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

2. Data Type
KMMCS 동서대학교

3. 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
KMMCS 동서대학교

4. 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
KMMCS 동서대학교

5. Approximation Methods
Polynomial Interpolation
Fourier Series
Spline
Radial Basis Function
(Moving) Least Square
Subdivision
Wavelets
KMMCS 동서대학교

6. Example
Consider the function
with the data on
KMMCS 동서대학교

7. Polynomial Interpolation
KMMCS 동서대학교

8. Shifts of One Basis Function
Approximation by shifts of one basis function :
How to choose ?
KMMCS 동서대학교

9. Gaussian Interpolation
KMMCS 동서대학교

10. Stationary and Non-stationary
Subdivision Scheme
Stationary and Non-stationary

11. Chainkin’s Algorithm : corner cutting
KMMCS 동서대학교

12. Deslauriers-Dubuc Algorithm
KMMCS 동서대학교

13. Subdivision
Non-stationary Butterfly Scheme
KMMCS 동서대학교

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

15. Subdivision Scheme
Formulation
KMMCS 동서대학교

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

17. 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
KMMCS 동서대학교

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

19. Basic Limit Function : DD-scheme
KMMCS 동서대학교

20. Basic Issues Convergence Smoothness Accuracy (Approximation Order)
KMMCS 동서대학교

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

22. 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.
KMMCS 동서대학교

23. Interpolatory Scheme
KMMCS 동서대학교

24. 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.
KMMCS 동서대학교

25. Reprod. Polynomials < L
Case 1 : L is Even, i.e., L=2N
KMMCS 동서대학교

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

27. Stencils of Masks
KMMCS 동서대학교

28. 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/ v,-55/512-5v,35/2048+v]
E(i)=O(7-i) for i=1:6 C3
<v< 5
O=[3,–25,150,150,–25,3]/256]
E=[-v,6v,–15v,1+20v,-15v,6v,-v]
<v<
KMMCS 동서대학교

29. 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
KMMCS 동서대학교

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

31. Example
KMMCS 동서대학교

32. Example
KMMCS 동서대학교

33. Laurent Polynomial
KMMCS 동서대학교

34. Smoothness
KMMCS 동서대학교

35. Smoothness : Comparison
KMMCS 동서대학교

36. Biorthogonal Wavelets
Let and be dual each other if
The corresponding wavelet functions are constructed by
KMMCS 동서대학교

37. Symmetric Biorthogonal Wavelets
KMMCS 동서대학교

38. Symmetric Biorthogonal Wavelets
KMMCS 동서대학교

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

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

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

42. 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
KMMCS 동서대학교

43. Thank You ! and Have a Good Tme in Busan!
KMMCS 동서대학교

44. Hope to see you in
KMMCS 동서대학교