Lifting

1. Lifting Part 2: Subdivision Ref: SIGGRAPH96

2. Subdivision Methods On constructing more powerful predictors …

3. Subdivision methods • Often referred to as the cascade algorithm • Systematic ways to build predictors • Concentrate on the P box • Types: • Interpolating subdivision • Average-interpolating • B-spline (more later) • …

4. Interpolating Subdivision First proposed by Deslauriers-Dubic

5. Basic Ideas • In general, use N (=2D) samples to build a polynomial of degree N-1 that interpolates the samples • Calculate the coefficient on the next finer level as the value of this polynomial • e.g., Lagrange polynomial (or Neville’s algorithm) • Order of the subdivision scheme is N • This can be extended to accommodate bounded interval and irregular sampling settings.

6. Math Review: Lagrange Polynomial • The unique n-th degree polynomial that passes through (n+1) points can be expressed as follows:

7. Linear and Cubic Interpolation Order = 2 Order = 4

8. … … … … Numerical Example: Cubic Interpolation Stencil -1/16 9/16 9/16 -1/16

9. Scaling Functions • All scaling functions at different levels are translates and dilates of one fixed function: • the fundamental solution (so named by the original inventor, Deslauriers-Dubuc) of the subdivision scheme • Obtained by cascade algorithm

10. 0,1,0,0 0, .5, 1, .5, 0,0,0,0 0,0,0,0 .5, .5, 0,0 Cascading (linear interpolation)

11. 0,1 0, .5,1, .5 0, .25, .5, .75, 1, .75, .5, .25 0,0 .5, .5 .25, .75, .75, .25

12. Cascading !

13. Compare with what we said before … • From forward transform • Hi-wire: coarsened signal • Lo-wire: difference signal • Subdivision: Inverse transform with zero detail • Cascading: apply delta sequence to get impulse response (literally) • Hi-wire: scaling functions • Lo-wire: wavelets

14. Interpolating Scaling Functions

15. Compact support [-N+1, N-1] Interpolating Smoothness N large, smoother … Polynomial reproduction Polynomials up to degree N-1 can be expressed as linear combinations of scaling functions Properties of Scaling Functions

16. Refinability Properties of Scaling Functions

17. Computing the filter coefficients N=2 N=4

18. upsampling sj+1 Refinement Relations sj

19. sj upsampling sj+1

20. Average-Interpolating Subdivision Proposed by Donoho (1993)

21. Basic Ideas Think of the signals as the intensity obtained from CCD

22. Meaning of Signal sj,k p(x) area = Sj,k (width) Sj,k: the average signal in this interval CCD sensor

23. Which (constant) polynomial would have produced these average? Subdivide according to the (implied) constant polynomial Averaging-interpolating subdivision (constant) Order = 1

24. defines the (implied) quadratic curve produce the finer averages accordingly Average-interpolating subdivision (quadratic) Order = 3

25. Average-Interpolating (N=3) The coefficient “2” is due to half width p(x) is the (implied) quadratic polynomial

26. 3rd degree polynomial Average-Interpolating (N=3) Define 4 conditions: P(x) can be determined

27. … … 0 1 2 3 1.5 Numeric Example (N=3) 4.875 5.125 Solve for P(1.5) =5.4375 using Lagrange polynomial (next page)

28. Details Lagrange Polynomial

29. Derive Weighting (N=3) Check: If sj,k-1 = sj, k= sj,k+1 = x, P(1.5) = 1.5x = 24x/16

30. Consider in-place Computation Problem: occupy the same piece of memory Solution 1 : compute sj+1,2k+1 first Not a good solution… dependent on execution sequence

31. Utilize inverse Haar transform ! Observe that …

32. Closed form of quadratic PAI

33. Three-Stage Lifting

34. Numerical Example (N=3) 2.75, 4.875, 4.25, 3.125 3, 5, 4, 3 0, 0, 0, 0 0.5, 0.25, –0.5, –0.25 3.25, 5.125, 3.75, 2.875 Merged Result: 2.75, 3.25, 4.875, 5.125, 4.25, 3.75, 3.125, 2.875

35. AI Subdivision

36. AI Scaling Function by Cascading (N=3) -0.125, 1, 0.125, 0 0, 1, 0, 0 0, 0, 0, 0 0.125, 1, -0.125, 0 0.25, 0, –0.25, 0 Merged Result: -0.125, 0.125, 1, 1, 0.125, -0.125, 0, 0

37. Remark • Recall inverse Haar preserves average … • Implying … • More about this later

38. Compact support [-N+1, N] Average-interpolating Polynomial reproduction Up to degree N-1 Smoothness: continuous of order R(N) Refinability: Obtained similarly as in interpolating subdivision Properties of Scaling Functions

39. Average-interpolating scaling functions

40. Types of Predictors: Interpolating Average-interpolating B-spline So far, we only considered subdivision in inverse transform. How about its role in forward transform? Roles of Predictors In inverse transform Subdivision In forward transform: Predict results to generate the difference signal (low-wire) More … On constructing more powerful P boxes Define “power”!? Summary

41. MRA and Lifting(part I)

42. MRA Properties • Scaling functions at all levels are dilated and translated copies of a single function

43. Order of an MRA • The order of MRA is N if every polynomial of degree < N can be written exactly as a linear combination of scaling functions of a given level • The order of MRA is the same as the order of the predictor used to build the scaling functions

44. Scaling functions: delta sequence on hi-wire Wavelets: delta sequence on lo-wire Graphing by Cascading More on this later

45. Homeworks • Derive the weights for cubic interpolation • Implement cascading to see scaling functions (and wavelets) at different levels • Use lifting to process audio data • Provide routines for read/write/plot data • denoising radio recordings (WAV)

46. undecided

47. Convention: • Smaller index, smaller data set (coarser) • 2D lifting the same as classical?! • Lifting and biorthogonality!?

48. Filter coefficient From lifting-2 Refinement relations follow from the fact that subdivision from level 0 with s0,k and level 1 with s1,k should be the same.