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

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.

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

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.

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 ¼ N + 1 f k + 1 2 i = ; f k + 1 2 i = 9 6 ( ) ¡ :

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

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

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 ]

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.

Ternary 4-point Interpolating Subdivision Scheme (Hassan)

where the weights are given by p 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

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

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

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

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

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

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

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.

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 ¡

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 +

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

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 + ]

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 =

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 ¡

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 ¼ = 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

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

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 ¹ f k + 1 3 i : [ ¡ 8 6 ¹ ; 2 7 ]

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 ¡

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 =

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

The mask of the butterfly scheme:

The mask of the butterfly scheme:

Butterfly subdivision scheme A = 2 6 4 ¢ ¡ w 1 3 7 5

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

Symmetric 8-point Butterfly SS A = 2 6 4 ¢ ° ¯ ® 1 3 7 5 ( 1 ; 2 = ® ) 3 4 ¯ 5 6 7 8 °

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 ° ¡

Symmetric 10-point Butterfly SS A = 2 6 4 ¢ w ° ¯ ® 1 3 7 5 ( 1 ; 2 = ® ) 3 4 ¯ 5 6 7 8 ° 9 w

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