An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes

Abstract

Subdivision schemes are equipped with some rules that take a polygon as an input and produce smooth curves or surfaces as an output. This presents the issue of how accurately the polygon approximates the limit curve and surface. What number of iterations/levels would be necessary to achieve the required shape at a user-specified error tolerance? In fact, several methods have been introduced in the case of stationary schemes to address the issue in terms of the error bounds (distance between polygon/polyhedron and limiting shape) and subdivision depth (the number of iterations required to obtain the result at a user-specified error tolerance). However, in the case of non-stationary schemes, this topic needs to be further studied to meet the requirements of new practical applications. This paper highlights a new approach based on a convolution technique to estimate error bounds and subdivision depth for non-stationary schemes. The given technique is independent of any condition on the coefficient of the non-stationary subdivision schemes, and it also produces the best results with the least amount of computational effort. In this paper, we first associated constants with the vectors generated by the given non-stationary schemes, then formulated an expression for the convolution product. This expression gives real values, which monotonically decrease with the increase in the order of the convolution in both the curve and surface cases. This convolution feature plays an important role in obtaining the user-defined error tolerance with fewer iterations. It achieves a trade-off between the number of iterations and user-specified errors. In practice, more iterations are needed to achieve a lower error rate, but we achieved this goal by using fewer iterations.

1. Introduction

Much research has been conducted on stationary and non-stationary subdivision schemes for data fitting and designing curves and surfaces. The stationary schemes (level-independent) are suitable for the fitting of data and the production of simple shapes [1,2], while non-stationary schemes (level-dependent) are better suited to generating conic sections. These schemes can modify the limiting subdivision shapes. The stationary schemes do not have these characteristics. In [3,4], the potentiality of these schemes was discussed in broad terms, as well as a comparison of these schemes. The non-stationary subdivision schemes [5,6] are useful in geometric modeling and biological imaging [7,8,9,10]. These schemes can also be used for the introduction of wavelets and framelets [11,12,13,14].

In fact, several methods have been presented in the case of stationary schemes to answer two important questions: How accurately does the polygon approximate the limit curve and surface? What number of iterations would be necessary to achieve the required shape at a user-specified error tolerance? These methods are based on the 1-norm, 2-norm (Euclidean distance), infinity-norm (i.e., to find the maximal differences of the control polygon and limiting shape), two-scale refinement equation, and convolution technique. The error bounds and subdivision depth play a vital role in answering the above questions. The bounds for subdivision surfaces have been used to develop an algorithm for interference detection [15]. It is also useful for curve and surface intersection, mesh generation, surface rendering, and other tasks. A survey of the work performed for the class of stationary scheme is given below.

Initially, Nairn et al. [16] and Karavelas et al. [17] presented the bounds for the Bezier curve generated by control polygons. Peters and Wu [18], Rief [19], and Floater [20] computed the bounds on the subdivision surface, approximation of polynomials, and cubic spline interpolant by their control structure. Cai [21] introduced the error estimation technique for the four-point interpolating scheme. Wang et al. [22,23,24], Zhou and Zeng [25], Zeng and Chen [26], and Huang and Wang [27,28,29,30,31] presented different versions of the bounds and subdivision depth for the Doo–Sabin [32], Loop [33], and Catmull–Clark [34] schemes based on the 1-norm, 2-norm, and infinity-norm. Further, Mustafa et al. [35] and Deng et al. [36] presented error bounds and subdivision depth based on the infinity-norm of stationary schemes of different arity (i.e., binary, ternary, etc.) and their generalizations. Moncayo and Amat [37] provided error bounds for binary stationary schemes based on the two-scale refinement equation. Their bounds were sharper than the bounds based on the 1-norm, 2-norm, and infinity-norm. Moncayo et al. [38], Mustafa et al. [39,40,41,42]. and Shahzad et al. [43] presented sharper bounds and depth for stationary schemes of varying arity using an advanced convolution technique [44]. The convolution involved a finite number of computations to achieve the required result. However, limited work has been performed in the case of non-stationary schemes. Thus far, only one article by Mustafa et al. [45] dealing with the non-stationary binary schemes based on the infinity-norm has been presented. Therefore, there is much space for work in this area. In this paper, a new technique is introduced to find the bound and depth of non-stationary binary subdivision schemes based on the convolution of any two vectors. This is a more advanced technique than all the existing techniques. It also removes the stringent condition imposed by [45]. The arrangement of the paper is as follows:

In Section 2, some basic preliminaries regarding non-stationary binary subdivision schemes are demonstrated. Section 3 is devoted to the convolution, error bounds, and subdivision depth results. In this section also, a few numerical experiments associated with the main results are described. Finally, the conclusion and future work are given in Section 4.

2. Fundamental Concepts

2.1. Non-Stationary Curves

Let $\{z_i^{\kappa };i\in \mathbb{Z}\}$ be a set of control points at the $\kappa$th level of the binary non-stationary subdivision scheme. The comprehensive definition of the non-stationary binary subdivision scheme [46] can be described as

$$z_{2i+\eta }^{\kappa +1}={\displaystyle \sum _{m=0}^{N-1}}{a}_{\eta ,m}^{\kappa }{z}_{i+m}^{\kappa },\eta =0,1,$$

(1)

with the convergent mask condition:

$$\begin{array}{c}\hfill {\displaystyle \sum _{m=0}^{N-1}}{a}_{\eta ,m}^{\kappa }=1,\eta =0,1.\end{array}$$

(2)

According to [45], a convergent non-stationary scheme satisfies

$$\begin{array}{ccc}\hfill b_{0,m}^{\kappa }& =& {\displaystyle \sum _{l=0}^m}\left(a_{0,l}^{\kappa }-a_{1,l}^{\kappa }\right),\hfill \\ \hfill b_{1,m}^{\kappa }& =& a_{0,m}^{\kappa }-b_{0,m}^{\kappa },\hfill \end{array}$$

(3)

such that

$${\displaystyle \sum _{m=0}^{N-1}}|b_{0,m}^{\kappa }|<1\mathrm{and}{\displaystyle \sum _{m=0}^{N-1}}|b_{1,m}^{\kappa }|<1.$$

Now, define the following relationship for $m=0,1,\dots ,N-1$:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{y}_{2m}^{\kappa }=b_{0,m}^{\kappa },\hfill \\ y_{2m+1}^{\kappa }=b_{1,m}^{\kappa }.\hfill \end{array}\right.\end{array}$$

(4)

The above symbolization will be used in finding the error bounds and subdivision depth of the non-stationary subdivision scheme for the univariate case.

2.2. Non-Stationary Surfaces

Let $\{z_{i,j}^{\kappa };i,j\in \mathbb{Z}\}$ be the set of control points for the $\kappa$th level surface case. Take the tensor product of the univariate subdivision curve (1) to obtain the general scheme for the bivariate case:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{z}_{2i,2j}^{\kappa +1}={\displaystyle \sum _{p_1=0}^{N-1}}{\displaystyle \sum _{p_2=0}^{N-1}}{a}_{0,p_1}^{\kappa }{a}_{0,p_2}^{\kappa }{z}_{i+p_1,j+p_2}^{\kappa },\hfill \\ z_{2i,2j+1}^{\kappa +1}={\displaystyle \sum _{p_1=0}^{N-1}}{\displaystyle \sum _{p_2=0}^{N-1}}{a}_{0,p_1}^{\kappa }{a}_{1,p_2}^{\kappa }{z}_{i+p_1,j+p_2}^{\kappa },\hfill \\ z_{2i+1,2j}^{\kappa +1}={\displaystyle \sum _{p_1=0}^{N-1}}{\displaystyle \sum _{p_2=0}^{N-1}}{a}_{1,p_1}^{\kappa }{a}_{0,p_2}^{\kappa }{z}_{i+p_1,j+p_2}^{\kappa },\hfill \\ z_{2i+1,2j+1}^{\kappa +1}={\displaystyle \sum _{p_1=0}^{N-1}}{\displaystyle \sum _{p_2=0}^{N-1}}{a}_{1,p_1}^{\kappa }{a}_{1,p_2}^{\kappa }{z}_{i+p_1,j+p_2}^{\kappa },\hfill \end{array}\right.\end{array}$$

(5)

with the condition of convergence:

$$\begin{array}{c}\hfill {\displaystyle \sum _{p_1=0}^{N-1}}{a}_{0,p_1}^{\kappa }={\displaystyle \sum _{p_1=0}^{N-1}}{a}_{1,p_1}^{\kappa }={\displaystyle \sum _{p_2=0}^{N-1}}{a}_{0,p_2}^{\kappa }={\displaystyle \sum _{p_2=0}^{N-1}}{a}_{1,p_2}^{\kappa }=1.\end{array}$$

(6)

By [45], we must have the following conditions to find the error bounds and subdivision depth:

$${\displaystyle \sum _{p_1=0}^{N-1}}|a_{0,p_1}^{\kappa }|{\displaystyle \sum _{p_2=0}^{N-1}}|b_{0,p_2}^{\kappa }|<1,{\displaystyle \sum _{p_1=0}^{N-1}}|a_{0,p_1}^{\kappa }|{\displaystyle \sum _{p_2=0}^{N-1}}|b_{1,p_2}^{\kappa }|<1,$$

$${\displaystyle \sum _{p_1=0}^{N-1}}|a_{1,p_1}^{\kappa }|{\displaystyle \sum _{p_2=0}^{N-1}}|b_{0,p_2}^{\kappa }|<1,{\displaystyle \sum _{p_1=0}^{N-1}}|a_{1,p_1}^{\kappa }|{\displaystyle \sum _{p_2=0}^{N-1}}|b_{1,p_2}^{\kappa }|<1,$$

where $b_{0,p_2}^{\kappa }$ and $b_{1,p_2}^{\kappa }$ for $p_2=0,1,\dots ,N-1$ are defined in (3). Now, consider the new expressions for $p_3,p_4=0,1,\dots ,N-1$, such that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_{2p_3}^{\kappa }=a_{0,N-p_3-1}^{\kappa }\hfill \\ u_{2p_3+1}^{\kappa }=a_{1,N-p_3-1}^{\kappa }{p}_3=0,...,N-1.\hfill \\ v_{2p_4}^{\kappa }=b_{0,N-p_4-1}^{\kappa }\hfill \\ v_{2p_4+1}^{\kappa }=b_{1,N-p_4-1}^{\kappa },p_4=0,...,N-1.\hfill \end{array}\right.\end{array}$$

(7)

3. Main Results and Numerical Applications

For both curve and surface cases, this section presents several important conclusions regarding convolution, error bounds, and subdivision depth. We applied the proposed work to a few examples to evaluate the effectiveness and authenticity of this work.

3.1. Subdivision Depth for Non-Stationary Subdivision Curves

Generalized inequalities are derived in this subsection to find the subdivision depth of binary non-stationary subdivision schemes for curve generation. Let $z^{\kappa }=\{z_n^{\kappa };n\ge 0\}$ be a vector of finite length and $y^{\kappa }={\left(y_n^{\kappa }\right)}_{n=0}^{2N-1}$ with $y_n^{\kappa }=0$ for $n\ge 2N$. Then, a single convolution product of $z^{\kappa }=z_n^{\kappa }$ and $y^{\kappa }=y_n^{\kappa }$ for the non-stationary binary curves is

$${({(z^{\left(0\right)})}^{\kappa }\star y^{\kappa })}_j={\displaystyle \sum _{n=0}^{\lfloor j/2\rfloor }}{z}_n^{\kappa }{y}_{j-2n}^{\kappa }.$$

(8)

Similarly, the ${\chi }_0^c$-convolution for the curve case is described as

$${({(\dots {(({({(z^{(0)})}^{\kappa }\star y^{\kappa })}^{(0)})\star y^{\kappa })}^{(0)}\star \dots \star y^{\kappa })}^{(0)}\star y^{\kappa })}_j={\displaystyle \sum _{m=0}^{\lfloor j/2^{{\chi }_0^c}\rfloor }}{z}_m^{\kappa }{C}_{m,j}^{{\chi }_0^c;y^{\kappa }},$$

(9)

with

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{C}_{m,j}^{1;y^{\kappa }}=y_{j-2m}^{\kappa },\hfill \\ C_{m,j}^{{\chi }_0^c;y^{\kappa }}={\displaystyle \sum _{n=2m}^{\lfloor j/2^{{\chi }_0^c-1}\rfloor }}{C}_{m,n}^{1;y^{\kappa }}{C}_{n,j}^{{\chi }_0^c-1;y^{\kappa }},{\chi }_0^c\ge 2.\hfill \end{array}\right.\end{array}$$

(10)

Hence, by (9), we obtain

$$\begin{array}{c}\hfill \parallel ({(\dots {(({({(z^{(0)})}^{\kappa }\star y^{\kappa })}^{(0)})\star y^{\kappa })}^{(0)}\star \dots \star y^{\kappa })}^{(0)}\star y^{\kappa }){\parallel }_{\infty }\\ \hfill \le {\parallel z\parallel }_{\infty }\underset{j}{max}\left\{{\displaystyle \sum _{m=0}^{\lfloor j/2^{{\chi }_0^c}\rfloor }}|C_{m,j}^{{\chi }_0^c;y^{\kappa }}|\right\}=\underset{j\in \mathsf{\Sigma }({\chi }_0^c,N)}{max}\left\{{\displaystyle \sum _{m=0}^{\lfloor j/2^{{\chi }_0^c}\rfloor }}|C_{m,j}^{{\chi }_0^c;y^{\kappa }}|\right\}\end{array}$$

(11)

with

$$\begin{array}{c}\hfill \mathsf{\Sigma }({\chi }_0^c,N)=\{\mathsf{\Omega }({\chi }_0^c,N)-2^{{\chi }_0^c}+1,\mathsf{\Omega }({\chi }_0^c,N)-2^{{\chi }_0^c}+2,\dots ,\mathsf{\Omega }({\chi }_0^c,N)\}\end{array}$$

(12)

and

$$\begin{array}{c}\hfill \mathsf{\Omega }({\chi }_0^c,N)=(2^{{\chi }_0^c}-1)(2N-1).\end{array}$$

(13)

The associated constants of the ${\chi }_0^c$-times convolution for non-stationary binary subdivision curves with $y^{\kappa }=\left\{y_0^{\kappa },y_1^{\kappa },\dots ,y_{2N-1}^{\kappa }\right\}$ can be assigned as follows:

$$\begin{array}{c}\hfill Y_{{\chi }_0^c}=\underset{j\in \mathsf{\Sigma }({\chi }_0^c,N)}{max}\left\{{\displaystyle \sum _{m=0}^{\lfloor j/2^{{\chi }_0^c}\rfloor }}|C_{m,j}^{{\chi }_0^c;y^{\kappa }}|\right\}.\end{array}$$

(14)

Following a discussion of the modified results for determining the error bounds in the non-stationary curve case, the following results describe an updated subdivision depth estimation technique based on the error bounds:

Theorem 1. 

Let $Z^0=\{z_i^0;i\in Z\}$ be an initial polygon and $\{z_i^{\kappa };\kappa \ge 0\}$ be a set of points recursively described by (1) with the condition (2). In addition, $Z^{\kappa }=\{z_i^{\kappa };i\in Z\}$ indicates the polygon’s points at the κth level. Then, after two consecutive subdivision refinements, the error bounds between the stages κ and $\kappa +1$ is

$$\begin{array}{c}\hfill \parallel Z^{\kappa +1}-Z^{\kappa }{\parallel }_{\infty }\le \alpha \beta {\left(Y_{{\chi }_0^c}\right)}^{\kappa },\beta =\underset{i}{max}∥▵p_i^0∥\end{array}$$

(15)

where $Y_{{\chi }_0^c},{\chi }_0^c\ge 1$ is defined in (14) and $\alpha =max\left({\sum }_{j=0}^{N-2}\left|{\tilde{a}}_{0,j}^{\kappa }\right|,{\sum }_{j=0}^{N-2}\left|{\tilde{a}}_{1,j}^{\kappa }\right|\right),$ such that

$$\begin{array}{c}\hfill {\tilde{a}}_{0,j}^{\kappa }={\displaystyle \sum _{i=j+1}^{N-1}}{a}_{0,i}^{\kappa },{\tilde{a}}_{1,0}^{\kappa }={\displaystyle \sum _{i=1}^{N-1}}{a}_{1,i}^{\kappa }-\frac{1}{2},and{\tilde{a}}_{1,j}^{\kappa }={\displaystyle \sum _{i=j+1}^{N-1}}{a}_{1,i}^{\kappa },j\ne 0.\end{array}$$

Proof. 

See [35], Theorem 1. □

Theorem 2. 

Assume $Z^{\infty }$ is a limit curve obtained by repeatedly applying a non-stationary subdivision scheme and $Z^{\kappa }$ is the curve obtained after the κth refinement. Then, using the same assumptions as given in Theorem 1,

$${∥Z^{\infty }-Z^{\kappa }∥}_{\infty }\le \alpha \beta \left(\frac{{\left(Y_{{\chi }_0^c}\right)}^{\kappa }}{1-Y_{{\chi }_0^c}}\right),$$

(16)

where $Y_{{\chi }_0^c}$ is defined in (14) and ${\chi }_0^c\ge 1$, such that $Y_{{\chi }_0^c}<1$.

Proof. 

See [35], Theorem 1. □

Theorem 3. 

Let $Z^{\infty }$ be the subdivision limit curve and $Z^{\kappa }$ be the κth level subdivision curve. If ${\rho }^{\kappa }$ is defined as the error bound between $Z^{\infty }$ and $Z^{\kappa }$ under the same conditions as in Theorem 2, then the following inequality holds for any error tolerance number $ϵ>0$:

$$\kappa \ge {log}_{Y_{{\chi }_0^c}}\left(\frac{ϵ(1-Y_{{\chi }_0^c})}{\alpha \beta }\right),$$

(17)

such that ${\rho }^{\kappa }\le ϵ$.

Proof. 

Let ${\rho }^{\kappa }$ be the distance defined in Theorem 2 such that

$${\rho }^{\kappa }={∥Z^{\infty }-Z^{\kappa }∥}_{\infty }\le \alpha \beta \left(\frac{{\left(Y_{{\chi }_0^c}\right)}^{\kappa }}{1-Y_{{\chi }_0^c}}\right).$$

Now, consider the following:

$$\alpha \beta \left(\frac{{\left(Y_{{\chi }_0^c}\right)}^{\kappa }}{1-Y_{{\chi }_0^c}}\right)\le ϵ,$$

which implies

$$\frac{\alpha \beta }{ϵ(1-Y_{{\chi }_0^c})}\le {\left(Y_{{\chi }_0^c}^{-1}\right)}^{\kappa }.$$

Taking the logarithm:

$$\kappa \ge {log}_{Y_{{\chi }_0^c}}\left(\frac{ϵ(1-Y_{{\chi }_0^c})}{\alpha \beta }\right),$$

then ${\rho }^{\kappa }\le ϵ$. This completes the proof. □

3.2. Numerical Applications for Curve Models

Now, a few numerical experiments are described to calculate the depths of non-stationary subdivision schemes for curves.

Experiment 1. 

Consider the three-point approximating non-stationary binary subdivision curve (NSBSC) defined in [5].

$$\begin{array}{ccc}\hfill z_{2i}^{\kappa +1}& =& {\gamma }_0^{\kappa }\left(\alpha \right)z_{i-1}^{\kappa }+{\gamma }_1^{\kappa }\left(\alpha \right)z_i^{\kappa }+{\gamma }_2^{\kappa }\left(\alpha \right)z_{i+1}^{\kappa },\hfill \\ \hfill z_{2i+1}^{\kappa +1}& =& {\gamma }_2^{\kappa }\left(\alpha \right)z_{i-1}^{\kappa }+{\gamma }_1^{\kappa }\left(\alpha \right)z_i^{\kappa }+{\gamma }_0^{\kappa }\left(\alpha \right)z_{i+1}^{\kappa },\hfill \end{array}$$

(18)

where

$$\begin{array}{c}\hfill {\gamma }_0^{\kappa }\left(\alpha \right)=\frac{si{n}^2\frac{3\alpha }{2^{\kappa +2}}}{sin\frac{\alpha }{2^{\kappa }}sin\frac{2\alpha }{2^{\kappa }}},{\gamma }_1^{\kappa }\left(\alpha \right)=\frac{sin\frac{5\alpha }{2^{\kappa +2}}sin\frac{3\alpha }{2^{\kappa +2}}+sin\frac{7\alpha }{2^{\kappa +2}}sin\frac{\alpha }{2^{\kappa +2}}}{sin\frac{\alpha }{2^{\kappa }}sin\frac{2\alpha }{2^{\kappa }}},{\gamma }_2^{\kappa }\left(\alpha \right)=\frac{si{n}^2\frac{\alpha }{2^{\kappa +2}}}{sin\frac{\alpha }{2^{\kappa }}sin\frac{2\alpha }{2^{\kappa }}}.\end{array}$$

If convolution coefficient $Y_{{\chi }_0^c}\ge 1$, then apply the ${\chi }_0^c$-times convolution until $Y_{{\chi }_0^c}<1$. Convolution may also be applied even $Y_{{\chi }_0^c}<1$ to obtain a smaller value of $Y_{{\chi }_0^c}$. A smaller value of $Y_{{\chi }_0^c}$ gives better results. In

Table 1. Convolution coefficients of the 3-point approximating NSBSC.

${\mathit{Y}}_{{\mathit{\chi }}_0^{\mathit{c}}}\to$	${\mathit{Y}}_1$	${\mathit{Y}}_2$	${\mathit{Y}}_3$	${\mathit{Y}}_4$	${\mathit{Y}}_5$
1st level ($\kappa$ = 1)	0.509795	0.254898	0.128345	0.064401	0.032343
2nd level ($\kappa$ = 2)	0.502419	0.251211	0.125831	0.062972	0.031522
3rd level ($\kappa$ = 3)	0.500603	0.250301	0.125207	0.062618	0.031318
4th level ($\kappa$ = 4)	0.500151	0.250075	0.125052	0.062529	0.031267
5th level ($\kappa$ = 5)	0.500038	0.250019	0.125013	0.062507	0.031254

, $Y_{{\chi }_0^c}$ are presented up to the fifth convolution and for distinct subdivision levels, as $\kappa =1,2,3,4,5.$ After using convolution, the coefficients for the proposed scheme at different levels κ are obtained. Here, Table 1 also shows that, for any refinement level, the numerical value of $Y_{{\chi }_0^c}$ decreases as the order of the convolution increases. The estimation of the subdivision depth at various levels κ is shown in

Table 2. Subdivision depth of the 3-point approximating NSBSC.

$ϵ$			$2.52e^{-3}$					$2.62e^{-6}$
$Y_{{\chi }_0^c}$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$
$Y_1$	6	6	6	6	6	16	16	16	16	16
$Y_2$	3	3	3	3	3	8	8	8	8	8
$Y_3$	2	2	2	2	2	5	5	5	5	5
$Y_4$	1	1	1	1	1	4	4	4	4	4
$Y_5$	1	1	1	1	1	3	3	3	3	3
$ϵ$			$2.67e^{-9}$					$2.86e^{-12}$
$Y_{{\chi }_0^c}$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$
$Y_1$	26	26	26	26	26	37	36	36	36	36
$Y_2$	13	13	13	13	13	18	18	18	18	18
$Y_3$	8	8	8	8	8	12	12	12	12	12
$Y_4$	6	6	6	6	6	9	9	9	9	9
$Y_5$	5	5	5	5	5	7	7	7	7	7

using Theorem 3.

Remark 1. 

There is a strong condition $\phi <1$ in [45] (Theorem 3.5, Equation (3.9)). The error bounds could not be calculated for $\phi \ge 1$ by using the technique of [45]; however, by using our technique, increasing the order of convolution ${\chi }_0^c$ causes the value of $Y_{{\chi }_0^c}$ to decrease until it is less than one. Therefore, the error bounds of non-stationary binary subdivision schemes with $Y_{{\chi }_0^c}\ge 1$ can be computed.

Remark 2. 

In Table 1 and Table 2: $\kappa =1,\kappa =2,\kappa =3,\kappa =4$, and $\kappa =5$ show the different behavior of the NSBSC presented in Experiment 1, while in Table 2, the rows of $Y_1,Y_2,Y_3,Y_4$, and $Y_5$ show various subdivision levels (number of iterations) corresponding to the user-defined error tolerance by Theorem 3. Table 2 shows that increasing the order of convolution requires fewer iterations to achieve the desired error. For a user-defined error tolerance of $2.86e^{-12}$, we need 37 iterations at the first convolution, but only 7 iterations after the fifth convolution to achieve the same error. This was the primary objective and goal of this study.

Experiment 2. 

Consider the four-point interpolating non-stationary binary subdivision curve defined in [6], where

$$\begin{array}{ccc}\hfill (a_{0,0}^{\kappa },a_{0,1}^{\kappa },a_{0,2}^{\kappa },a_{0,3}^{\kappa })& =& \left(0,1,0,0\right),\hfill \\ \hfill (a_{1,0}^{\kappa },a_{1,1}^{\kappa },a_{1,2}^{\kappa },a_{1,3}^{\kappa })& =& \left(-w,\frac{1}{2}+w,\frac{1}{2}+w,-w\right),\hfill \end{array}$$

(19)

where

$$\begin{array}{c}\hfill w=\frac{1}{16co{s}^2(\frac{\alpha }{2^{\kappa +2}})cos\left(\frac{\alpha }{2^{\kappa +1}}\right)},\alpha =\frac{\pi }{4}.\end{array}$$

If convolution coefficient $Y_{{\chi }_0^c}\ge 1$, then apply the ${\chi }_0^c$-times convolution until $Y_{{\chi }_0^c}<1$. The numerical values of $Y_{{\chi }_0^c}$ are shown in

Table 3. Convolution coefficients of the 4-point approximating NSBSC.

${\mathit{Y}}_{{\mathit{\chi }}_0^{\mathit{c}}}\to$	${\mathit{Y}}_1$	${\mathit{Y}}_2$	${\mathit{Y}}_3$	${\mathit{Y}}_4$	${\mathit{Y}}_5$
1st level ($\kappa$ = 1)	0.564343	0.378685	0.229794	0.139262	0.084417
2nd level ($\kappa$ = 2)	0.562954	0.375908	0.227357	0.137343	0.082985
3rd level ($\kappa$ = 3)	0.562613	0.375226	0.22676	0.136874	0.082635
4th level ($\kappa$ = 4)	0.562528	0.375056	0.226612	0.136757	0.082548
5th level ($\kappa$ = 5)	0.562507	0.375014	0.226575	0.136728	0.082527

for ${\chi }_0^c$: 1, 2, 3, 4, and 5 at different levels κ: 1, 2, 3, 4, and 5. The subdivision depths of the four-point interpolating NSBSC at different levels κ are estimated by using Theorem 3, which are shown in

Table 4. Subdivision depth of the 4-point interpolating NSBSC.

$ϵ$			$9.22e^{-3}$					$6.57e^{-3}$
$Y_{{\chi }_0^c}$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$
$Y_1$	6	6	6	6	6	14	14	14	14	14
$Y_2$	3	3	3	3	3	8	8	8	8	8
$Y_3$	2	2	2	2	2	5	5	5	5	5
$Y_4$	1	1	1	1	1	4	4	4	4	4
$Y_5$	1	1	1	1	1	3	3	3	3	3
$ϵ$			$4.68e^{-7}$					$3.33e^{-9}$
$Y_{{\chi }_0^c}$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$
$Y_1$	23	23	23	23	23	32	31	31	31	31
$Y_2$	13	13	13	13	13	18	18	18	18	18
$Y_3$	9	8	8	8	8	12	12	12	12	12
$Y_4$	6	6	6	6	6	9	9	9	9	9
$Y_5$	5	5	5	5	5	7	7	7	7	7

.

3.3. Subdivision Depth for Non-Stationary Subdivision Surfaces

Here, some notations and results of the convolutions for the bivariate case are presented, then the results for the error bounds and subdivision depths of non-stationary subdivision surfaces are introduced.

Let $\{z_{m,n}^{\kappa };m,n\ge 0\}$ be a two-dimensional vector and $\{u_n^{\kappa };n\ge 0\}={\left\{u_n^{\kappa }\right\}}_{n=0}^{2N-1}$, $\{v_n^{\kappa };n\ge 0\}={\left\{v_n^{\kappa }\right\}}_{n=0}^{2N-1}$ with $u_n^{\kappa }=v_n^{\kappa }=0$ for $n\ge 2N$. Then, the convolution product of $z^{\kappa }=z_n^{\kappa }$, $u^{\kappa }=u_n^{\kappa }$ and $v^{\kappa }=v_n^{\kappa }$ for non-stationary binary subdivision surfaces is as follows:

$$z_{i,j}^{{\chi }_0^s;\kappa }={\left(z^{{\chi }_0^s-1;0;\kappa }\star u^{\kappa }{v}^{\kappa }\right)}_{i,j}={\displaystyle \sum _{m=0}^{⌊i/2⌋}}{\displaystyle \sum _{n=0}^{⌊j/2⌋}}{z}_{m,n}^{{\chi }_0^s-1;\kappa }{u}_{i-2m}^{\kappa }{v}_{j-2n}^{\kappa }.$$

(20)

Similarly, the ${\chi }_0^s$ convolution reformulation for the surface case is described as

$$\begin{array}{c}\hfill z_{i,j}^{{\chi }_0^s;\kappa }={(\dots (((z^{{\chi }_0^s-1;0}\star u^{\kappa }{v}^{\kappa })\star u^{\kappa }{v}^{\kappa })\star \dots \star u^{\kappa }{v}^{\kappa })\star u^{\kappa }{v}^{\kappa })}_{i,j}={\displaystyle \sum _{m=0}^{\lfloor i/2^{{\chi }_0^s}\rfloor }}{\displaystyle \sum _{n=0}^{\lfloor j/2^{{\chi }_0^s}\rfloor }}{z}_{m,n}^{0;\kappa }{C}_{m,i}^{{\chi }_0^s,u^{\kappa }}{C}_{n,j}^{{\chi }_0^s,v^{\kappa }},\end{array}$$

(21)

with

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{C}_{m,i}^{1;u^{\kappa }}=u_{i-2m}^{\kappa },\hfill \\ C_{m,i}^{{\chi }_0^s;u^{\kappa }}={\displaystyle \sum _{p_1=2m}^{\lfloor i/2^{{\chi }_0^s-1}\rfloor }}{C}_{m,p_1}^{1;u^{\kappa }}{C}_{p_1,i}^{{\chi }_0^s-1;u^{\kappa }},\hfill \\ C_{n,j}^{1;v^{\kappa }}=v_{j-2n}^{\kappa },\hfill \\ C_{n,j}^{{\chi }_0^s;v^{\kappa }}={\displaystyle \sum _{p_2=2n}^{\lfloor j/2^{{\chi }_0^s-1}\rfloor }}{C}_{n,p_2}^{1;v^{\kappa }}{C}_{p_2,j}^{{\chi }_0^s-1;v^{\kappa }},{\chi }_0^s\ge 2.\hfill \end{array}\right.\end{array}$$

(22)

From (21), we obtain

$$\underset{i,j}{max}|z_{i,j}^{{\chi }_0^s;\kappa }|\le U_{{\chi }_0^s}{V}_{{\chi }_0^s}\underset{m,n}{max}|z_{m,n}^0|,$$

(23)

where

$$U_{{\chi }_0^s}=\underset{i}{max}\left\{{\displaystyle \sum _{m=0}^{\lfloor i/2^{{\chi }_0^s}\rfloor }}|C_{m,i}^{{\chi }_0^s,u^{\kappa }}|\right\}$$

(24)

and

$$V_{{\chi }_0^s}=\underset{j}{max}\left\{{\displaystyle \sum _{n=0}^{[j/2^{{\chi }_0^s}]}}|C_{n,j}^{{\chi }_0^s,v^{\kappa }}|\right\}.$$

(25)

Furthermore,

$$\underset{i,j}{max}\left\{{\displaystyle \sum _{m=0}^{\lfloor i/2^{{\chi }_0^s}\rfloor }}{\displaystyle \sum _{n=0}^{\lfloor j/2^{{\chi }_0^s}\rfloor }}|C_{m,i}^{{\chi }_0^s,u^{\kappa }}||C_{n,j}^{{\chi }_0^s,v^{\kappa }}|\right\}=\underset{i,j\in \mathsf{\Sigma }({\chi }_0^s,N)}{max}\left\{{\displaystyle \sum _{m=0}^{\lfloor i/2^{{\chi }_0^s}\rfloor }}{\displaystyle \sum _{n=0}^{\lfloor j/2^{{\chi }_0^s}\rfloor }}|C_{m,i}^{{\chi }_0^s,u^{\kappa }}||C_{n,j}^{{\chi }_0^s,v^{\kappa }}|\right\},$$

(26)

where $\mathsf{\Sigma }({\chi }_0^s,N)$ is defined in (12).

The following results were established for finding the error bounds and subdivision depths of the non-stationary bivariate case, then a modified subdivision depth estimation technique based on the established error bounds is presented.

Theorem 4. 

Consider $z_{i,j}^0,i,j\in \mathbb{Z}$ as the initial control polygon and $z_{i,j}^{\kappa }$ as the recursively generated control polygon by (5) and (6). Let $Z^{\kappa }$ be the polygon formed by the points $z_{i,j}^{\kappa }$. Then, using a similar technique described in [45], the error bounds between two successive refinement levels κ and $\kappa +1$ are obtained.

$$\begin{array}{c}\hfill \parallel Z^{\kappa +1}-Z^{\kappa }{\parallel }_{\infty }\le \left(\rho {\beta }_1+\tau {\beta }_2+\sigma {\beta }_3\right)U_{{\chi }_0^s}{V}_{{\chi }_0^s},\end{array}$$

(27)

where $U_{{\chi }_0^s},V_{{\chi }_0^s},{\chi }_0^s\ge 1$ is defined in (24)–(25) and ρ, τ, σ, and ${\beta }_t$ are defined in [45].

Proof. 

The proof in [45] is analogous. □

Theorem 5. 

Let $Z^{\infty }$ be the limit surface generated by the subdivision iterative process. Then, using the same assumptions as given in Theorem 4,

$$\parallel Z^{\infty }-Z^{\kappa }{\parallel }_{\infty }\le \nu \left(\frac{{\left(U_{{\chi }_0^s}{V}_{{\chi }_0^s}\right)}^{\kappa }}{1-U_{{\chi }_0^s}{V}_{{\chi }_0^s}}\right),$$

(28)

where ${\chi }_0^s\ge 1$, such that $U_{{\chi }_0^s}{V}_{{\chi }_0^s}<1$ and ν is described in [45].

Proof. 

The proof in [45] is analogous. □

Theorem 6. 

Let κ represent the number of iterations, and suppose ${\zeta }^{\kappa }$ is the estimation of the error between non-stationary surface $Z^{\infty }$ and the ${\kappa }^{th}$ stage control polygon $Z^{\kappa }$. If all the conditions of Theorem 5 hold, then the following inequality holds for any error tolerance number $\delta >0$:

$$\kappa \ge {log}_{\left(U_{{\chi }_0^s}{V}_{{\chi }_0^s}\right)}\left(\frac{\delta (1-U_{{\chi }_0^s}{V}_{{\chi }_0^s})}{\nu }\right),$$

(29)

such that ${\zeta }^{\kappa }\le \delta$.

Proof. 

The proof in Theorem 3 is analogous. □

3.4. Numerical Application for Surface Models

Here, a few test experiments for non-stationary subdivision surfaces are presented.

Experiment 3. 

Consider the tensor product of the three-point approximating non-stationary binary subdivision curve scheme given in (18). The five-times convolution ${\chi }_0^s$ was applied to find the five convolution coefficients $U_{{\chi }_0^s}{V}_{{\chi }_0^s}$ defined by (24) and (25). The numerical values of these coefficients are shown in

Table 5. Convolution coefficients of the 3-point approximating non-stationary subdivision surface.

${\mathit{U}}_{{\mathit{\chi }}_0^{\mathit{s}}}{\mathit{V}}_{{\mathit{\chi }}_0^{\mathit{s}}}\to$	${\mathit{U}}_1{\mathit{V}}_1$	${\mathit{U}}_2{\mathit{V}}_2$	${\mathit{U}}_3{\mathit{V}}_3$	${\mathit{U}}_4{\mathit{V}}_4$	${\mathit{U}}_5{\mathit{V}}_5$
1st level ($\kappa$ = 1)	0.509795	0.254898	0.128345	0.064401	0.032368
2nd level ($\kappa$ = 2)	0.502419	0.25121	0.12583	0.062972	0.031528
3rd level ($\kappa$ = 3)	0.500603	0.250301	0.125207	0.062618	0.031319
4th level ($\kappa$ = 4)	0.500151	0.250075	0.125052	0.062529	0.031267
5th level ($\kappa$ = 5)	0.500038	0.250019	0.125013	0.062507	0.031254

. Using Theorem 6, the subdivision depths for different iteration levels κ are shown in

Table 6. Subdivision depth of the 3-point approximating non-stationary subdivision surface.

$ϵ$			$6.51e^{-3}$					$6.84e^{-6}$
$U_{{\chi }_0^s}{V}_{{\chi }_0^s}$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$
$U_1{V}_1$	6	6	6	6	6	16	16	16	16	16
$U_2{V}_2$	3	3	3	3	3	8	8	8	8	8
$U_3{V}_3$	2	2	2	2	2	5	5	5	5	5
$U_4{V}_4$	1	1	1	1	1	4	4	4	4	4
$U_5{V}_5$	1	1	1	1	1	3	3	3	3	3
$ϵ$			$7.17e^{-9}$					$7.52e^{-12}$
$U_{{\chi }_0^s}{V}_{{\chi }_0^s}$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$
$U_1{V}_1$	26	26	26	26	26	37	36	36	36	36
$U_2{V}_2$	13	13	13	13	13	18	18	18	18	18
$U_3{V}_3$	8	8	8	8	8	12	12	12	12	12
$U_4{V}_4$	6	6	6	6	6	9	9	9	9	9
$U_5{V}_5$	5	5	5	5	5	7	7	7	7	7

.

Experiment 4. 

Consider the four-point interpolating non-stationary binary subdivision surfaces (NSBSSs) obtained by the tensor product of the curve (19). Now, use (24) and (25) to calculate the convolution coefficients $U_{{\chi }_0^s}{V}_{{\chi }_0^s}$ at the ${\chi }_0^s$th convolution. These coefficients are shown in

Table 7. Convolution coefficients of the 4-point approximating NSBSSs.

${\mathit{U}}_{{\mathit{\chi }}_0^{\mathit{s}}}{\mathit{V}}_{{\mathit{\chi }}_0^{\mathit{s}}}\to$	${\mathit{U}}_1{\mathit{V}}_1$	${\mathit{U}}_2{\mathit{V}}_2$	${\mathit{U}}_3{\mathit{V}}_3$	${\mathit{U}}_4{\mathit{V}}_4$	${\mathit{U}}_5{\mathit{V}}_5$
1st level ($\kappa$ = 1)	0.79049	0.402995	0.208033	0.105486	0.052898
2nd level ($\kappa$ = 2)	0.783521	0.398725	0.205803	0.104247	0.052272
3rd level ($\kappa$ = 3)	0.781815	0.397683	0.205258	0.103946	0.05212
4th level ($\kappa$ = 4)	0.781391	0.397425	0.205123	0.10387	0.052082
5th level ($\kappa$ = 5)	0.781285	0.39736	0.205089	0.103852	0.052073

.

The subdivision depths for different levels κ of the proposed scheme are presented in

Table 8. Subdivision depth of the 4-point interpolating NSBSSs.

$ϵ$			$1.67e^{-2}$					$4.68e^{-5}$
$U_{{\chi }_0^s}{V}_{{\chi }_0^s}$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$
$U_1{V}_1$	19	18	18	18	18	44	42	42	42	42
$U_2{V}_2$	4	4	4	4	4	10	10	10	10	10
$U_3{V}_3$	2	2	2	2	2	6	6	6	6	6
$U_4{V}_4$	1	1	1	1	1	4	4	4	4	4
$U_5{V}_5$	1	1	1	1	1	3	3	3	3	3
$ϵ$			$1.31e^{-7}$					$3.67e^{-10}$
$U_{{\chi }_0^s}{V}_{{\chi }_0^s}$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$	$\kappa =1$	$\kappa =2$	$\kappa =3$	$\kappa =4$	$\kappa =5$
$U_1{V}_1$	69	66	66	66	65	94	90	90	89	89
$U_2{V}_2$	17	16	16	16	16	23	23	23	23	23
$U_3{V}_3$	9	9	9	9	9	13	13	13	13	13
$U_4{V}_4$	7	7	7	7	7	9	9	9	9	9
$U_5{V}_5$	5	5	5	5	5	7	7	7	7	7

.

4. Conclusions and Future Work

In this paper, the results were furnished for the estimation of the errors and subdivision depths of non-stationary binary curves and surfaces at distinct subdivision stages. These results are a very useful, modified, and efficient version of the work given in [45]. It can be clearly seen that the method given in [45] failed to find the error bounds and subdivision depths of those non-stationary subdivision schemes where the strong condition was not satisfied. In this situation, the proposed method works because the associated constants decrease as the order of convolution increases. This was the best outcome of this proposed work. Moreover, it can also be observed from Table 2, Table 4, Table 6 and Table 8 that this technique is efficient because it needed fewer iterations to obtain the subdivision depths of the scheme within the user-defined error tolerance. Furthermore, other possible extensions are in image processing and scattered data interpolation [47,48,49].