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Wavelets Shai Avidan Tel Aviv University Slide Credits (partial - PowerPoint PPT Presentation

Wavelets Shai Avidan Tel Aviv University Slide Credits (partial list) Rick Szeliski Steve Seitz Alyosha Efros Yacov Hel-Or Miki Elad Hagit Hel-Or Marc Levoy Bill Freeman Fredo Durand Sylvain


  1. Wavelets Shai Avidan Tel Aviv University

  2. Slide Credits � (partial list) • Rick Szeliski • Steve Seitz • Alyosha Efros • Yacov Hel-Or • Miki Elad • Hagit Hel-Or • Marc Levoy • Bill Freeman • Fredo Durand • Sylvain Paris • Andrew Adams

  3. Laplacian Pyramid • Make the coarse level by downsampling • Make the fine level by upsampling the coarse layer, and taking the difference with the original • Reconstruct by upsampling the coarse layer and adding the fine layer

  4. Problem • Lapalacian Pyarmid is a redundant representation. Coarse level is blurry and redundant. What about the fine layer?

  5. The basic 2x2 transform Lets start with a simple transform Given signal [a,b] we obtain [c,d] Given [c,d] we can recover [a,b] Observe that c is sum of elements and d is their difference The scaling function: [1 1] The wavelet function: [-1 1]

  6. Extending to even length vector Given sequence v[M] Example: v=[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16] c = 1/sqrt(2) [ (1+2) (3+4)…(15+16)] = 1/sqrt(2) [3 7 11 15 19 23 27 31] d =1/sqrt(2) [(1-2) (3-4)…(15-16)] = 1/sqrt(2) [-1 -1 … -1]

  7. To reconstruct Reconstruct: c = 1/sqrt(2)[0 3 0 7 0 11 0 15 0 19 0 23 0 27 0 31] d = 1/sqrt(2)[0 -1 0 -1 … 0 -1] LPF(c) = 1/2[(0+3) (3+0) (0+7) (7+0)…] = 1/2[3 3 7 7 … 31 31] HPF(d) = 1/2[(0+1) (-1-0) …] = 1/2[1 -1 …] LPF(c)-HPF(d) = 1/2([3 3 7 7 … 31 31]-[1 -1 …]) = 1/2([(3-1) (3+1) (7-1) (7+1)…]) = [1 2 3 4 …] Recall: LPF is: 1/sqrt(2)[1 1] HPF is: 1/sqrt(2)[1 -1]

  8. Block diagrams

  9. In Matrix form A matrix representation of the transform on vector v:

  10. Multi-Resolution Repeat the process on the coarse level

  11. And in Matrix form The right matrix is the one we saw before, the left matrix keeps the last two elements and transforms the first two

  12. 2D Extension

  13. Wavelets • Memory efficient (unlike pyramids) • Time efficient (O(n) and not O(nlogn) like Fourier) • Simple to compute • Orthogonal • Not redundant

  14. Wavelet basis functions

  15. The Scaling function

  16. The inner product

  17. The wavelet function

  18. The wavelet function

  19. Example I I = [9 7 3 5]

  20. Example II

  21. Example III

  22. Wavelet Lifting Here’s another way to build a wavelet transform: Given vector X of size N=2^n Denote P_G(0) = X P_G(1) = Reduce(P_G(0)) – Select even elements Use P_G(1) to predict P^odd_G(0) Now let: P_w(0) = P^odd_G(0)-P_G(1) And we can repeat this in a multi-scale fashion.

  23. Version I

  24. Version I - Example

  25. Lifting Scheme (Version I) So far we’ve used simple Reduce. Can we use a more complicated reduce function? The answer is yes and it’s called Lifting Scheme. It was proposed by Sweldens in the mid 90’s. Given X, split to even X_e and odd X_o. Use X_e to predict X_o, so: X^new_o = X_o – Pred(X_e) The advantage is that we can use better prediction schemes with larger support

  26. Lifting Scheme (ExampleI) Observe we have many more zeros

  27. Lifting Scheme

  28. Lifting Scheme

  29. Lifting Scheme

  30. Lifting Scheme Predict and Update don’t have to be linear…

  31. Wavelet example

  32. Wavelet example

  33. Wavelet Shrinkage Denoising Noisy image Clean image

  34. Wavelet Shrinkage Denoising

  35. Wavelet Shrinkage Denoising New Wavelet coef. value Wavelet coef. value

  36. Wavelet Shrinkage Pipeline

  37. Wavelet Compression (I)

  38. Wavelet Compression (II)

  39. Wavelet Compression (III)

  40. Wavelet Compression (Result)

  41. JPEG 2000 Wavelet transform used in JPEG 2000

  42. JPEG 2000

  43. JPEG 2000

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