Subdivision: From Stationary to Non-stationary scheme.

transcript

Subdivision: From Subdivision: From Stationary to Non-Stationary to Non-stationary scheme.stationary scheme.

Jungho YoonDepartment of Mathematics

Ewha W. University

2006.01.09 KMMCS 동서대학교

Data Type

Sampling/Reconstruction How to Sample/Re-sample ? - From Continuous object to a finite point

How to handle the sampled data - From a finite sampled data to a continuous

representation

Error between the reconstructed shape and the original shape

Subdivision SchemeSubdivision Scheme A simple local averaging rule to build curves and

surfaces in computer graphics

A progress scheme with naturally built-in Multiresolution Structure

One of the most im portant tool in Wavelet Theory

Approximation Methods

Polynomial Interpolation Fourier Series Spline Radial Basis Function (Moving) Least Square Subdivision Wavelets

Example

Consider the function

with the data on

Polynomial Interpolation

Shifts of One Basis Function Approximation by shifts of one basis

function :

How to choose ?

Gaussian Interpolation

Subdivision Scheme

Stationary and Non-stationary

Chainkin’s Algorithm : Chainkin’s Algorithm : corner cuttingcorner cutting

Deslauriers-Dubuc AlgorithmDeslauriers-Dubuc Algorithm

SubdivisionSubdivision Non-stationary Butterfly Scheme

Subdivision SchemeSubdivision Scheme Types

► Stationary or Nonstationary

► Interpolating or Approximating

► Curve or Surface

► Triangular or Quadrilateral

Subdivision SchemeSubdivision Scheme Formulation

Subdivision SchemeSubdivision Scheme Stationary Scheme, i.e.,

Curve scheme (which consists of two rules)

Subdivision : The Limit Subdivision : The Limit FunctionFunction

: the limit function of the subdivision Let Then is called the basic limit funtio

n. In particular, satisfies the two scale relation

Basic Limit Function : B-splinesBasic Limit Function : B-splines

B_1 spline Cubic spline

Basic Limit FunctionBasic Limit Function : DD- : DD-schemescheme

Basic IssuesBasic Issues

Convergence

Smoothness

Accuracy (Approximation Order)

BBmm-spline subdivision scheme-spline subdivision scheme

Laurent polynomial :

Smoothness Cm-1 with minimal support.

Approximation order is two for all m.

Interpolatory SubdivisionInterpolatory Subdivision

The general form

4-point interpolatory scheme :

The Smoothness is C1 in some range of w. The Approximation order is 4 with w=1/16.

Interpolatory SchemeInterpolatory Scheme

GoalGoal Construct a new scheme which combines the ad

vantages of the aforementioned schemes, while overcoming their drawbacks. Construct Biorthogonal Wavelets

This large family of Subdivision Schemes includes the DD interpolatory scheme and

B-splines up to degree 4.

Reprod. Polynomials < LReprod. Polynomials < L

Case 1 : L is Even, i.e., L=2N

Reprod. Polynomials < LReprod. Polynomials < L Case 2 : L is Odd, i.e., L=2N+1

Stencils of MasksStencils of Masks

Quasi-interpolatory subdivisionQuasi-interpolatory subdivision General case

L Mask set Sm.

Range of tension

1 O=[v, 1-v] (* If v=1/4, quad spline) E= [1-v, v]

C1 1/4

2 O=[v, 1-2v, v] (* If v= 1/8, cubic spline) E= [1/2, 1/2]

C2 1/8

3 O=[-1/16,9/16,9/16,-1/16] E= [-v, 4v,1-6v,4v,-v]

C2 0.0208<v<0.0404

4 O=[-v,–77/2048+5v,385/512-10v, 385/1024+10v,-55/512-5v,35/2048+v] E(i)=O(7-i) for i=1:6

C3 -0.0106<v<-0.0012

5 O=[3,–25,150,150,–25,3]/256] E=[-v,6v,–15v,1+20v,-15v,6v,-v]

C3 -0.0084<v<-0.0046

Quasi-interpolatory subdivisionQuasi-interpolatory subdivision

Comparison

CubicB-spline

4-pts interpolatoryscheme

SL Where L=4 (4-5)-scheme

Support of limit ftn [-2, 2] [-3, 3] [-4, 4]

MaximalSmoothness C2 C1 C3

Approximation

Order2 4 4

Quasi-interpolatory subdivisionQuasi-interpolatory subdivision Basic limit functions for the case L=4

ExampleExample

Laurent Polynomial

Smoothness

Smoothness : Comparison

Biorthogonal WaveletsBiorthogonal Wavelets

Let and be dual each other if

The corresponding wavelet functions are constructed by

Symmetric Biorthogonal WaveletsSymmetric Biorthogonal Wavelets

Nonstationary SubdivisionNonstationary Subdivision

Varying masks depending on the levels, i.e.,

AdvantagesAdvantages

Design Flexibility

Higher Accuracy than the Scheme based on Polynomial

Nonstationary SubdivisionNonstationary Subdivision

Smoothness

Accuracy

Scheme (Quasi-Interpolatory)

Non-Stationary Wavelets

Schemes for Surface

Current Project Construct a new compactly supported biorthogon

al wavelet systems based on Exponential B-splines

Application to Signal process and Medical Imaging (MRI or CT data) Wavelets on special points such GCL points for Numerical PDE

Thank You !and

Have a Good Tme in Busan!

Hope to see you in

Subdivision: From Stationary to Non-stationary scheme.

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