Wavelets and their applications in CG&CAGD

1. Wavelets and their applications in CG&CAGD Speaker: Qianqian Hu Date: Mar. 28, 2007

2. Outline • Introduction • 1D wavelets (eg, Haar wavelets) • 2D wavelets (eg, spline wavelets) • Multiresolution analysis • Applications in CG&CAGD • Fairing curves • Deformation of curves

3. References • L.M. Reissell, P. Schroder, M.F. Cohen. A wavelets and their applications in Computer Graphics, Sig 94 • E.J. Stollnitz, T.D. DeRose, D.H., Salesin. Wavelets for Computer Graphics: A Primer.IEEE Computer Graphics and Applications, 1995, 15. • G. Amati. A multi-level filtering approach for fairing planar cubic B-spline curves, CAGD, 2007 (24) 53-66 • S. Hahmann, B. Sauvage, G.P., Bonneau. Area preserving deformation of multiresolution curves, CAGD, 2005 (22) 359-367. • M, Bertram. Single-knot wavelets for non-uniform Bsplines. CAGD, 2005 (22) 849-864.

4. Background • In 1974, French engineer J.Morlet put forward the concept of wavelet transform. • A wavelet basis is constructed by Y.Meyer in 1986. • <<Ten lectures on wavelets>> by I.Daubechies

5. Applications • Math: numerical analysis, curve/surface construction, solve PDE, control theory • Signal analysis: filtering, denoise, compression, transfer • Image process: compression, classification, recognition and diagnosis • Medical imaging: reduce the time of MRI, CT, B-ultrasonography

6. Applications in CG&CAGD • Image editing • Image compression • Automatic LOD control for editing • Surface construction for contours • Deformation • Fairing curves

7. What is wavelets analysis? • A method of data analysis, similar to Taylor expansion, Fourier transform a coarse function A complex function detail coefficients

8. Haar wavelet transform(I) • The simplest wavelet basis [8 4 1 3] detail coefficients [6 2] 8 = 6 + 2 1 = 2 + (-1) 4 = 6 –2 3 = 2 – (-1) [2 -1]

9. Haar wavelet transform(II) The wavelet transform is given by [4 2 2 -1]

10. Advantages • (1) reconstruct any resolution of the function • (2) many detail coefficients are very small in magnitude.

11. Haar wavelet basis functions • The vector space V j • The spaces V j are nested • The basis for V j is given by

12. Example • The four basis functions for V 2

13. Wavelets • The orthogonal space • The properties: • together with form a basis for • Orthogonal property:

14. Haar wavelets • Definition:

15. 2D Haar wavelet transforms(I) • The standard decomposition

16. 2D Haar wavelet transforms(II) • The non-standard decomposition

17. 2D Haar basis functions(I) • The standard construction

18. 2D Haar basis functions(II) • The non-standard construction

19. Haar basis • Advantages: • Simplicity • Orthogonality • Very compact supports • Non-overlapping scaling functions • Non-overlapping wavelets • Disadvantages: • Lack of continuity

20. B-spline wavelets • Define the scaling functions • 1) endpoint interpolation • 2) For , choose k=2j+d-1 to produce 2j equally-spaced interior intervals.

21. B-spline scaling functions

22. Multiresolution analysis • A nested set of vector spaces {Vj}: • Wavelet spaces {Wj}: for each j

23. Refinement equations • For scaling functions • For wavelets

24. Filter bank • For a funcion in Vn with the coefficients A low-resolution version Cn is The lost detail is

25. Analysis & synthesis • Analysis: Splitting Cn into Cn-1 and Dn-1 Analysis filters: An and Bn • Synthesis: recovering Cn from Cn-1 and Dn-1 Synthesis filters: Pn and Qn

26. Framework • Step1: select the scaling functions Φj(x) for each j =0,1… • Step2: select an inner product defined on the functions in V0 ,V1 … • Step3: select a set of wavelets Ψj(x) that span Wj for each j=0,1,…

27. Image compression in L2 • Description of problem Suppose we are given a function f(x) expressed as and a user-specified error tolerance ε. We are looking for such that for L2 norm.

28. L2 compression • For a function ,σis a permutation of 0,…,M-1. the approximation error is

29. Main steps • Step 1: compute coefficients in a normalized 2D Haar basis. • Step 2: Sort the coefficients in order of decreasing magnitude • Step 3: Starting with M’ = M, find the least M’ with

30. Example

31. Multiresolution curves • Change the overall “sweep” of a curve while maintaining its characters • Change a curve’s characters without affecting its overall “sweep” • Edit a curve at any continuous level of detail • Continuous levels of smoothing • Curve approximation within a prescribed error.

32. Example

33. Editing “character” • For multiresolution decomposition C0 ,...,Cn-1, D0 ,…,Dn-1, replacing Dj ,…,Dn-1 with Ďj ,…, Ďn-1

34. Fairing curves • Main idea: wavelet transform • Imperfections: • undesired inflections • curvature bumps • curvature discontinuities • non-monotonic curvature

35. Multi-level representation • A cubic planar B-spline curve with a uniform knot sequence and a multiplicity vector

36. Two scale relations Synthesis filters Definition of wavelets • Vj ={Njk,m(u)=Φjk(u)}, Wj ={Ψjk(u)} satisfy where Pj={pjk,l}, Qj={qjk,l}

37. Decomposition • Function fj+1(u) is decomposed into fj(u) and gj(u). where Aj={ajk,l}, Bj={bjk,l}

38. Local fairness global fairness Curvature • For a planar curve fj(u)=(x(u),y(u)), curvature: curvature derivative: fairness indicators:

39. Thresholding • Hard thresholding σ:(Rn×R) --->Rn with detail functions Dj=(dj1, dj2,…, djk), a threshold value λ∈[0,1] σ(Dj,λ) = Dj-λDj

40. Algorithm

41. Example 1

42. Example 1

43. Example 2

44. Example 2

45. Curve deformation • Multiresolution editing • Area preserving

46. Multiresolution curve • For a curve c(t) Decomposition: Reconstruction:

47. Example

48. Area of a MR-curve • The signed area: • For any level of resolution L, where

49. Area matrix(I)

50. Area matrix(II)