Subdivision: From Stationary to Non-stationary scheme. Jungho Yoon Department of Mathematics Ewha W. University

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 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 2006.01.09 KMMCS 동서대학교

Stationary and Non-stationary Subdivision Scheme Stationary and Non-stationary

Chainkin’s Algorithm : corner cutting 2006.01.09 KMMCS 동서대학교

Deslauriers-Dubuc Algorithm 2006.01.09 KMMCS 동서대학교

Subdivision Non-stationary Butterfly Scheme 2006.01.09 KMMCS 동서대학교

Subdivision Scheme Types ► Stationary or Nonstationary ► Interpolating or Approximating ► Curve or Surface ► Triangular or Quadrilateral 2006.01.09 KMMCS 동서대학교

Subdivision Scheme Formulation 2006.01.09 KMMCS 동서대학교

Subdivision Scheme Stationary Scheme, i.e., Curve scheme (which consists of two rules) 2006.01.09 KMMCS 동서대학교

Subdivision : The Limit Function : the limit function of the subdivision Let Then is called the basic limit funtion. In particular, satisfies the two scale relation 2006.01.09 KMMCS 동서대학교

Basic Limit Function : B-splines B_1 spline Cubic spline 2006.01.09 KMMCS 동서대학교

Basic Limit Function : DD-scheme 2006.01.09 KMMCS 동서대학교

Basic Issues Convergence Smoothness Accuracy (Approximation Order) 2006.01.09 KMMCS 동서대학교

Bm-spline subdivision scheme Laurent polynomial : Smoothness Cm-1 with minimal support. Approximation order is two for all m. 2006.01.09 KMMCS 동서대학교

Interpolatory 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 Scheme 2006.01.09 KMMCS 동서대학교

Goal Construct a new scheme which combines the advantages 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 < L Case 1 : L is Even, i.e., L=2N 2006.01.09 KMMCS 동서대학교

Reprod. Polynomials < L Case 2 : L is Odd, i.e., L=2N+1 2006.01.09 KMMCS 동서대학교

Stencils of Masks 2006.01.09 KMMCS 동서대학교

Quasi-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] 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] -0.0084<v<-0.0046 2006.01.09 KMMCS 동서대학교

Quasi-interpolatory subdivision Comparison Cubic B-spline 4-pts interpolatory scheme SL Where L=4 (4-5)-scheme Support of limit ftn [-2, 2] [-3, 3] [-4, 4] Maximal Smoothness C2 C1 C3 Approximation Order 2 4 2006.01.09 KMMCS 동서대학교

Quasi-interpolatory subdivision Basic limit functions for the case L=4 2006.01.09 KMMCS 동서대학교

Example 2006.01.09 KMMCS 동서대학교

Example 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 Wavelets Let and be dual each other if The corresponding wavelet functions are constructed by 2006.01.09 KMMCS 동서대학교

Symmetric Biorthogonal Wavelets 2006.01.09 KMMCS 동서대학교

Symmetric Biorthogonal Wavelets 2006.01.09 KMMCS 동서대학교

Nonstationary Subdivision Varying masks depending on the levels, i.e., 2006.01.09 KMMCS 동서대학교

Advantages Design Flexibility Higher Accuracy than the Scheme based on Polynomial 2006.01.09 KMMCS 동서대학교

Nonstationary Subdivision Smoothness Accuracy Scheme (Quasi-Interpolatory) Non-Stationary Wavelets Schemes for Surface 2006.01.09 KMMCS 동서대학교

Current Project Construct a new compactly supported biorthogonal 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 2006.01.09 KMMCS 동서대학교