Abstract
This paper presents 6-point subdivision schemes with cubic precision. We first derive a relation between the 4-point interpolatory subdivision and the quintic B-spline refinement. By using the relation, we further propose the counterparts of cubic and quintic B-spline refinements based on 6-point interpolatory subdivision schemes. It is proved that the new family of 6-point combined subdivision schemes has higher smoothness and better polynomial reproduction property than the B-spline counterparts. It is also showed that, both having cubic precision, the well-known Hormann-Sabin’s family increase the degree of polynomial generation and smoothness in exchange of the increase of the support width, while the new family can keep the support width unchanged and maintain higher degree of polynomial generation and smoothness.
1. Introduction
Subdivision is an efficient method for generating curves and surfaces in computer aided geometric design. In general, subdivision schemes can be divided into two categories: interpolatory schemes and approximating schemes. Interpolatory schemes get better shape control while approximating schemes have better smoothness. The most well-known interpolatory subdivision scheme is the classical 4-point binary scheme proposed by Dyn et al. [1]. In 1989, it was extended to the 6-point binary interpolatory scheme by Weissman [2]. Most approximating schemes were developed from splines. Two of the most famous approximating schemes are Chaikin’s algorithm [3] and cubic B-spline refinement algorithm [4], which actually generate uniform quadratic and cubic B-spline curves with continuity and continuity, respectively.
The deep connection between interpolatory schemes and approximating schemes has been studied in many literatures [5–15]. In 2001, Maillot and Stam [5] introduced a push-back operation which is applied at each round of approximating refinement to progressively interpolate the control vertices. In 2007, Li and Ma [6] observed a relation between 4-point interpolatory subdivision and cubic B-spline curve refinement, and, motivated by this relation, they proposed a universal method for constructing interpolatory subdivision through the addition of weighted averaging operations to the mask of approximating subdivision. In 2008, Lin et al. [7] found another relation between 4-point interpolatory subdivision and cubic B-spline refinement and constructed interpolatory subdivision from approximating subdivision based on the relation. The deep connection between interpolatory and approximating schemes was also studied in [8–12] which exploited the generating functions of approximating subdivision and interpolatory subdivision. In 2012, Pan et al. [13] provided a combined ternary approximating and interpolatory subdivision scheme with continuity. Li and Zheng [14] constructed interpolatory subdivision from primal approximating subdivision with a new observation of the link between interpolatory and approximating subdivision. In 2013, Luo and Qi [15] made some theoretical analysis from the generation polynomial perspective and constructed some new interpolatory schemes from approximating schemes.
Our work is motivated by a new observation about the 4-point interpolatory subdivision and the quintic B-spline curve refinement. The observation gives us heuristics to construct combined subdivision schemes from existing subdivision schemes. The idea is to construct the counterparts of cubic and quintic B-spline refinements and make the relations between the 6-point interpolatory subdivision and the counterparts of cubic and quintic B-spline refinements similar to those between the 4-point interpolatory subdivision and the cubic, quintic B-spline curve refinements. Since the 6-point interpolatory subdivision from which the new subdivision scheme is deduced has good properties such as high smoothness and high accuracy, we are interested in studying which properties of the new subdivision scheme are better than their counterparts.
The new family of 6-point combined subdivision schemes is defined as follows:
(1) is called the 6-point combined interpolatory and approximating binary subdivision scheme. If , (1) generates 6-point interpolatory subdivision; otherwise, (1) produces approximating subdivision. It is proved that when suitably setting the tension parameter, all schemes from (1) are able to generate curves with continuity and reproduce cubic polynomials, whereas the B-spline refinements attain only linear precision. Moreover, we also make a comparison of properties between our family and famous Hormann-Sabin’s family [16] which has the same cubic precision.
2. Preliminaries
In this section, we recall some fundamental definitions and results that are necessary to the development of the subsequent results.
Given a set of initial control points , the set of control points at level are recursively defined by the following binary subdivision rules: The finite set is called mask. The iterative algorithm based on the repeated application of (2) is termed subdivision scheme and is denoted by . The symbol of the scheme is defined as .
Theorem 1 (see [17]). Let a binary subdivision scheme be convergent. Then the mask satisfies
Theorem 2 (see [17]). Let subdivision scheme with mask satisfy (3). Then there exists a subdivision scheme (first-order divided difference scheme of ) with the property where and . The symbol of is . Generally, if (the th-order divided difference scheme of ) exists with mask , then the symbol of is .
Theorem 3 (see [17]). (a) Let subdivision scheme have mask , and its th-order divided difference scheme exists with mask satisfying If there exists an integer , such that , then the subdivision scheme is continuous, where In particular, when ,(b) Let with being contractive (i.e., maps any initial data to zero). Then, is convergent and continuous.
Theorem 4 (see [18, 19]). Let denote the space of all univariate polynomials with real coefficients up to degree . Then a univariate subdivision scheme (i)generates if and only if (ii)reproduces with respect to the parametrization with and denoting the subdivision level, if and only if it generates and
3. Construction of the New Family
This section first explains a new observation about the relation between 4-point interpolatory subdivision and quintic B-spline refinement. Then, a new family of 6-point combined subdivision schemes is deduced.
3.1. A New Observation
Given an initial control polygon with vertices , as shown in Figure 1, the rules of 4-point interpolatory subdivision for generating level vertices are and quintic B-spline refinement for generating level vertices is
(a) The relation between 4-point scheme and quintic -spline scheme
(b) The relation between 4-point scheme and cubic -spline scheme
Denote by , the displacements of vertices from quintic B-spline refinement to 4-point interpolatory subdivision after one step of refinement, as shown in Figure 1(a), where the black lines represent the initial control polygon, the magenta lines represent the control polygon after one step of 4-point interpolatory subdivision, and the green lines represent the control polygon after one step of quintic B-spline refinement. Then, from (10) and (11), we can get
A new observation is which shows that the relation between 4-point interpolatory subdivision and quintic B-spline refinement is similar to the one between 4-point interpolatory subdivision and cubic B-spline refinement discovered by Lin et al. in [7]; that is, , as shown in Figure 1(b), where the blue lines represent the control polygon after one step of cubic B-spline refinement. So, from the point of view of displacements, 4-point interpolatory scheme has the same connections with cubic B-spline and quintic B-spline.
We further found that though 6-point interpolatory subdivision is also constructed from polynomial interpolation just like 4-point interpolatory subdivision, analogous connection does not exist between 6-point interpolatory subdivision and quintic B-spline refinement.
3.2. Construction of the New 6-Point Combined Scheme
As is shown in [2], the rules of 6-point interpolatory subdivision for generating level vertices are
Suppose the new subdivision have the following rule:where are tension parameters, and thenUsing relation (13), it can be deduced that So, we obtain and then the new subdivision can be concluded from (15) and (18) aswhich is the form of (1) in Section 1.
The mask and symbol of subdivision (1) are respectively. When , symbol (21) can be written asIn particular, when ,
Denote the family of subdivision (1) by and subfamily (22) by . We call them the counterparts of cubic and quintic B-spline refinements based on the 6-point interpolatory subdivision. Figure 2 illustrates the limit curves of some members of . In Section 4, we will prove that the family generates curves with continuity, and the subfamily attains continuity when and reproduces cubic polynomials.
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4. Analysis of the New Family
4.1. Smoothness Analysis
Proposition 5. The scheme defined by (1) converges and has smoothness when and , or and ; and when , the subfamily generates continuous limit curves.
Proof. The symbol of can be written aswhereLet denote the coefficients of Laurent polynomial . By Theorem 3(b), if is contractive, then is . Whenwhich shows that is contractive; hence, whenWhen , the symbol of the subfamily is which can also be written as , where When , .
Hence, by Theorem 3(b), is contractive and is when .
4.2. Generation Degree and Reproduction Degree
Polynomial generation and polynomial reproduction are desirable properties because any convergent subdivision scheme that reproduces polynomials of degree has approximation order [18]. The polynomial generation of degree is the capability of subdivision schemes to generate the full space of polynomials of degree [20]. The polynomial reproduction is the capability of subdivision schemes to produce in the limit exactly the same polynomial from which the initial data is sampled. The generation degree is not less than the reproduction degree. For example, the generation degree of degree-n B-spline refinement is , but the reproduction degree of degree-n B-spline refinement only attains 1. Hormann and Sabin [16] proposed a family of subdivision schemes which is defined by the product of the symbol of B-spline refinement with a degree-2 polynomial and increased the degree of polynomial reproduction of B-spline schemes from 1 to 3.
Let and suppose . Using Theorem 4, we get the following results.
Proposition 6. The subdivision scheme generatesIn particular, when , generates .
Proof. The symbol of can be written as where and .
Then, , and .
Moreover, when , and when , Hence, according to Theorem 4(i), we get that when , the subdivision scheme generates ; when generates and when , generates .
Proposition 7. If applying the parameter shift , the subdivision scheme reproduceswith respect to the parametrization , where denotes the subdivision level. In particular, when , reproduces .
Proof. To consider the reproduction degree of the subdivision scheme , in view of Theorem 4(ii), we just need to consider . Using the notation in Proposition 6, we get that so , and when ,Then , and when , Hence, using Theorem 4(ii), we conclude that when , the subdivision scheme reproduces ; when reproduces and when , reproduces .
As the new family of subdivision schemes is deduced from the 6-point interpolatory scheme using the relation between 4-point interpolatory scheme and cubic, quintic B-spline, the properties of all of them are summarized in Table 1. For the new subfamily and Hormann-Sabin’s family and both have cubic precision, we list corresponding properties for a comparison in Table 2.
5. Conclusions
In this paper, we present a new family of 6-point combined subdivision schemes which provides the representation of wide variety of shapes and a subfamily of subdivision schemes with high smoothness and cubic precision. All these properties are required in many applications, such as computer aided geometric design and geometric modeling. The subfamily attains cubic precision whereas the B-spline schemes have linear precision (see Figure 3). On the other hand, both having cubic precision, Hormann-Sabin’ family increases the degree of polynomial generation and smoothness in exchange of the increase of the support width, while can keep the support width unchanged and maintain higher degree of polynomial generation and smoothness. Moreover, the tension parameter makes able to provide more choices in applications (see Figures 4 and 5).
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Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant nos. 61472466 and 61070227, the NSFC-Guangdong Joint Foundation Key Project under Grant no. U1135003, and the Fundamental Research Funds for the Central Universities under Grant no. JZ2015HGXJ0175.