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Subdivision Schemes: Sampling to Non-stationary Reconstruction

Explore the evolution from stationary to non-stationary subdivision schemes for accurate data sampling and error minimization, focusing on techniques like polynomial interpolation, Fourier series, and wavelets. Learn to handle sampled data for smooth reconstruction in computer graphics and Wavelet Theory applications.

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Subdivision Schemes: Sampling to Non-stationary Reconstruction

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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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