Abstract
In the field of subdivision, the smoothness increases as the arity of schemes increases. The family of high arity schemes gives high smoothness comparative to low arity schemes. In this paper, we propose a simple and generalized formula for a family of multiparameter quaternary subdivision schemes. The conditions for convergence of subdivision schemes are also presented. Moreover, we derive subdivision schemes after substituting the different values of parameters. We also analyzed the important properties of the proposed family of subdivision schemes. After comparison with existing schemes, we analyze that the proposed family of subdivision schemes gives better smoothness and approximation compared with the existing subdivision schemes.
1. Introduction
Subdivision schemes are the backbone of Computer Aided Geometric Design (CAGD). Subdivision schemes are used for the generation of smooth curves from the initial polygon. If the rules of subdivision schemes are four, then subdivision schemes are called quaternary subdivision schemes.
In 2009, a 4-point quaternary scheme is presented in [1]. The purposed scheme has -continuity. A family of quaternary schemes is presented in [2]. They used the Cox–De Boor recursion formula for the construction of quaternary schemes. In 2013, Ghaffar et al. [3] presented a generalized formula for the generation of 4-point subdivision schemes of binary, ternary, and quaternary subdivision schemes. In the same year, Amat and Liandrat [4] presented a 4-point scheme for the elimination of the Gibbs phenomenon.
In 2018, Pervaz [5] presented a 4-point quaternary scheme. They discuss the shape preserving properties of the subdivision scheme. Ashraf et al. [6, 7] presented and analyzed the geometrical properties of four point interpolating subdivision schemes. Hameed et al. [8] presented a 4-point subdivision scheme for regular curves and surfaces design. Hussain et al. [9] presented a generalized formula for 5-point subdivision schemes of any arity. Khan et al. [10] presented a computational method for the generation of subdivision schemes. Conti and Romani [11] presented an algebraic technique for the generation of -ary subdivision schemes. Romani [12] presented an algorithm for the generation of dual interpolating -ary subdivision schemes. Romani and Viscardi [13] presented a new class of univariate stationary interpolating subdivision schemes of arity . Recently, Mustafa et al. [14] presented a family of integer-point ternary parametric subdivision schemes.
1.1. Our Contributions
In the field of subdivision, as arity increases, the smoothness also increases. The main purpose of this work is to present a simple and generalized formula for derivation of multiparametric quaternary subdivision schemes based on Laurent polynomial. The conditions for the construction of subdivision schemes are also presented. Our schemes give better approximation and smoothness compared to the same type of existing subdivision schemes (see Figures 1 and 2).
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The paper is organized as follows. In Section 2, we present the general formula with different cases of a family of quaternary subdivision schemes. Analysis of the proposed family is presented in Section 3. Section 4 is for the comparison of the proposed family of subdivision schemes with existing subdivision schemes. Conclusions are drawn in Section 5.
2. General Formula for Multiparameter Family of Quaternary Subdivision Schemes
In this section, we present a general formula for the multiparameter family of quaternary approximating subdivision schemes based on Laurent polynomial. The general formula is
The value of controls the complexity and that of controls the parameters in subdivision schemes. By using different values of and , we get the Laurent polynomial of family of -point quaternary parametric subdivision schemes. Here, we will discuss the different cases and conditions for family of quaternary subdivision schemes.
Case 1. By putting and , with , in (1), we can obtain the Laurent polynomial of subdivision schemeThe mask of the scheme corresponding to the Laurent polynomial isThe scheme corresponding to mask (3) is
Case 2. By setting , and with in (2), we get the Laurent polynomial of 3-point schemeThe mask of the scheme corresponding to the Laurent polynomial (5) isThe scheme corresponding to mask (6) is
Case 3. By setting , , and , with in (2), we can obtain the Laurent polynomial of 4-point schemeThe mask of the scheme corresponding to the Laurent polynomial (8) isThe scheme corresponding to mask (9) is
Case 4. By putting , and , with in (2), we can obtain the Laurent polynomial of 4-point schemeThe mask of the scheme corresponding to the Laurent polynomial (11) isThe scheme corresponding to the mask isScheme (13) is the general 4-point quaternary scheme with 4 parameters.
Similarly for different values of and , we get the -point quaternary approximating subdivision schemes having parameters. In Table 1, we present the mask of family members of quaternary schemes for different values of and .
3. Analysis of the Unified Family of Quaternary Curve Subdivision Schemes
This section contains the analysis of important properties of the proposed subdivision schemes. For this, we consider the 4-point scheme. After substituting the values of , , , and in (13), we get a 4-point parametric scheme
The Laurent polynomial corresponding to scheme (14) is
The Laurent polynomial method [15] is used to compute the degree of generation, degree of reproduction, and continuity analysis. Moreover, Rioul’s method [16] is used to compute lower and upper bounds on Hölder regularity of scheme (14). The analysis of other schemes is similar.
Theorem 1. A 4-point quaternary subdivision scheme (14) has cubic reproduction with respect to the dual parametrization for .
Proof. By taking the derivative of (15) with respect to , we getAfter substituting in (15) and (16), we get and . The value of shift parameter . Hence, by [15], the subdivision scheme (14) has dual parametrization. Further, we can easily verify thatHence, by [15], the scheme corresponding to has cubic reproduction with respect to the dual parametrization.
Table 2 summarizes the results of degree of generation, values of parameters, shift parameter, and parametrization of a proposed family of quaternary subdivision schemes. Here, , , , values of , , and parametrization denote the positive integer, degree of generation, parameter values, shift parameter, and parametrization of the scheme, respectively.
Theorem 2. A 4-point quaternary subdivision scheme (14) has continuity for .
Proof. Consider the Laurent polynomialwhere is defined in (15). This impliesAfter simplification, we getLet be the mask of the scheme corresponding to , then we haveThe scheme corresponding to is continuous if ; for this, we have to check thatIf , then . Then, by [15], the scheme corresponding to has continuity, which completes the proof.
Theorem 3. The Hölder regularity of a 4-point scheme (14) is , where is defined as
Proof. The Laurent polynomial (15) can be written aswhereFrom (25), the coefficients of in are , , , and . The number of factors in is . The matrices has order where and 3. The elements of the matrices , and can be derived by , for , = 1, 2, and 3; then, we haveThe eigenvalues of , and are , and , respectively. For bounds on Hölder regularity, we calculate , with denoting the infinity norm, since is bounded from below by the spectral radii and from above by the infinity norm of the matrices . So and . Then by [16], we have . So Hölder regularity of the scheme is computed by , where is defined aswhich completes the proof.
Corollary 1. The 4-point scheme (14) is continuous if and only if , i.e., if and only if .
Theorem 4. The limit stencils providing the evaluations of the basic limit function of the 4-point scheme (14) at integers and half integers are and , respectively.
Proof. The local subdivision matrices for limit stencils of 4-point scheme (14) at integers and half integers are and , respectively, withThe eigenvalues of the matrix are . The eigenvectors of local subdivision matrix corresponding to eigenvalues areThe inverse of isFor the decomposition of matrix , we need , where is the scalar matrix in which eigenvalues are arranged diagonally; therefore, we now compute ; therefore, . This impliesSince the subdivision scheme is dual, after computing the limit stencil at half integers by the local matrix , the limit stencil at integers must be computed asThe matrix of limit stencils at half integers isAfter multiplying the matrix of limit stencil at half integers with local subdivision matrix , we getwithHence, the limit stencils providing the evaluations of the basic limit functions of the 4-point scheme (14) at integers and half integers arerespectively, which completes the proof.
In Figure 3, we present the basic limit functions of the proposed 4-point quaternary approximating subdivision scheme for different values of and show its evaluations at integers and half integers which coincide with the limit stencils computed in Theorem 4.
4. Comparison with Existing Schemes
Here we will present the comparison of our proposed family of quaternary subdivision schemes with existing quaternary subdivision schemes in visual performance. In Figure 1, we present the comparison of proposed 4-point scheme with 4-point scheme presented in [1] and 4-point scheme presented in [4] , respectively. Here, black dotted lines show the initial polygon, red solid lines are the limit curves of 4-point scheme , and blue solid lines are the limit curves of 4-point scheme presented in [1] and 4-point scheme presented in [4]. We see that, our proposed schemes give maximum smoothness and best approximation compared with the schemes presented in [1, 4].
In Figure 2, we present the comparison of proposed 4-point scheme with 4-point scheme presented in [5]. Here, black doted lines show the initial polygon, red solid lines are the limit curve of 4-point scheme , and blue solid lines are the limit curve of 4-point scheme presented in [5]. We see that the approximating scheme presented in [5] gives interpolating behavior, but our proposed schemes give maximum smoothness and best approximation compared with the schemes presented in [5].
5. Conclusions
In this paper, we have presented a general formula for the derivation of multiparametric family of quaternary subdivision schemes. We present the complete analysis of the proposed family of the multiparametric quaternary subdivision schemes. We also present the comparison with exiting quaternary subdivision schemes. The comparison shows that our proposed family gives maximum smoothness compared with existing quaternary subdivision schemes.
Data Availability
The data used to support the findings of the study are available within this paper.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this study.