Wavelets and their applications in CG&CAGD

Speaker: Qianqian HuDate: 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 L.M. Reissell, P. Schroder, M.F. Cohen. A wavelets a

nd their applications in Computer Graphics, Sig 94 E.J. Stollnitz, T.D. DeRose, D.H., Salesin. Wavelets f

or 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.

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

Applications Math: numerical analysis, curve/surfac

e 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

Applications in CG&CAGD

Image editing Image compression Automatic LOD control for editing Surface construction for contours Deformation Fairing curves

What is wavelets analysis?

A method of data analysis, similar to Taylor expansion, Fourier transform

a coarse functionA complex function detail

coefficients

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]

Haar wavelet transform(II)

The wavelet transform is given by [4 2 2 -1]

Advantages

(1) reconstruct any resolution of the function

(2) many detail coefficients are very small in magnitude.

Haar wavelet basis functions The vector space V j

The spaces V j are nested

The basis for V j is given by

Example

The four basis functions for V 2

Wavelets The orthogonal space

The properties: together with form a basis for Orthogonal property:

ji

1,j j j j jiW V W V

ji 1jV

1

0| 0, , .j j j j j j j j

i i i i i idx W V

Haar wavelets Definition:

2D Haar wavelet transforms(I) The standard decomposition

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

2D Haar basis functions(I) The standard construction

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

Haar basis Advantages:

Simplicity Orthogonality Very compact supports Non-overlapping scaling functions Non-overlapping wavelets

Disadvantages: Lack of continuity

B-spline wavelets Define the scaling functions

1) endpoint interpolation 2) For , choose k=2j+d-1

to produce 2j equally-spaced interior intervals.

B-spline scaling functions

Multiresolution analysis A nested set of vector spaces {Vj}:

Wavelet spaces {Wj}: for each j

1j j jW V V

Refinement equations

For scaling functions

For wavelets

Filter bank For a funcion in Vn with the coefficients

A low-resolution version Cn isThe lost detail is

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

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) th

at span Wj for each j=0,1,…

Image compression in L2

Description of problemSuppose we are given a function f(x) expressed as

and a user-specified error tolerance ε. We are lookingfor

such that for L2 norm.

L2 compression

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

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

Example

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.

Example

Editing “character”

For multiresolution decomposition C0 ,...,Cn-

1, D0 ,…,Dn-1, replacing Dj ,…,Dn-1 with Ďj ,…, Ďn-1

Fairing curves

Main idea: wavelet transform Imperfections:

undesired inflections curvature bumps curvature discontinuities non-monotonic curvature

Multi-level representation

A cubic planar B-spline curve

with a uniform knot sequence

and a multiplicity vector

Definition of wavelets

Vj ={Njk,m(u)=Φj

k(u)}, Wj ={Ψjk(u)} satisfy

where Pj={pjk,l}, Qj={qj

k,l}

Two scale relations

Synthesis filters

Decomposition

Function fj+1(u) is decomposed into fj

(u) and gj(u).

where Aj={ajk,l}, Bj={bj

k,l}

Curvature For a planar curve fj(u)=(x(u),y(u)), curvature:

curvature derivative:

fairness indicators:Local fairness

global fairness

Thresholding Hard thresholding σ:(Rn×R) --->Rn wit

h detail functions Dj=(dj1, dj

2,…, djk), a t

hreshold value λ∈[0,1]

σ(Dj, λ) = Dj-λDj

Algorithm

Example 1

Example 1

Example 2

Example 2

Curve deformation

Multiresolution editing Area preserving

Multiresolution curve For a curve c(t)

Decomposition:Reconstruction:

Example

Area of a MR-curve The signed area:

For any level of resolution L,

where

Area matrix(I)

Area matrix(II)

Efficient computation of ML

By

(P)-filter:(Q)-filter:

By symmetry:

Illustration

ML for Chaikin MR curves The scaling function: quadratic unifor

m B-splines

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,Y

0. (3) Area preservation: compute X,Y such that A=Aref.

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.

Smoothness criteria

Minimization the bending energy

For a MR-curve at L level, the energy can be expressed as

where

Area preserving deformation

The optimization problem

where

Linearization(I) Using Lagrange multiplyers,

Linearizing the area constraint

For , there isIf 0, then

Linearization(II) The minimization problem with linear

ized area constraint:

The equivalent equation

Algorithm

Influence of α

Example

Localized deformation

Selection of index subset

{1,2,…,2n}=I J,∪ I: modified coefficients; J: unchanged coefficients

The linear system of equations:

Local deformation

Upholding moved point

Modification of detail coefficients

Example

Multiresolution surfaces(I) Using tensor products of B-spline scal

ing functions and wavelets

Multiresolution surfaces(II) Wavelets based on subdivision surfaces for

arbitrary topology typeM. Lounsbery, T.D. DeRose, J. Warren. Multiresolution analysis for surfa

ces of arbitrary topological type. TOG 1997, 16(1): 34-73

Thanks a lot!