Some Non-Stationary Subdivision Schemes - IEEE Xplore

Some Non-Stationary Subdivision Schemes Sunita Daniel and P. Shunmugaraj Department of Mathematics and Statistics Indian Institute of Technology, Kanpur 208016, India [email protected], [email protected]

Abstract

2. Notation, definitions and preliminary results

In this paper we present some non-stationary binary and ternary subdivision schemes with parameters for generating curves. Some of these schemes reproduce some trigonometric functions and, in particular, circles. One of these is a non-stationary version of the Chaikin’s scheme and it generates trigonometric splines.

In this section we recall some definitions and present some results which will be used in the next section. Given a set of control points P 0 = {p0i ∈ Rd : i ∈ Z} at level 0, a subdivision scheme {Sk }k∈Z+ generates recursively a new set of control points P k+1 = {pk+1 : i ∈ Z} at the i (k + 1)th level by the subdivision rule: for i ∈ Z,

1. Introduction In a recent paper of Hassan and Dodgson [8], a family of interpolating and approximating 3-point binary and ternary subdivision schemes were introduced. These schemes were compared with the established schemes such as 2-point, 4point and 6-point binary and ternary subdivision schemes. All the schemes mentioned above are stationary schemes. Sometimes, in computer graphics and geometric modelling, it is required to have schemes to construct circular parts or parts of conics. It seems that (linear) stationary schemes cannot generate circles and non-stationary schemes have such a capability. Some non-stationary subdivision schemes reproducing or generating circles are available in the literature [3, 5, 6, 10, 12, 15, 19]. We refer to the recent survey article of Sabin [14] for an excellent review of progress in the area of subdivision. Recently some non-stationary subdivision schemes corresponding to 4-point binary and ternary and 6-point binary stationary schemes have been introduced [5, 10, 16, 17]. It was observed that these non-stationary subdivision schemes reproduce circles. In this paper we present non-stationary counterparts to the 2-point, 3-point and 4point interpolating and approximating stationary schemes and compare these non-stationary schemes with some established schemes. Some of these schemes reproduce some trigonometric functions and, in particular, circles. The 2point binary non-stationary approximating scheme is a nonstationary version of the Chaikin’s scheme and it generates trigonometric splines.

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= {Sk P k }i = pk+1 i

ski−N j pkj ,

(2.1)

j∈Z

where the set {ski : i ∈ Z, ski = 0} is finite for every k ∈ Z+ . When N = 2, we deal with the binary subdivision scheme and when N = 3, the ternary subdivision scheme. For k = 0, 1, ... the points pki , i ∈ Z are assigned to the mesh points N −k i, i ∈ Z respectively. The set sk := {ski : i ∈ Z} of coefficients is called the mask at the k th level of the subdivision scheme. The set {i ∈ Z : ski = 0} is called the support of the mask sk . If the mask is independent of k, then the scheme is called stationary, otherwise it is called non-stationary. Since the subdivision scheme is applied componentwise, it is sufficient to state the subdivision scheme for the initial points p0i ∈ R. Definition 2.1. A binary (respectively, ternary) subdivision scheme {Sk } is said to be C m if for every initial data P 0 ∈ l∞ there exists a limit function f ∈ C m (R) such that limk→∞ supi∈Z |pki − f (2−k i)| = 0 (respectively, limk→∞ supi∈Z |pki −f (3−k i)| = 0) and f is not identically 0 for some initial data P 0 . The function f is called the limit function of the scheme {Sk } for the data P 0. Definition 2.2. Let {Sk } be a binary subdivision k scheme with mask We define {si : i ∈ Z}. Sk ∞ = max{ j∈Z |sk2j |, j∈Z |sk2j+1 |}. If {Sk } is a ternary subdivision scheme, then we define Sk ∞ = max{ i∈Z |sk3i |, i∈Z |sk3i+1 |, i∈Z |sk3i+2 |}.

Theorem 2.1. [2] Let {Sk } and {S} be two binary subdivision schemes having finite masks with the same support. Suppose {Sk } is non-stationary and {S} is stationary. If ∞ mk 2 Sk − S∞ < ∞ and {S} is C m then the nonk=0 stationary scheme {Sk } is C m .

[..., 0, 0, 14 , 34 , 34 , 14 , 0, 0, ...] and for convenience we write it as [ 14 , 34 , 34 , 14 ]. The non-stationary counterpart to this algorithm is given as follows

The proof of the following theorem is exactly similar to the proof of Theorem 2.1 given in [2, Theorem 8].

where

Theorem 2.2. Let {Sk } and {S} be two ternary subdivision schemes having finite masks with the same support. Suppose {Sk } is non-stationary and {S} is stationary. If ∞ mk 3 Sk − S∞ < ∞ and {S} is C m then the nonk=0 stationary scheme {Sk } is C m .

and

Definition 2.3. The basic limit function of the scheme {Sk } is defined as the limit function of the scheme for the data {p0i : p00 = 1, p0i = 0 ∀i = 0}. Suppose we have a data set D = {(xi , f (xi )) : i = 0, 1, ..., n} and 0 < α < L(x) =

n

π 2.

Consider the function L(x) where

f (xj ) cos(

j=0

and L(x) =

n

α(x − xj ) )Q(x), when n = 2m−1 2

f (xj )Q(x) when n = 2m,

j=0

for some m > 0 and Q(x) =

n k=0,k=j

((sin(

α(xj − xk ) α(x − xk ) )/(sin )). 2 2

It is known [11] that the function L(x) ∈ Tn where Tn = span{cos(jαx), sin(jαx) : j = 0, 1, 2, ..., m} and L(x) interpolates D. It is well known [11] that when n is even L(x) is the unique function in Tn which interpolates D. When n is odd, there exist several functions in Tn interpolating D. But L(x) is the unique function in Tn which interpolates D and has the minimum amplitude [11] among the other interpolants from Tn . We call the function L(x) a Lagrange like interpolant of the above data.

3. A 2-point binary non-stationary approximating scheme Chaikin first proposed a binary 2-point stationary approximating scheme in [1]. It was shown in [13] that this scheme produces a quadratic B-spline. Its mask is


k k k k pk+1 = ak0 pki + ak1 pki+1 , pk+1 2i 2i+1 = a1 pi + a0 pi+1 (3.1)

ak0 = (sin

3α α )/(sin k−1 ) 2k+1 2

ak1 = (sin

α α )/(sin k−1 ) 2k+1 2

for 0 < α < π3 . Its mask is [ak1 , ak0 , ak0 , ak1 ]. This scheme is a particular case (n = 3) of the non-stationary scheme introduced in [9] for the evaluation of the trigonometric spline of order n, n > 2. This scheme is C 1 and it produces a trigonometric spline of order 3 [9]. Interestingly, the scheme (3.1) reproduces circles. This can be illustrated as follows. Let us start with the initial control points p0i = cos(2iα), i = 0, 1, ..., n, n > 4 and 0 < α < π3 . At the first level of the algorithm (k = 0), we have p12i

= = = =

sin α2 sin 3α 2 cos(2iα) + cos((i + 1)2α) sin(2α) sin(2α) sin α2 sin(2α − α2 ) cos(2iα) + cos((i + 1)2α) sin(2α) sin(2α) α α cos cos(2iα) − sin sin(2iα) 2 2 1 cos((2i + )α) 2

Similarly we can also show that p12i+1 = cos((2i + 32 )α). Therefore at the first level, the algorithm (3.1) generates the set of points 1 3 {cos((2i + )α), cos((2i + )α) : i = 0, 1, 2, ..., n − 1}. 2 2 It can be easily verified that the points generated at the second level of the algorithm belong to {cos((2i + 12 + 1 1 3 3 1 4 )α), cos((2i + 2 + 4 )α), cos((2i + 2 + 4 )α), cos((2i + 3 3 2 + 4 )α) : i = 0, 1, ..., n − 1}. The points generated at the k th level belong to {cos((2i +

k aj j=1

2j

)α) : aj = 1 or 3, i = 0, 1, ..., n − 1}.

Similarly if we start with the intial control points, p0i = sin(2iα), i = 0, 1, ..., n, n > 4, we can show that the points generated at the k th level of the algorithm belong to k a {sin((2i + j=1 2jj )α) : aj = 1 or 3, i = 0, 1, ..., n − 1}. For i = 0, 1, ..., n2 when n is even and for i = 0, 1, ..., n+1 2 when n is odd, if we choose a set of equidistant 4π 2π points p0i = (cos(i 4π n ), sin(i n )) on a circle and α = n ,

then the limit curve of the scheme (3.1) is the original unit circle. The reconstruction of the circle is illustrated in Figure 4(a). The basic limit functions of the Chaikin’s scheme and the scheme (3.1) with α = π4 are presented in Figure 1(a) and Figure 1(b) respectively.

4. A 3-point ternary non-stationary interpolating scheme Hassan and Dodgson [8] introduced a 3-point interpolating stationary ternary subdivision scheme {SH } in [8]. Its mask is

Proof. In order to prove the above inequalities, we make use of the following three inequalities: sin a a π ≥ for 0 < a < b < , sin b b 2 π θ csc θ < t csc t for 0 < θ < t < 2 and

π sin x for 0 < x < . x 2 To prove (i), note that cos x ≤

−δ0 bk0

4 4 1 1 [a − , 0, a, − 2a, 1, − 2a, a, 0, a − ] 3 3 3 3

k+1

[δ0 bk0 , 0, δ2 bk2 , δ1 bk1 , 1, δ1 bk1 , δ2 bk2 , 0, δ0 bk0 ]

≥ δ0 2·3α

2·3k

= =

bk0 =

sin sin

and bk2 =

sin

, bk1 =

2α 2·3k+1 2

−δ0 bk0

(4.1)

sin

sin

4α 2·3k+1

α 2·3k

=

δ1 bk1

− a ≤ −δ0 bk0 ≤ ( 13 − a) cos21

(ii)

( 43

− 2a) cos 3k+1 ≤ α

(iii) a ≤ δ2 bk2 ≤ a cos21

δ1 bk1

α 3k


≤

2α 4α α csc2 k+1 k+1 2·3 2·3 2 · 3k 8δ1 1 = α 9 cos2 2·3 k 1 4 = ( − 2a) 2 α 3 cos 3k ≤ δ1

and δ1 bk1

=

= α 3k

( 43

2α 2α sin 2·32α k+1 (2 sin 2·3k+1 cos 2·3k+1 )

sin2

≥ 2δ1 cos

Lemma 4.1. 1 3

2α α α )( )2 csc2 2 · 3k+1 3k+1 2 · 3k α2 1 δ0 2k+2 2α 2 2α 3 ( 2·3k ) cos2 2·3 k 1 δ0 2α . 9 cos2 2·3 k δ0 (

The proof of (i) follows from the two inequalities given above. Similarly, to prove (ii), note that

2 9,

(i)

≤ ≤

α sin 2·34α k+1 sin 2·3k+1 . α 2α sin 2·3k sin 2·3k

The scheme (4.1) is an interpolating scheme. When a = δ0 = δ1 = δ2 = 1. It can be shown that when a = 29 the point ( 3i+1 , pk+1 3i+1 ) lies on the Lagrange like interpolant 3k+1 of the points {( 3jk , pkj ) : j = i − 1, i, i + 1} and the point ( 3i+2 , pk+1 3i+2 ) lies on the Lagrange like interpolant of the 3k+1 points {( 3jk , pkj ) : j = i, i + 1, i + 2}. We now present the proof of the convergence of the scheme (4.1). In order to prove the convergence, we need some estimates of δi bki , i = 0, 1, 2 which are given in the following two lemmas.

δ0 9 1 −a 3

2α 2·3k+1 2α 2·3k

and

where δ0 = 3 − 9a, δ1 = 32 − 94 a, δ2 = 92 a for a > 0 and for some α such that 0 < α < π2 , 2α 2·3k+1 2α 2·3k

sin 2·3αk+1 sin 2·32α k+1 )( α 2α ) sin 2·3 sin k 2·3k α

where a > 0. This scheme is C 1 when 29 < a < 39 . It can k+1 be shown that when a = 29 , the point ( 3i+1 3k+1 , p3i+1 ) lies on j the Lagrange interpolant of the points {( 3k , pkj ) : j = i − k+1 1, i, i + 1} and the point ( 3i+2 3k+1 , p3i+2 ) lies on the Lagrange j interpolant of the points {( 3k , pkj ) : j = i, i + 1, i + 2}. In this section we introduce a non-stationary counterpart to this scheme. Consider the non-stationary ternary scheme {Sk } with mask

sin 2·3αk+1 − α sin 2·3 k

= δ0 (

2

α 2·3k 2α 2α k+1 ( 2·3α k+1 ·3 2·3k

8δ1 2α cos . 9 2 · 3k+1

The proof of (iii) is similar to the proof of (i). 1

− 2a) cos2

α 3k

Lemma 4.2. (i) |δ0 bk0 + ( 13 − a)| ≤ C0 312k

2α

k+1

)( 2·3α

2·3k

)

1 (ii) |δ1 bk1 − ( 43 − 2a)| ≤ C1 32k

(iii) |δ2 bk2 − a| ≤ C2 312k for some constants C0 , C1 and C2 independent of k. Proof. By (i) of Lemma 4.1, |δ0 bk0

1 + − a| ≤ 3 ≤ =

where C0 =

( 13 −a)α2 cos2 α .

1 1 |( − a) 2 3 cos

α 3k sin2 3αk a) cos2 α

1 − ( − a)| 3

1 ( − 3 1 C0 2k 3

This proves (i). Similarly,

1 α 4 4 |( − 2a) 2 α − ( − 2a) cos k+1 | 3 cos 3k 3 3 1 − cos3 3αk 4 ≤ ( − 2a)( ) 3 cos2 3αk 1 − cos4 3αk 4 ≤ ( − 2a)( ) 3 cos2 3αk (sin2 3αk ) 4 ≤ 2( − 2a) 3 cos2 α3 2

α 4 2k ≤ 2( − 2a) 3 2 α 3 cos 3 1 = C1 2k 3 2α2 ( 4 −2a)

Theorem 4.1. The non-stationary scheme {Sk } is C 1 for 2 3 9 < a < 9. Proof. We use Theorem 2.2 to prove the theorem. We k claim that ∞ k=0 3 Sk − SH ∞ < ∞, where {SH } is the 3-point interpolating stationary ternary scheme introduced in the beginning of this section. Let {si : i ∈ Z} and {ski : i ∈ Z} denote the masks of {SH } and {Sk } respectively. Note that Sk − SH ∞ = max{ j∈Z |ski−3j − si−3j | : i = 0, 1, 2}. Since j∈Z |sk−3j − s−3j | = 0 k and j∈Z |sk1−3j − s1−3j | = j∈{1,−2,4} |sj − sj | = |skj − sj | = j∈Z |sk2−3j − s2−3j |, we have, j∈{−1,2,−4} ∞ k k ∞ k k k=0 3 Sk − SH ∞ = k=0 3 (|s4 − s4 | + |s1 − s1 | + k |s−2 − s−2 |). From (i) of Lemma 4.2, it follows that 3k |sk4 − s4 | =

k=0

≤

∞ k=0 ∞ k=0


1 3k |δ0 bk0 + ( − a)| 3 3k C0

The support of the basic limit function of the scheme (4.1) is [−2, 2]. Suppose 0 < α < π2 , a = 29 , k > 0 and m < n for some m and n. If the initial data lie on a graph of a function f (x) ∈ span{1, cos(αx), sin(αx)} and the values are given on a set of equidistant points { 3ik : m ≤ i ≤ n}, then the limit function of the scheme (4.1) exactly reproduces the original function f . This follows from the fact that the Lagrange like interpolant L(x) is unique in T2 = span{1, cos(αx), sin(αx)} which interpolates the given data D = (x0 , f (x0 )), (x1 , f (x1 )), (x2 , f (x2 )). In particular, if we choose a set of equidistant points p0i = 2π (cos(i 2π n ), sin(i n )), i = 0, 1, 2, ..., n, on a circle and α = 2π n , then the limit curve is the original unit circle. In general, π if we take α such that 0 < α < 2n , then the scheme (4.1) reproduces functions spanned by {cos(jαx), sin(jαx) : j = 0, 1, 2, .., n}. The reconstruction of the circle by the scheme (4.1) is illustrated in Figure 4(b). The basic limit functions of the 3-point stationary interpolating scheme with a = 0.27 and the scheme (4.1) with a = 0.27 and α = π4 are presented in Figure 2(a) and Figure 2(b) respectively.

5.

where C1 = cos32 α . Hence by (ii) of Lemma 4.1, the 3 inequality (ii) follows. The proof of (iii) is similar to the proof of (i).

∞

∞ k k Similarly,we can also prove that k=0 3|s1 − s1 | < ∞ ∞ k k k ∞ and 3 |s − s | < ∞. Hence −2 −2 k=0 k=0 3 Sk − 2 3 1 SH ∞ < ∞. Since {SH } is C for 9 < a < 9 , the scheme {Sk } is C 1 for 29 < a < 39 .

1 < ∞. 32k

Non-stationary schemes

interpolating

2-point

The mask of the 2-point interpolating ternary stationary scheme is [ 13 , 23 , 1, 23 , 13 ]. It is observed in [8] that this scheme is C 0 . The non-stationary counterpart to this scheme has the mask [γ1k , γ0k , 1, γ0k , γ1k ]

(5.1)

where γ0k = (cos(

α 2α α ) sin( ))/(sin ) 2 · 3k+1 2 · 3k+1 2 · 3k

γ1k = (cos(

2α α α ) sin( ))/(sin ) 2 · 3k+1 2 · 3k+1 2 · 3k

and

for some α such that 0 < α < π2 . This scheme is C 0 . The proof of the convergence is similar to the proof of the convergence of the 3-point ternary scheme (4.1). The support of the basic limit function of this scheme is [-1,1]. The k+1 3i+2 points ( 3i+1 , pk+1 3i+1 ) and ( 3k+1 , p3i+2 ) lie on the Lagrange 3k+1 j like interpolant of the points {( 3k , pkj ) : j = i, i + 1}. It can be easily seen that the 2-point binary nonstationary interpolating scheme coincides with its stationary counterpart.

6 A 4-point binary non-stationary interpolating scheme Dyn and Levin introduced a 4-point binary interpolating stationary subdivision scheme in [4]. Its mask is [−w, 0, 12 + w, 0, 1, 12 + w, 0, −w]. For 0 < w < 18 , the k+1 1 scheme is C 1 . When w = 16 , the point ( 2i+1 2k+1 , p2i+1 ) lies j on the Lagrange interpolant of the points {( 2k , pkj ) : j = i − 1, i, i + 1, i + 2}. We present here a non-stationary counterpart to the stationary scheme mentioned above. Its mask is 1 1 [−β0 wk , 0, β1 ( +wk ), 0, 1, β1 ( +wk ), 0, −β0 wk ] (6.1) 2 2 where β0 = 16w, β1 = and

8 16 + w 9 9

α α sin k+1 ) k 2 2 for some w > 0 and for some α such that 0 < α < π 1 2 . This is an interpolating scheme and for 0 < w < 8 , 1 the scheme is C . The proof of the convergence of this scheme is similar to the proof of the convergence of the 3point ternary scheme (4.1). The support of the basic limit function is [−3, 3]. This scheme coincides with the scheme 1 introduced in [10] and [5, page 36 ] when w = 16 . k+1 1 2i+1 For w = 16 , the point ( 2k+1 , p2i+1 ) lies on the Lagrange like interpolant of the points {( 2jk , pkj ) : j = i − 1, i, i + 1, i + 2}. It is known [10] that for w = 1 16 , the scheme reproduces circles and functions spanned by {1, x, cos(α·), sin(α·)}. In fact, for 0 < α < π 2n , the scheme (6.1) reproduces functions spanned by {x, cos(jαx), sin(jαx) : j = 0, 1, 2, .., n}. We illustrate the reconstruction of astroid and cardioid 1 by the scheme (6.1) with w = 16 in Figure 5. In Figure 5(a), we take 12 points on the parametric curve astroid: x(t) = 3 cos t+cos 3t, y(t) = 3 sin t−sin 3t and α = π6 . In Figure 5(b), we take 12 points on the parametric curve cardioid: x(t) = 12 (1 + 2 cos t + cos 2t), y(t) = 12 (2 sin t + sin 2t) and α = π6 . wk = (sin2

α

2k+2

)/(2 sin

7. Conclusion In [16], a 4-point ternary interpolating non-stationary subdivision scheme with a parameter corresponding to a stationary subdivision scheme [7] is introduced. The support of this scheme is [−3, 3]. A 6-point non-stationary subdivision scheme with a parameter, a counterpart to the scheme of Weismann [18], is introduced in [17]. The support of this scheme is [−5, 5]. These two schemes are C 2 for certain ranges of parameters and the schemes reproduce circles for


some parameters. One can observe from the various nonstationary schemes discussed in this paper that the ternary interpolating schemes have higher order smoothness and smaller supports than the corresponding binary interpolating schemes. Moreover, the supports of the non-stationary interpolating schemes are same as their stationary counterparts. In Figure 3(a) and Figure 3(b), we present the basic limit functions of the scheme (6.1) and the 4-point ternary non-stationary interpolating scheme respectively. As we derived a 2-point binary non-stationary approximating scheme, we can also derive a 3-point binary nonstationary approximating scheme from the scheme introduced in [9] by taking n = 5. This non-stationary 3-point approximating scheme is C 3 and generates a trigonometric spline of order 5. Deriving a ternary subdivision scheme for trigonometric spline of order n, n > 2 and in particular deriving a 3-point ternary approximating scheme is left for future work. The subdivision schemes discussed in this paper can be naturally used for the design of surfaces using a tensor product approach. Further generalizations of some of the schemes for generating surfaces from arbitrary meshes are beyond the scope of this paper.

(a)

(b)

Figure 1. Basic limit functions of 2-point binary approximating stationary and nonstationary schemes.

(a)

(b)

Figure 2. Basic limit functions of the 3point ternary interpolating stationary and non-stationary schemes.

References [1] G. M. Chaikin. An algorithm for high-speed curve generation. Computer Graphics and Image Processing, 3 : 346–349, 1974.

(a)

(b)

Figure 3. Basic limit functions of the 4-point binary and 4-point ternary non-stationary in1 terpolating schemes when w = 16 , α = π4 and 1 π μ = 11 , α = 4 respectively.

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1 -1

-0.5

0

0.5

1

-1

-0.5

(a)

0

0.5

1

(b)

Figure 4. Reconstruction of circles by the schemes (3.1) and (4.1). 2

4

1.5

3 2

1

1

0.5

0

0

-1

-0.5

-2

-1

-3

-1.5

-4 -4 -3 -2 -1 0

1

2

3

4

-2 -0.5

0

0.5

1

1.5

2

2.5

Figure 5. Reconstruction of astroid and cardioid.

[2] N. Dyn and D. Levin. Analysis of aymptotically equivalent binary subdivision schemes. Journal of Mathematical Analysis and Applications, 193 : 594–621, 1995. [3] N. Dyn and D. Levin. Stationary and non-stationary binary subdivision schemes. In Computer Aided Geometric Design, Tom Lyche and L. Schumaker, editors, Academic Press, New York, pages 209–216, 1992. [4] N. Dyn, D. Levin and J.A. Gregory. 4-point interpolatory subdivision scheme for curve design. Computer Aided Geometric Design, 4 : 257–268, 1987. [5] N. Dyn and D. Levin. Subdivision schemes in geometric modelling. Acta Numerica, 11 : 73–144, 2002. [6] N. Dyn, D. Levin and A. Luzzatto. Exponential reproducing subdivision schemes. Foundations of Computational Mathematics, 3 : 187-206, 2003.


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