -1/2, 0, 0 0, 0, 0, 0 0, 0, -1/2, 1/2, 0, 0, 0, 0 0, 1, 0, 0 0, 1/2, 0, 0 0, 1/16, -9/16, 7/16, 1/16, 0, 0, 0 0, -1/16, -7/16, 9/16, -1/16, 0, 0, 0 0, -1/8, 1/8, 1/8, -1/8, 0, 0, 0 AI Wavelets by Cascading (N=3) Merged result: 0, 0, 1/16, -1/16, -9/16, -7/16, 7/16, 9/16, 1/16, -1/16, 0, 0, 0

  31. Wavelets from Average-Interpolation

  32. MRA (Part 2)

  33. Function Projection • Consider an initial signal, its coarser approximation: • Their difference lies in the space spanned by the wavelet functions

  34. There are many things related to the definition of dn-1(x) how dn-1,l is computed (the forward transform) The basis (wavelets) If the order of MRA is N, then the wavelet transform started from any polynomial sn(x) with degree less than N will yield zero dn-1,l That is, …

  35. Dual Order of MRA • We say the dual order is if the wavelets have vanishing moments: • All wavelets (translated and dilated) have the same vanishing moments • Recall: moment, time shifting, time scaling

  36. Dual Order of MRA • As a result, all detail functions dj(x), represented as linear combination of the wavelet basis, have the same vanishing moments • And, all coarser versions of sj(x) have the moments independent of j quantitatively stating what properties are preserved during coarsening

  37. Homeworks • derive update operator for cubic interpolation • Implement cascading for graphing wavelets • Verify biorthogonality of linear lifting • Compare interpolating and AI

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