Analysis of stationary subdivision schemes for curve design based on radial basis function interpolation

Applied Mathematics and Computation 215 (2010) 3851–3859

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Analysis of stationary subdivision schemes for curve design based on radial basis function interpolation Yeon Ju Lee a, Jungho Yoon b,*,1 a b

Division of Applied Mathematics, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejeon 305-701, South Korea Department of Mathematics, Ewha W. University, Seoul 120-750, South Korea

a r t i c l e

i n f o

a b s t r a c t This paper provides a large family of interpolatory stationary subdivision schemes based on radial basis functions (RBFs) which are positive definite or conditionally positive definite. A radial basis function considered in this study has a tension parameter k > 0 such that it provides design flexibility. We prove that for a sufficiently large k P k0 , the proposed 2L-point ðL 2 NÞ scheme has the same smoothness as the well-known 2L-point Deslauriers–Dubuc scheme, which is based on 2L  1 degree polynomial interpolation. Some numerical examples are presented to illustrate the performance of the new schemes, adapting subdivision rules on bounded intervals in a way of keeping the same smoothness and accuracy of the pre-existing schemes on R. We observe that, with proper tension parameters, the new scheme can alleviate undesirable artifacts near boundaries, which usually appear to interpolatory schemes with irregularly distributed control points. Ó 2009 Elsevier Inc. All rights reserved.

Keywords: Stationary subdivision Radial basis function Interpolation Smoothness Gaussian Multiquadric Inverse multiquadric

1. Introduction Subdivision schemes are important and efficient tools for generating smooth curves and surfaces from a finite set of control points. If the subdivision rule is the same at all levels and positions of iteration, the scheme is called a stationary uniform (hereafter, stationary) subdivision scheme. For planar curves, this rule which is specified by a mask a :¼ fan : n 2 Zg is used iteratively, at each recursion step, for generation of new discrete values on a finer grid. Specifically, given an initial control points f 0 ¼ ffn0 : n 2 Zg, a subdivision scheme S defines new sets of points f k ¼ ffnk : n 2 Zg; k 2 Zþ , by k

f kþ1 ¼ Sf ; where k

ðSf Þj ¼

X

aj2n fnk :

ð1Þ

n2Z

We view S as a linear operator on bounded sequences and say that a subdivision scheme S is C m if for the initial data d ¼ ffn0 ¼ d0;n : n 2 Zg with d0;n the Kronecker delta, there exists a limit function u 2 C m ðRÞ; uX0, satisfying

lim sup jfnk  uð2k nÞj ¼ 0;

k!1 n2Z

where f k ¼ Sk d:

ð2Þ

The function u is called the basic limit function of S, which is formally defined by u ¼ S1 d. * Corresponding author. E-mail addresses: [email protected] (Y.J. Lee), [email protected] (J. Yoon). 1 Jungho Yoon was Supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2009-0084583). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.11.028

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The equation in (1) reveals that there are two rules, that is, even and odd rules, which extend the initial control points to finer points: In particular, if the even rule satisfies the relation a2n ¼ dn;0 with n 2 Z, the subdivision scheme is called interpolatory. This interpolatory rule guarantees that the initially given point set belongs to the limit function such that they can control the shape of the resulting subdivision curves (or surfaces) in a more intuitive way than approximating schemes. Well-known examples are the 4-point and 6-point schemes [8,9] which have been generalized to the Deslauriers–Dubuc scheme [6] and the butterfly scheme on triangular mesh for surface design [8,10]. All these schemes are based on polynomial interpolation. In addition, interpolatory non-stationary schemes were studied in [1,2,11,12]. Typical examples of non-interpolatory subdivision schemes are the B-spline schemes [5]. Moreover, a general treatment of stationary subdivision schemes can be found in [3–6,8,9]. In [14], using the multiquadrics (MQs) /k ðxÞ ¼ ðx2 þ k2 Þb=2 with k > 0 and b 2 Rþ n 2Zþ , a new class of 2L-point (L 2 N) interpolatory subdivision schemes has been introduced, where Rþ indicates the set of all non-negative real numbers. The new values are obtained by multiquadric interpolation to the 2L symmetric points surrounding the new point. In this study, we introduce a large family of interpolatory schemes by employing smooth RBFs which are conditionally positive definite of order g P 0. Well-known examples are Gaussians (GAs) /k ðxÞ ¼ expðjxj2 =k2 Þ and inverse multiquadrics (IMQs) /k ðxÞ ¼ ðjxj2 þ k2 Þ1=2 , k > 0 as well as multiquadrics. It is worth pointing out that each scheme based on these RBFs has a tension parameter k > 0, which provides design flexibility. Thus, this paper focuses on these three families of subdivision schemes (based on MQ, IMQ and GA) and analyzes their smoothness and tension properties. In this paper, we prove that for sufficiently large k P k0 , the proposed 2L-point scheme has the same smoothness as the 2L-point Deslauriers–Dubuc scheme. Furthermore, we present some numerical results by adapting modified subdivision rules for bounded intervals, while keeping the same smoothness of the pre-existing scheme defined for R. We see that, with proper tension parameters, the new scheme can alleviate undesirable artifacts near boundaries, which usually appear to interpolatory schemes with irregularly distributed control points. 2. Stationary subdivision schemes using RBF interpolation 2.1. Subdivision scheme using RBF interpolation We now introduce the 2L-point interpolatory subdivision schemes based on RBF interpolation. Let P
X 0 :¼ fL þ 1; . . . ; Lg: Let /k be a smooth RBF with a tension parameter k > 0 and conditionally positive definite of order g P 0. Let un ; n 2 X 0 , be the Lagrange functions from spanf/k ð  nÞ : n 2 X 0 g þ P
a12n :¼ un ð21 Þ;

a2n ¼ d0;n ;

n 2 X0:

ð3Þ

It is necessary to point out that this mask is independent of the spatial location j 2 Z and the refinement level k 2 Zþ , i.e., stationary uniform. A reader is referred to [14] for the specific construction of the mask in (3). Moreover, letting fdn : n 2 Zg be the mask of the 2L-point Deslauriers–Dubuc interpolatory scheme, the general form of an can be written as

an ¼ dn þ cn xðkÞ;

n 2 X0;

for some constants cn 2 R, where xðkÞ is a function k. The explicit forms of an ; n 2 X 0 , are very lengthy and complicated. An example of the MQ-based 4-point interpolatory scheme can be found in [14]. 2.2. Laurent polynomials To simplify the presentation of a subdivision scheme and its analysis, it is convenient to assign to each rule, defined by a mask a, the Laurent polynomial

aðzÞ :¼

X

an zn :

ð4Þ

n2Z

Indeed, since the mask of the 2L-point subdivision scheme is finitely supported, aðzÞ is a Laurent polynomial of finite degree. Further, the Laurent polynomial corresponding to the iterated scheme S‘ ; ‘ > 0, is given by

a½‘ ðzÞ ¼

‘1 Y

j

aðz2 Þ ¼

X

n a½‘ n z ;

ð5Þ

n2Z

j¼0

½‘

where the scheme corresponding to fan g is a 2‘ different rule mapping f k to f kþ‘ . The norm of the scheme S‘ is defined by ‘

kS k1 ¼ max

( X a2Z

)

½‘ jacþ2‘ a j

‘

: c ¼ 0; . . . ; 2  1 :

ð6Þ

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Let S1 be the subdivision rule for the divided differences of the original control points so that it has the property

df

kþ1

k

¼ S1 df ; kþ1

k where f k ¼ Sk f 0 and ðdf Þn ¼ 2k ðfnk  fn1 Þ. In fact, the characteristic Laurent polynomial a1 ðzÞ for S1 is given by a1 ðzÞ ¼ aðzÞ2z=ð1 þ zÞ. For more details about the above discussion, the readers are referred to the paper [8].

2.3. Smoothness analysis The main tool for the analysis of convergence of a stationary subdivision scheme is given as follows: Theorem 1. (Smoothness of stationary subdivision scheme [8]). Let S be a stationary binary subdivision scheme with the P characteristic Laurent polynomial aðzÞ ¼ n2Z an zn . (a) If aðzÞ ¼ 12 ð1 þ zÞa1 ðzÞ and the scheme S1 corresponding to a1 ðzÞ is C c , the scheme S is a convergent, and the basic limit function u of S is in C cþ1 ðRÞ. P (b) An interpolatory subdivision scheme S converges to C c -limit function only if it reproduces P
Table 1 The smoothness and corresponding ranges of k of 2L-point interpolatory subdivision scheme based on GAs. 2L

Smoothness

m ¼ ð0; 1Þ

4-pt

C0

ð0; 1Þ

C1

½6:3754; 1Þ

6-pt

8-pt

0

m ¼ ð2; 3Þ

m ¼ ð4; 5Þ

ð0; 1Þ

ð0; 1Þ

C1

½0:6503; 1Þ

½0:4897; 1Þ

C2

½10:2309; 1Þ

½0:7242; 1Þ

ð0; 1Þ

ð0; 1Þ

ð0; 1Þ

C1

½0:6749; 1Þ

½0:5712; 1Þ

½0:3772; 1Þ

C2

½5:9237; 1Þ

½0:7819; 1Þ

½0:6585; 1Þ

C3

½8:8628; 1Þ

½2:7504; 1Þ

½0:8543; 1Þ

C

C

0

Table 2 The smoothness and corresponding ranges of k of 2L-point interpolatory subdivision scheme based on IMQs. 2L

Smoothness

m ¼ ð0; 1Þ

4-pt

C0

ð0; 1Þ

C1

½6:8409; 1Þ

6-pt

8-pt

0

m ¼ ð2; 3Þ

ð0; 1Þ

ð0; 1Þ

C1

½0:5483; 1Þ

½0:3016; 1Þ

C2

½18:7013; 1Þ

½0:7018; 1Þ

C

0

m ¼ ð4; 5Þ

ð0; 1Þ

ð0; 1Þ

ð0; 1Þ

C1

½0:5889; 1Þ

½0:4165; 1Þ

½0:1493; 1Þ

C2

½11:8112; 1Þ

½0:8272; 1Þ

½0:5737; 1Þ

C3

½18:3080; 1Þ

½5:0963; 1Þ

½1:0110; 1Þ

C

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Table 3 Smoothness of 2L-point interpolatory subdivision scheme using the MQ /k ðxÞ ¼ ðx2 þ k2 Þb=2 with b 2 2N  1. For each chosen b and smoothing factor m, the smoothness and its corresponding range of k are given. 2L

b

4-pt

b¼1

6-pt

b¼1

b¼3

8-pt

b¼1

b¼3

b¼5

Smoothness

m ¼ ð0; 1Þ

m ¼ ð2; 3Þ

m ¼ ð4; 5Þ

0

ð0; 1Þ

C1

ð0; 1Þ

0

ð0; 1Þ

C1

ð0; 1Þ

ð0; 1Þ

C2

½15:2283; 1Þ

ð0:3042; 1Þ

C0

ð0; 1Þ

ð0; 1Þ

C1

ð0; 1Þ

ð0; 1Þ

C2

½10:1401; 1Þ

ð0; 1Þ

C0

ð0; 1Þ

ð0; 1Þ

ð0; 1Þ

C1

ð0; 1Þ

ð0; 1Þ

ð0; 1Þ

C2

½9:8132; 1Þ

½0:3911; 1Þ

½0:1756; 1Þ

C3

½15:3802; 1Þ

½4:2311; 1Þ

½0:6361; 1Þ

C0

ð0; 1Þ

ð0; 1Þ

ð0; 1Þ

C1

ð0; 1Þ

ð0; 1Þ

ð0; 1Þ

C2

½6:8161; 1Þ

ð0; 1Þ

ð0; 1Þ

C3

½11:0839; 1Þ

½3:1116; 1Þ

ð0; 1Þ

C

C

ð0; 1Þ

C0

ð0; 1Þ

ð0; 1Þ

ð0; 1Þ

C1

ð0; 1Þ

ð0; 1Þ

ð0; 1Þ

C2

ð0; 1Þ

ð0; 1Þ

ð0; 1Þ

C3

½8:3582; 1Þ

½1:0367; 1Þ

ð0; 1Þ

Proof. In this proof, denote by fak;n g (instead of fan g) the mask of Sk with the tension parameter k > 0. Since ak;2n ¼ d2n ¼ d0;n with n 2 X 0 , we need to estimate only the difference ak;12n  d12n ; n 2 X 0 . Recall from (3) that ak;12n ¼ un ð21 Þ; n 2 X 0 , with un the Lagrange function of the RBF interpolation on X 0 from the space spanf/k ð  ‘Þ : ‘ 2 X 0 g þ P 0 such that for any k P k0 , the 2Lpoint scheme Sk based on the /k -interpolation has the same smoothness as the 2L-point Deslauriers–Dubuc interpolatory scheme SD . Proof. Suppose that SD is C c with c 2 N. Letting ak ðzÞ be the Laurent polynomial corresponding to Sk , consider the subdivision scheme 12 Sk;cþ1 with its characteristic Laurent polynomial

 cþ1 1 1X 1 2z ak;cþ1;n zn :¼ ak ðzÞ: ak;cþ1 ðzÞ ¼ 2 2 n2Z 2 1þz

Then, in order to prove this theorem, it is sufficient to show that the scheme 12 Sk;cþ1 converges uniformly to zero. To this end, applying (5), write the Laurent polynomial corresponding to the M-iterated scheme ð12 Sk;cþ1 ÞM as follows: M1 Y ‘¼0

‘ 1 1 X ½M ak;cþ1;n zn : ak;cþ1 ðz2 Þ ¼ M 2 2 n2Z

On the other hand, since the scheme SD is C c , there exists a subdivision scheme polynomial

1 S , 2 D;cþ1

corresponding to the Laurent

 cþ1 1 1X 1 2z dcþ1;n zn :¼ dðzÞ dcþ1 ðzÞ ¼ 2 2 n2Z 2 1þz such that the scheme 12 SD;cþ1 converges uniformly to zero for all initial control points. It means that for some M 2 N,

 M    1    SD;cþ1  < 1:   2 1

Applying (5), we see that the Laurent polynomial corresponding to the M-iterated scheme ð12 SD;cþ1 ÞM takes the form M1 Y ‘¼0

‘ 1 1 X ½M n dcþ1;n z : dcþ1 ðz2 Þ ¼ M 2 2 n2Z

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It is immediate from (6) that

 ( ) M    1 1 X ½M   M jd j : j ¼ 0; . . . ; 2  1 < 1:  SD;cþ1  ¼ max M   2 2M n2Z cþ1;jþ2 n

ð7Þ

1

½M

½M

Note here that dcþ1;n and ak;cþ1;n are linear combinations of dcþ1;n and ak;cþ1;n with the same coefficients, respectively. Thus, ½M ½M due to Lemma 1, it is obvious that ak;cþ1;n converges to dcþ1;n as k tends to 1. It clearly deduces from (7) that for some k0 > 0,

 ( ) M    1 1 X ½M   M ja  1 < 1;  Sk;cþ1  ¼ max M j : j ¼ 0; . . . ; 2   2 2M n2Z k;cþ1;jþ2 n 1

for any k P k0 . Due to [8, Theorem 3.2], we find that the scheme Sc is uniformly convergent for k P k0 and by Theorem 1, the scheme Sk with k > k0 has the same smoothness C c as SD . h The remainder of this section is devoted to providing the smoothness of the 2L-point interpolatory scheme Sk . Here, depending on the choice of /k and m, various subdivision rules are generated. One interesting fact is that for each pair m ¼ 2n; 2n þ 1 with n 2 Zþ and 0 6 m < 2L  2, the same mask is obtained. The specific smoothness of Sk can be obtained by using the known technique (e.g., [8, Theorem 3.2] and Theorem 1). Although this is algorithmic in principle, it is almost impossible to analyze it without the help of a computer program because the algebraic manipulations are too much involved. Thus, the MAPLE program is utilized for this computation. A reader who is interested in this algorithm is referred to [7,14]. Tables 1 and 2 display the smoothness of the 2L-point scheme Sk based on GAs and IMQs, respectively. The specific ranges of k corresponding the smoothness of Sk are presented. In fact, these radial functions are positive definite so that /k -interpolation can be implemented without the polynomial space P
kþ1 f2jþ1

8 2L1 P > > j;12n fnk ; > a 0 6 j 6 L  2; > > > n¼0 > > > L < P k j;12n fjþn a ; L  1 6 j 6 2k N  L; ¼ > n¼Lþ1 > > > k > 2P N > > > j;12n fnk ; 2k N  L þ 1 6 j 6 2k N  1; a > : k

ð8Þ

n¼2 N2Lþ1

where

j;12n ¼ a

8 j1 P > > n ð‘ þ jÞa12‘ þ a12ðnjÞ ; 0 6 j 6 L  2; 0 6 n 6 j þ L; u > > > > ‘¼Lþ1 > > > > j1 P > > < n ð‘ þ jÞa12‘ ; 0 6 j 6 L  2; L þ j þ 1 6 n 6 2L  1; u ‘¼Lþ1

> > > a12n ; > > > > > 2k N1j;12ð2k NnÞ ; > a > > > :

ð9Þ

L  1 6 j 6 2k N  L; L þ 1 6 n 6 L; 2k N  L þ 1 6 j 6 2k N  1; 2k N  2L þ 1 6 n 6 2k N:

j;n g. From now on, let Sk denote the modified subdivision scheme on the interval ½0; N with the adapted mask fa Having performed numerical experimentations with several alternatives of /k and L ¼ 4; 6, we found that the advantages of the proposed scheme are more noticeable in the case of 6-point scheme than the case of 4-point scheme (we will discus it at the end of this section). Thus, we will concentrate on the comparison of the proposed 6-point scheme to the known 6-point Deslauriers–Dubuc scheme. First, the curves in Fig. 1 are generated by the different 6-point schemes: the RBF-based schemes and the Deslauriers–Dubuc scheme. The solid curves are obtained by the RBF-based schemes and the dash-dotted curves are by the Deslauriers–Dubuc scheme. In this comparison, differences around boundaries are apparent. The 6-point Deslauriers–Dubuc scheme produces

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(a)GA:6-pointscheme

(b)IMQ:6-pointscheme

(c)MQ:6-pointscheme

Fig. 1. A comparison between the scheme based on RBF interpolation (solid lines) and the Deslauriers–Dubuc scheme (dash-dotted lines). Here, k ¼ 2 for GA and IMQ, and k ¼ 1:5 for MQ.

undesirable undulations near the boundaries. However, by choosing a suitable parameter k, the proposed scheme can improve the shape of the limit curves and alleviate undesirable artifacts. Specifically, the curves in Fig. 1(a) and (b) are generated by 2 2 using /k ðxÞ ¼ ex =k (GA) and /k ðxÞ ¼ ðx2 þ k2 Þ1=2 (IMQ) with k ¼ 2:0, respectively, and the curves in Fig. 1(c) are by using 2 b 2 /k ðxÞ ¼ ðx þ k Þ (MQ) with k ¼ 1:5 and b ¼ 3. The RBF-based schemes provide visually better curves than the classical

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Fig. 2. The effect of the tension k in the MQ-based 6-point interpolatory scheme Sk . The curves in the box on the left-hand side are generated by using k ¼ 0:01; 1; 2; 4; 6; 8; 15 from the top.

Fig. 3. The curvatures of the basic limit functions of the 6-point MQ-based scheme with b ¼ 3 and m ¼ 3 (bold-dotted line). For any choice of k 2 ð0; 1Þ, the corresponding scheme is C 2 . The straight line on the top indicates the curvature of the classical 6-point scheme (dash-dotted line).

6-point scheme. Moreover, these numerical results demonstrates that the MQ-based scheme provides more pleasant results than the schemes based on the GA and IMQ. The effect of the tension parameter k is demonstrated in detail in Fig. 2, where the MQ-based 6-point interpolatory scheme is used. The curves in Fig. 2, from the top in the box on the left-hand side, correspond to the values of the tension parameters k ¼ :01; 1; 2; 4; 6; 8; 15. As k decreases, the limit curve tends to become looser to the initial polyline. But, as k increases to 1, the limit curve tends to the result of Deslauriers–Dubuc scheme (see Lemma 1), yielding a twisting artifact for a large value k ¼ 15. It is also of interest to see that as k approaches to 0, the subdivision mask based on the MQ-interpolation converges to the mask of the interpolatory scheme based on the function jxjb (i.e., piecewise polynomial if b is odd integer). Having performed numerical experiments, we found that the range 0 < k 6 3 is suggested as a good choice of k. Fig. 3 describes the maximum curvature of the basic limit function (say, uk ) of the MQ-based 6-point scheme with b ¼ 3 and m ¼ 3. When 0 < k 6 3, the maximum curvature of uk becomes smaller. Another advantage of the MQ-based scheme is that it provides a certain smoothness in a wider range of k than the other two cases (see Table 3). For instance, the MQ-based 6-point scheme provides C 2 smoothness for any k 2 ð0; 1Þ with m ¼ 3, but the GA and IMQ-based schemes are C 2 for k away from zero, i.e, for k P k0 with k0 > 0. Fig. 4 shows another interesting results of the RBF-based scheme Sk . The control points in Fig. 4 are highly non-uniform. It is known that the Deslauriers–Dubuc scheme has an inherent drawback of interpolatory subdivision schemes. If the given control points are irregular, its limit curve may result in an unpleasant artifact as shown in Fig. 4. However, by choosing a proper tension parameter k ¼ 1 the MQ-based scheme can obtain curves without twisting artifact (similar results can be generated by GA and IMQ). It is worthwhile to note that the (maximum) curvature of uk reaches the smallest when k ¼ 1. Remark 1. After having numerical experimentations, we observed that the MQ-based 4-point scheme provides better results than the schemes based on GA and IMQ. However, the advantage of our 4-point scheme is less obvious than the case of 6-point scheme. In Fig. 5, we compare the MQ-based 4-point and the other 6-point schemes. Differences around boundaries are more noticeable in the 6-point scheme.

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Fig. 4. Interpolating curves generated by the 6-point scheme Sk based on the MQ with k ¼ 1 (solid lines) and the 6-point Deslauriers–Dubuc scheme (dashdotted lines).

(a) MQ : 4-point scheme

(b) MQ : 6-point scheme

(c) IMQ : 6-point scheme

(d) GA : 6-point scheme

Fig. 5. A comparison between the scheme based on RBF interpolation (solid lines) and the Deslauriers–Dubuc scheme (dash-dotted lines). Here, k ¼ 2 for GA and IMQ, and k ¼ 1:5 for MQ.

Remark 2. For the initial data d½n ¼ fd‘;n : ‘ ¼ 0; . . . ; Ng with n ¼ 0; . . . ; N, the limit function uk;n :¼ S1 k fd‘;n g is no more than a shift of one basic limit function. It can be written as follows:

Y.J. Lee, J. Yoon / Applied Mathematics and Computation 215 (2010) 3851–3859

uk;n ðxÞ ¼

8 > > > < uk ðx  nÞ þ

1 P

a¼2Lþ2

> uk ðx  nÞ; > > : uk;Nn ðn  xÞ;

3859

 n ðaÞuk ðx  aÞ; 0 6 n 6 2L  1; u 2L 6 n 6 N  2L; N  2L þ 1 6 n 6 N;

where uk is the basic limit function of Sk . Thus, the modified subdivision scheme Sk for bounded intervals has the same smoothness as the scheme Sk . Acknowledgements The authors are grateful to the anonymous referee for many valuable suggestions for this paper. References [1] C. Beccari, G. Casciola, L. Romani, A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics, Comput. Aided Geomet. Des. 24 (2007) 1–9. [2] C. Beccari, G. Casciola, L. Romani, An interpolating 4-point C 2 ternary non-stationary subdivision scheme with tension control, Comput. Aided Geomet. Des. 24 (2007) 210–219. [3] A. Cavaretta, W. Dahmen, C.A. Micchelli, Stationary subdivision, Mem. Am. Math. Soc. 93 (1991) 1–186. [4] S. Choi, B. Lee, Y. Lee, J. Yoon, Stationary subdivision schemes reproducing polynomials, Comput. Aided Geomet. Des. 23 (2006) 351–360. [5] E. Cohen, T. Lyche, R. Riesenfeld, Discrete B-spline and subdivision techniques in Computer-Aided Geometric Design and Computer Graphics, Comput. Graphics Image Process. 14 (1980) 87–111. [6] G. Deslauriers, S. Dubuc, Symmetric iterative interpolation, Constr. Approx. 5 (1989) 49–68. [7] N. Dyn, Interpolation and approximation by radial and related functions, in: C.K. Chui, L.L. Schumaker, J. Ward (Eds.), Approximation Theory VI, Academic press, 1989, pp. 211–234. [8] N. Dyn, Subdivision schemes in computer-aided geometric design, in: W.A. Light (Ed.), Advances in Numerical Analysis, Wavelets, Subdivision Algorithms and Radial Basis Functions, vol. II, Oxford University Press, 1992, pp. 36–104. [9] N. Dyn, J.A. Gregory, D. Levin, A four-point interpolatory subdivision scheme for curve design, Comput. Aided Geomet. Des. 4 (1987) 257–268. [10] N. Dyn, J.A. Gregory, D. Levin, A Butterfly subdivision scheme for surface interpolation with tension control, ACM Trans. Graph. 9 (1990) 160–169. [11] N. Dyn, D. Levin, A. Luzzatto, Exponential reproducing subdivision scheme, Found. Comput. Math. 3 (2003) 187–206. [12] N. Dyn, D. Levin, J. Yoon, Analysis of univariate non-stationary subdivision schemes with application to Gaussian-based interpolatory schemes, SIAM J. Math. Anal. 39 (2) (2007) 470–488. [13] Y. Lee, G. Yoon, J. Yoon, Convergence property of increasingly flat radial basis function interpolation to polynomial interpolation, SIAM J. Math. Anal. 39 (2) (2007) 537–553. [14] B. Lee, Y. Lee, J. Yoon, Stationary binary subdivision schemes using radial basis function interpolation, Adv. Comput. Math. 25 (2006) 57–72. [15] J. Yoon, Construction of adapted subdivision schemes for bounded intervals based on radial basis function interpolation, J. Appl. Math. Comput. 24 (2007) 95–104.