Univariate approximating schemes and their non-tensor product generalization

Ghulam Mustafa; Robina Bashir

doi:10.1515/math-2018-0126

Enjoy 40% off

academic books on De Gruyter Brill *

Article Open Access

Univariate approximating schemes and their non-tensor product generalization

Ghulam Mustafa and Robina Bashir

Published/Copyright: December 31, 2018

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

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.

Keywords: Subdivision scheme; continuity; polynomial reproduction; monotonicity; non-tensor product

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 C^k⁺ⁿ by multiplying the factor 1+z2kwith 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 W_q is refine stage and S_q is smoothing stage.

Hormann and Sabin [11] proposed a family of subdivision schemes with symbol a_k(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 K_k(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 a_odd(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 m_even(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(z)=1+z2is smoothing stage while r_n_,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 α_odd(z) is the extracted odd sub-symbol of [4], β_even(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 C^m⁺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]

(1)αodd(z)=1+z2.

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

(2)βeven(z)=1+z2−18z2+108z−18.

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

(3)γμ(z)=1+z238μz2+(2−16μ)z+8μ.

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

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

Substituting (1), (2) and (3) in (4), we get the symbol of the scheme P_{q_m,n,μ}

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

where m and n are non-negative integers. As it is apparent that the symbol of the scheme P_{q_m,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 P_{q_m,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 P_{q_m,0,μ}.

Theorem 2.1

The schemeP_{q_m,0,μ}is C^m⁺²for μ ϵ (0, 0.125).

Proof. Symbol of the scheme P_{q_m,0,μ} is given by

(6)qm,0,μ(z)=1+z2ma(z),

where

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

and

b(z)=8μz2+(2−16μ)z+8μ.

Let S_b 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 S_b is contractive. Therefore, by Corollary 4.11 of [22], the scheme S_a is C² for μ ϵ (0, 0.125). So by (6) scheme P_{q_m,0,μ} is C^m⁺² for μ ϵ (0, 0.125).

Similarly, we can easily find out the continuity of other schemes P_{q_m,n,μ} by taking into account the same formalism. The order of continuity of some proposed univariate subdivision schemes P_{q_m,0,μ} , P_{q_m,1,μ} , P_{q_m,2,μ} and P_{q_m,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 P_{q_m,0,μ}.

Table 1

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

n	Scheme	Ranges	O(C)	n	Scheme	Ranges	O(C)
0	P_{q_m,0,μ}	−0.375 < μ < 0.625	C^m⁺⁰	2	P_{q_m,2,μ}	−0.195 < μ < 0.445	C^m⁺⁰
	...	−0.125 < μ < 0.375	C^m⁺¹		...	−0.194 < μ < 0.442	C^m⁺¹
	...	0 < μ < 0.125	C^m⁺²		...	−0.034 < μ < 0.282	C^m⁺²
					...	−0.026 < μ < 0.235	C^m⁺³
					...	0.045 < μ < 0.09	C^m⁺⁴
1	P_{q_m,1,μ}	−0.275 < μ < 0.525	C^m⁺⁰	3	P_{q_m,3,μ}	−0.356 < μ < 0.618	C^m⁺⁰
	...	−0.075 < μ < 0.3	C^m⁺¹		...	−0.131 < μ < 0.380	C^m⁺¹
	...	−0.068 < μ < 0.295	C^m⁺²		...	−0.128 < μ < 0.375	C^m⁺²
	...	0.025 < μ < 0.104	C^m⁺³		...	−0.002 < μ < 0.235	C^m⁺³
					...	0.003 < μ < 0.191	C^m⁺⁴
					...	0.006 < μ < 0.081	C^m⁺⁵

Theorem 2.2

The Hölder continuity of the schemeP_{q_m,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

(8)M0=2−16μ08μ8μ,M1=8μ8μ02−16μ.

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 P_{q_m,0,μ} is C^m⁺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(kbk^l)/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 P_{q_m,n,μ} increase.

Table 2

Continuity of some members of the family of schemes

n	μ	Continuity	Lower bound on Hölder continuity	Upper bound on Hölder continuity
0	0.0625	C^m⁺²	C^m⁺³	C^m⁺³
1	0.0375	C^m⁺³	C^m^+3.255	C^m^+3.2603
2	0.0676	C^m⁺³	C^m^+4.478	C^m⁺⁵

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 π_d. 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 schemeP_{q_m,n,μ}generates π_m_+n+2for all m, n ϵ N. Moreover, ifμ=116, then P_{q_m,n,μ}generates π_m_+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 q_m_,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 μ=116two more terms (1+z) can be factored out from q_m_,n,μ(z), then we have D^(k+1)q_m_,n,μ(−1) = D^(k+2)q_m_,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 π_d 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 shiftτ=5+m+3n2,then the subdivision schemeP_{q_m,n,μ}reproduces π₁with respect to the parametrization in [15] for all m, n ϵ ℕ and μ ϵ ℝ. Moreover, ifμ=−3+m32,thenP_{q_m,n,μ}reproduces π₃for all m, n ϵ ℕ.

Proof. Since the condition D⁽¹⁾q_m,_n,_μ(1) = 5 + m + 3n is verified by the symbol q_m,_n,_μ(z) for all μ ϵ ℝ, so polynomial reproduction of P_{q_m,n,μ} is π₁ with the parameter shift τ=5+m+3n2.We observe that when μ=−3+m32,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 P_{q_m,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 (x_i , f_i), i = 0, 1, 2, . . . , n is monotonically increasing if f_i < f_i₊₁8 i = 0, 1, 2, . . . , n and the derivative at the data points obey the condition d_i > 0 ∀ i = 0, 1, 2, . . . , n.

In the following, we examine monotonicity preservation of binary scheme P_{q_1,0,μ}.

Theorem 2.5

Let{fi0}i∈Zsatisfy

…f−10<f00<f10<…<fn−10<fn0<fn+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 andξ=−1μ,ξ∈R.If1ξ≤R0≤ξ,{fik}is defined by the subdivision schemeP_{q_1,0,μ}, then

(9)dik>0,1ξ≤Rk≤ξ,k≥0,k∈Z,i∈Z.

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

di0=fi+10−fi0>0,1ξ≤R0≤ξ,then (9) is true.

Suppose that (9) holds for k,dik=fi+1k−fik>0,1ξ≤Rk≤ξ.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 dik>0and

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

This implies that d2ik+1>0.Similarly, we see that d2i+1k+1>0for 0.1 ≤ μ ≤ 0.9 and ξ=−1μ.Now we prove that 1ξ≤Rk+1≤ξ.First we show that r2ik+1−ξ≤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 ξ=−1μ.

This implies that

r2ik+1−ξ≤0.

It implies further that r2ik+1≤ξ.Now we show that 1r3ik+1−ξ<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 ξ=−1μ.

This implies that

1r2ik+1−ξ≤0.

It implies further that 1r2ik+1≤ξ.In the same way, we can get r2i+1k+1≤ξand 1r2i+1k+1≤ξ.So R^k⁺¹ ≤ ξ. Since Rk+1=maxi{rik,1rik},it is obvious that Rk+1≥1ξ.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 P_{q_1,0,μ} , P_{q_1,1,μ} , P_{q_1,2,μ} and P_{q_1,3,μ} respectively.

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

Figure 1

The curves (a), (b), (c) and (d) are generated by the schemes P_{q_1,0,μ} , P_{q_1,1,μ} , P_{q_1,2,μ} and P_{q_1,3,μ} respectively, using the monotone data set given below.

Table 3

Monotone data set [24].

i	1	2	3	4	5	6	7	8	9	10	11
x_i	0.1	4	6.5	10	15	25	40	50	62	65	66
y_i	1	1	2	3.5	5.5	5.5	10	10	12.5	18	20

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 P_q_2,0,μ , P_q_1,2,μ and P_q_2,2,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 P_q_2,0,μ , P_q_1,2,μ respectively. Figure 3(c) shows the non-interpolating behaviour of [15] at any parametric value. The proposed schemes P_q_2,0,μ and P_q_1,2,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 2

Most expanded and most shrinked curves: The curves (a), (b), (c) and (d) are generated by the schemes P_{q_2,0,μ} , P_{q_1,2,μ}, P_{q_2,2,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 3

Interpolating behaviour: The curves (a) , (b) and (c) are generated by the schemes P_{q_2,0,μ} , P_{q_1,2,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 4

Most expanded and most shrinked curves: The curves (a), (b) and (c) are generated by the schemes P_{q_1,0,μ} , P_{q_1,1,μ} and [15] respectively.

$Figure 5 Interpolating behaviour: The curves (a), (b) and (c) are generated by the schemes Pq1,0,μ , Pq1,1,μ and [15] respectively.$

Figure 5

Interpolating behaviour: The curves (a), (b) and (c) are generated by the schemes P_{q_1,0,μ} , P_{q_1,1,μ} and [15] respectively.

The Figures 4(a)-4(c) show the most expanded and most shrinked curves that are generated by the schemes P_{q_1,0,μ} , P_{q_1,1,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 P_{q_1,0,μ} , P_{q_1,1,μ} and [15] respectively. The schemes P_{q_1,0,μ} and P_{q_1,1,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 P_{q_m,n,μ} be the family of non-tensor product bivariate subdivision schemes then we propose the symbol of this family as

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

By substituting m = 1 and n = 0 in (10), we get symbol of the scheme P_{q_1,0,μ} as follows:

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

The bivariate subdivision scheme P_{q_1,0,μ} has the mask

(12)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.

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 schemeP_{q_1,0,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 schemeP_{q_1,0,μ}have C¹and 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 P_{q_1,0,μ} by checking the contractivity of the scheme. Since the scheme with symbol 121+z1231+z223b1,0,μ(z1,z2),121+z1241+z222b1,0,μ(z1,z2)is contractive for μ ϵ (−0.2215, 0.4785) and then scheme P_{q_1,0,μ} is convergent for μ ϵ (−0.2215, 0.4785). In the same way, the scheme with symbol 121+z1221+z223b1,0,μ(z1,z2),121+z1231+z222b1,0,μ(z1,z2),121+z124

1+z22b1,0,μ(z1,z2)is contractive for μ ϵ (−0.05178, 0.3017) therefore the scheme P_{q_1,0,μ} is C¹-continuous.

Again since, the scheme with symbol 121+z121+z223

b1,0,μ(z1,z2),121+z1221+z222b1,0,μ(z1,z2),121+z1231+z22b1,0,μ(z1,z2),121+z124

b_1,0,μ(z₁, z₂) is contractive for μ ϵ (−0.0517, 0.25), so the scheme P_{q_1,0,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.

Scheme	Type	O(C)
Binary non-tensor product [15]	Interpolating	C¹
Binary non-tensor product [15]	Approximating	C¹
Binary non-tensor product [6]	Approximating	C¹
Proposed binary non-tensor product P_q_1,0,μ	Approximating	C²
Proposed binary non-tensor product P_q_1,1,μ	Approximating	C³

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 P_{q_1,0,μ} and P_{q_1,1,μ} of generating and reproducing polynomials as well as monotonicity preservation of the data.

Theorem 3.2

The subdivision schemeP_{q_1,0,μ}generates π₂for all μ ϵ ℝ and generates π₄forμ=116.

Proof. Let w₁ = (1, −1), w₂ = (−1, 1), w₃ = (−1, −1) and let D_j with j ϵ ℕ², denote a directional derivative. Since q_1,0,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 P_{q_1,0,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 P_{q_1,0,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 P_{q_1,0,μ} generates π₃ for μ=116.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 P_{q_1,0,μ} generates π₄ for μ=116.This completes the proof.

Theorem 3.3

For the parameter shift(τ1,τ2)=(124,104),the subdivision schemeP_{q_1,0,μ}reproduces π₁with respect to the parametrization defined in [15] for all μ ϵ ℝ.

Proof. Let D_j with j ϵ ℕ², denote a directional derivative. Since the symbol q_1,0,μ(z₁, z₂) satisfies the conditions in Theorem 3.2. Since q_1,0,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 P_{q_1,0,μ} produced π₁ for all μ ϵ ℝ.

Similarly, we can prove the following theorems.

Theorem 3.4

The subdivision schemeP_{q_1,1,μ}generates π₃for all μ ϵ ℝ and generates π₄forμ=116.

Theorem 3.5

If applying the parameteric shift (τ₁, τ₂) = (3, 4), the subdivision schemeP_{q_1,1,μ}reproduces π₁with 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 P_{q_1,0,μ}.

Definition 3.1

[18] Bivariate data (x_i , y_j , f_i_,j), i = 0, 1, 2, . . . , n and j = 0, 1, 2, . . . , m, where x₁ < x₂ < . . . < x_n and y₁ < y₂ < . . . < y_m are said to be monotonically increasing if f_i_,j < f_i_+1,j and f_i_,j < f_i_,j+18 i = 0, 1, 2, . . . , n and j = 0, 1, 2, . . . , m, if the derivative at the data points obey the condition d_i_,j > 0 8 i = 0, 1, 2, . . . , n and j = 0, 1, 2, . . . , m.

Theorem 3.6

Suppose that the initial data{fi,j0}=(xi0,yj0,fi,j0)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 andδ=−1μ,δ∈R.If1δ≤Yi,j+t0,Yi+1,j+t0≤δ,{fi,jk}is defined by the subdivision schemeP_{q_1,0,μ}, then

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

Proof. We use mathematical induction to prove (13). When k=0,di,j0>0,1δ≤Yi,j+t0,Yi+1,j+t0≤δ,then (13) is true.

Suppose that (13) holds for k i.e. di,jk>0,1δ≤Yi,j+tk,Yi+1,j+tk≤δ.Next we will prove that (13) holds for k + 1.

First we show that d2i,2jk+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 δ=−1μ,we get

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

We know that di,j+3k>0and

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

This implies that d2ik+1>0.Similarly, we see that d2i+1,2jk+1>0,d2i,2j+1k+1>0andd2i+1,2j+1k+1>0for0.1≤μ≤0.9and δ=−1μ.

Now we prove that 1δ≤Yi,j+tk,Yi+1,j+tk≤δ.First we show that y2i,2jk+1−δ≤0.

For this, consider

y2i,2jk+1−δ=d2i+1,2jk+1d2i,2jk+1−δ.

After some simplification and substituting δ=−1μ,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 y2i,2jk+1≤δ.Now we show that 1y2i,2jk+1−δ<0.

For this, consider

1y2i,2jk+1−δ=d2i,2jkd2i,2j+1k−δ.

After some simplification and substituting δ=−1μ, 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 1y2i,2jk+1≤δ.In the same way, we can get y2i,2j+1k+1≤δ,y2i+1,2jk+1≤δ,y2i+1,2j+1k+1≤δ,1y2i,2j+1k+1≤δ,1y2i+1,2jk+1≤δand1y2i+1,2j+1k+1≤δ. SoYi,j+tk,Yi+1,j+tk≤δ. Since Yi,j+tk=maxi,j{yi,j+tk,1yi,j+tk}andYi+1,j+tk=maxi,j{yi+1,j+tk,1yi+1,j+tk}, it is obvious that Yi,j+tk,Yi+1,j+tk≥1δ. 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 P_{q_1,0,μ} and P_{q_1,1,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 P_{q_1,0,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 P_{q_1,0,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 P_{q_1,1,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 6

(a) Initial monotone data. (b) A monotonicity preserving surface obtained by the proposed scheme P_{q_1,0,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 7

(a) Control mesh. (b)-(d) Surfaces obtained by the proposed schemes P_{q_1,0,μ} at first, second and third subdivision levels respectively.

$Figure 8 (a) Control mesh. (b)-(d) Surfaces obtained by the proposed schemes Pq1,1,μ at first, second and third subdivision levels respectively.$

Figure 8

(a) Control mesh. (b)-(d) Surfaces obtained by the proposed schemes P_{q_1,1,μ} at first, second and third subdivision levels respectively.

Table 5

Monotone data set [25].

x/y	1	100	200	300
1	0.6931	9.2104	10.5967	11.4076
100	9.2104	9.9035	10.8198	11.5129
200	10.5967	10.8198	11.2898	11.7753
300	11.4076	11.5129	11.7753	12.1007

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

References

[1] Rham G. de, Un peude Mathematiques a proposed’ une courbe plane, Revwe de Mathematiques elementry II, Oevred Compl., 1947, 678-689Search in Google Scholar

[2] Chaikin G. M., An algorithm for high speed curve generation, Comp. Graph. Imag. Proces., 1974, 3, 346-34910.1016/0146-664X(74)90028-8Search in Google Scholar

[3] Mustafa G., Khan F. and Ghaffar A., The m-point approximating subdivision scheme, Lobachevskii J.Math., 2009, 30, 138-14510.1134/S1995080209020061Search in Google Scholar

[4] Zheng H., Huang S., Guo F. and Peng G., Integer-point binary approximating subdivision schemes, J. Info. Comput. Sci., 2014, 10, 3387-339810.12733/jics20104000Search in Google Scholar

[5] Mustafa G. , Ghaffar A. and Bari M., (2n-1)-point binary approximating scheme, Digital Information Management (ICDIM), International Conference, 363-368 (2013)10.1109/ICDIM.2013.6694036Search in Google Scholar

[6] Khan F. and Mustafa G., A new non-tensor product C¹ subdivision scheme for regular quad meshes, World Appl. Sci. J., 2013, 24, 1635-1641Search in Google Scholar

[7] Zheng H. C., Huang S. C., Guo F. and Peng G. H., Designing multi-parameter curve subdivision schemes with high continuity, Appl. Math. Comput., 2014, 243, 197-20810.1016/j.amc.2014.05.113Search in Google Scholar

[8] Lane J. M. and Riesenfeld R. F., A theoretical development for computer generation and display of piecewise polynomial surfaces, IEEE Tran. Patt. Anal. Mach. Intel., 1980, 2, 35-4610.1109/TPAMI.1980.4766968Search in Google Scholar PubMed

[9] Cashman T. J., Hormann K. and Reif U., Generalized Lane-Riesenfeld algorithms, Comp. Aid. Geomet. Des., 2013, 30, 398-40910.1016/j.cagd.2013.02.001Search in Google Scholar

[10] Ashraf P., Mustafa G. and Deng J., A six-point variant on the Lane-Riesenfeld algorithm, J. Appl. Math., 2014, 1-710.1155/2014/628285Search in Google Scholar

[11] Hormanm K. and Sabin M., A family of subdivision schemes with cubic precision, Comp. Aid. Geomet. Des., 2008, 25, 41-5210.1016/j.cagd.2007.04.002Search in Google Scholar

[12] Conti C. and Romani L., Dual univariate m-ary subdivision schemes of de Rham-type, J.Math. Anal. Appl., 2013, 407, 443-45610.1016/j.jmaa.2013.05.009Search in Google Scholar

[13] Mustafa G., Ashraf P. and Aslam M., Binary univariate dual and primal subdivision schemes, SeMA, 2014, 65, 23-3510.1007/s40324-014-0017-6Search in Google Scholar

[14] Dyn N., Levin D. and Gregory J. A., A 4-point interpolatory subdivision scheme for curve design, Comp. Aid. Geomet. Des., 1987, 4, 257-26810.1016/0167-8396(87)90001-XSearch in Google Scholar

[15] Romani L., A Chaikin-based variant of Lane-Riesenfeld algorithm and its non-tensor product extension, Comp. Aid. Geomet. Des., 2015, 32, 22-4910.1016/j.cagd.2014.11.002Search in Google Scholar

[16] Cai Z., Convergence, error estimation and some properties of four-point interpolation subdivision scheme, Comp. Aid. Geomet. Des., 1995, 12, 459-46810.1016/0167-8396(94)00024-MSearch in Google Scholar

[17] Hussan M. Z. and Hussan M., Visualization of data preserving monotonicity, Appl. Math. Comput., 2007, 190, 1353-136410.1016/j.amc.2007.02.022Search in Google Scholar

[18] Hussan M. Z., Sarfraz M. and Shaikh T. S., Monotone data visualization using rational functions, World Appl. Sci. J., 2012, 16, 1496-1508Search in Google Scholar

[19] Kuijt F. and Damme R. V., Monotonicity preserving interpolatory subdivision schemes, J. Comput. Appl. Math., 1999, 101, 203-22910.1016/S0377-0427(98)00220-9Search in Google Scholar

[20] Shalom Y., Monotonicity preserving subdivision scheme, J. Approx. Theo., 1993, 74, 41-5810.1006/jath.1993.1052Search in Google Scholar

[21] Tan J., Zhuang X. and Zhang L., A new four-point shape-preserving C³ subdivision scheme, Comp. Aid. Geomet. Des., 2014, 31, 57-6210.1016/j.cagd.2013.12.003Search in Google Scholar

[22] Dyn N. and Levin D., Subdivision schemes in the geometric modelling, Act. Numer., 2002, 11, 73-14410.1017/S0962492902000028Search in Google Scholar

[23] Rioul O., Simple regularity for subdivision schemes, SIAM J. Math. Anal., 1992, 23, 1544-157610.1137/0523086Search in Google Scholar

[24] Abbas M., Majid A. A., Awang M. N. H. and Ali J. M., Monotonicity-preserving rational bi-cubic spline surface interpolation, Sci. Asia, 2014, 40S, 22-3010.2306/scienceasia1513-1874.2014.40S.022Search in Google Scholar

[25] Hussain M. Z. and Hussain M., Visualization of data preserving monotonicity, Appl. Math. Comput., 2007, 190, 1353-136410.1016/j.amc.2007.02.022Search in Google Scholar

Received: 2017-05-31

Accepted: 2018-11-16

Published Online: 2018-12-31

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0126

Keywords for this article

Subdivision scheme; continuity; polynomial reproduction; monotonicity; non-tensor product

Creative Commons

BY-NC-ND 4.0