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. > 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] [6 2] detail coefficients 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=2 j +d-1 to produce 2 j equally-spaced interior intervals.

21. B-spline scaling functions

22. Multiresolution analysis A nested set of vector spaces {V j }: Wavelet spaces {W j }: for each j

23. Refinement equations For scaling functions For wavelets

24. Filter bank For a funcion in V n with the coefficients A low-resolution version C n is The lost detail is

25. Analysis & synthesis Analysis: Splitting C n into C n-1 and D n-1 Analysis filters: A n and B n Synthesis: recovering C n from C n-1 and D n-1 Synthesis filters: P n and Q n

26. Framework Step1: select the scaling functions Φ j (x) for each j =0,1… Step2: select an inner product defined on the functions in V 0,V 1 … Step3: select a set of wavelets Ψ j (x) that span W j for each j=0,1, …

27. Image compression in L 2 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 L 2 norm.

28. L 2 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 C 0,...,C n-1, D 0, …,D n-1, replacing D j, …,D n-1 with Ď j, …, Ď n-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. Definition of wavelets V j ={N j k,m (u)=Φ j k (u)}, W j ={Ψ j k (u)} satisfy where P j ={p j k,l }, Q j ={q j k,l } Two scale relations Synthesis filters

37. Decomposition Function f j+1 (u) is decomposed into f j (u) and g j (u). where A j ={a j k,l }, B j ={b j k,l }

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

39. Thresholding Hard thresholding σ:(R n ×R) --->R n with detail functions D j =(d j 1, d j 2,…, d j k ), a threshold value λ ∈ [0,1] σ(D j, λ) = D j -λD j

40. Algorithm

41. Example 1

43. 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)

51. Efficient computation of M L By (P)-filter: (Q)-filter: By symmetry:

52. Illustration

53. M L for Chaikin MR curves The scaling function: quadratic uniform B-splines

54. Overview of deformation (1) Decomposition: express curve c(t) in a multiresolution basis at level L. (2) Deformation: bend the coarse polygon to get the coordinates X 0,Y 0. (3) Area preservation: compute X,Y such that A=A ref.

55. Optimization method Minimize a smoothness term and a distance term. The smoothness term: prevent the curve to have unwanted wiggles. The distance term: respect the defined deformation as much as possible.

56. Smoothness criteria Minimization the bending energy For a MR-curve at L level, the energy can be expressed as where

57. Area preserving deformation The optimization problem where

58. Linearization(I) Using Lagrange multiplyers, Linearizing the area constraint For, there is If  0, then

59. Linearization(II) The minimization problem with linearized area constraint: The equivalent equation

60. Algorithm

61. Influence of α

62. Example

63. Localized deformation Selection of index subset {1,2,…,2 n }=I ∪ J, I: modified coefficients; J: unchanged coefficients The linear system of equations:

64. Local deformation

65. Upholding moved point

66. Modification of detail coefficients

67. Example

68. Multiresolution surfaces(I) Using tensor products of B-spline scaling functions and wavelets

69. Multiresolution surfaces(II) Wavelets based on subdivision surfaces for arbitrary topology type M. Lounsbery, T.D. DeRose, J. Warren. Multiresolution analysis for surfaces of arbitrary topological type. TOG 1997, 16(1): 34-73

70. Thanks a lot!