A Class of Refinement Schemes With Two Shape Control Parameters

A subdivision scheme defines a smooth curve or surface as the limit of a sequence of successive refinements of given polygon or mesh. These schemes take polygons or meshes as inputs and produce smooth curves or surfaces as outputs. In this paper, a class of combine refinement schemes with two shape control parameters is presented. These even and odd rules of these schemes have complexity three and four respectively. The even rule is designed to modify the vertices of the given polygon, whereas the odd rule is designed to insert a new point between every edge of the given polygon. These schemes can produce high order of continuous shapes than existing combine binary and ternary family of schemes. It has been observed that the schemes have interpolating and approximating behaviors depending on the values of parameters. These schemes have an interproximate behavior in the case of non-uniform setting of the parameters. These schemes can be considered as the generalized version of some of the interpolating and B-spline schemes. The theoretical as well as the numerical and graphical analysis of the shapes produced by these schemes are also presented.


I. INTRODUCTION
The refinement schemes also known as subdivision schemes are widely used in the design of curves and surfaces. Initially, schemes were introduced without parameters. Later on, to improve the flexibility of designing the curves, some of the schemes were introduced with parameters. Here we present a brief survey of the schemes with parameters. The first scheme with parameter was introduced by Dyn et al. [21] in 1987. Then in 1990, Dyn et al. [1] noticed that the smoothness of the curves can be increased by using parameters in the scheme. Later on, a class of 4-point subdivision schemes with two parameters was presented in 2004 by [2]. In 2007, Siddiqi and Ahmad [3] presented a 3-point approximating C 2 scheme with single parameter. Shen and Huang [4] introduced a class of curve schemes with several parameters in 2007. Siddiqi and Rehan [5] also presented a 4-point approximating scheme The associate editor coordinating the review of this manuscript and approving it for publication was Yeliz Karaca . with one parameter in 2010. Mustafa et al. [6] presented a 3-point scheme with three parameters in 2011. In 2012, Ghaffar et al. [7] presented a class of 3-point a-ary scheme with one parameter. In 2013, Cao and Tan [8] presented a binary 5-point relaxation scheme with one parameter. Tan et al. [9] presented a 4-point C 3 scheme with two parameters in 2014. In 2014, Tan et al. [10] also presented a 5-point scheme with one parameter. Zheng et al. [11] introduced a scheme with multi-parameters in 2014. Mustafa et al. [12] introduced the families of interpolating schemes with parameters in 2014. In 2017, Feng et al. [13] presented a family of non-uniform schemes with variable parameters. Tan et al. [14] presented a new 5-point binary approximating scheme with two parameters in 2017. In 2018, Asghar and Mustafa [15] presented a family of a-ary univariate subdivision schemes with single parameter.
Another trend to introduce the combined schemes was evoked. These schemes have interpolating and approximating behaviors. Beccaria et al. [16] introduced a unified  framework for interpolating and approximating schemes with a parameter in 2010. Rehan and Siddiqi [17] presented a combined binary 6-point scheme with a parameter in 2015. Hameed and Mustafa [18] presented a class of schemes with non-uniform setting of the parameter in 2016. In 2012, Pan et al. [19] presented a C 2 continuous combined ternary approximating and interpolating schemes. The new family of ternary 6-point combined schemes with C 3 -continuity was derived by Shi et al. [20] in 2018.

A. OUR CONTRIBUTION
The main purpose of this work is to increase the number of choices, for end user, of the scheme for curve modeling with less complexity and maximum smoothness in the shapes. Our scheme has less complexity and generates curve of maximum degree of smoothness than its counterpart (i.e., 4-point) schemes. It is proved in the last section of this paper. It is evidenced that there are situations or data when our scheme is more appropriate to use than the well-known schemes. This type of data is represented by Figure 1(a). It is also called initial sketch. Figure 1(b) and Figure 1(c) are generated by [22] and B-spline of degree 5 respectively, while the fitted curve by presented refinement scheme for α = −0.2e-1 and β = 0.088542 is shown in Figure 1(d). The curve generated by [22] is not compatible with the initial sketch due to its oscillating behaviour, while the gentle behaviour of the B-spline of degree 5 pushes the limit curve away from its initial sketch. The presented refinement scheme gives good result in this case. This paper is divided into following sections. In Section 2, we present a class of schemes with two parameters. We also discuss the theoretical analysis of the shapes produced by these schemes in this section. In Section 3, we present the numerical and graphical analysis of the shapes produced by the schemes. Section 4, is devoted for the comparison and conclusion.

II. A CLASS OF REFINEMENT SCHEMES
If we have an initial sketch of any shape obtained by joining the 2D points f 0 i , i ∈ Z then to refine the sketch, we suggest the following refinement scheme where k represents the refinement level while α and β are the shape parameters. The refinement scheme consists of two refinement rules. One of the rules (called even rule) is used to update the vertices of the initial sketch and it uses three points of the initial sketch to insert a new one, hence its complexity is three. While the other rule (called odd rule) is used to subdivide the edges and it uses four points of the initial sketch to insert a new one, hence its complexity is four. Since there are two rules in this scheme, therefore it is called binary scheme. Graphical representation is shown in Figure 2. We can get a class of refinement schemes from (1) by assigning different values to the parameters. If we arrange the points involved in odd and even rules of (1) as then the sequence of coefficients of these points in odd and even rules is This sequence can be represented in terms of the following Laurent polynomial Generally, Laurent polynomial is expressed in the form . . . + a −n z −n + a −(n−1) z −(n−1) + . . . + a −1 z −1 + a 0 + a 1 z + . . . + a n−1 z n−1 +a n z n +. . ., where a i are constants and only finitely many a i are nonzero. The detailed information about Laurent polynomial can be find in [25], [26]. Laurent polynomial or z-transform is a main tool to analyze the schemes.
Since there are two parameters in the scheme therefore we may express the one in terms of the other to explore the properties of the scheme. Classically, the schemes are analyzed by checking their degree of continuity and degrees of polynomial generation and reproduction under some conditions. These conditions can be used to express the parameters in terms of the others. The parameters α and β can be expressed as α We get subclasses of (1) by substituting values of α and β in (1). Hence the refinement rules corresponding to β = and its Laurent polynomial is While the refinement rules corresponding to α = 1 and its Laurent polynomial is The following special cases show that the scheme (1) is the generalized version of B-spline of degree 1, 3 and 5. It is also the generalized version of the interpolatory schemes of [21] and [22].
• For α = 3 16 and β = 1 26 , we get B-spline scheme of degree-5. Through out the paper, S u , S v S c r , S c L r and S b r are the schemes corresponding to the polynomials u(z), v(z), c r (z), c L r (z) and b r (z) for r ≥ 0, L ≥ 2 respectively. Some of these polynomials are defined in coming section.

A. THEORETICAL ANALYSIS OF THE SHAPES
In the similar way to the arguments in [25], we can identify the ranges of parameters to get the shapes of different degree of smoothness (i.e. order of continuity) produced by the scheme.
Lemma 1: Let {f 0 i } i∈Z be the initial sketch of the shape then the refinement scheme S u defined by (2) produces the C 0 -continuous shape for the parametric interval 7 16 − 1 16

91.
Proof: To find out the C 0 -continuity of the refinement scheme (2), we rewrite equation (3) as This implies that The infinity norm of c 0 is calculated as This implies It is clear that ||c 0 || ∞ < 1 for − 1 4 < α < 3 4 . So, the scheme S c 0 is contractive. We may improve the range of parameter by taking c L 0 (z) = c 0 (z)c 0 (z 2 ) . . . c 0 (z 2 L−1 ), where L > 0. For simplicity, we may take L = 2 (6) and c 0 (z 2 ) is obtained by replacing z 2 in the place of z in (6). Hence we get This implies that where R i 0 are the coefficients of z i for i = −9, −8, . . . , 6 in (7) respectively. The infinity norm of c 2 0 is This implies that Since Therefore, ||c 2 0 || ∞ < 1 for common range of the parameter So S c 2 0 is contractive and the scheme S b 0 is convergent and the scheme S u is C 0 -continuous.
Lemma 2: Let {f 0 i } i∈Z be the initial sketch of the shape then the refinement scheme S u defined by (2) produces the C 1 -continuous shape for the parametric interval 5 16 −

33.
Proof: To find out the C 1 -continuity of the refinement scheme (2), we rewrite equation (3) as This implies that The infinity norm of c 1 is calculated as So we get Hence ||c 1 || ∞ < 1 for 0 < α < 1 2 . So, the scheme S c 1 is contractive. Now we use c 2 1 (z) = c 1 (z)c 1 (z 2 ) to improve the range of parameter. This implies This implies that where R i 1 are the coefficients of z i for i = −9, −8, . . . , 3 in (8) respectively. The infinity norm of c 2 1 is Hence we get Hence ||c 2 1 || ∞ < 1 for Hence the common range for C 1 -continuity of the scheme S u is This completes the proof. Lemma 3: Let {f 0 i } i∈Z be the initial sketch of the shape then the refinement scheme S u produces the C 2 -continuous shape for the parametric interval 0 < α < 1 2 . Proof: To find out the C 2 -continuity of the refinement scheme (2), we rewrite equation (3) as We take infinity norm of c 2 It is to be noted that ||c 2 || ∞ < 1 for 0 < α < 1 2 . Therefore, the scheme S c 2 is contractive and for further improvement take c 2 2 (z) = c 2 (z)c 2 (z 2 ). This implies Now by taking infinity norm of c 2 2 , we get Which is less than 1 for The common range is In this case, further improvement in the range has not been seen. This completes the proof. Similarly, wet get the following result Lemma 4: Let {f 0 i } i∈Z be the initial sketch of the shape then the refinement scheme S u produces the C 3 -continuous shape for the interval 1 8 < α < 3 16 + 1 16 √ 5. From Lemmas 2 -4, we get the following Theorem 5: Let {f 0 i } i∈Z be the initial sketch of the shape then the refinement scheme S u produces the C 0 -continuous shape for the parametric interval 7 where b i (z) = (1 + z)c i (z): i = 0, 1, 2, 3, 4. We further calculate This implies that and if we put i = 4 and β = 1 26 in (10), we get Thus we have For the special value of α = 0 in (1), we get the following interpolating scheme The Laurent polynomial of (11) is Theorem 8: The interpolating scheme defined by (11) produces C 0 and C 1 -continuous shapes for 1 where m i (z) = (1 + z)n i (z): i = 1, 2. We further calculate the expression n 2 i (z) = n i (z)n i (z 2 ), i = 0, 1. The infinity norm ||n 2 0 || ∞ and ||n 2 1 || ∞ are less than one for For β = 0 in (1), we have the following scheme.

B. POLYNOMIAL GENERATION AND REPRODUCTION
In this section, we discuss another feature of the scheme. If the initial data is sampled from the polynomial of degree d then we are interested to see whether or not the new data obtained from the scheme lie on the graph of same polynomial. If the new data lie on the graph of same polynomial then we say that the scheme reproduces polynomial of degree d. If the new data lie on the graph of another polynomial but with degree d then we say that the scheme generates polynomial of same degree. Mathematically, polynomial generation of degree d is equivalent to u(z) This means the scheme has primal parametrization. In the similar way to the arguments in [23], we can get the degree of polynomial generation and reproduction with respect to the primal parametrization of the scheme.
Lemma 10: The degree of polynomial generation of the scheme (2) is 3.
Proof: Since Laurent polynomial u(z) of the scheme (2) is This completes the proof. 98322 VOLUME 8, 2020 Theorem 11: A refinement scheme (2) reproduces polynomials of degree 1 with respect to the primal parameterizations with τ = 0 if and only if Proof: The Laurent polynomial (3) of the scheme (2) and its derivative with respect to z are Taking z = −1 in all above, we get It is easy to see that This further implies that Thus and This completes the proof. Proof: The Laurent polynomial (3) of the scheme (2) and its derivative with respect to z are Taking z = −1 in all above, we get It is easy to see that This further implies that where v (k) (1) is the kth derivative of v(z) at z = 1.
Lemma 17: The degree of polynomial generation of the scheme (11) is 1.

VOLUME 8, 2020
Theorem 18: A refinement scheme (11) reproduces polynomials of degree 1 with respect to the primal parameterizations with τ = 0 if and only if where w (k) (z) is the kth derivative of the Laurent polynomial of the scheme (11).
Lemma 19: The degree of polynomial generation of the scheme (13) is 1.
Theorem 20: A refinement scheme (13) reproduces polynomials of degree 1 with respect to the primal parameterizations with τ = 0 if and only if where y (k) (z) is the kth derivative of the Laurent polynomial of the scheme (13).

C. LIMIT STENCILS OF THE SCHEMES
Since the limit curves produced by the refinement schemes do not have closed form so the traditional methods fail to compute the points on the curve. In this case, we compute the limit stencils of the schemes. This is just the sequence of scalers. If we consider the initial points of the polygon as a sequence of vectors. Then making the linear combination of these vector and scalars, we get the point on the limit curve produced by the refinement scheme.
Theorem 21: The limit stencil of the scheme (2) is Proof: By taking i = −1 and 0 in even and odd rules and i = 1 in even rule of the scheme (2), we get Its matrix representation is: Eigenvalues of this matrix are The matrix of eigenvectors V corresponding to these eigenvalues is Its inverse V −1 is, as shown at the bottom of the next page. The diagonal matrix D of the eigenvalues can be written as So we have, f j+1 Hence the limit stencil is 1 3 Similarly, we get the following results. Theorem 22: The limit stencil of the scheme (4) is

IV. COMPARISON AND CONCLUSION
In this paper, we have presented a unified refinement scheme with two parameters. This scheme has unified interpolating, approximating and interproximate schemes. One shape parameter controls the interpolating property of the refinement scheme while the other controls the approximating property of the scheme. Interproximate scheme can be • C 2 -continuous shape for − 1 16 < β < 3 8 . • C 3 -continuous shape for 0 < β < 9 164 + 7 164 √ 5.
• C 4 -continuous shape for β = 1 26 . The scheme for α = 0 produces • C 0 -continuous shape for 1 8 − 1 8 √ 13 < β < 3 8 . • C 1 -continuous shape for 1 8 − 1 8 √ 5 < β < 0. The scheme for β = 0 produces • C 0 -continuous shape for − 1 4 < α < 3 4 . • C 1 -continuous shape for 0 < α < 1 8 + 1 8 √ 5. In Table 1, we summarize the properties of the schemes for different values of shape parameters α and β. Since the degree of polynomial generation of all the schemes is one therefore by [26] the approximation order of all schemes is two. Since the odd and even rules of our schemes have complexity 3 and 4 therefore we have compared the order of continuity of the shapes produced by our schemes and existing schemes having the complexity 3 and 4. The comparison is presented in Table 2. The schemes presented in this table with labels * and ** have complexity 4 in both odd and even rules but in our schemes the complexity in odd rule is less than 4. The schemes with labels * and ** can produce one order extra continuous shapes but with the high cost of complexity comparative to our approximating schemes. Moreover, the difference between the shapes with order of continuities 4 and 5 can not be observed by our naked eyes so one extra order of continuity in the shapes with high cost of complexity has no use. Furthermore, our schemes produce higher order continuous shape then ternary combined family of schemes [19], [20]. Our interpolating and exiting schemes in the combined family produce the same order of continuous shapes. Overall, we conclude that the proposed schemes are better in the sense of complexity and continuity than the combined binary and ternary family of schemes as well as other individual schemes. In this paper, we have also suggested the method to compute the points on the limit curves. In future, we will study that how the theoretical results of the presented refinement schemes are potentially applicable to the design of graph filters [24].