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Subdivision: From Stationary to Non-stationary scheme. Jungho Yoon Department of Mathematics Ewha W. University. Data Type. Sampling/Reconstruction. How to Sample/Re-sample ? - From Continuous object to a finite point set How to handle the sampled data - PowerPoint PPT Presentation
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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
2006.01.09 KMMCS 동서대학교
Example
Consider the function
with the data on
2006.01.09 KMMCS 동서대학교
Polynomial Interpolation
2006.01.09 KMMCS 동서대학교
Shifts of One Basis Function Approximation by shifts of one basis
function :
How to choose ?
2006.01.09 KMMCS 동서대학교
Gaussian Interpolation
Subdivision Scheme
Stationary and Non-stationary
2006.01.09 KMMCS 동서대학교
Chainkin’s Algorithm : Chainkin’s Algorithm : corner cuttingcorner cutting
2006.01.09 KMMCS 동서대학교
Deslauriers-Dubuc AlgorithmDeslauriers-Dubuc Algorithm
2006.01.09 KMMCS 동서대학교
SubdivisionSubdivision Non-stationary Butterfly Scheme
2006.01.09 KMMCS 동서대학교
Subdivision SchemeSubdivision Scheme Types
► Stationary or Nonstationary
► Interpolating or Approximating
► Curve or Surface
► Triangular or Quadrilateral
2006.01.09 KMMCS 동서대학교
Subdivision SchemeSubdivision Scheme Formulation
2006.01.09 KMMCS 동서대학교
Subdivision SchemeSubdivision Scheme Stationary Scheme, i.e.,
Curve scheme (which consists of two rules)
2006.01.09 KMMCS 동서대학교
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
2006.01.09 KMMCS 동서대학교
Basic Limit Function : B-splinesBasic Limit Function : B-splines
B_1 spline Cubic spline
2006.01.09 KMMCS 동서대학교
Basic Limit FunctionBasic Limit Function : DD- : DD-schemescheme
2006.01.09 KMMCS 동서대학교
Basic IssuesBasic Issues
Convergence
Smoothness
Accuracy (Approximation Order)
2006.01.09 KMMCS 동서대학교
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.
2006.01.09 KMMCS 동서대학교
Interpolatory SchemeInterpolatory Scheme
2006.01.09 KMMCS 동서대학교
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
2006.01.09 KMMCS 동서대학교
Reprod. Polynomials < LReprod. Polynomials < L Case 2 : L is Odd, i.e., L=2N+1
2006.01.09 KMMCS 동서대학교
Stencils of MasksStencils of Masks
2006.01.09 KMMCS 동서대학교
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
2006.01.09 KMMCS 동서대학교
Quasi-interpolatory subdivisionQuasi-interpolatory subdivision Basic limit functions for the case L=4
2006.01.09 KMMCS 동서대학교
ExampleExample
2006.01.09 KMMCS 동서대학교
ExampleExample
2006.01.09 KMMCS 동서대학교
Laurent Polynomial
2006.01.09 KMMCS 동서대학교
Smoothness
2006.01.09 KMMCS 동서대학교
Smoothness : Comparison
2006.01.09 KMMCS 동서대학교
Biorthogonal WaveletsBiorthogonal Wavelets
Let and be dual each other if
The corresponding wavelet functions are constructed by
2006.01.09 KMMCS 동서대학교
Symmetric Biorthogonal WaveletsSymmetric Biorthogonal Wavelets
2006.01.09 KMMCS 동서대학교
Symmetric Biorthogonal WaveletsSymmetric Biorthogonal Wavelets
2006.01.09 KMMCS 동서대학교
Nonstationary SubdivisionNonstationary Subdivision
Varying masks depending on the levels, i.e.,
2006.01.09 KMMCS 동서대학교
AdvantagesAdvantages
Design Flexibility
Higher Accuracy than the Scheme based on Polynomial
2006.01.09 KMMCS 동서대학교
Nonstationary SubdivisionNonstationary Subdivision
Smoothness
Accuracy
Scheme (Quasi-Interpolatory)
Non-Stationary Wavelets
Schemes for Surface
2006.01.09 KMMCS 동서대학교
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!
2006.01.09 KMMCS 동서대학교
Hope to see you in