Subdivision: From Stationary to Non-stationary scheme.

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

2. Data Type

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

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

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

6. Example • Consider the function with the data on

7. Polynomial Interpolation

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

9. Gaussian Interpolation

10. Subdivision Scheme Stationary and Non-stationary

11. Chainkin’s Algorithm : corner cutting

12. Deslauriers-Dubuc Algorithm

13. Non-stationary Butterfly Scheme Subdivision

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

15. Subdivision Scheme • Formulation

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

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

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

19. Basic Limit Function : DD-scheme

20. Basic Issues • Convergence • Smoothness • Accuracy (Approximation Order)

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

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.

23. Interpolatory Scheme

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.

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

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

27. Stencils of Masks

28. Quasi-interpolatory subdivision • General case

29. Quasi-interpolatory subdivision • Comparison

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

31. Example

32. Example

33. Laurent Polynomial

34. Smoothness

35. Smoothness : Comparison

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

37. Symmetric Biorthogonal Wavelets

38. Symmetric Biorthogonal Wavelets

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

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

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

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

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