Subdivision: From Stationary to Non-stationary scheme. Jungho Yoon Department of Mathematics Ewha W. University
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
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
Approximation Methods • Polynomial Interpolation • Fourier Series • Spline • Radial Basis Function • (Moving) Least Square • Subdivision • Wavelets
Example • Consider the function with the data on
Shifts of One Basis Function • Approximation by shifts of one basis function : • How to choose ?
Subdivision Scheme Stationary and Non-stationary
Non-stationary Butterfly Scheme Subdivision
Subdivision Scheme • Types ► Stationary or Nonstationary ► Interpolating or Approximating ► Curve or Surface ► Triangular or Quadrilateral
Subdivision Scheme • Formulation
Subdivision Scheme • Stationary Scheme, i.e., • Curve scheme (which consists of two rules)
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
Basic Limit Function : B-splines B_1 spline Cubic spline
Basic Issues • Convergence • Smoothness • Accuracy (Approximation Order)
Bm-spline subdivision scheme • Laurent polynomial : • Smoothness Cm-1 with minimal support. • Approximation order is two for all m.
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.
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.
Reprod. Polynomials < L • Case 1 : L is Even, i.e., L=2N
Reprod. Polynomials < L • Case 2 : L is Odd, i.e., L=2N+1
Quasi-interpolatory subdivision • General case
Quasi-interpolatory subdivision • Comparison
Basic limit functions for the case L=4 Quasi-interpolatory subdivision
Biorthogonal Wavelets • Let and be dual each other if • The corresponding wavelet functions are constructed by
Nonstationary Subdivision • Varying masks depending on the levels, i.e.,
Advantages • Design Flexibility • Higher Accuracy than the Scheme based on Polynomial
Nonstationary Subdivision • Smoothness • Accuracy • Scheme (Quasi-Interpolatory) • Non-Stationary Wavelets • Schemes for Surface
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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