1 / 70

Wavelets and their applications in CG&CAGD

Wavelets and their applications in CG&CAGD. Speaker: Qianqian Hu Date: Mar. 28, 2007. Outline. Introduction 1D wavelets (eg, Haar wavelets) 2D wavelets (eg, spline wavelets) Multiresolution analysis Applications in CG&CAGD Fairing curves Deformation of curves. References.

Audrey
Download Presentation

Wavelets and their applications in CG&CAGD

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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)

More Related