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Wavelets and their applications in CG

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A wavelets and their applications in Computer Graphics, Sig 94 ... where Pj={pjk,l}, Qj={qjk,l} Two scale relations. Synthesis filters. Decomposition ... – PowerPoint PPT presentation

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Title: Wavelets and their applications in CG


1
Wavelets and their applications in CGCAGD
  • Speaker Qianqian Hu
  • Date Mar. 28, 2007

2
Outline
  • Introduction
  • 1D wavelets (eg, Haar wavelets)
  • 2D wavelets (eg, spline wavelets)
  • Multiresolution analysis
  • Applications in CGCAGD
  • 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.
  • ltltTen lectures on waveletsgtgt 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 CGCAGD
  • 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 k2jd-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
  • Fj(x) for each j 0,1
  • Step2 select an inner product defined on the
    functions in V0 ,V1
  • Step3 select a set of wavelets ?j(x) that span
    Wj for each j0,1,

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

28
L2 compression
  • For a function ,sis 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 curves 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 Dj ,, Dn-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
  • Vj Njk,m(u)Fjk(u), Wj ?jk(u) satisfy
  • where Pjpjk,l, Qjqjk,l

37
Decomposition
  • Function fj1(u) is decomposed into fj(u) and
    gj(u).
  • where Ajajk,l, Bjbjk,l

38
Curvature
  • For a planar curve fj(u)(x(u),y(u)), curvature
  • curvature derivative
  • fairness indicators

39
Thresholding
  • Hard thresholding s(RnR) ---gtRn with detail
    functions Dj(dj1, dj2,, djk), a threshold value
    ??0,1
  • s(Dj, ?) Dj-?Dj

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)
51
Efficient computation of ML
  • By
  • (P)-filter
  • (Q)-filter
  • By symmetry

52
Illustration
53
ML 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
    X0,Y0.
  • (3) Area preservation
  • compute X,Y such that AAref.

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 a
62
Example
63
Localized deformation
  • Selection of index subset
  • 1,2,,2nI?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!
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