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

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

3. Subdivision scheme • Useful method to construct smooth curves and surfaces in CAGD • The rule :

4. Subdivision scheme • Rule : • Interpolatory s.s. & Non-interpolatory s.s • Stationary s.s. & Non-stationary s.s

5. B-spline subdivision scheme • It has maximal smoothness Cm-1 with minimal support. • It has approximation order only 2 for all m. • Cubic-spline :

6. Interpolatory subdivision scheme • 4-point interpolatory s.s. : • The Smoothness is C1 in some range ofw. • The Approximation order is 4 with w=1/16.

7. Goal We want to construct a new scheme which has good smoothness and approximation order.

8. Quasi-interpolatory subdivision scheme • Construction

9. 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.

10. 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

11. Quasi-interpolatory subdivision scheme • Various basic limit function which start with d

12. Quasi-interpolatory subdivision scheme

13. Quasi-interpolatory subdivision scheme

14. Quasi-interpolatory subdivision scheme • Comparison of schemes which use cubic

15. Example

16. Comparison with some example • Example < cubic-spline > < Sa > E=0.8169 E=0.1428

17. Quasi-interpolatory subdivision scheme • General case

18. Exponential quasi-interpolatory s.s. • Construction

19. Analysis of non-stationary s.s.

20. Exponential quasi-interpolatory s.s.

21. Exponential quasi-interpolatory s.s. • Example E=7.7716e-016 E=0.1434

22. Next Study