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Lifting Scheme

Lifting Scheme. The Lifting Scheme - Topics . Reminder :approximations,details Haar wavelet transform Lifting scheme Update Higher order extensions. One Stage Filtering . Approximations and details:

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Lifting Scheme

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  1. Lifting Scheme

  2. The Lifting Scheme - Topics • Reminder:approximations,details • Haar wavelet transform • Lifting scheme • Update • Higher order extensions

  3. One Stage Filtering Approximations and details: • The low-frequency content is the most important part in many applications, and gives the signal its identity. This part is called “Approximations” • The high-frequency gives the ‘flavor’, and is called “Details”

  4. LPF Input Signal HPF Approximations and Details: • Approximations:low-frequency components of the signal • Details: high-frequency components A D

  5. A* LPF Input Signal D* HPF Decimation • The former process produces twice the data • To correct this, we Down sample (or: Decimate) the filter output by two. A complete one stage block :

  6. Example*: * Wavelet used: db2

  7. Low pass filter high pass filter Multi-level Decomposition • Iterating the decomposition process, breaks the input signal into many lower-resolution components: Wavelet decomposition tree:

  8. A Simple Example: The Haar Wavelet • Consider two neighboring samples a and b of a sequence. - a and b have some correlation. A simple linear transform: • High correlation - small |d|, fewer bits representation. (i.e. a=b,d=0)

  9. The Haar Wavelet Con’t • No loss of any information • Reconstruction formula of a and b: The key behind Haar Wavelet Transform: these reconstruction formulas can be found by inverting a 2x2 matrix.

  10. The Haar Wavelet Con’t • Signal Sn of 2nsample values Sn,l: Sn= {Sn,l | 0=< l <= 2n} • Apply average (Sn-1) and difference (dn-1) transform onto each pair: a = S2l, b = S2l+1Sn-1,l = Sn,2l+Sn,2l - 2n-1 pairs (l=0…2n-1) 2 dn-1,l = Sn,2l+1-Sn,2l • Recover the original signal Sn from Sn-1 anddn-1

  11. The Haar Wavelet Con’t • Sn-1 as Approximations • dn-1 as Details • Signal with local coherence - approximations closely resembles the original signal - detail is very small (efficient representation)

  12. The Haar Wavelet Con’t • Applying the same transform (averages and differences) to Sn-1 itself. • Split Sn-1 to (yet) coarser signal Sn-2 and another difference signal dn-2,each of them contain 2n-2 samples. • We can repeat this transform n times till S0contains only one sample S0,0. This is the Haar transform

  13. The Haar transform Sn Structure of the wavelet transform: recursively split into averages and differences Sn-1 dn-1 Sn-2 dn-2 S1 d1 S0 d0

  14. The Haar transform Con’t • We end up with: - n detail signals dj(0<=j<=n-1), each with 2j coefficients - one signal s0 containing only one sample s0,0 - the average of all the samples of the original signal.

  15. Inverse Haar transform • Start from S0 and dj for 0<=j<n and obtain Sn again. * note that the total number of coefficients after transform is 1 for S0 plus 2j for each dj. (the exact number of samples of the original signal). n-1 1+∑ 2j = 2n j=0

  16. The Haar transform d0 d1 dn-2,l dn-1,l S0 S1 S2 … Sn-1,l Sn,l • Structure of the inverse wavelet transform: recursively merge averages and differences.

  17. The Haar transform Con’t • The Haar transform can be thought of as applying a NxN matrix (N = 2n) to the signal Sn. - The cost of computing the transform is O(N). (FFT cost is O(NlogN) and linear transformation of an N vector is O(N2)).

  18. Haar Transform in-place • We want to overwrite old values with new values. • First step: compute the difference: d=b-a and store it in b location. • Second step: We use a and d to compute the average: S=a+d/2 and store it in a location. a+ = b/2 ; b -= a

  19. Haar Transform in-place Con’t • Inverse: run code backwards! a- = b/2 ; b+ = a • This particular scheme is a simple instance of the lifting scheme

  20. The Lifting Scheme • Consider a signal Sn with 2nsamples . • Goal: transform the signal into: - a coarser signal Sn-1 - a detail signal dn-1 • 3 steps: - Split - Predict - Update

  21. The Lifting Scheme - Split • divide the input data into: - even indexed samples Sn. - odd indexed samples Sn+1. Lazy wavelet transform

  22. The Lifting Scheme - Predict • predict the odd elements from the even elements-output detail. Sn-1 detail

  23. Predict - - - Example –Predict using Haar • The even sample is the prediction for the odd sample. • The detail dn-1is the difference between the odd sample and its predictor: oddsample left neighbouring even sample - predictor dn-1= Sn,2l+1- Sn,2l detail Note: perfect if function is constant, detail coefficients zero

  24. General Case - Predict • Detail = difference between odd element and predict function calculated from the even elements dn-1=oddn- P(evenn)

  25. The Lifting Scheme - Update • follows the predict phase. • The approximations Sn-1(the signal for next step) should maintain the average of the original signal Sn. • smoother input for the next step of the wavelet transform. Sn-1 = evenj,i + U( dn-1)

  26. Example – Update Haar • Sets Sn-1 to be the average of the even/odd pair (e.g., the even element si and its odd successor, si+1): Sn-1= even j,i + odd j, i 2

  27. Haar transform update step: • recover the original value (from predict step) : dn-1= oddj,i - evenj,i=> oddj,i= evenj,i + dn-1 • Substituting this into the average (prev), we get : Sn-1= even j,i + even j,i + dn-1 2 Sn-1= even j,I + dn-1= even + U(dn-1) 2

  28. The Lifting Scheme • The averages (even elements) become the input for the next recursive step of the forward transform. Sn-1 Sn-2 Sn dn-2 dn-1

  29. The Lifting Scheme • In place computation: - (oddj-1,evenj-1): = Split(Sj) - oddj-1- = P(evenj-1) - evenj-1+ = U(oddj-1)

  30. Inverse Lifting Scheme

  31. Inverse Lifting Scheme-3 stepsUndo Update • dn,S0 given recover even samples by subtructing the update info: • Sn-1 = evenj,i + U( dn-1) evenj,i = Sn-1-U(dn-1) Haar: Sn,2l= Sn-1,l - dn-1,l/2

  32. Inverse Lifting Scheme - Undo Predict • evenn-1,dn-1 given recover odd samples by adding prediction info: dn-1 =oddn-1- P(evenn-1) oddn-1= dn-1 + (evenn-1) • Haar: Sn,2l+1=dn-1,l+Sn,2l

  33. Inverse Lifting Scheme-Merge • zipper odd and even samples recover original signal-inverse Lazy wavelet: Sn=Merge(evenn-1,oddn-1) • Inverse in place • Evenj-1 -= U(oddj-1) • Oddj-1 += P(evenj-1) • Sj:= Merge(oddj-1,evenj-1) • Inverse transform: reversing the order of the operations and flipping the signs

  34. Inverse Lifting Scheme Even values - Undo Update Undo Predict Merge + Odd values

  35. The Lifting Scheme-Example Sn = X = [ 1 2 3 4 5 6 7 8 ] 2n=8 , n=3 Split: Xe=[2 4 6 8] Xo=[1 3 5 7] Pred:averaging neighboures (edges:simple subtruction of one neighbour-can fix by zero padding or wrap around or reflection) Pred{Xe}=[2 3 5 7] dn-1= d2 = Xo- Pred{Xe} = [-1 0 0 0]

  36. The Lifting Scheme-Example Update:Sn-1=even j,I + dn-1,l 2 Xe=[2 4 6 8] dn-1= [-1 0 0 0] S2 = even+d2/2 = [1.5 4 6 8] We repeat recursively: • Split:S2 is splitted simillarly: Xe=[4 8] Xo=[1.5 6] Pred{Xe} = [4 6] dn-2= d1 = Xo- Pred{Xe} = [-2.5 0] S1 = even+d1/2 = [2.75 8] Xe=[8] Xo=[2.75] Pred{Xe} = [8] dn-3= d0 = Xo- Pred{Xe} = [-5.25] S0 = even+d0/2 = [5.375]

  37. The Lifting Scheme-Example • The pyramid will be: d2= [-1 0 0 0] d1 = [-2.5 0] d0 = [-5.25] S0 = [5.375] Good result(most of elements are 0).

  38. The Lifting Scheme-Inverse Example We will recover the original signal from the pyramid: The pyramid : d2= [-1 0 0 0] , d1 = [-2.5 0], d0 = [-5.25] , S0 = [5.375] Inverse transform: Xe =S0 – d0/2 = [8] Pred{Xe}= [8] Xo=d0+Pred{Xe} = [2.75] S1 = [2.75 8]

  39. The Lifting Scheme- Inverse Example Xe=S1 – d1/2 = [4 8] Pred{Xe} = 4 6 Xo=d1+ Pred{Xe} = [1.5 6] S2 = [1.5 4 6 8] Xe=S2– d2/2 = [2 4 6 8] Pred{Xe} = 2 3 5 7 Xo=d0+ Pred{Xe} = [1 3 5 7] S3 = [1 2 3 4 5 6 7 8] Perfect reconstruction!

  40. The Lifting Scheme • Suppose Predict and Update are linear. description of their operation as matrices P and U: Xonew=Xo-PXe Xenew =Xe+UXonew=Xe+UXo-UPXe or: Xonew I -P Xo Xenew U I-UP Xe

  41. The Lifting Scheme - Example 2 • Assume: P=I and U=0.5I Xonew I -p Xo I -I Xo Xenew U I-UP Xe 0.5I 0.5I Xe Odd elementsEven elements differencesaverage of neighbouring pairs

  42. An example:linear wavelet transform • Haar - simple and fast wavelet transform • Limitations - not smooth enough: blocky • Erasing Haar Coefficients: • Fourier analysis not always applicable

  43. The Lifting Scheme • Lifting Build more powerful versions • An example linear wavelet transform

  44. linear wavelet transform –Prediction • Linear Prediction - Use even on either side - Predictor for odd sample Sn,2l+1: average of neighboring samples:on left Sn,2l, on right Sn,2l+2

  45. linear wavelet transform - Prediction

  46. linear wavelet transform • Even values are subsampled

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