Mask of interpolatory symmetric
subdivision schemes
Kwan Pyo Ko
Dongseo Univ.
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
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.
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
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
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Chaikin’s Algorithm
pk2i+ 1 =
3 k 1 k
pi + pi + 1
4
4
pk2i++11 =
1 k 3 k
pi + pi + 1
4
4
-converges to the quadratic B-Spline.
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Dubuc-Deslauriers Scheme
• D-D obtained the mask by using polynomial
reproducing property
X
aj ¡
k
j
p(k)
=
p(
);
2k
2
j 2 Z; p 2 ¼2N + 1
• 4-point D-D scheme:
f 2ik + 1 = f ik ;
f 2ik ++ 11 =
9 k
1 k
(f i + f ik+ 1 ) ¡
(f
16
16 i ¡
KMMCS, August 25~26, 2006, Ewha Univ.
k ):
+
f
1
i+ 2
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Mask of D-D scheme
2 (n= 1)
4 (n= 2)
6 (n= 3)
8 (n= 4)
10 (n= 5)
1
2
9
16
150
256
1225
2048
19845
32768
KMMCS, August 25~26, 2006, Ewha Univ.
0
1
¡ 16
25
¡ 256
245
¡ 2048
2250
¡ 16384
0
0
3
256
49
2048
567
16384
0
0
0
¡
¡
5
2048
405
16384
0
0
0
0
35
65536
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
4-point Interpolating Scheme (Dyn)
pk2i+ 1 = pki
µ
pk + 1 =
2i + 1
¶
1
+ w (pk + pk ) ¡ w(pk
i
i+ 1
i¡
2
C1 for jwj <
KMMCS, August 25~26, 2006, Ewha Univ.
1
+ pk )
i+ 2
1
4
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Mask of binary 2n-point SS
• We can obtain the mask of interpolating symmetric
SS by using symmetry and the necessary condition
for smoothness.
1
[w + ; ¡ w]
2
9
1
[2w +
; ¡ 3w ¡
; w]
16
16
[5w +
[14w +
75
25
3
; ¡ 9w ¡
; 5w ¡
; ¡ w]
128
256
256
1225
245
49
5
; ¡ 28w ¡
; 20w ¡
; ¡ 7w ¡
; w]
2048
2048
2048
2048
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
4-point Approximating Scheme
(Dyn, Floater, Hormann)
f 2ik + 1 = ¡
7 k
f
128 i ¡
1+
105 k
35 k
5 k
fi +
f i+ 1 ¡
f
128
128
128 i + 2
f 2ik ++ 11 = ¡
5 k
f
128 i ¡
1+
35 k 105 k
7 k
fi +
f i+ 1 ¡
f
128
128
128 i + 2
2
- C curve.
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Ternary 4-point Interpolating
Subdivision Scheme (Hassan)
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
pj3j+ 1 = pij
pj3j++11 = apij ¡
1
+ bpij + cpij + 1 + dpij + 2
pj3j++12 = dpij ¡
1
+ cpij + bpij + 1 + apij + 2
where the weights are given by
1
1
13 1
7
1
1
1
a= ¡
¡ w; b =
+ w; c =
¡ w; d = ¡
+ w:
18 6
18 2
18 2
18 6
C 2 for
KMMCS, August 25~26, 2006, Ewha Univ.
1
15
< w<
1
9
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Ternary 4-point Interpolating
Subdivision Scheme (Ko)
f 3ik + 1
55 k
= ¡
f
1296 i ¡
f 3ik ++ 11 = ¡
1 k
f i¡
16
f 3ik ++ 12 = ¡
35 k
f
1296 i ¡
KMMCS, August 25~26, 2006, Ewha Univ.
385 k
77 k
35 k
f +
f
¡
f
1+
432 i
432 i + 1 1296 i + 2
1+
9 k
9 k
1 k
fi +
f i+ 1 ¡
f i+ 2
16
16
16
1+
77 k 385 k
55 k
fi +
f i+ 1 ¡
f
432
432
1296 i + 2
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Ternary 3-point Interpolating
Subdivision Scheme (Hassan)
f 3ik + 1 = f ik
f 3ik ++ 11 = af ik¡
1
+ (1 ¡ a ¡ b)f ik + bf ik+ 1
f 3ik ++ 12 = bf ik¡
1
+ (1 ¡ a ¡ b)f ik + af ik+ 1
-
C1
for
KMMCS, August 25~26, 2006, Ewha Univ.
2
1
< b< ;
9
3
1
a = b¡
3
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Ternary 3-point Approximating
Subdivision Scheme (Hassan)
a=
1
[1; 4; 10; 16; 19; 16; 10; 4; 1]
27
-
KMMCS, August 25~26, 2006, Ewha Univ.
C2
curve
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Binary 3-point Approximating
Subdivision Scheme (Hassan)
a = [a; b; 1 ¡ a ¡ b; 1 ¡ a ¡ b; b; a]
1
for b = a +
4
1
1
C2
b
=
a
+
;
0
<
a
<
for
4
8
3
1
C
a
=
[1; 5; 10; 10; 5; 1]
for
16
C1
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Generalization of Mask
2N + 3
a
=
f
a
g
j
• Consider the problem of finding mask
j = ¡ 2N ¡
3
reproducing polynomial of degree (2N+1)
X
k
j
aj ¡ 2k p(k) = p( );
2
j 2 Z;
p 2 P2N + 1 :
We let
v = a2N + 2 ;
KMMCS, August 25~26, 2006, Ewha Univ.
w = a2N + 3
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
we get the mask
µ
a2j = ±j ;0 + (¡ 1) j + N + 1
¶
2N + 2
v:
N+ j + 1
µ
a2j + 1
=
+
¶
µ
¶
j
N + 1 2N + 1 (¡ 1)
2N + 1
24N + 1
N
2j + 1 N + j + 1
µ
¶
2N + 1
(2N + 2)(2j + 1)
(¡ 1) j + N + 1 w
N + j + 1 (N + j + 2)(N ¡ j + 1)
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
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.
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Analysis of convergence and
smoothness of binary SS by the
formalism of Laurent polynomials
The general form of an interpolatory SS:
f 2ik + 1
=
f 2ik ++ 11
=
f ik
X
a1+ 2j f ik¡
j
j 2Z
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
For each scheme S, we define the symbol
X
a(z) =
ai zi
i2Z
Theorem 1 Let S be a convergent SS, then
X
X
(¤)
a2j =
j 2Z
a2j + 1 = 1
j 2Z
a(¡ 1) = 0;
a(1) = 2
This condition guarantees the existence of associated
Laurent polynomial a1 (z) which can be defined as
follows:
a1 (z) =
KMMCS, August 25~26, 2006, Ewha Univ.
2z
a(z)
1+ z
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Theorem 2 Let S denote a SS with symbol a(z)
satisfying (*). Then there exists a SS S1 with the
property
df
k
= S1 df
k¡ 1
k
k
k
k
k
k
k 0
where f = S f and df = f (df ) i = 2 (f i + 1 ¡ f i )g
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
4-point interpolatory SS
• The 4 –point interpolatory SS:
f 2ik + 1
f 2ik ++ 11
f ik
a3 f ik¡
=
=
1
+ a1 f ik a¡ 1 f ik+ 1 + a¡ 3 f ik+ 2
• The Laurent-polynomial of this scheme is
a(z) = a¡ 3 z¡
3
+ a¡ 1 z¡
1
+ 1 + a1 z + a3 z3
• By symmetry and necessary condition a(¡ 1) = 0
• we get mask
a1 + a3 =
1
2
KMMCS, August 25~26, 2006, Ewha Univ.
[¡ w; 12 + w; 12 + w; ¡ w]
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
6-point interpolatory SS
• The 6 –point interpolatory SS:
f 2ik + 1
f 2ik ++ 11
f ik
a5 f ik¡
=
=
2
+ a3 f ik¡
1
+ a1 f ik a¡ 1 f ik+ 1 + a¡ 3 f ik+ 2 + a¡ 5 f ik+ 3
• The Laurent-polynomial of this scheme is
a(z) = a5 z¡
5
+ a¡ 3 z¡
3
+ a¡ 1 z¡
1
+ 1 + a1 z + a3 z3 + a¡ 5 f ik+ 3
• By symmetry and necessary condition, a(¡ 1) = a2 (¡ 1) = 0
we get mask
a1 + a3 + a5 =
24a5 + 8a3 +
1
2
1
2
= 0
KMMCS, August 25~26, 2006, Ewha Univ.
[w; ¡
1
16
9 + 2w; 9 + 2w; ¡
¡ 3w; 16
16
1
16
¡ 3w; w]
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Ternary interpolatory SS
• A ternary SS
f ik + 1
X
=
ai ¡
3j
f jk
j 2Z
• The general form of an interpolatory SS:
f 3ik + 1
=
f 3ik ++ 11
=
f ik
X
a1+ 3j f ik¡
k
a2+ 3j f ik¡
k
j 2Z
f 3ik ++ 12
X
=
j 2Z
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
For each scheme S, we define the symbol
X
ai zi
a(z) =
i2Z
Theorem 5 Let S be a convergent SS, then
X
X
(¤)
X
a3j =
j 2Z
a3j + 1 =
a3j + 2 = 1
j 2z
j 2Z
a(e2i ¼=3 ) = a(e4i ¼=3 ) = 0;
a(1) = 3
Theorem 6 Let S denote a SS with symbol a(z)
satisfying (*). Then there exists a SS S1 with the
property
df
where
k
= S1 df
3z2
a1 (z) =
a(z)
1 + z + z2
KMMCS, August 25~26, 2006, Ewha Univ.
k¡ 1
and df
k
= f (df k ) i = 3k (f ik+ 1 ¡ f ik )g
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Ternary 4-point interpolatory SS
• The ternary 4 –point interpolatory SS:
f 3ik + 1
f 3ik ++ 11
=
=
f ik
a4 f ik¡
1
+ a1 f ik a¡ 2 f ik+ 1 + a¡ 5 f ik+ 2
f 3ik ++ 12
=
a5 f ik¡
1
+ a2 f ik + a¡ 1 f ik+ 1 + a¡ 4 f ik+ 2
• The Laurent-polynomial of this scheme is
a(z) = a¡ 5 z¡
5
+ a¡ 4 z¡
4
+ a¡ 2 z¡
KMMCS, August 25~26, 2006, Ewha Univ.
2
+ a¡ 1 z¡
1
+ 1 + a1 z + a2 z2 + a4 z4 + a5 z5
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
• By symmetry and necessary condition, we get
mask
a1 + a2 + a4 + a5
3a2 ¡ 3a4 + 6a5
a4 + a5
• Let a5 = ¡
f 3ik ++ 11
f 3ik ++ 12
1
18
+
=
=
1
1
=
1
¡
9
a(1) = 3
a1 (1) = 3
a2 (1) = 3
1¹
6
1
¡
18
1
: [¡
+
18
: [¡
KMMCS, August 25~26, 2006, Ewha Univ.
1 13
¹;
+
6 18
1 7
¹;
¡
6 18
1 7
¹;
¡
2 18
1 13
¹;
+
2 18
1
¹ ;¡
2
1
¹ ;¡
2
1
+
18
1
¡
18
1
¹]
6
1
¹]
6
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Theorem Let S be a bivariate SS with a compactly
supported mask corresponding to its symbol
X
a(z1 ; z2 ) =
ai j z1i z2j Then we have
i ;j
1) For i=1,2,
a(z1 ; z2 )
has 1 +
zi
a(z1 ; z2 )j z i = ¡
2)
a(z1 ; z2 )
has
1 + z1 z2
= 0;
as a factor if and only if
a(z1 ; t=z2 )j t = ¡
KMMCS, August 25~26, 2006, Ewha Univ.
1
as a factor if and only if
1
= 0;
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Geometrical view: the condition a(¡ 1; z2 ) = 0 if and
only if the sums of even masks and of odd masks
along each horizontal line are same, that is
X
(¡ 1) i ai ;k = 0
i
And we can see that
X
a(z1 ; t=z2 )j t = ¡
1
= 0;
if and only if
(¡ 1) i ai ;i + k = 0
i
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Butterfly scheme
8
X
1 k
k
k
k
k
qe = (pe;1 + pe;2 ) + 2w(pe;3 + pe;4 ) ¡ w
pke;j
2
j=5
pki + 1 = pki [ qek
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
The mask of the butterfly scheme:
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
The mask of the butterfly scheme:
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Butterfly subdivision scheme
2
6
6
6
6
A= 6
6
6
6
4
¢
¢
¢
¢
¡ w
¡ w
¢
¢
¢
¡ w
¢
2w
¢
¡ w
KMMCS, August 25~26, 2006, Ewha Univ.
¢
¡ w
2w
1
2
1
2
2w
¡ w
¢ ¡ w
¢ 2w
1
2
1
1
2
1
2
1
2
2w
¢ ¡ w
¢
¢
¡ w
¢
2w
¢
¡ w
¢
¢
¢
¡ w
¡ w
¢
¢
¢
¢
3
7
7
7
7
7
7
7
7
5
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
• From the matrix, we see that the mask satisfies
X
(¡
1) i ai ;k
i
X
=
(¡
i
1) i ak ;i
X
=
(¡ 1) i ai ;i + k = 0
i
• The symbol of the butterfly scheme:
X3
X3
a(z1 ; z2 ) =
ai ;j z1i z2j
i= ¡ 3 j = ¡ 3
a(z1 ; z2 ) =
c( z1 ; z2 )
=
+
1
(1 + z1 )(1 + z2 )(1 + z1 z2 )(1 ¡ wc(z1 ; z2 ))(z1 z2 ) ¡ 1 ;
2
2z1¡ 2 z2¡ 1 + 2z1¡ 1 z2¡ 2 ¡ 4z1¡ 1 z2¡ 1 ¡ 4z1¡ 1 ¡ 4z2¡ 1 + 2z1¡ 1 z2
2z1 z2¡ 1 + 12 ¡ 4z1 ¡ 4z2 ¡ 4z1 z2 + 2z12 z2 + 2z1 z22
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Symmetric 8-point Butterfly SS
2
6
6
6
6
A= 6
6
6
6
4
¢ ¢
¢ ¢
¢ °
¢ ¢
° ¯
° ¢
¢ °
3
¢ ¢ ° ° ¢
° ¢ ¯ ¢ ° 7
7
¯ ® ® ¯ ° 7
7
® 1 ® ¢ ¢7
7
® ® ¯ ° ¢7
7
¯ ¢ ° ¢ ¢5
° ¢ ¢ ¢ ¢
(1; 2 = ®); (3; 4) = ¯); (5; 6; 7; 8 = ° )
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
• The bivariate symbol of this scheme is assumed to
be factorizable:
a(z1 ; z2 ) = (1 + z1 )(1 + z2 )(1 + z1 z2 )b(z1 ; z2 )
• Factorization implies
X
(¡
1) i ai ;k
X
=
i
(¡
1) i ak ;i
X
=
i
(¡ 1) i ai ;i + k = 0
i
° = ¡ w
2° + ¯ = ¡ 2® + 1 = 0:
• We get
If we set
we obtain the same mask of the butterfly scheme
®=
KMMCS, August 25~26, 2006, Ewha Univ.
1
;
2
¯ = 2w;
° = ¡ w
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
Symmetric 10-point Butterfly SS
2
6
6
6
6
A= 6
6
6
6
4
¢ ¢
¢ ¢
¢ °
w ¢
° ¯
° ¢
w °
¢
°
¯
®
®
¯
°
w
¢
®
1
®
¢
w
° °
¯ ¢
® ¯
® ¢
¯ °
° ¢
¢ ¢
w
°
°
w
¢
¢
¢
3
7
7
7
7
7
7
7
7
5
(1; 2 = ®); (3; 4 = ¯); (5; 6; 7; 8 = ° ); (9; 10 = w)
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko
• From the factorization, we have
2° + ¯ = 0; ¡ 2® ¡ 2w + 1 = 0
• We find the mask of 10-point butterfly scheme:
®=
1
¡ w
2
¯ = 2°
5; 6; 7; 8 = ¡ °
9; 10 = w
• This mask is exact with a modified butterfly scheme
1
° =
+ w
16
KMMCS, August 25~26, 2006, Ewha Univ.
Mask of interpolatory symmetric subdivision schemes, Kwan Pyo Ko