A New 4-Point Quaternary Approximating Subdivision Scheme
A new 4-point quaternary approximating subdivision scheme with one shape parameter is proposed and analyzed. Its smoothness and approximation order are higher but support is smaller in comparison with the existing binary and ternary 4-point subdivision schemes.
Computer Aided Geometric Design (CAGD) is a branch of applied mathematics concerned with algorithms for the design of smooth curves/surfaces. One common approach to the design of curves/surfaces related to CAGD is the subdivision scheme. It is an algorithm to generate smooth curves and surfaces as a sequence of successively refined control polygons. At each refinement level, new points are added into the existing polygon and the original points remain existed or discarded in all subsequent sequences of control polygons. The number of points inserted at level between two consecutive points from level is called arity of the scheme. In the case when a number of points inserted are the subdivision schemes are called binary, ternary-ary, respectively. An important review of the different subdivision schemes, which range from binary to any arity, can be found in [1–3]. Due to good properties of the 4-point binary and ternary subdivision schemes [4–9], much attention has been given to extend their ability in modelling curves and surfaces. For example, the 4-point ternary interpolating subdivision scheme  can generate higher smoothness than the 4-point binary one  by using the same number of control points.
Now a days, the variety of subdivision schemes investigated; our interest is in the direction of quaternary schemes. The goal of this paper is to construct 4-point quaternary subdivision scheme having the higher smoothness and approximation order but smaller support than existing 4-point binary and ternary schemes.
Here we present a 4-point quaternary approximating subdivision scheme. A polygon is mapped to a refined polygon by applying the following four subdivision rules: where the weights are given by The paper is organized as follows. In Section 2 we list all the basic facts about quaternary subdivision schemes needed in the paper. Sections 3 and 4 are devoted for analysis of proposed scheme and its properties, respectively. Finally, in Section 5, comparison of our scheme with other existing 4-point schemes is presented. Some examples reflecting the performance of our scheme by setting the shape parameter to various values are also offered.
A general compact form of univariate quaternary subdivision scheme which maps a polygon to a refined polygon is defined by where the set of coefficients is called mask of the scheme. A necessary condition for the uniform convergence of the subdivision scheme (2.1) is that A subdivision scheme is uniformly convergent if for any initial data , there exists a continuous function , such that for any closed interval , that satisfies Obviously .
For the analysis of subdivision scheme with mask , it is very practical to consider the -transform of the mask, which is usually called the of the scheme. Since the scheme has mask of finite support, the corresponding is Laurent polynomial, namely, polynomial in positive and negative powers of the variables. From (2.2) and (2.4) the Laurent polynomial of a convergent subdivision scheme satisfies This condition guarantees existence of a related subdivision scheme for the divided differences of the original control points and the existence of associated Laurent polynomial: The subdivision scheme with is related to scheme with by the following theorem.
Theorem 2.1 (see ). Let denote a subdivision scheme with symbol satisfying (2.2). Then there exists a subdivision scheme with the property where and . Furthermore, is a uniformly convergent if and only if converges uniformly to the zero function for all initial data , in the sense that
Theorem 2.1 indicates that for any given subdivision scheme , with a mask satisfying (2.2), we can prove the uniform convergence of by first deriving the mask of and then computing for , where is the first integer for which If such an exists, converges uniformly. Since there are four rules for computing the values at next refinement level, we define the norm where Theorem 2.2 (see ). Let be subdivision scheme with a characteristic -polynomial . If the subdivision scheme , corresponding to the -polynomial , converges uniformly then for any initial control polygon .
Corollary 2.3. If is a subdivision scheme of the form above and converges uniformly to the zero function for all initial data then for any initial control polygon .
Corollary 2.3 indicates that for any given quaternary subdivision scheme , we can prove by first deriving the mask of and then computing for , where is the first integer for which . If such an exists, then .Theorem 2.4 (see ). The approximation order of a convergent subdivision scheme which is exact for (set of polynomials at most degree n) is .
3. Smoothness Analysis of Proposed Scheme
This section is devoted for analysis of 4-point quaternary approximating subdivision scheme by using Laurent polynomial method. The following result shows that scheme is continuous.Theorem 3.1. The 4-point quaternary approximating subdivision scheme (1.1) is for any in .
Proof. For the given mask of proposed scheme the Laurent polynomial is where From Laurent polynomial (2.10) for and (3.2), we have For continuity of we require that the Laurent polynomial satisfy (2.5), which it does, and The norm of scheme is This implies for where Therefore converges uniformly. Hence, by Corollary 2.3, . By (3.4) the Laurent polynomial of scheme can be written as where Utilizing (2.10) for and and (3.8) we get For continuity of it needs that satisfy (2.5), which it does, and The norm of scheme is This implies for where Therefore is uniformly convergent. Hence, by Corollary 2.3, . Now from (3.10) Laurent polynomial of scheme is where With the choice of and , and by (3.14) For continuity, it is necessary that Laurent polynomial satisfy (2.5), which is incidentally true, and also for first integer value of for which . This implies for where Therefore converges uniformly. Hence, by Corollary 2.3, . Now from (3.16) Laurent polynomial of scheme can be written as where With the choice of , we have following by (3.20) For continuity, it is necessary that Laurent polynomial satisfy (2.5), which is incidentally true, and also for first integer value of for which : This implies for where Therefore converges uniformly. Hence, by Corollary 2.3, .
3.1. Hölder Exponent
From the above discussion, we conclude that our scheme is . In the following paragraph we generalize its smoothness based on Rioul's method  and Hassan et al.  (in generalize sense). We conclude that scheme has Hölder regularity for all , where is defined by For the convenience of computation, we set . Since by (3.24) we obtain that Hölder regularity against of the 4-point quaternary scheme is Figure 1(a) shows a graph of the Hölder exponent against . Notice that the highest smoothness of the 4-point quaternary scheme is achieved at , and its Hölder exponent is Figure 1(b) shows the result of proposed scheme (1.1) after four subdivision levels. In this figure the control polygon is drawn by dotted lines. The thin solid line is produced by setting , and the bold solid line is produced by setting .
3.2. Subdivision Rule for Open Polygon
When dealing with open initial polygon , it is not possible to refine the first and last edges by rule (1.1). However, the extension of this strategy to deal with open polygon requires a well-defined neighborhood of end points. Since the first and last edges can be treated analogously, it will be sufficient to derive the rules only for one side of the open polygon. To this aim, we see that if we define just one auxiliary point as extrapolatory rule in the initial polygon . Then the first edge of the nonrefined polygon can be refined by the following rules:
Remark 3.2. Subdivision rule (3.29) for last edges does not affect the convergence of the proposed scheme to a continuously differentiable limit. It is sufficient to show that, taken and , and refining the polygon by (1.1), after steps of subdivision the expression of the point turns out to coincide with .
4. Basic Properties of the Scheme
In this section, we discuss approximation order and support of basic limit function of 4-point quaternary approximating scheme.
4.1. Approximation Order
Here we show that the approximation order of proposed scheme is five. The following lemma based on the technique of Sabin  is needed to follow up the claim.
Lemma 4.1. The proposed 4-point quaternary subdivision scheme reproduces all the cubic polynomials for and quartic at .
Proof. We carry out this result by taking our origin the middle of an original span with ordinate . If , then we have,
where are defined by (1.1).
If , then where represents the differences of the vertices.
If , then Taking further differences, we get
If , then This implies that
If , then By taking differences, we have Thus the proposed scheme has cubic in all and quartic precision at .
4.2. Support of Basic Limit Function
The basic function of a subdivision scheme is the limit function of proposed scheme for the following data: Figure 2(a) shows the basic limit function of proposed scheme. The following theorem is related to the support of limit function.
(a) Basic limit functions
(b) Close curves
(c) Open curves
Theorem 4.3. The basic limit function of proposed 4-point scheme has support width , which implies that it vanishes outside the interval .
Proof. Since the basic function is the limit function of the scheme for the data (4.7), its support width can be determine by computing how for the effect of the nonzero vertex will propagate along by. As the mask of the scheme is a 16-long sequence by centering it on that vertex, the distances to the last of its left and right nonzero coefficients are equal to and 7, respectively. At the first subdivision step we see that the vertices on the left and right sides of at & are the furthest nonzero new vertices. At each refinement, the distance on both sides is reduced by the factor . At the next step of the scheme this will propagate along by on the left and on the right. Hence after subdivision steps the furthest nonzero vertex on the left will be at and on the right will be at . So the total support width is .
5. Comparison and Application
In Table 1, we compare some properties of proposed 4-point subdivision scheme with those of other 4-point schemes having smaller arity.
In Figures 2(b) and 2(c), we illustrate performance of our scheme by setting the shape parameter to various values, which illustrate how this parameter affect the shape of limit curve. Moreover, Figures 3 and 4 show the comparison of support width and approximation order of proposed scheme with other existing 4-point approximating schemes.
(a) Binary 4-point 
(b) Ternary 4-point 
(a) Level 1
(b) Level 2
(c) Level 3
The authors are pleased to acknowledge the anonymous referees whose precious and enthusiastic comments made this manuscript more constructive. First author pays special thanks to Professor Deng Jian Song (University of Science and Technology of China) for his continuous research assistance. This work is supported by the Indigenous PhD scholarship scheme of Higher Education Commission Pakistan.
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