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Subdivision: From Stationary to Non-stationary scheme.

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

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Subdivision: From Stationary to Non-stationary scheme.

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  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

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

  44. Hope to see you in

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