A New Computational Approach to Estimate the Subdivision Depth of n-Ary Subdivision Scheme

The <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-ary subdivision scheme has traditionally been designed to generate smooth curve and surface from control polygon. In this paper, we propose a new subdivision depth computation technique for <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-ary subdivision scheme. The existing techniques do not ensure the computation of subdivision depth unless some strong condition is assumed on the mask of the scheme. But our technique relaxes the effect of strong condition assumed on the mask of the scheme by increasing the number of convolution steps. Consequently, a more precise subdivision depth technique for a given error tolerance is presented in this paper.


I. INTRODUCTION
The n-ary subdivision scheme (nASS) is defined as the set of n-rules with respect to a sequence of control polygons. The nASS takes a polygon as an input and produces a refined polygon by applying n-rules on each edge of a coarse polygon. The repeated applications of these n-rules on the polygons produce a sequence of refined polygons. The sequence of refined polygons converges to a smooth shape. Let X 0 , X 1 , . . . , X k , . . . be a sequence of polygons and X ∞ be a limiting shape then the distance (also called error) between polygon and limiting shape approaches to zero as k approaches to ∞. In literature, there are different types of nASS but few of them are listed in [1]- [9]. Most of the researchers have discussed the some of well known properties of nASS such as smoothness/continuity, Hölder regularity, approximation order and support of the scheme. But few work has been done on error and subdivision depths of nASS. Let the designer has error tolerance and the polygon is divided k-times. If the distance between refined polygon at kth level The associate editor coordinating the review of this manuscript and approving it for publication was Ailong Wu . and the limiting shape is less than then k is called the subdivision depth of the limiting shape with respect to . In other words, subdivision depth tells us the number of subdivision steps needed to meet the designer error tolerance.
A first attempt to find the distance between polygon and the limiting shape is done in [10] for binary subdivision schemes. Its generalization for ternary and quaternary schemes was done in [11], [12]. Its further generalization for n-ary scheme was done in [13]. In this reference, the technique to compute the subdivision depth has also been introduced. The distance of a subdivision surface to its control polyhedron has been computed in [14]. Generally, the above techniques do not ensure the computations of subdivision depth unless some strong condition is assumed on the mask of the schemes. The condition for curve case is δ 1 < 1 while for surface case is δ 2 < 1, where δ 1 and δ 2 are defined in ( [13], Equations (5) and (6)).  The generalizations of the work of [10], [11] is done by [15]- [17] by using the convolution technique.
The error bounds of Doo-Sabin surfaces have been computed by Huawei et al. [18] in 2002. The different versions of the error bounds and subdivision depths of Catmull-Clark surfaces have been presented by [19]- [21]. The error bounds and subdivision depths of Loop subdivision surfaces have been computed by [22], [23]. But all these techniques have not been extended for the computation of error bounds and subdivision depths for n-ary, n > 2 schemes yet.
In this paper, we attempt to find the generalized version of the work done in [13] by using the convolution technique. Our technique can relax the effect of strong condition assumed on the mask of the schemes by increasing the number of convolution steps. Using the proposed technique, very less number of iterations (subdivision depths) are required to reach the user given error tolerance. So, this method reduces the burden of computational cost.
Rest of the paper is arranged as: In Section 2, we find the subdivision depth of the n-ary schemes for the generation of curves. Section 3 is devoted for the generalization of the work presented in Section 2 for surface case. The applications of our computational technique are given in Section 4. Section 5 is for conclusion.

II. SUBDIVISION DEPTH OF n-ARY SCHEME FOR CURVE
As is usually the case in subdivision depth papers, we will first describe our subdivision depth technique in a curve setting and then generalize it for a surface.

A. PRELIMINARY RESULTS FOR CURVE
Let X k = {x k i ; i ∈ Z} be a control polygon with points in R N , where N ≥ 2 and k be an integer (non-negative), which indicates the number of iterations (subdivision level). A generalized n-ary subdivision scheme for curve is described as By [13], we have We introduce the coefficients for u = 0, 1, . . . , N − 1, such that In the field of science, mathematics and engineering the convolution product has been used. It is a process that can be used for various branches of signal processing, edge detection and data smoothing. Here in the following section, we define some important results regarding the convolution of n-ary subdivision scheme for curve.

B. ONE DIMENSIONAL CONVOLUTION REFORMULATION
Let (x l ) l≥0 be a limited length vector and (t l ) l∈N = (t l ) nN −1 l=0 , with t l = 0 if l ≥ nN . The one time convolution product of x = (x l ) and t = (t l ) of n-ary subdivision scheme for curve is given by  where . denotes the integer part. Similarly, we get the reformulation for c 0 th convolutions with By (5), we get Lemma 1: The term E c 0 l,f given in the right hand side of inequality (7) has the following relation for n-ary subdivision scheme (8) Proof: Here, we begin the induction process over c 0 . similarly We suppose that for an integer M , it is true for Now, we will prove for By using (11), we acquire Using (10) and (11), we have Similarly, in the following lemma, we can deduce an another relation for the same term E c 0 l,f . Lemma 2: The term E c 0 l,f has the following relation for n-ary subdivision scheme

(12)
Proof: Here, we start the induction process over c 0 . similarly We suppose that it is true for Now, we will prove for By using (15), we acquire Using (14) and (15), we have . This completes the proof.

Corollary 3:
The term sup f f /n c 0 l=0 |E c 0 l,f | presented in the right hand side of the inequality (7) has the following alternate form . Then for f > (c 0 , N ) and by using (6), we acquire Now by using (17) and (18), we get (16).

C. SUBDIVISION DEPTH OF THE SCHEME FOR CURVE
Now firstly, we present some results for computing the distance between two consecutive polygons. Secondly, we compute the distance between kth level polygon and limiting curve. Then we describe an important theorem regarding the subdivision depth. Theorem 4: Let X k and X k+1 be two consecutive polygons obtained from the subdivision scheme (1) then the distance between these polygons is where T c 0 , c 0 ≥ 1 defined in (16) Proof: Similar to the proof given in [13]. Theorem 5: Let X k and X ∞ be kth level polygon and limiting curve respectively obtained from the subdivision scheme (1) then the distance between them is where c 0 ≥ 1, such that T c 0 < 1.
Proof: Similar to the proof given in [13]. Theorem 6: Let k be the subdivision depth and θ k be the distance between X k and X ∞ . For arbitrary > 0, if then θ k ≤ . Proof: Since by (20) therefore to attain given error tolerance > 0, consider then θ k ≤ . This completes the proof.

III. SUBDIVISION DEPTH OF n-ARY SCHEME FOR SURFACE
The surface case is the generalization of the curve case: We perform two dimensional convolution followed by the computation of distance between polygons to compute the subdivision depth.

A. PRELIMINARY RESULTS FOR SURFACE
Let X k = {x k i,j ; i, j ∈ Z} be a polygon at kth level with points in R N , where N ≥ 2. A tensor product of n-ary subdivision scheme (1) is described as where a α,r and a β,s satisfies (2).  We introduce the coefficients for u, v = 0, 1, . . . , N − 1 such that (24) Similarly, we acquire the reformulation for c 0 -th convolutions Also and

C. SUBDIVISION DEPTH OF THE SCHEME FOR SURFACE
In this section, we first estimate the distance between two successive polygon X k and X k+1 obtained from (22) then we estimate the distance between polygon X k and the limiting surface X ∞ . After that, we present the subdivision Depth of the scheme for surface. Theorem 7: Let X k and X k+1 be two consecutive polygons obtained from the subdivision scheme (22) then the distance between these polygons is where Y c 0 and Z c 0 for c 0 ≥ 1 are defined in (29) − (30) and η h α,β , ξ h , α, β = 0, 1, . . . , n − 1 are defined as Proof: Similar to the proof given in [13]. Theorem 8: Let X k and X ∞ be kth level polygon and limiting surface respectively obtained from the subdivision scheme (22) then the distance between them is where c 0 ≥ 1, such that Y c 0 Z c 0 < 1 and ν is defined as Proof: Similar to the proof given in [13].  Theorem 9: Let k be the subdivision depth and ϑ k be the distance between X k and X ∞ . For arbitrary > 0, if then ϑ k ≤ . Proof: Since by (32) To obtain given tolerance > 0, consider then ϑ k ≤ . This completes the proof.
• Surface case: The convolution constants Y c 0 Z c 0 for c 0 ≥ 1 of the tensor product of the scheme (34) are presented in Table 3. In Table 4, subdivision depths are shown and their graphical view is shown in Figure 1(b).

It has been observed that the subdivision depth decreases with the increase of convolution steps. That is we need less number of iterations to get the required result by increasing the number of convolution steps.
Example 11: If we take w = − 35 24 in ( [6], Equation 9), we get 4-point ternary approximating scheme with following coefficients.
• Curve case: The convolution constants T c 0 of the scheme (35) are gathered in Table 5. In Table 6, subdivision depths are shown and their graphical view is shown in Figure 2(a).
• Surface case: The convolution constants Y c 0 Z c 0 of the tensor product of the scheme (35) are presented in Table 7. In Table 8, subdivision depths are shown and their graphical view is shown in Figure 2 Table 9. In Table 10, subdivision depths are shown and their graphical view is shown in Figure 3(a).
• Surface case: The convolution constants of the tensor product of the scheme (36) are presented in Table 11.    In Table 12, subdivision depths are shown and their graphical view is shown in Figure 3 Table 13. In Table 14, subdivision depths are shown and their graphical view is shown in Figure 4(a).
• Surface case: The convolution constants of the tensor product of the scheme (37) are presented in Table 15.
In Table 16, subdivision depths are shown and their graphical view is shown in Figure 4(b).

V. CONCLUSION AND FUTURE WORK
The main purpose of this research was to provide an optimal technique to compute the subdivision depth. In other word, the aim was to predict the number of subdivision steps required to get an error-tolerant shape. In this paper, we have presented the technique to compute the depth of n-ary subdivision scheme. The advantage of this technique over the existing technique is that the strong condition imposed on the mask/coefficients of the scheme can be knocked out by increasing the number of convolution steps. We have also presented the subdivision depths of binary, ternary, quaternary and quinary schemes in this paper. These examples sentence that the proposed technique is valid and applicable for the computation of depth. The authors are looking, as a future work, to extend the computational technique of the subdivision depth of n-ary subdivision scheme for the generation of the shapes in higher dimensions.

AVAILABILITY OF DATA AND MATERIAL
''Data sharing not applicable to this article as no datasets were generated or analysed during the current study.'' Professor and the Dean of the Department of Mathematics, Huzhou University, Huzhou. His current research interests include special functions, functional analysis, numerical analysis, operator theory, ordinary differential equations, partial differential equations, inequalities theory and applications, and robust filtering and control. VOLUME 8, 2020