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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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**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**Thank You !**and Have a Good Tme in Busan!

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