Four-Point -Ary Interpolating Subdivision Schemes
We present an efficient and simple algorithm to generate 4-point n-ary interpolating schemes. Our algorithm is based on three simple steps: second divided differences, determination of position of vertices by using second divided differences, and computation of new vertices. It is observed that 4-point n-ary interpolating schemes generated by completely different frameworks (i.e., Lagrange interpolant and wavelet theory) can also be generated by the proposed algorithm. Furthermore, we have discussed continuity, Hölder regularity, degree of polynomial generation, polynomial reproduction, and approximation order of the schemes.
In general, subdivision schemes can be divided into two categories: approximating and interpolating. For interpolating curve subdivision, new vertices are computed and added to the old polygons for each time of subdivision and the limit curve passes through all the vertices of the original control polygon. Interpolating subdivision schemes are more attractive than approximating schemes in computer aided geometric designs because of their interpolation property. In addition, the interpolation subdivisions are more intuitive to the users.
Initial work on interpolating subdivision schemes was started by Dubuc . Later on, Deslauriers and Dubuc  have introduced a family of schemes by using Lagrange polynomials indexed by the size of the mask and the arity. In , Dyn et al. have studied a family of interpolating schemes with mask size of four. Consequent to this, the research communities are interested in introducing higher arity schemes (i.e., ternary, quaternary-ary) which give better results and less computational cost. Lian  has constructed both the -point -ary for any and -point -ary for any odd interpolatory subdivision schemes for curve design by using wavelet theory. Mustafa and Rehman  have presented general formulae for the mask of -point -ary interpolating and approximating schemes for any integer and . These formulae provide mask of higher arity schemes and generalize lower arity schemes. Mustafa et al.  have presented an explicit formula for the mask of odd points -ary, for any odd , interpolating subdivision schemes.
In , it has been proved that the large support and higher arity schemes may outperform than small support and lower arity schemes. Even though these schemes are not in practice. It has been suggested that the research on large support and higher arity schemes may continue.
The multistage approach is very handy to construct subdivision schemes. This idea is variously used by others. Lane and Riesenfeld  have presented two fast subdivision algorithms for the evaluation of B-spline and Bernstein curves and surfaces. They have expressed new idea, in which, after a single duplication stage in which the number of control points is doubled by just taking each point twice, a sequence of smoothing operators is applied. Catmull and Clark  have used this technique to present the original description of subdivision in which each refinement is expressed in three stages. Zorin and Schröder  have considered the construction of an increasing sequence of alternating primal/dual quadrilateral subdivision schemes based on a simple averaging approach. Oswald and Schröder  used the same motif to generate families of subdivision schemes.
Augsdörfer et al.  first describe the original 4-point binary subdivision scheme and then apply six variations on the scheme which are obtained by tuning the local stages in various ways, producing some interesting subdivision schemes all of which are improvements on the original. We generalize the same technique to build the family of four-point -ary interpolating subdivision schemes. It is observed that 4-point -ary interpolating schemes introduced by [2, 4] can also be constructed by our generalized technique, even though these schemes have been constructed by different frameworks.
This paper is organized as follows: Section 2 presents some preliminary results. Section 3 consists of multistep algorithm which generates 4-point -ary interpolating subdivision schemes. Analysis of two schemes is also presented in this section. In Section 4, Hölder regularity, polynomial generation, polynomial reproductions and approximation order of ternary and quaternary subdivision schemes have been discussed. Numerical examples and conclusion are presented in Section 5.
A general compact form of univariate -ary subdivision scheme S  which maps polygon to a refined polygon is defined by where the set of coefficients is called the mask at th level of refinement. A necessary condition for the uniform convergence of subdivision scheme (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 , it satisfies Obviously, .
A symbol called Laurent polynomial of the mask plays an efficient role to analyze the convergence and smoothness of subdivision scheme. From (2) and (4) the Laurent polynomial of convergent subdivision scheme satisfies where are the th root of unity. This condition guarantees the existence of a related subdivision scheme for the divided differences of the original control points and the existence of an associated Laurent polynomial The subdivision scheme with Laurent polynomial is related to the scheme with Laurent polynomial by the following theorem.
Theorem 1 (see ). Let denote a subdivision scheme with Laurent polynomial satisfying (5). Then there exists a subdivision scheme with the property where and . Furthermore, is a uniformly convergent if and only if converges uniformly to zero function for all initial data , in the sense that
The above theorem indicates that for any given scheme , with the mask satisfying (2), we can prove the uniform convergence of by deriving the mask of and computing for , where is the first integer for which . If such an exists, then converges uniformly. Since there are “” rules for computing the values at the next refinement level, so we define the norm where
Theorem 2 (see ). Let be the subdivision scheme with a characteristic -polynomial , . If the subdivision scheme , corresponding to the -polynomial , converges uniformly, then for any initial control polygon .
Corollary 3 (see ). 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 .
The above Corollary 3 indicates that for any given -ary 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 .
Definition 4 (see ). For any subdivision scheme we denote by the corresponding parametric shift and attach the data for , to the parameter values
Theorem 6 (see ). A convergent subdivision scheme that reproduces polynomial (set of polynomials at most degree ) has an approximation order of .
3. Multistep Algorithm
We construct 4-point -ary interpolating subdivision schemes by using three-step algorithm instead of using Lagrange polynomial and wavelets theory, and so forth. These three steps are as follows. (i) Calculate second divided differences.
At each old vertex compute the second divided difference ; that. is is the second divided difference at point and is the second divided difference at point (see Figure 1): where .(ii) Determine the position of vertices by using divided differences.
In -ary subdivision scheme each segment is divided into subsegments at each refinement level. First point is inserted at the position and second point at the position and proceeding in the same way the -th point at the position . By using divided differences and , we calculate the position of -th newly inserted points between two old vertices and by (iii) Computation new vertices.
Finally, we calculate positions of new vertices , ,, by using , ,, , respectively, by where . By solving, above set of equations, we get the position of new vertices , .
A 4-Point Ternary Interpolating Scheme. In ternary subdivision scheme each segment is divided into three subsegments at each refinement level. One point is inserted at the position and another point at the position (see Figure 2). For in (14), we get second divided differences and at points and For in (15), we get By using (17), we get For in (16), we have This implies By using (19), we get Now 4-point ternary scheme can be written as The above scheme was introduced by Deslauriers and Dubuc  in 1989 by using Lagrange interpolant. Later on, this scheme was also reconstructed by  by using wavelet theory.
A 4-Point Quaternary Interpolating Scheme. In quaternary subdivision scheme each segment is divided into four subsegments at each refinement level. First, second and third points are inserted at the positions , , and , respectively (see Figure 3). For in (14)–(16), we get the following 4-point quaternary interpolating scheme of Deslauriers and Dubuc This scheme was also reconstructed by  in 2009.
Remark 7. By substituting in (14)–(16), we get the mask of 4-point -ary interpolating scheme generated by two different frameworks, that is, Lagrange interpolation  and wavelet theory . In this paper, we propose alternative approach completely different from these approaches. In coming section, we discuss two existing schemes also produced by our framework.
3.2. Analysis of Subdivision Schemes
Here we present the analysis of 4-point ternary and quaternary interpolating subdivision schemes. Analysis of other schemes can be done in a similar way.
3.2.1. Analysis of 4-Point Ternary Subdivision Scheme
The Laurent polynomial for the scheme (23) is Using (11) for , and , we get If is the scheme corresponding to , then by (10) Using (9), (26), and (27), we get As we see , then by Theorem 1 the scheme is . Similarly, then by Corollary 3 the scheme is .
Remark 8. Similarly, we can prove that quaternary 4-point subdivision scheme is .
4. Properties of Subdivision Schemes
In this section, we show that how limit curve of 4-point ternary and 4-point quaternary subdivision schemes give response to initial polynomial data. For this we discuss Hölder regularity, degree of polynomial generation, polynomial reproduction and approximation order of the schemes (23) and (24).
4.1. Hölder Regularity
Hölder regularity is an extension of the notion of continuity which gives more information about any scheme. A function is defined to be regular of order and if it is time continuously differentiable and is Lipschitz of order for all and in and some constant .
According to Dyn and Levin  and Rioul , the Hölder regularity of subdivision scheme with symbol can be computed in the following way. Let , without loss of generality we can assume to be the nonzero coefficients of and let , be the matrices with elements Then the Hölder regularity is given by , where is the joint spectral radius of the matrices , , that is, Since is bounded from below by the spectral radii and from above by the norm of the metrics , ,, then
Theorem 9. The Hölder regularity of scheme (23) is .
Proof. The Laurent polynomial (25) of the scheme (23) can be written as where From (31) and (34), , , , , and , thus , and then , , and are the matrices with elements where . This implies From (33) and (37) we have Since the largest eigenvalue and the max-norm of the metrics are , so
Theorem 10. The Hölder regularity of scheme (24) is .
Proof. The Laurent polynomial of scheme (24) can be written as where From (31) and (40), , , , , and , thus , and then , , and are the matrices with elements where . This implies From (33) and (43) we have Thus the largest eigenvalue and the max-norm of the metrics are , so
Remark 11. It is generally observed that as we decrease the arity of the scheme the Hölder exponent increases. For example, Cashman et al.  have proved that the Hölder exponent for binary scheme is 2 while from above theorems we see that Hölder exponents for ternary and quaternary schemes are 1.8173 and 1.7077, respectively. Trivially, the Hölder exponent approaches to 1 for large arity scheme.
4.2. Polynomial Generation
The generation degree of a subdivision scheme is the maximum degree of polynomials that can potentially be generated by the scheme, provided that the initial data is chosen correctly. Suppose is polynomial of degree of initial data and symbol of the scheme is then the limit curve of the refined data at any level is polynomial of degree . So the condition is necessary and sufficient for the scheme being able to generate polynomial of degree .
Theorem 12. The degree of polynomial generation of scheme (23) is .
Proof. Since the Laurent polynomial of the scheme (23) is where then degree of polynomial generation is .
Theorem 13. The degree of polynomial generation of scheme (24) is .
Proof. Since the Laurent polynomial of (24) can be written as where then degree of polynomial generation of scheme is .
4.3. Polynomial Reproduction and Approximation Order
The polynomial reproduction property has its own importance, as the reproduction property of the polynomials up to a certain degree implies that the scheme has approximation order. Polynomial reproduction of degree requires polynomial generation of degree . For this, polynomial reproduction can be made from initial data which has been sampled from some polynomial function. In the view , the polynomial reproduction property of the proposed scheme can be obtained after having the parameterizations given in (12).
Proof. By taking first derivative of (25) and substituting in it, we get This implies that So from (12), the scheme (23) has primal parametrization. For , , and from (25), we get Similarly, for and ( denotes the order of derivative) By (25), we get . Also , which implies that . Similarly for , we can easily show that which completes the proof.
Theorem 15. A 4-point ternary interpolating scheme (23) has an approximation order of .
Proof of the following theorem is similar to the proof of Theorem 14.
Again by Theorem 6, we get the following theorem.
Theorem 17. A 4-point quaternary interpolating scheme (24) has an approximation order of .
Remark 18. The considered schemes (i.e., 3-point ternary and 4-point quaternary) are exactly the same as obtained by using imputation from Lagrange interpolation at four consecutive points ; therefore, 3-point ternary and 4-point quaternary schemes have polynomial reproduction of degree 3 and approximation properties are obvious (said by the referee). These schemes also have been generated by  using wavelet framework and by our algorithm, so by construction these schemes do no inherit these properties and that is only the reason to include the above theorems.
5. Numerical Examples and Conclusion
Six examples are depicted to show the usefulness of 4-point 2-ary, 3-ary, 4-ary, 5-ary, 6-ary, and 7-ary interpolating subdivision schemes at 1st subdivision level in Figure 4. In this figure the control polygons are drawn by dotted lines while the subdivision curves are drawn by solid lines. From Figure 4, it is clear that the initial polygon converges rapidly to limit curve as we increase the arity of the subdivision scheme.
(a) 4-point 2-ary
(b) 4-point 3-ary
(c) 4-point 4-ary
(d) 4-point 5-ary
(e) 4-point 6-ary
(f) 4-point 7-ary
For many subdivision levels with any of these schemes the limit curves of the -ary scheme with large may exhibit sharper singularities at the initial control points compared to the schemes with small (also mentioned by the referee). But if the initial control points come from noisy source, then -ary scheme with large is the better choice. The scheme with small exhibit overfitting (see ). The main purpose to give comparison at first level is to provide the clear visual differences among the refined polygons produced by different schemes.
In this paper, we have presented a multistep algorithm which generates 4-point -ary interpolating subdivision schemes. We have also observed that the 4-point -ary schemes generated by Lagrange polynomials and wavelet theory can also be generated by proposed multistep algorithm. Some significant properties like Hölder regularity, degree of polynomial generation, degree of polynomial reproduction, and approximation order have been also discussed in this paper.
This work was supported by Indigenous Ph.D. Scholarship Scheme of HEC Pakistan. The authors would like to thank the referees for their helpful suggestions and comments which show the way to improve this work.
G. Mustafa and I. P. Ivrissimtzis, “Model selection for the Dubuc-Deslauriers family of subdivision scheme,” in Proceedings of the 14th IMA Conference on Mathematics of Surfaces, University of Birmingham, Birmingham, UK, September2013.View at: Google Scholar
J. M. Lane and R. F. Riesenfeld, “A theorical development for computer generation and display of piecewise polynomial surfaces,” IEEE Transaction on Pattern Analysis and Machine Intelligence, vol. 2, no. 1, pp. 35–46, 1980.View at: Google Scholar
E. Catmull and J. Clark, “Recursively generated B-spline surfaces on arbitrary topological meshes,” Computer Aided Geometric Design, vol. 10, no. 6, pp. 183–188, 1978.View at: Google Scholar
N. Aspert, Non-linear subdivision of univariate signals and discrete surfaces [EPFL Thesis], École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 2003.
M. F. Hassan and N. A. Dodgson, “Ternary and three-point univariate subdivision schemes,” in Curve and Surface Fitting, A. Cohen, J. L. Marrien, and L. L. Schumaker, Eds., pp. 199–208, Nashboro Press, Sant-Malo, France, 2002.View at: Google Scholar
N. Dyn, Tutorials on Multiresolution in Geometric Modelling, Summer School Lecture Notes, Mathematics and Visualization, Springer, Berlin, Germany, 2002.View at: MathSciNet