Exponential Quasi-interpolatory Subdivision Scheme Yeon Ju Lee and Jungho Yoon Department of Mathematics, Ewha W. University Seoul, Korea

Exponential quasi-interpolatory s.s. Contents  Subdivision scheme – several type of s.s.  Quasi-interpolatory subdivision scheme  Construction  Smoothness & accuracy  Example  Exponential quasi-interpolatory subdivision scheme  Construction  Smoothness  Example

Exponential quasi-interpolatory s.s. Subdivision scheme  Useful method to construct smooth curves and surfaces in CAGD  The rule :

Exponential quasi-interpolatory s.s. Subdivision scheme  Rule :  Interpolatory s.s. & Non-interpolatory s.s  Stationary s.s. & Non-stationary s.s

Exponential quasi-interpolatory s.s. B-spline subdivision scheme   It has maximal smoothness C m-1 with minimal support.  It has approximation order only 2 for all m.  Cubic-spline :

Exponential quasi-interpolatory s.s. Interpolatory subdivision scheme   4-point interpolatory s.s. :  The Smoothness is C 1 in some range of w.  The Approximation order is 4 with w=1/16.

Exponential quasi-interpolatory s.s.  Goal We want to construct a new scheme which has good smoothness and approximation order.

Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme  Construction

Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme  Advantage  L : odd (L+1,L+2)-scheme. So in even pts case, it has tension.  L : even (L+2,L+2)-scheme. It has tension in both case.  This scheme has good smoothness.  It has approximation order L+1.

Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme  The mask set of cubic case In cubic case, the mask can reproduce polynomials up to degree 3. odd case : use 4-pts even case : use 5-pts with tension v

Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme  Various basic limit function which start with 

Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme

Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme

Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme  Comparison of schemes which use cubic Cubic-spline4-pts interp. s.s.S a Where L=3 Support of limit ftn[-2, 2][-3, 3][-4, 4] Maximal SmoothnessC2C2 C1C1 C3C3 Approximatio n Order 244

Exponential quasi-interpolatory s.s. Example

Comparison with some example  Example E= E=0.1428

Exponential quasi-interpolatory s.s. Quasi-interpolatory subdivision scheme  General case LMask setSm.Range of tension 3 O=[-1/16,9/16,9/16,-1/16] E= [-v, 4v,1-6v,4v,-v] C3C <v< O=[-v, – 77/2048+5v,385/512-10v, 385/ v,-55/512-5v,35/2048+v] E(i)=O(7-i) for i=1:6 C3C <v< O=[3, – 25,150,150, – 25,3]/256] E=[-v,6v, – 15v,1+20v,-15v,6v,-v] C4C <v< O=[-v,385/ v, – 2079/ v, 51975/ v,5775/ v, -7245/ v,945/ v,-231/65536+v] E(i)=O(9-i) for i=1:8 C4C <v< O=[-5,49, – 245,1225,1225, – 245,49, – 5]/2048 E=[-v, 8v, – 28v,56v,1-70v,56v, – 28v,8v, – v] C5C <v<0.0015

Exponential quasi-interpolatory s.s.  Construction

Exponential quasi-interpolatory s.s. Analysis of non-stationary s.s.

Exponential quasi-interpolatory s.s.

 Example E=7.7716e-016 E=0.1434

Exponential quasi-interpolatory s.s. Next Study