The Family of Multiparameter Quaternary Subdivision Schemes

In the field of subdivision, the smoothness increases as the arity of schemes increases. /e 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. /e 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.


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]. e purposed scheme has C 3 -continuity. A family of quaternary schemes is presented in [2]. ey 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. ey 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 m-ary subdivision schemes. Romani [12] presented an algorithm for the generation of dual interpolating m-ary subdivision schemes. Romani and Viscardi [13] presented a new class of univariate stationary interpolating subdivision schemes of arity m. Recently, Mustafa et al. [14] presented a family of integer-point ternary parametric subdivision schemes.

Our Contributions.
In the field of subdivision, as arity increases, the smoothness also increases. e main purpose of this work is to present a simple and generalized formula for derivation of multiparametric quaternary subdivision schemes based on Laurent polynomial. e 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). e 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.

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. e general formula is λ l,q (z) � 1 + z + z 2 + z 3 l+1 a 0 + a 1 z + a 2 z 2 + · · · + a q z q .
(1) e value of l controls the complexity and that of q controls the parameters in subdivision schemes. By using different values of l and q, we get the Laurent polynomial of family of (l + 1)-point quaternary (q + 1) parametric subdivision schemes. Here, we will discuss the different cases and conditions for family of quaternary subdivision schemes.
Similarly for different values of l and q, we get the l + 1-point quaternary approximating subdivision schemes having (q + 1) parameters. In Table 1, we present the mask of family members of quaternary schemes for different values of l and q.

Analysis of the Unified Family of Quaternary Curve Subdivision Schemes
is 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 a 0 � w, a 1 � 1/128 − w, a 2 � 1/128 − w, and a 3 � w in (13), we get a 4-point parametric scheme e Laurent polynomial corresponding to scheme (14) is e 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). e analysis of other schemes is similar. (14) has cubic reproduction with respect to the dual parametrization for w � − (21/1024).
Proof. e Laurent polynomial (15) can be written as (25) 256w. e number of factors in λ(z) is k � 4. e matrices B n has order 3 × 3 where n � 0, 1, 2 and 3. e elements of the matrices B 0 , B 1 , B 2 , and B 3 can be derived by (B n ) ij � b (3+n)+i− 4j , for i, � 1, 2, and 3; then, we have denoting the infinity norm, since μ is bounded from below by the spectral radii and from above by the infinity norm of the matrices B 0 , B 1 , B 2 en by [16], we have μ � max(|2 − 256w|, |256w|). So Hölder regularity of the scheme S is computed by r � 4 − log 4 (μ), where μ is defined as which completes the proof.  Proof. e local subdivision matrices for limit stencils of 4point scheme (14) at integers and half integers are P k+1 I � S I P k I and P k+1 I/2 � S I/2 P k I/2 , respectively, with For the decomposition of matrix S I/2 , we need Δ I/2 , where Δ I/2 is the scalar matrix in which eigenvalues are arranged diagonally; therefore, we now compute lim k⟶∞ Δ k respectively, which completes the proof.

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In Figure 3, we present the basic limit functions of the proposed 4-point quaternary approximating subdivision scheme for different values of w and show its evaluations at integers and half integers which coincide with the limit stencils computed in eorem 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 λ 3,3 with 4-point scheme α 4 4 presented in [1] ((a), (b)&(c)) and 4-point scheme presented in [4] ((d), (e)&(f)), respectively. Here, black dotted lines show the initial polygon, red solid lines are the limit curves of 4-point scheme λ 3,3 , and blue solid lines are the limit curves of 4-point scheme α 4 4 presented in [1] and 4point scheme presented in [4]. We see that, our proposed schemes λ 3,3 give maximum smoothness and best approximation compared with the schemes presented in [1,4].
In Figure 2, we present the comparison of proposed 4point scheme λ 3,3 with 4-point scheme KP presented in [5]. Here, black doted lines show the initial polygon, red solid lines are the limit curve of 4-point scheme λ 3,3 , and blue solid lines are the limit curve of 4-point scheme KP presented in [5]. We see that the approximating scheme KP presented in [5] gives interpolating behavior, but our proposed schemes λ 3,3 give maximum smoothness and best approximation compared with the schemes presented in [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. e comparison shows that our proposed family gives maximum smoothness compared with existing quaternary subdivision schemes.

Data Availability
e data used to support the findings of the study are available within this paper.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this study.  (14) for different values of parameter w. e red lines denote the initial polyline, and the blue, black, and green lines represent the basic limit functions produced with w � − 1/256, 0, 1/256 respectively. e circles denote the evaluations of the basic limit function at integers, and the asterisks denote the evaluations of the basic limit function at half integers.