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Univariate approximating schemes and their non-tensor product generalization

Ghulam Mustafa
/ Robina Bashir
Published Online: 2018-12-31 | DOI: https://doi.org/10.1515/math-2018-0126

Abstract

This article deals with univariate binary approximating subdivision schemes and their generalization to non-tensor product bivariate subdivision schemes. The two algorithms are presented with one tension and two integer parameters which generate families of univariate and bivariate schemes. The tension parameter controls the shape of the limit curve and surface while integer parameters identify the members of the family. It is demonstrated that the proposed schemes preserve monotonicity of initial data. Moreover, continuity, polynomial reproduction and generation of the schemes are also discussed. Comparison with existing schemes is also given.

MSC 2010: 65D17; 65D07; 65D05

1 Introduction

One of the important areas of study in Computer Aided Geometric Design is subdivision. Subdivision schemes have become very important for providing smooth curves and surfaces through an iterative process from a finite set of control points. At each step of iteration, a new set of points is created from the old points. In general, approximating subdivision schemes produce smoother curves and surfaces as compared to interpolating subdivision schemes.

Approximating schemes were first developed by Rham [1]. A famous corner cutting linear approximation scheme was introduced by Chaikin [2], which can generate the piecewise continuous C1 limiting curves. Consequent to this, a lot of work has been done by different authors in the area of binary approximating subdivision schemes. Mustafa et al. [3] presented the m-point binary approximating subdivision scheme. Zheng et al. [4] introduced a general formula to generate a family of integer-point binary approximating sub-division schemes with a parameter. Mustafa et al. [5] presented a family of (2n−1)-point binary approximating subdivision schemes with free parameters for describing curves. Khan and Mustafa [6] introduced a new approach to construct a non-tensor product C1 subdivision scheme for quadrilateral meshes. Zheng et al. [7] devised a multi-parameter method which generates a class of existing binary subdivision schemes. By using their method continuity of existing schemes can be increased up to Ck+n by multiplying the factor ${\left(\frac{1+z}{2}\right)}^{k}$with the symbol of the existing scheme.

Lane and Riesenfeld [8] then presented a unified framework to represent the uniform B-spline curves and their tensor product extensions by a subdivision process. This framework consists of two stages, the first

stage doubles the control point by taking each point twice and the second stage is the midpoint averaging of these points.

Cashman et al. [9] presented the generalized Lane-Riesenfeld algorithm with 4-point variant. A subdivision step T is therefore

$T=SkR,$

where R is refine stage and S is smoothing stage.

Ashraf et al. [10] applied a six point variant on the Lane-Riesenfeld algorithm to generate a family of subdivision schemes by defining

$Qq=SqmWq,$

where Wq is refine stage and Sq is smoothing stage.

Hormann and Sabin [11] proposed a family of subdivision schemes with symbol ak(z) by convolution of uniform B-spline with kernel given by

$ak(z)=2σ(z)kKk(z),$

where σ(z) is a smoothing operator of the B-spline and Kk(z) is a convolution of the order-k B-spline with the kernel.

Conti and Romani [12] proposed a strategy for constructing dual m-ary approximating subdivision schemes of de Rham-type, starting from two primal schemes of arity m and 2 respectively. Symbol of their scheme is

$c(z)=aodd(z)b(z),$

where aodd(z) is the odd sub-symbol of a primal binary scheme and b(z) is the symbol of a primal m-ary scheme. Mustafa et al. [13] presented an algorithm that generates a family of binary univariate dual and primal approximating subdivision schemes, starting with two binary schemes, defined as

$Pl(z)=(meven(z))ln(z),$

where meven(z) is the even sub-symbol of [14] and n(z) is the symbol of [11]. Romani [15] introduced an algorithm which generates the univariate and bivariate non-tensor product subdivision schemes with tension parameter. The symbol of the scheme is defined as

$an,w(z)=(s(z))nrn,w(z),$

where $s\left(z\right)=\frac{1+z}{2}$is smoothing stage while rn,w(z) is refine stage.

1.1 Motivation

All the above algorithms are also called Refine-Smooth algorithms. In these algorithms there is one smoothing operator followed by one refining operator. But in the proposed algorithm there are two smoothing operators followed by one refining operator. That is, we propose an algorithm which uses symbols of well known subdivision schemes, starting with three binary schemes i.e.

$qm,n,μ(z)=(αodd(z))m(βeven(z))nγμ(z),$

where αodd(z) is the extracted odd sub-symbol of [4], βeven(z) is the extracted even sub-symbol of [14] and γμ(z) is the symbol of [5]. The schemes produced by this algorithm are continuous up to Cm+n+2, where m and n are parameters that identify members of the family and play a crucial role in the continuity of the proposed schemes. The parameter μ controls the shape of the limit curves of the schemes. Moreover, this algorithm produces higher order continuous schemes compared with to the existing algorithms. This algorithm can easily be generalized to produce non-tensor product binary approximating schemes for surface generation. Furthermore, monotonicity preservation is also an important shape preserving property of subdivision schemes. In [16, 17, 18, 19, 20, 21] the monotonicity of univariate schemes has been discussed. In this paper, we examine monotonicity preservation of univariate schemes and non-tensor product schemes.

The remainder of this article is organized into 3 sections. In Section 2, firstly we present an algorithm which generates a family of univariate binary approximating subdivision schemes with a tension parameter. Secondly, we discuss the smoothness analysis of univariate schemes and finally we discuss the monotonicity, polynomial generation and reproduction of the schemes. Section 3 extends the ideas presented in Section 2 to design a new family of non-tensor product subdivision schemes for quadrilateral meshes. The smoothness analysis of non-tensor product schemes is also discussed in the same section. In Section 3, we also discuss the monotonicity, polynomial generation and reproduction properties of non-tensor product subdivision schemes. Applications and conclusion are also given in this section.

2 Algorithm for univariate schemes

In this section, we present an algorithm for the construction of a family of binary approximating subdivision schemes.

For this, we consider the odd sub-symbol of cubic B-spline scheme [4]

$αodd(z)=1+z2.$(1)

Similarly, the even sub-symbol of 4-point binary interpolating scheme [14] is

$βeven(z)=1+z2−18z2+108z−18.$(2)

The symbol of the three point scheme [5] is given by

$γμ(z)=1+z238μz2+(2−16μ)z+8μ.$(3)

Let us denote the family of the binary approximating subdivision scheme by Pqm,n,μ, where the general member of the proposed family has the symbol of the form

$qm,n,μ(z)=(αodd(z))m(βeven(z))nγμ(z).$(4)

Substituting (1), (2) and (3) in (4), we get the symbol of the scheme Pqm,n,μ

$qm,n,μ(z)=1+z2m+n+3−18z2+108z−18n8μz2+(2−16μ)z+8μ,$(5)

where m and n are non-negative integers. As it is apparent that the symbol of the scheme Pqm,n,μ is dependent on the parameter μ and on two other parameters m and n. The parameter μ controls the shape of the limit curves of the schemes while m and n characterize the elements of the scheme Pqm,n,μ.

2.1 Smoothness analysis of univariate schemes

In this section, we discuss the continuity and Hölder continuity of the schemes. We use the theory of generating function [22] for continuity and Rioul’s [23] method for Hölder continuity. In the following theorem, we examine the convergence and smoothness of the scheme Pqm,0,μ.

Theorem 2.1

The scheme Pqm,0,μ is Cm+2 for μ ϵ (0, 0.125).

Proof. Symbol of the scheme Pqm,0,μ is given by

$qm,0,μ(z)=1+z2ma(z),$(6)

where

$a(z)=1+z23b(z),$(7)

and

$b(z)=8μz2+(2−16μ)z+8μ.$

Let Sb be the scheme corresponding to the symbol b(z). Since

$12Sb∞=max12∑j∈Z|b2j|,12∑j∈Z|b2j+1|,$

then for μ ϵ (0, 0.125), we have

$12Sb∞=max8μ2+8μ2,2−16μ2<1.$

Hence Sb is contractive. Therefore, by Corollary 4.11 of [22], the scheme Sa is C2 for μ ϵ (0, 0.125). So by (6) scheme Pqm,0,μ is Cm+2 for μ ϵ (0, 0.125).

Similarly, we can easily find out the continuity of other schemes Pqm,n,μ by taking into account the same formalism. The order of continuity of some proposed univariate subdivision schemes Pqm,0,μ , Pqm,1,μ , Pqm,2,μ and Pqm,3,μ for certain ranges of parameter is shown in Table 1. Hölder continuity is an extension to the notion of continuity. In the following theorem, we compute the Hölder continuity of the scheme Pqm,0,μ.

Table 1

The order of continuity O(C) of proposed binary approximating schemes for certain ranges of parameter.

Theorem 2.2

The Hölder continuity of the scheme Pqm,0,μ is 3.

Proof. From (7), let b0 = 8μ, b1 = 2 − 16μ, b2 = 8μ, then M0, M1 are the matrices with elements

$(M0)ij=b2+i−2j,(M1)ij=b2+i−2j+1,$

where i, j = 1, 2. This implies

$M0=2−16μ08μ8μ,M1=8μ8μ02−16μ.$(8)

From (8) and [23], the spectral radius λ of the metrics M0 and M1 can be express as follows

$max2−16μ,2−16μ≤λ≤max2−16μ,2−16μ.$

Since the largest eigenvalue and the max-norm of the metrics is 1 for μ = 0.0625, where μ ϵ (0, 0.125), so the Hölder continuity h = 2 − log2(1) = 3. So by (6), the Hölder continuity of the scheme Pqm,0,μ is Cm+3.

Similarly, we can compute the Hölder continuity of other members of the family. If the largest eigenvalue and the max-norm of the metrics are not equal then we calculate the lower and upper bounds of the Hölder continuity. The lower bound of the Hölder continuity is h = 2 − log2(kbkl)/l for some integer l and the upper bound of the Hölder continuity is h = 2 − log2(λ). It is clear from Table 2 that as we increase n, the level of continuity and the Hölder continuity of the schemes Pqm,n,μ increase.

Table 2

Continuity of some members of the family of schemes

2.2 Response of univariate schemes to polynomial and monotone data

In this section, we examine the response of schemes to polynomial data by taking into account the polynomial generation and reproduction. We also examine the behaviour of the schemes for monotone data. We use the techniques developed in [15] to discuss polynomial generation and polynomial reproduction.

2.2.1 Polynomial generation

The polynomial generation of degree d is the ability of subdivision scheme to generate the full space of polynomials up to degree d denoted by πd. The generation degree of a subdivision scheme is the maximum degree of a polynomial that can potentially be generated by the scheme.

Theorem 2.3

The subdivision scheme Pqm,n,μ generates πm+n+2 for all m, n ϵ N. Moreover, if $\mu =\frac{1}{16},$ then Pqm,n,μ generates πm+n+4.

Proof. Since conditions

$qm,n,μ(1)=2,qm,n,μ(−1)=0,D(k)qm,n,μ(−1)=0,k=1,2,…,m+n+2,$

are verified by qm,n,μ(z) for all μ ϵ ℝ and D(k) denotes the kth derivative. Thus, in view of Proposition 2.1 of [15] degree of polynomial generation is m + n + 2 for all μ ϵ ℝ. Moreover, by setting $\mu =\frac{1}{16}$two more terms (1+z) can be factored out from qm,n,μ(z), then we have D(k+1)qm,n,μ(−1) = D(k+2)qm,n,μ(−1) = 0. So the degree of polynomial generation is m + n + 4.

2.2.2 Polynomial reproduction

Polynomial reproduction is an attractive property for a subdivision scheme. For a subdivision scheme to reproduce πd it must be able to generate polynomials of the same degree as the limit functions for some initial data. The degree of polynomial reproduction can never exceed the degree of polynomial generation.

Theorem 2.4

If applying the parameter shift $\tau =\frac{5+m+3n}{2},$then the subdivision scheme Pqm,n,μ reproduces π1 with respect to the parametrization in [15] for all m, n ϵand μ ϵ. Moreover, if $\mu =-\frac{3+m}{32},$then Pqm,n,μ reproduces π3 for all m, n ϵ ℕ.

Proof. Since the condition D(1)qm,n,μ(1) = 5 + m + 3n is verified by the symbol qm,n,μ(z) for all μ ϵ ℝ, so polynomial reproduction of Pqm,n,μ is π1 with the parameter shift $\tau =\frac{5+m+3n}{2}.$We observe that when $\mu =-\frac{3+m}{32},$the following two more conditions

$D(2)qm,n,μ(z)|z=1=2τ(τ−1),D(3)qm,n,μ(z)|z=1=2τ(τ−1)(τ−2),$

are satisfied for all m, n ϵ ℕ. Thus reproduction of Pqm,n,μ is π3.

2.3 Monotonicity preservation

Monotonicity preserving plays a key role in the shape preserving properties of subdivision schemes.

Definition 2.1

[18] Univariate data (xi , fi), i = 0, 1, 2, . . . , n is monotonically increasing if fi < fi+1 8 i = 0, 1, 2, . . . , n and the derivative at the data points obey the condition di > 0 ∀ i = 0, 1, 2, . . . , n.

In the following, we examine monotonicity preservation of binary scheme Pq1,0,μ.

Theorem 2.5

Let $\left\{{f}_{i}^{0}{\right\}}_{i\in \mathbb{Z}}$satisfy

$…f−10

Denote

$dik=fi+1k−fik,rik=di+1kdik,Rk=maxi{rik,1rik},k≥0,k∈Z,i∈Z.$

Furthermore, let 0.1 ≤ μ ≤ 0.9 and $\xi =-\frac{1}{\mu },\xi \in \mathbb{R}.If\frac{1}{\xi }\le {R}^{0}\le \xi ,\left\{{f}_{i}^{k}\right\}$is defined by the subdivision scheme Pq1,0,μ , then

$dik>0,1ξ≤Rk≤ξ,k≥0,k∈Z,i∈Z.$(9)

Proof. We use mathematical induction to prove (9). When k = 0,

${d}_{i}^{0}={f}_{i+1}^{0}-{f}_{i}^{0}>0,\frac{1}{\xi }\le {R}^{0}\le \xi ,$then (9) is true.

Suppose that (9) holds for $k,{d}_{i}^{k}={f}_{i+1}^{k}-{f}_{i}^{k}>0,\frac{1}{\xi }\le {R}^{k}\le \xi .$Next we will prove that (9) holds for k + 1. Consider

$d2ik+1=18+12μdik+38−μdi+1k+12μdi+2k.$

This implies

$d2ik+1=dik18+12μ+38−μrik+12μri+1krik.$

This further implies

$d2ik+1≥dik18+12μ+38−μ1ξ+12μ1ξ2.$

We know that ${d}_{i}^{k}>0$and

$dik18+12μ+38−μ1ξ+12μ1ξ2>0,for0.1≤μ≤0.9andξ=−1μ.$

This implies that ${d}_{2i}^{k+1}>0.$Similarly, we see that ${d}_{2i+1}^{k+1}>0$for 0.1 ≤ μ ≤ 0.9 and $\xi =-\frac{1}{\mu }.$Now we prove that $\frac{1}{\xi }\le {R}^{k+1}\le \xi .$First we show that ${r}_{2i}^{k+1}-\xi \le 0$.Since

$r2ik+1=d2i+1kd2ik=12μdik+38−μdi+1k+18+12μdi+2k18+12μdik+38−μdi+1k+12μdi+2k,$

then

$r2ik+1−ξ≤dik38+12μξ2+14−32μξ+32μ−38+−12μ1ξdi+1k18+μξ+38μ.$

The denominator and numerator of the right hand side of the above expression are less than and greater than zero respectively for 0.1 ≤ μ ≤ 0.9 and $\xi =-\frac{1}{\mu }.$

This implies that

$r2ik+1−ξ≤0.$

It implies further that ${r}_{2i}^{k+1}\le \xi .$Now we show that $\frac{1}{{r}_{3i}^{k+1}}-\xi <0.$

$1r2ik+1=d2ikd2i+1k=18+12μdik+38−μdi+1k+12μdi+2k12μdik+38−μdi+1k+18+12μdi+2k.$

This implies

$1r2ik+1−ξ≤dik{12μξ2−14ξ−(12μ+14)}di+1k{(18+μ)ξ+(38−μ)}.$

The denominator and numerator of the right hand side of the above expression are less than and greater than zero respectively for 0.1 ≤ μ ≤ 0.9 and $\xi =-\frac{1}{\mu }.$

This implies that

$1r2ik+1−ξ≤0.$

It implies further that $\frac{1}{{r}_{2i}^{k+1}}\le \xi .$In the same way, we can get ${r}_{2i+1}^{k+1}\le \xi$and $\frac{1}{{r}_{2i+1}^{k+1}}\le \xi .$So Rk+1ξ. Since ${R}^{k+1}=\underset{i}{max}\left\{{r}_{i}^{k},\frac{1}{{r}_{i}^{k}}\right\},$it is obvious that ${R}^{k+1}\ge \frac{1}{\xi }.$This completes the proof.

2.4 Numerical experiments of univariate schemes

In this section, we present the performance, geometrical behaviour and effect of a parameter on the limit curves of the schemes. We also present the response of the limit curves produced by the schemes towards the initial data.

Figure 1 is produced by using the monotone data set given in Table 3. Figures 1(a)-1(d) are monotone curves obtained by the schemes Pq1,0,μ , Pq1,1,μ , Pq1,2,μ and Pq1,3,μ respectively.

Figure 1

The curves (a), (b), (c) and (d) are generated by the schemes Pq1,0,μ , Pq1,1,μ , Pq1,2,μ and Pq1,3,μ respectively, using the monotone data set given below.

Table 3

Monotone data set [24].

The Figure 2, 3, 4, 5 shows a comparison of proposed schemes with existing schemes [15]. Dashed dotted lines indicate the initial polygon. Solid lines show the most expanded curves and dashed lines show the most shrinked curves. Arrows show the distance between most expanded and most shrinked curves. Figures 2(a)-2(c) show that the most expanded and most shrinked curves are obtained by the schemes Pq2,0,μ , Pq1,2,μ and Pq2,2,μ at different parametric values and Figure 2(d) shows the behaviour of existing scheme of [15]. We can see that the Figures 3(a)-3(b) represent the interpolating behaviour of proposed scheme Pq2,0,μ , Pq1,2,μ respectively. Figure 3(c) shows the non-interpolating behaviour of [15] at any parametric value. The proposed schemes Pq2,0,μ and Pq1,2,μ show the approximating behaviour as well as the interpolating behaviour at different parametric values.

Figure 2

Most expanded and most shrinked curves: The curves (a), (b), (c) and (d) are generated by the schemes Pq2,0,μ , Pq1,2,μ, Pq2,2,μ and [15] respectively.

Figure 3

Interpolating behaviour: The curves (a) , (b) and (c) are generated by the schemes Pq2,0,μ , Pq1,2,μ and [15] respectively.

Figure 4

Most expanded and most shrinked curves: The curves (a), (b) and (c) are generated by the schemes Pq1,0,μ , Pq1,1,μ and [15] respectively.

Figure 5

Interpolating behaviour: The curves (a), (b) and (c) are generated by the schemes Pq1,0,μ , Pq1,1,μ and [15] respectively.

The Figures 4(a)-4(c) show the most expanded and most shrinked curves that are generated by the schemes Pq1,0,μ , Pq1,1,μ and [15] at different parametric values respectively. The limit curves presented in Figures 5(a)-5(c) show the interpolating behaviour by of schemes Pq1,0,μ , Pq1,1,μ and [15] respectively. The schemes Pq1,0,μ and Pq1,1,μ have both approximating and interpolating behaviour while the scheme in [15] gives only interpolating behaviour.

3 Algorithm for non-tensor product schemes

By generalizing the algorithm as devised in Section 2, we get a family of non-tensor product approximating schemes with tension parameter μ for quadrilateral meshes. Let Pqm,n,μ be the family of non-tensor product bivariate subdivision schemes then we propose the symbol of this family as

$qm,n,μ(z1,z2)=(αodd(z1))m(βeven(z2))nγμ(z1)γμ(z2).$(10)

By substituting m = 1 and n = 0 in (10), we get symbol of the scheme Pq1,0,μ as follows:

$q1,0,μ(z1,z2)=1+z1241+z2238μz12+(2−16μ)z1+8μ×8μz22+(2−16μ)z2+8μ.$(11)

The bivariate subdivision scheme Pq1,0,μ has the mask

$q1,1(z1,z2)=−132μ2−1128μ+14μ2364μ+58μ2−14μ2+33128μ−1916μ2+1332μ−1916μ2+1332μ−14μ2+33128μ364μ+58μ2−1128μ+14μ2−132μ2−116μ2−1128μ−1512+364μ+12μ23256+14μ+54μ2−12μ2+33512+2964μ3364μ+13128−198μ23364μ+13128−198μ2−12μ2+33512+2964μ3256+14μ+54μ2−1512+364μ+12μ2−116μ2−1128μ132μ2−132μ−1128+33128μ−14μ2364+3764μ−58μ214μ2+33128−65128μ−5132μ+1332+1916μ2−5132μ+1332+1916μ214μ2+33128−65128μ364+3764μ−58μ2−1128+33128μ−14μ2132μ2−132μ18μ2−364μ−3256+1332μ−μ29128+34μ−52μ2μ2+99256−4532μ−10932μ+3964+194μ2−10932μ+3964+194μ2μ2+99256−4532μ9128+34μ−52μ2−3256+1332μ−μ218μ2−364μ132μ2−132μ−1128+33128μ−14μ2364+3764μ−58μ214μ2+33128−65128μ−5132μ+1332+1916μ2−5132μ+1332+1916μ214μ2+33128−65128μ364+3764μ−58μ2−1128+33128μ−14μ2132μ2−132μ−116μ2−1128μ−1512+364μ+12μ23256+14μ+54μ2−12μ2+33512+2964μ3364μ+13128−198μ23364μ+13128−198μ2−12μ2+33512+2964μ3256+14μ+54μ2−1512+364μ+12μ2−116μ2−1128μ−132μ2−1128μ+14μ2364μ+58μ2−14μ2+33128μ−1916μ2+1332μ−1916μ2+1332μ−14μ2+33128μ364μ+58μ2−1128μ+14μ2−132μ2.$(12)

3.1 Smoothness analysis of bivariate proposed schemes

Here, we use the theory of generating function [22] to derive continuity of non-tensor product schemes.

Theorem 3.1

If μ ϵ (−0.2215, 0.4785) then the subdivision scheme Pq1,0,μ converges to a continuous surface when starting from any regular quadrilateral mesh. Moreover, if μ ϵ (−0.05178, 0.3017) and μ ϵ (−0.0517, 0.25), then the limit surfaces generated by scheme Pq1,0,μ have C1 and C2-continuity respectively.

Proof. From (11), we have

$b1,0,μ(z1,z2)=8μz12+(2−16μ)z1+8μ8μz22+(2−16μ)z2+8μ.$

In view of [22](Theorem 4.30), we can determine the range of the parameter μ which guarantees the convergence of the scheme Pq1,0,μ by checking the contractivity of the scheme. Since the scheme with symbol $\frac{1}{2}{\left(\frac{1+{z}_{1}}{2}\right)}^{3}{\left(\frac{1+{z}_{2}}{2}\right)}^{3}{b}_{1,0,\mu }\left({z}_{1},{z}_{2}\right),\frac{1}{2}{\left(\frac{1+{z}_{1}}{2}\right)}^{4}{\left(\frac{1+{z}_{2}}{2}\right)}^{2}{b}_{1,0,\mu }\left({z}_{1},{z}_{2}\right)$is contractive for μ ϵ (−0.2215, 0.4785) and then scheme Pq1,0,μ is convergent for μ ϵ (−0.2215, 0.4785). In the same way, the scheme with symbol $\frac{1}{2}{\left(\frac{1+{z}_{1}}{2}\right)}^{2}{\left(\frac{1+{z}_{2}}{2}\right)}^{3}{b}_{1,0,\mu }\left({z}_{1},{z}_{2}\right),\frac{1}{2}{\left(\frac{1+{z}_{1}}{2}\right)}^{3}{\left(\frac{1+{z}_{2}}{2}\right)}^{2}{b}_{1,0,\mu }\left({z}_{1},{z}_{2}\right),\frac{1}{2}{\left(\frac{1+{z}_{1}}{2}\right)}^{4}$

$\left(\frac{1+{z}_{2}}{2}\right){b}_{1,0,\mu }\left({z}_{1},{z}_{2}\right)$is contractive for μ ϵ (−0.05178, 0.3017) therefore the scheme Pq1,0,μ is C1-continuous.

Again since, the scheme with symbol $\frac{1}{2}\left(\frac{1+{z}_{1}}{2}\right){\left(\frac{1+{z}_{2}}{2}\right)}^{3}$

${b}_{1,0,\mu }\left({z}_{1},{z}_{2}\right),\frac{1}{2}{\left(\frac{1+{z}_{1}}{2}\right)}^{2}{\left(\frac{1+{z}_{2}}{2}\right)}^{2}{b}_{1,0,\mu }\left({z}_{1},{z}_{2}\right),\frac{1}{2}{\left(\frac{1+{z}_{1}}{2}\right)}^{3}\left(\frac{1+{z}_{2}}{2}\right){b}_{1,0,\mu }\left({z}_{1},{z}_{2}\right),\frac{1}{2}{\left(\frac{1+{z}_{1}}{2}\right)}^{4}$

b1,0,μ(z1, z2) is contractive for μ ϵ (−0.0517, 0.25), so the scheme Pq1,0,μ is C-continuous.

In Table 4, we compare the continuity of proposed non-tensor product schemes with some existing binary non-tensor product schemes. It is observed that the continuity of proposed schemes is better than the continuity of existing schemes.

Table 4

The order of continuity O(C) of proposed non-tensor product schemes with some existing non-tensor product schemes.

3.2 Response of non-tensor product schemes to polynomial and monotone data

In this section, we investigate the capability of the non-tensor product approximating subdivision schemes Pq1,0,μ and Pq1,1,μ of generating and reproducing polynomials as well as monotonicity preservation of the data.

Theorem 3.2

The subdivision scheme Pq1,0,μ generates π2 for all μ ϵand generates π4 for $\mu =\frac{1}{16}.$

Proof. Let w1 = (1, −1), w2 = (−1, 1), w3 = (−1, −1) and let Dj with j ϵ2, denote a directional derivative. Since q1,0,μ(1, 1) = 4 and

$D(1,0)q1,0,μ(w1)=0,D(1,0)q1,0,μ(w2)=0,D(1,0)q1,0,μ(w3)=0,D(0,1)q1,0,μ(w1)=0,D(0,1)q1,0,μ(w2)=0,D(0,1)q1,0,μ(w3)=0,$

then scheme Pq1,0,μ generates π1 for all μ ϵ ℝ. Again since

$D(1,1)q1,0,μ(w1)=0,D(1,1)q1,0,μ(w2)=0,D(1,1)q1,0,μ(w3)=0,D(2,0)q1,0,μ(w1)=0,D(2,0)q1,0,μ(w2)=0,D(2,0)q1,0,μ(w3)=0,D(0,2)q1,0,μ(w1)=0,D(0,2)q1,0,μ(w2)=0,D(0,2)q1,0,μ(w3)=0,$

then the scheme Pq1,0,μ generates π2 for all μ ϵ ℝ. Further

$D(2,1)q1,0,μ(w1)=0,D(2,1)q1,0,μ(w2)=0,D(2,1)q1,0,μ(w3)=0,D(1,2)q1,0,μ(w1)=0,D(1,2)q1,0,μ(w2)=0,D(1,2)q1,0,μ(w3)=0,D(3,0)q1,0,μ(w1)=0,D(3,0)q1,0,μ(w2)=0,D(3,0)q1,0,μ(w3)=0,D(0,3)q1,0,μ(w1)=48μ−3,D(0,3)q1,0,μ(w2)=0,D(0,3)q1,0,μ(w3)=0,$

so the scheme Pq1,0,μ generates π3 for $\mu =\frac{1}{16}.$Further more

$D(2,2)q1,0,μ(w1)=0,D(2,2)q1,0,μ(w2)=0,D(2,2)q1,0,μ(w3)=0,D(3,1)q1,0,μ(w1)=0,D(3,1)q1,0,μ(w2)=0,D(3,1)q1,0,μ(w3)=0,D(1,3)q1,0,μ(w1)=144μ−9,D(1,3)q1,0,μ(w2)=0,D(1,3)q1,0,μ(w3)=0,D(4,0)q1,0,μ(w1)=0,D(4,0)q1,0,μ(w2)=96μ−6,D(4,0)q1,0,μ(w3)=0,D(0,4)q1,0,μ(w1)=48μ−3,D(0,4)q1,0,μ(w2)=0,D(0,4)q1,0,μ(w3)=0,$

so the scheme Pq1,0,μ generates π4 for $\mu =\frac{1}{16}.$This completes the proof.

Theorem 3.3

For the parameter shift $\left({\tau }_{1},{\tau }_{2}\right)=\left(\frac{12}{4},\frac{10}{4}\right),$the subdivision scheme Pq1,0,μ reproduces π1 with respect to the parametrization defined in [15] for all μ ϵ ℝ.

Proof. Let Dj with j ϵ2, denote a directional derivative. Since the symbol q1,0,μ(z1, z2) satisfies the conditions in Theorem 3.2. Since q1,0,μ(1, 1) = 4 and

$D(1,0)q1,0,μ(1,1)−4τ1=0,D(0,1)q1,0,μ(1,1)−4τ2=0,$

then the scheme Pq1,0,μ produced π1 for all μ ϵ ℝ.

Similarly, we can prove the following theorems.

Theorem 3.4

The subdivision scheme Pq1,1,μ generates π3 for all μ ϵand generates π4 for $\mu =\frac{1}{16}.$

Theorem 3.5

If applying the parameteric shift (τ1, τ2) = (3, 4), the subdivision scheme Pq1,1,μ reproduces π1 with respect to the parametrization in [15] for all μ ϵ ℝ.

Now, we examine monotonicity preservation of the binary non-tensor product approximating subdivision scheme Pq1,0,μ.

Definition 3.1

[18] Bivariate data (xi , yj , fi,j), i = 0, 1, 2, . . . , n and j = 0, 1, 2, . . . , m, where x1 < x2 < . . . < xn and y1 < y2 < . . . < ym are said to be monotonically increasing if fi,j < fi+1,j and fi,j < fi,j+1 8 i = 0, 1, 2, . . . , n and j = 0, 1, 2, . . . , m, if the derivative at the data points obey the condition di,j > 0 8 i = 0, 1, 2, . . . , n and j = 0, 1, 2, . . . , m.

Theorem 3.6

Suppose that the initial data $\left\{{f}_{i,j}^{0}\right\}=\left({x}_{i}^{0},{y}_{j}^{0},{f}_{i,j}^{0}\right)$are strictly monotonically increasing for all i, j ϵ ℤ.

Denote

$di,jk=fi+1,j+1k−fi+1,jk−fi,j+1k+fi,jk,yi,j+tk=di+1,j+tkdi,j+tk,yi+1,j+tk=di+2t,j+t+1kdi+1,j+tk,Yi,j+tk=maxi,j{yi,j+tk,1yi,j+tk},Yi+1,j+tk=maxi,j{yi+1,j+tk,1yi+1,j+tk},$

$wheret=0,1andk≥0,k∈Z,i,j∈Z.$

Furthermore, let 0.1 ≤ μ ≤ 0.9 and $\delta =-\frac{1}{\mu },\delta \in \mathbb{R}.If\frac{1}{\delta }\le {Y}_{i,j+t}^{0},{Y}_{i+1,j+t}^{0}\le \delta ,\left\{{f}_{i,j}^{k}\right\}$is defined by the subdivision scheme Pq1,0,μ , then

$di,jk>0,1δ≤Yi,j+tk,Yi+1,j+tk≤δ,k≥0,k∈Z,i,j∈Z.$(13)

Proof. We use mathematical induction to prove (13). When $k=0,{d}_{i,j}^{0}>0,\frac{1}{\delta }\le {Y}_{i,j+t}^{0},{Y}_{i+1,j+t}^{0}\le \delta ,$then (13) is true.

Suppose that (13) holds for k i.e. ${d}_{i,j}^{k}>0,\frac{1}{\delta }\le {Y}_{i,j+t}^{k},{Y}_{i+1,j+t}^{k}\le \delta .$Next we will prove that (13) holds for k + 1.

First we show that ${d}_{2i,2j}^{k+1}>0.$Consider

$d2i,2jk+1=f2i+1,2j+1k+1−f2i+1,2jk+1−f2i,2j+1k+1+f2i,2jk+1.$

After some simplification and substituting $\delta =-\frac{1}{\mu },$we get

$d2i,2jk+1=di,j+3k−272μ11+1538μ10−36316μ9+78132μ8−83132μ7+88132μ6−71164μ5+37764μ4−10532μ3+2μ2−1932μ+532.$

We know that ${d}_{i,j+3}^{k}>0$and

$−272μ11+1538μ10−36316μ9+78132μ8−83132μ7+88132μ6−71164μ5+37764μ4−10532μ3+2μ2−1932μ+532>0.$

This implies that ${d}_{2i}^{k+1}>0.$Similarly, we see that ${d}_{2i+1,2j}^{k+1}>0,{d}_{2i,2j+1}^{k+1}>0\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}{d}_{2i+1,2j+1}^{k+1}>0\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}0.1\le \mu \le 0.9$and $\delta =-\frac{1}{\mu }.$

Now we prove that $\frac{1}{\delta }\le {Y}_{i,j+t}^{k},{Y}_{i+1,j+t}^{k}\le \delta .$First we show that ${y}_{2i,2j}^{k+1}-\delta \le 0.$

For this, consider

$y2i,2jk+1−δ=d2i+1,2jk+1d2i,2jk+1−δ.$

After some simplification and substituting $\delta =-\frac{1}{\mu },$we get

$y2i,2jk+1−δ≤ψ1ψ2,$

where

$ψ1=−98μ3+54932μ2−2878μ+288564+516μ8−5332μ7+28532μ6−34116μ5+154332μ4−82116μ3+346164μ2−167932μ,$

and

$ψ2=−98μ3+51332μ2−63532μ+161564−532μ7+34μ6−6916μ5+414μ4−78332μ3+85932μ2−174364μ.$

The denominator and numerator of the right hand side of the above expression are less than and greater than zero respectively for 0.1 ≤ μ ≤ 0.9. This implies that

$1y2i,2jk+1−δ≤0.$

Further this implies that ${y}_{2i,2j}^{k+1}\le \delta .$Now we show that $\frac{1}{{y}_{2i,2j}^{k+1}}-\delta <0.$

For this, consider

$1y2i,2jk+1−δ=d2i,2jkd2i,2j+1k−δ.$

After some simplification and substituting $\delta =-\frac{1}{\mu },$ we get

$1y2i,2jk+1−δ≤χ1χ2,$

where

$χ1=−98μ3+54932μ2−2878μ+288564532μ9+2932μ8−194μ7+18916μ6−44916μ5+111132μ4−82116μ3+346164μ2−167932μ,$

and

$χ2=−98μ3+51332μ2−63532μ+161564+532μ8−2932μ7+14732μ6−17716μ5+954μ4−78332μ3+85932μ2−174364μ.$

The denominator and numerator of the right hand side of the above expression are greater than and less than zero respectively for 0.1 ≤ μ ≤ 0.9. This implies that

$1y2i,2jk+1−δ≤0.$

Further this implies that $\frac{1}{{y}_{2i,2j}^{k+1}}\le \delta .$In the same way, we can get ${y}_{2i,2j+1}^{k+1}\le \delta ,\phantom{\rule{thinmathspace}{0ex}}{y}_{2i+1,2j}^{k+1}\le \delta ,\phantom{\rule{thinmathspace}{0ex}}{y}_{2i+1,2j+1}^{k+1}\le \delta ,\phantom{\rule{thinmathspace}{0ex}}\frac{1}{{y}_{2i,2j+1}^{k+1}}\le \delta ,$$\frac{1}{{y}_{2i+1,2j}^{k+1}}\le \delta \phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\frac{1}{{y}_{2i+1,2j+1}^{k+1}}\le \delta .\phantom{\rule{thinmathspace}{0ex}}\text{\hspace{0.17em}So}\phantom{\rule{thinmathspace}{0ex}}{Y}_{i,j+t}^{k},\phantom{\rule{thinmathspace}{0ex}}{Y}_{i+1,j+t}^{k}\le \delta .$ Since ${Y}_{i,j+t}^{k}=\underset{i,j}{max}\left\{{y}_{i,j+t}^{k},\frac{1}{{y}_{i,j+t}^{k}}\right\}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{Y}_{i+1,j+t}^{k}=\underset{i,j}{max}\left\{{y}_{i+1,j+t}^{k},\frac{1}{{y}_{i+1,j+t}^{k}}\right\},$ it is obvious that ${Y}_{i,j+t}^{k},{Y}_{i+1,j+t}^{k}\ge \frac{1}{\delta }.$ This completes the proof.

3.3 Numerical experiments of non-tensor product schemes

In this section, we show the performance, geometrical behaviour and effect of a parameter on the limit surfaces of the schemes Pq1,0,μ and Pq1,1,μ.

The monotone data set given in Table 5 has been used to produce monotone surfaces. Figure 6(a) is the initial mesh of the monotone data. Figure 6(b) is the monotone surface generated by the scheme Pq1,0,μ for μ = 0.5. Figure 7(a) is the initial control mesh while Figures 7(b)-7(d) are the surfaces produced by the proposed scheme Pq1,0,μ at first, second and third subdivision levels with μ = 0.1 respectively. Figure 8(a) is the initial control mesh while Figures 8(b), Figures 8(c), 8(d) are the surfaces produced by the proposed scheme Pq1,1,μ at first, second and third subdivision levels with μ = 0.15 respectively.

Figure 6

(a) Initial monotone data. (b) A monotonicity preserving surface obtained by the proposed scheme Pq1,0,μ.

Figure 7

(a) Control mesh. (b)-(d) Surfaces obtained by the proposed schemes Pq1,0,μ at first, second and third subdivision levels respectively.

Figure 8

(a) Control mesh. (b)-(d) Surfaces obtained by the proposed schemes Pq1,1,μ at first, second and third subdivision levels respectively.

Table 5

Monotone data set [25].

3.4 Conclusion

In this paper, we have proposed two algorithms to generate the families of univariate and bivariate approximating subdivision schemes with one tension and two integer parameters. The integer parameters identify

members of the proposed family. It has been shown that the proposed schemes have higher continuity and Hölder continuity compared with existing schemes. Comparison of the continuity of proposed non-tensor product schemes with some of the existing non-tensor schemes has also been given. It has been demonstrated through several examples that geometrical behaviour of the univariate and bivariate subdivision schemes depends on the tension parameter. Monotonicity preservation of proposed univariate and bivariate schemes has been proved. Moreover, polynomial reproduction and generation of the proposed schemes have also been discussed.

Acknowledgement

This work is supported by the Indigenous Ph.D. Scholarship Scheme of Higher Education Commission (HEC) Pakistan and NRPU No-3183

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Accepted: 2018-11-16

Published Online: 2018-12-31

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1501–1518, ISSN (Online) 2391-5455,

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