1. Mask of interpolatory symmetric subdivision schemes
Kwan Pyo Ko
Dongseo Univ.

2. Objective
we rebuild the masks of well-known interpolating symmetric subdivision schemes -binary 2n-point scheme, ternary 4-point scheme using symmetry
and necessary condition for smoothness and the butterfly scheme, the modified butterfly scheme
using the factorization property.

3. Schemes for curves DD scheme
Dyn 4-point scheme (4-point interpolating scheme)
4-point approximating scheme
Ternary 4-point interpolating scheme
Ternary 4-point approximating scheme
Ternary 3-point interpolating scheme
Ternary 3-point approximating scheme
Binary 3-point approximating scheme
Class (family) of subdivision scheme

4. Chaikin’s Algorithm -converges to the quadratic B-Spline. 3 p = 4 3 p+ 1 2 i = 3 4 p k + 1 2 i = 4 3
-converges to the quadratic B-Spline.

5. Dubuc-Deslauriers Scheme
D-D obtained the mask by using polynomial reproducing property
4-point D-D scheme: X k a j 2 p ( ) = ; Z N + 1 f k + 1 2 i = ; f k + 1 2 i = 9 6 ( ) :

6. Mask of D-D scheme 2 ( n = 1 ) 4 9 6 3 5 8 7

7. 4-point Interpolating Scheme (Dyn)k + 1 2 i = p k + 1 2 i = w ( )
C1 for j w
< 1 4

8. Mask of binary 2n-point SS
We can obtain the mask of interpolating symmetric SS by using symmetry and the necessary condition for smoothness. [ w + 1 2 ; ] [ 2 w + 9 1 6 ; 3 ] [ 5 w + 7 1 2 8 ; 9 6 3 ] [ 1 4 w + 2 5 8 ; 9 7 ]

9. 4-point Approximating Scheme (Dyn, Floater, Hormann)k + 1 2 i = 7 8 5 3 f k + 1 2 i = 5 8 3 7
C2 curve.

10. Ternary 4-point Interpolating Subdivision Scheme (Hassan)

11. where the weights are given byp j + 1 3 = i p j + 1 3 = a i b c d 2 p j + 1 3 2 = d i c b a
where the weights are given by a = 1 8 6 w ; b 3 + 2 c 7 d : C 2 f o r 1 5
< w 9

12. Ternary 4-point Interpolating Subdivision Scheme (Ko)f k + 1 3 i = 5 2 9 6 8 4 7 f k + 1 3 i = 6 9 2 f k + 1 3 i 2 = 5 9 6 7 4 8

13. Ternary 3-point Interpolating Subdivision Scheme (Hassan)f k + 1 3 i = f k + 1 3 i = a ( b ) f k + 1 3 i 2 = b ( a ) 2 9
< b 1 3 ; a =
for C 1

14. Ternary 3-point Approximating Subdivision Scheme (Hassan)= 1 2 7 [ ; 4 6 9 ]
curve C 2

15. Binary 3-point Approximating Subdivision Scheme (Hassan)= [ ; b 1 ]
for b = a + 1 4 C 1
for b = a + 1 4 ;
< 8 C 2
for a = 1 6 [ ; 5 ] C 3

16. Generalization of Mask
Consider the problem of finding mask
reproducing polynomial of degree (2N+1)
We let a = f j g 2 N + 3 X k a j 2 p ( ) = ; Z P N + 1 : v = a 2 N + ; w 3

17. we get the mask a = ± + ( ¡ 1 ) µ ¶ v : a = N µ ¶ ( ¡ ) w 2 j ; N 2 j+ ( 1 ) N v : a 2 j + 1 = N 4 ( ) w

18. Remark
We obtain symmetric SS which reproduce all polynomial degree (2N+1) and which is not interpolatory.
In case v=0, it becomes (2N+4) interpolatory SS.
In case v=w=0, it becomes (2N+2) D-D scheme.

19. Analysis of convergence and smoothness of binary SS by the formalism of Laurent polynomials
The general form of an interpolatory SS: f k + 1 2 i = X j Z a

20. For each scheme , we define the symbol
Theorem 1 Let be a convergent SS, then
This condition guarantees the existence of associated Laurent polynomial which can be defined as follows: S a ( z ) = X i 2 Z S ( ) X j 2 Z a = + 1 a ( 1 ) = ; 2 a 1 ( z ) a 1 ( z ) = 2 +

21. Theorem 2 Let denote a SS with symbol satisfying (. )
Theorem 2 Let denote a SS with symbol satisfying (*). Then there exists a SS with the property
where and a ( z ) S S 1 d f k = S 1 f k = S d f k = ( ) i 2 + 1 g

22. 4-point interpolatory SS
The 4 –point interpolatory SS:
The Laurent-polynomial of this scheme is
By symmetry and necessary condition
we get mask f k + 1 2 i = a 3 a ( z ) = 3 + 1 a ( 1 ) = a 1 + 3 = 2 [ w ; 1 2 + ]

23. 6-point interpolatory SS
The 6 –point interpolatory SS:
The Laurent-polynomial of this scheme is
By symmetry and necessary condition, we get mask f k + 1 2 i = a 5 3 a ( z ) = 5 + 3 1 f k i a ( 1 ) = 2 a 1 + 3 5 = 2 [ w ; 1 6 3 9 + 2 ] 2 4 a 5 + 8 3 1 =

24. Ternary interpolatory SS
A ternary SS
The general form of an interpolatory SS: f k + 1 i = X j 2 Z a 3 f k + 1 3 i = X j 2 Z a

25. d f = S For each scheme , we define the symbol
Theorem 5 Let be a convergent SS, then
Theorem 6 Let denote a SS with symbol satisfying (*). Then there exists a SS with the property
where and S a ( z ) = X i 2 Z S ( ) X j 2 Z a 3 = + 1 z a ( e 2 i = 3 ) 4 ; 1 S a ( z ) S 1 d f k = S 1 a 1 ( z ) = 3 2 + d f k = ( ) i 3 + 1 g

26. Ternary 4-point interpolatory SS
The ternary 4 –point interpolatory SS:
The Laurent-polynomial of this scheme is f k + 1 3 i = a 4 2 5 a ( z ) = 5 + 4 2 1

27. By symmetry and necessary condition, we get mask
Let a 1 + 2 4 5 = 3 6 9 a ( 1 ) = 3 a 1 ( ) = 3 a 2 ( 1 ) = 3 a 5 = 1 8 + 6 f k + 1 3 i : [ 8 6 ; 2 7 ]

28. For i=1,2, has as a factor if and only if
Theorem Let be a bivariate SS with a compactly supported mask corresponding to its symbol
Then we have
For i=1,2, has as a factor if and only if
has as a factor if and only if S a ( z 1 ; 2 ) = X i j a ( z 1 ; 2 ) 1 + z i a ( z 1 ; 2 ) j i = a ( z 1 ; 2 ) 1 + z 2 a ( z 1 ; t = 2 ) j

29. And we can see that if and only if
Geometrical view: the condition if and only if the sums of even masks and of odd masks along each horizontal line are same, that is
And we can see that if and only if a ( 1 ; z 2 ) = X i ( 1 ) a ; k = a ( z 1 ; t = 2 ) j X i ( 1 ) a ; + k =

30. Butterfly scheme q = 1 2 ( p + ) w ¡ X p = [ q k e ; 3 4 8 j 5 k + 1 i

31. The mask of the butterfly scheme:

32. The mask of the butterfly scheme:

33. Butterfly subdivision schemeA = 2 6 4 w 1 3 7 5

34. From the matrix, we see that the mask satisfies
The symbol of the butterfly scheme: X i ( 1 ) a ; k = + a ( z 1 ; 2 ) = 3 X i j a ( z 1 ; 2 ) = + w c c ( z 1 ; 2 ) = + 4

35. Symmetric 8-point Butterfly SSA = 2 6 4 1 3 7 5 ( 1 ; 2 = ) 3 4 5 6 7 8

36. The bivariate symbol of this scheme is assumed to be factorizable:
Factorization implies
We get If we set we obtain the same mask of the butterfly scheme a ( z 1 ; 2 ) = + b X i ( 1 ) a ; k = + 2 + = 1 : = w = 1 2 ; w

37. Symmetric 10-point Butterfly SSA = 2 6 4 w 1 3 7 5 ( 1 ; 2 = ) 3 4 5 6 7 8 9 w

38. From the factorization, we have
We find the mask of 10-point butterfly scheme:
This mask is exact with a modified butterfly scheme 2 + = ; w 1 = 1 2 w 5 ; 6 7 8 9 = 1 6 + w