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ISRN Computational Mathematics
Volume 2012 (2012), Article ID 924839, 10 pages
A New 5-Point Ternary Interpolating Subdivision Scheme and Its Differentiability
Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan
Received 25 July 2012; Accepted 18 September 2012
Academic Editors: L. Hajdu, L. Pan, and Q.-W. Wang
Copyright © 2012 Ghulam Mustafa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A new 5-point ternary interpolating scheme with a shape parameter is introduced. The resulting curve is for a certain range of parameters. The differentiable properties of the proposed scheme to extend its application in the generation of smooth curves are explored. Application of the proposed scheme is given to show its visual smoothness. The scheme is also extended to a 5-point tensor product ternary interpolating scheme, and its numerical examples are also included.
Geometric modeling plays a significant role to cover up the gap between computer and industry. It has a pivotal importance in the fields of aircraft manufacturing, automobile industry, and general product design.
One of the most important tools of computer aided geometric design is “Subdivision.” Subdivision is a well flourished field. It is a process of taking unrefined shape and to polish it up to produce another shape that is more visually tempting. Due to the comprehensibility and simplicity of this method, it is used in the fields of 3D geometrical measurement, computer graphics, computer animation, and computer aided geometric design.
In 1986, Dubuc  presented a interpolation through an iterative scheme. Dyn et al.  introduced a 4-point interpolating subdivision scheme for curve design. Later on, Deslauriers and Dubuc  introduced a symmetric iterative interpolation process. Weissman  also offered a 6-point interpolating scheme in 1990. In 2002, Hassan et al. [5, 6] gave ternary three-point and 4-point interpolatory schemes. Further analysis of ternary three-point univariate scheme was given in technical report by Hassan and Dodgson  in 2004. Dyn  has given the analysis of the convergence and smoothness of interpolating and approximating schemes by Laurent’s polynomial method.
In 2007, Beccari et al.  presented an interpolating 4-point ternary nonstationary scheme with tension control. They also offered a nonstationary uniform tension controlled interpolating 4-point scheme reproducing conics  in 2007. Ko  in his Ph.D. thesis presented a detailed study on subdivision scheme. Zheng et al.  presented the method to find the differentiability of a four-point ternary scheme. Lian  extended 3-point and 5-point interpolating schemes into -ary subdivision scheme for curve design. Conti et al.  derived symmetric subdivision masks of the Hurwitz type to the interpolating scheme masks. In this paper, we present a new 5-point ternary interpolating subdivision scheme with one parameter.
Let , , denote a sequence of points in , , where is a nonnegative integer, then a univariate ternary subdivision scheme which maps coarse points to refined points is defined by where and the set of coefficients is called mask of the scheme. The -transform of the mask of subdivision scheme can be given as which is called the symbol or the Laurent polynomial of the scheme. The Laurent polynomial of a convergent subdivision scheme satisfies The existence of associated subdivision scheme to for the divided differences of the original control polygon and of related Laurent polynomial is assured by the following condition: where By Hassan and Dodgson , continuity of the scheme requires that satisfy (4) and , for the first integer value of .
3. A 5-Point Ternary Interpolating Scheme
In this section, we construct a 5-point ternary interpolating subdivision scheme and give its analysis.
3.1. Construction of the Scheme
Consider the following three recursive relations which refine given the th level polygon to th level polygon We have the following sequence from the above recurrence relation: The Laurent polynomial of this sequence is Let be the divided difference subdivision schemes of corresponding to the Laurent polynomial , then by Hassan and Dodgson , the necessary condition for continuity of scheme is that satisfy (3), which gives for and by (4) Solving the system of liner equation (9) by taking , we get , , , and .
Now we propose the following 5-point ternary interpolating subdivision scheme: where , , , , and .
3.2. Smoothness Analysis Necessary Conditions
By using the matrix formalism of Hassan and Dodgson , we derive necessary conditions for a scheme to be , , and by using the eigenvalues of the subdivision matrices for, mid-point rule and the vertex rule. By (14), we have the following subdivision matrices for mid-point and the vertex rules: where , , , , and . The above subdivision matrices have the following eigenvalues, respectively: Now we find the bounds on the parameter for continuity of the scheme. From , the necessary condition for 5-point scheme to be is that all eigenvalues must be less than 1. This implies that the proposed scheme is continuous for . The necessary condition for continuity is that all eigenvalues except 1 must be less than 1/3. This is true for . The necessary condition for the scheme to be is that all eigenvalues except 1 and 1/3 must be less than 1/9. This gives the combine range for continuity of the proposed scheme.
3.3. Smoothness Analysis Sufficient Conditions
In this section, we calculate sufficient conditions for , , and continuity of our proposed scheme by the Laurent polynomial method.
By substituting values of , , in and then by (5), we get Since for , , then scheme is . Similarly for , , then scheme is , and for , then scheme is .
-continuous scheme must satisfy the condition ; this is true for and , but there is no combine range of for which . Similarly we can prove that for , we have ; therefore the scheme is not -continuous. By summarizing the above discussion, we have the following theorem.
Theorem 1. Let be defined by (14) for , where are given initial control points. The values correspond to , and let be the limit function of this process, then is , , and -continuous for the improved ranges , , and , respectively.
Remark 2. From the Laurent polynomial method, we have the improved ranges for continuity as compared to the ranges derived by using eigenanalysis method.
3.4. Basic Properties of the Scheme
The proposed 5-point scheme has the following properties. (i)It is exact for cubic interpolation. (ii)Its approximation order is four. (iii)Its highest smoothness is achieved at and its Hölder exponent is . (iv)The basic limit function of proposed scheme has support width 7, which implies that it vanishes outside the interval [−7/2, 7/2].
3.5. A Modified 5-Point Ternary Interpolating Scheme
In this section, we present a modified form of the 5-point ternary interpolating scheme which interpolates end-points without using any auxiliary control points in the case of open polygon.
Theorem 3. Let be the values corresponding to for at level and the set of initial control points of initial polygon , then new values at level can be defined recursively as
4. Differentiability of the 5-Point Ternary Scheme
In this section, we find exact expressions for the first and the second derivatives of limit function of the proposed scheme.
Theorem 4. For given initial control points , let be the values defined by the ternary 5-point interpolating subdivision scheme (14) corresponding to and with ; be the corresponding limit function, then for arbitrarily fixed , the limit function has the derivatives
Proof. Let us denote
As the scheme is interpolatory, we have . At each subdivision step, new control points are obtained by using the rule where is vertex subdivision matrix with , , and , , . Matrix has seven eigenvalues , , , , , , and and has seven orthogonal eigenvectors. Let and be the right and left eigenvectors of matrix corresponding to the eigenvalues , , then direct computation leads to
and left eigenvectors are If the limit curve generated by 5-point ternary interpolating subdivision scheme defines a -continuous function for , then necessarily
So we have where , . This implies that where . Since matrix has linearly independent eigenvectors, there exist scalars , such that can be written as This implies that ; therefore can be written as By using (43) in (40), we have This implies that In view of and using (43) in (41) we have By simplification, we get So, we have Since we have By using the above facts in (45) and (49), we get Now by multiplying (43) with the left eigenvector , corresponding to eigenvalue , and in view of we have This implies that So, we have Using the above result in (52), we have Now by (29) and using the value of , we have By simplification, we get Similarly, by multiplying (43) with the left eigenvector , corresponding to , we get This implies that Using the above equation in (53), we have So by (29) and using the value of, , we have After simplification, we get
5. Comparison and Application of the Scheme
We give the comparison of some properties of the proposed 5-point ternary scheme with other existing binary and ternary schemes in Table 1.
In this section, we also display the performance of the proposed schemes (14) and the scheme introduced in Theorem 3, by applying these schemes on open and closed polygons. Figure 1(a) shows initial open polygon and its limit curve. In this figure we also show the initial closed polygon and the results after first and 2nd subdivision level with parametric value .
6. A 5-Point Tensor Product Ternary Scheme
In this section, we present a 5-point tensor product ternary interpolating subdivision scheme. Some numerical examples of this tensor product scheme are also included. By taking the tensor product of scheme (14), we have the following nine rules representing a 5-point tensor product ternary interpolating scheme:
Figure 2 shows how does the proposed 5-point tensor product ternary interpolating scheme work on control polygon with control points , , ? In this figure, solid lines show the faces of initial polygon while dotted lines show the faces of refined polygon.
Figure 3(a) shows the outline of the cup like shape which is considered to be an initial control polygon of proposed scheme. Figures 3(b) and 3(c) show the results of the proposed scheme after first and second subdivision levels. This figure shows that the results of a 5-point tensor product ternary interpolating scheme are visually smooth.
In this paper, we have presented a new 5-point ternary interpolating subdivision scheme with one parameter. The differentiability of the proposed scheme is also calculated. Some important properties of the scheme such as support, approximation order, conditions of -, -, -continuity, and Hölders exponent are derived. A modified 5-point ternary scheme to deal with open polygons is also proposed to enhance the modeling ability of the scheme. Comparison of the proposed scheme with the other existing scheme is given. Some numerical examples are presented to show the visual performance of our scheme. In this paper, we have also introduced a 5-point tensor product interpolating scheme.
This work is supported by the Indigenous Ph.D. Scholarship Scheme of the Higher Education Commission (HEC), Pakistan.
- S. Dubuc, “Interpolation through an iterative scheme,” Journal of Mathematical Analysis and Applications, vol. 114, no. 1, pp. 185–204, 1986.
- N. Dyn, D. Levin, and J. A. Gregory, “A 4-point interpolatory subdivision scheme for curve design,” Computer Aided Geometric Design, vol. 4, no. 4, pp. 257–268, 1987.
- G. Deslauriers and S. Dubuc, “Symmetric iterative interpolation processes,” Constructive Approximation, vol. 5, no. 1, pp. 49–68, 1989.
- A. Weissman, A 6-point interpolatory subdivision scheme for curve design [M.S. thesis], Tel-Aviv University, Tel Aviv, Israel, 1990.
- M. F. Hassan, I. P. Ivrissimitzis, N. A. Dodgson, and M. A. Sabin, “An interpolating 4-point ternary stationary subdivision scheme,” Computer Aided Geometric Design, vol. 19, no. 1, pp. 1–18, 2002.
- M. F. Hassan and N. A. Dodgson, “Ternary three point univariate subdivision scheme,” in Curve and Surface Fitting: Saint-Malo, A. Cohen, J. Laouis Merrien, and L. L. Schumaker, Eds., pp. 199–208, 2002.
- M. F. Hassan and N. A. Dodgson, “Further analysis of ternary three point univariate subdivision scheme,” Tech. Rep. 599, University of Cambridge Computer Laboratory, 2004.
- N. Dyn, Tutorials on Multiresolution in Geometric Modelling, Summer School Lecture Notes Series: Mathematics and Visualization, Springer, Berlin, Germany, 2002.
- C. Beccari, G. Casciola, and L. Romani, “An interpolating 4-point ternary non-stationary subdivision scheme with tension control,” Computer Aided Geometric Design, vol. 24, no. 4, pp. 210–219, 2007.
- C. Beccari, G. Casciola, and L. Romani, “A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics,” Computer Aided Geometric Design, vol. 24, no. 1, pp. 1–9, 2007.
- K. P. Ko, A Study on Subdivision Scheme, Dongseo University, Busan, Republic of Korea, 2007.
- H. Zheng, H. Zhao, Z. Ye, and M. Zhou, “Differentiability of four point ternary subdivision scheme and its application, LEANG,” International Journal of Computer Science, vol. 36, no. 1, pp. 1–4, 2007.
- J.-A. Lian, “On a-ary subdivision for curve design: II. 3-point and 5-point interpolatory schemes,” Application and Applied Mathematics, vol. 3, no. 1, pp. 176–187, 2008.
- C. Conti, L. Gemignani, and L. Romani, “From symmetric subdivision masks of hurwitz type to interpolatory subdivision masks,” Linear Algebra and Its Applications, vol. 431, no. 10, pp. 1971–1987, 2009.
- M. A. Sabin, “Eigenanalysis and artifacts of subdivision curves and surfaces,” in Tutorials on Multiresolution in Geometric Modelling, A. Iske, E. Quak, and M. S. Floater, Eds., chapter 4, pp. 69–92, Springer, Berlin, Germany, 2002.