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

2006.01.09 KMMCS 동서대학교

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

set

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

representation

Error between the reconstructed shape and the original shape

2006.01.09 KMMCS 동서대학교

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

2006.01.09 KMMCS 동서대학교

Approximation Methods

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

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Example

Consider the function

with the data on

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Polynomial Interpolation

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Shifts of One Basis Function Approximation by shifts of one basis

function :

How to choose ?

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Gaussian Interpolation

Subdivision Scheme

Stationary and Non-stationary

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

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Deslauriers-Dubuc AlgorithmDeslauriers-Dubuc Algorithm

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SubdivisionSubdivision Non-stationary Butterfly Scheme

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Subdivision SchemeSubdivision Scheme Types

► Stationary or Nonstationary

► Interpolating or Approximating

► Curve or Surface

► Triangular or Quadrilateral

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Subdivision SchemeSubdivision Scheme Formulation

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Subdivision SchemeSubdivision Scheme Stationary Scheme, i.e.,

Curve scheme (which consists of two rules)

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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

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

B_1 spline Cubic spline

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

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Basic IssuesBasic Issues

Convergence

Smoothness

Accuracy (Approximation Order)

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BBmm-spline subdivision scheme-spline subdivision scheme

Laurent polynomial :

Smoothness Cm-1 with minimal support.

Approximation order is two for all m.

2006.01.09 KMMCS 동서대학교

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.

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Interpolatory SchemeInterpolatory Scheme

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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.

2006.01.09 KMMCS 동서대학교

Reprod. Polynomials < LReprod. Polynomials < L

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

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

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Stencils of MasksStencils of Masks

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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

2006.01.09 KMMCS 동서대학교

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

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

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ExampleExample

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ExampleExample

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Laurent Polynomial

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Smoothness

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Smoothness : Comparison

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Biorthogonal WaveletsBiorthogonal Wavelets

Let and be dual each other if

The corresponding wavelet functions are constructed by

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Symmetric Biorthogonal WaveletsSymmetric Biorthogonal Wavelets

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Symmetric Biorthogonal WaveletsSymmetric Biorthogonal Wavelets

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Nonstationary SubdivisionNonstationary Subdivision

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

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AdvantagesAdvantages

Design Flexibility

Higher Accuracy than the Scheme based on Polynomial

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Nonstationary SubdivisionNonstationary Subdivision

Smoothness

Accuracy

Scheme (Quasi-Interpolatory)

Non-Stationary Wavelets

Schemes for Surface

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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

2006.01.09 KMMCS 동서대학교

Thank You !and

Have a Good Tme in Busan!

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Hope to see you in