Variations in 2D and 3D models by a new family of subdivision schemes and algorithms for its analysis

A new family of combined subdivision schemes with one tension parameter is proposed by the interpolatory and approximating subdivision schemes. The displacement vectors between the points of interpolatory and approximating subdivision schemes provide the flexibility in designing the limit curves and surfaces. Therefore, the limit curves generated by the proposed subdivision schemes variate in between or around the approximating and interpolatory curves. We also design few analytical algorithms to study the properties of the proposed schemes theoretically. The efficiency of these algorithms are analyzed by calculating their time complexity. The graphical representations and graphical properties of the proposed schemes are also analyzed.


Introduction
A common problem in computer aided geometric design consists of drawing smooth curves and surfaces that either interpolate or approximate a given shape. Subdivision schemes are one of the most successful approaches in this area. Subdivision schemes are used to construct smooth curves and surfaces from a given set of control points through iterative re nements. Due to its clarity and simplicity, subdivision schemes have been esteemed in many elds such as image processing, computer graphics, and computer animation. Generally, subdivision schemes can be categorized as interpolatory and approximating subdivision schemes. Interpolatory schemes get better shape control while approximating schemes have better smoothness.
e most important interpolatory 4-point binary subdivision scheme was proposed by Dyn et al. [1] which gives the curves with C 1 continuity. is scheme was extended to the 6-point binary interpolatory scheme by Weissman [2]. e 6-point scheme presented by Weissman [2] gives the limiting curves with C 2 continuity. Most of the approximating schemes were developed from splines. Two of the most famous approximating schemes are Chaikin's subdivision scheme [3] and cubic B-spline subdivision scheme [4], which actually generate uniform quadratic and cubic B-spline curves with C 1 continuity and C 2 continuity, respectively.
ere also exists a class of parametric subdivision schemes, called combined subdivision schemes. Combined subdivision schemes can be regarded either as a member of the approximating schemes or of the interpolatory schemes according to the speci c values assumed by the parameters. Pan et al. [5] presented a combined ternary 3-point relaxed subdivision scheme with three tension parameters. Novara and Romani [6] extended their technique to construct a 3-point relaxed ternary combined subdivision scheme with three tension parameters to generate curves up to C 3 continuity. Recently, Zhang et al. [7] also used similar technique to construct a 4-point relaxed ternary scheme with four tension parameters to generate the curves up to C 4 smoothness. e nice performance of combined subdivision schemes motivates us to the direction of constructing the binary combined subdivision scheme. Our contribution in this manner is presented in this research. e remainder of this article is organized as follows: Section 2 deals with some basic definition and basic results. Section 3 provides the framework for constructing a family of combined binary subdivision schemes. In Section 4, we provide the analytical properties of the proposed family of schemes. Section 5 gives the graphical behaviors and graphical properties of the proposed family of schemes. Concluding remarks are presented in Section 6.

Preliminaries
In this section, we present some basic notations and definition relating to the results which we have used in this article.
A general compact form of linear, uniform, and stationary binary univariate subdivision scheme S a which maps a polygon P k � P k i , i ∈ Z to a refined polygon P k+1 � P k+1 i , i ∈ Z is defined as e symbol of the above subdivision scheme is given by the Laurent polynomial where a � a i , i ∈ Z is called the mask of the subdivision scheme. e necessary condition for the binary subdivision scheme (1) to be convergent is that its mask satisfies the basic sum rule j∈Z a i− 2j � 1, i � 0, 1, which is equivalent to the following relation: a(− 1) � 2 and, a(− 1) � 0.
Definition 1. An algorithm is a procedure for solving a mathematical problem in a finite number of steps that frequently involves repetition of an operation.
Definition 2. In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm. It is denoted by T(n), where n is the input of the given algorithm. e time complexity indicates the number of times the operations are executed in an algorithm. e number of instructions executed by a program is affected by the size of the input and how their elements are arranged.

Definition 3.
Step count method considers each and every step to the given algorithm for calculating the final time complexity of the given algorithm. So, each and every step in the given algorithm contributes to the final complexity of given algorithm.

Family of the Combined Primal Schemes
In this section, we discuss the construction of the family of (2n + 2)-point binary relaxed schemes. ese schemes are the binary schemes, so are the combination of two refinement rules. One of these rules, consists of the affine combination of (2n + 1)-point of level k to insert a new point at level k + 1, is used to update the vertex of the given polygon; hence, it is called the vertex rule. e other refinement rule, consists of the affine combination of (2n + 2)-points of level k to insert a new point at level k + 1, is used for subdividing each edge of the given polygon and so is called the edge rule. Before going to the construction process of the family of combined subdivision schemes, we discuss our motivation behind this research here.

Motivation and Contribution.
In this article, we define a novel technique for constructing combined subdivision schemes by using interpolatory and approximating subdivision schemes with optimal performance in curves and surface fitting. Since the family of interpolatory schemes by Deslauriers and Dubuc [8] gives optimal performance in reproducing polynomials and the family of approximating B-spline schemes gives good performance in fitting curves with higher continuity, so we are motivated to unify them into a single family of schemes. e (2n + 2)-point interpolatory subdivision schemes defined by Deslauriers and Dubuc [8] can reproduce polynomials up to degree 2n + 1 and generate the limiting curves with C 1 . Whereas, the (2n + 2)-point relaxed B-spline approximating schemes are C 4n -continuous, while the degree of polynomial reproduction of these schemes is only one. Since the discussed two schemes are the primal schemes, hence each of them is the combination of one vertex and one edge rules. e vertex rule of interpolatory schemes keeps the initial vertices at every level of subdivision, while the vertex rule of the approximating schemes updates the initial vertices at each level of subdivision. Similarly, the edge rule of each scheme subdivides every edge of the given polygon into two edges at each level of subdivision. e limit curves generated by the interpolatory and approximating schemes after 4 subdivision levels are shown in Figure 1. In this article, we construct a family of combined subdivision schemes with one parameter which controls the shape of the limit curve.
We construct the new family of schemes by the family of interpolatory schemes defined by Deslauriers and Dubuc [8] and the family of approximating B-spline schemes. erefore, the proposed family of schemes carry the properties of these two types of schemes and give optimal numerical and mathematical results. From Figure 2. Figure 3, we can see the better performance of the proposed schemes than that of their parent schemes (schemes which are used in their construction). Figure 2 shows that if we choose the value of tension parameter for the proposed 4-point scheme between − 2.5 and − 2, then this scheme is good for fitting noisy data. Figure 3(c) shows that the curves fitted by the proposed 6point scheme do not give the artifacts as presented by the 6point scheme [8] in Figure 3(a). is figure also shows that the curve fitted by the proposed scheme preserve the shape of the initial polygon if we take value of tension parameter between − 0.5 and 0, while the curve fitted by the 6-point B-spline scheme does not have this ability, as shown in Figure 3 Construction of the family of schemes is discussed in coming subsections in detail.

Framework for the Construction of Family of Subdivision
Schemes.

e Geometrical and Mathematical Interpretation of the Family of Schemes.
e family of proposed schemes (9) is obtained by moving the points R k+1 2i,2n+2 and R k+1 2i+1,2n+2 of the family of schemes (5) to the new position according to the displacement vectors αD k+1 2i,2n+2 and αD k+1 2i+1,2n+2 , respectively. In order to make the construction of the family of schemes understandable, we explain the following steps by fixing n � 1 and k � 0.
Step 1. We take the points P 0 i+4,4 � P k i+4 , and P 0 i+5,4 � P k i+5 as the initial control points, i.e. at zeroth level of subdivision. e points are shown by red bullets in Figure 4. In this figure, the combination of red lines makes the initial polygon.
Step 2. Now, we calculate points R k+1 2i and R k+1 2i+1 which are denoted by blue solid squares in Figure 4. We calculate fifteen points (blue solid squares) by applying the following refinement rules of the 4-point interpolatory scheme of Deslauriers and Dubuc [8] on the initial points All the points inserted by the above subdivision scheme are shown in Figure 4; however, we restrict our next calculations to only two points R k+1 2i and R k+1 2i+1 . In this figure, blue polygon is the initial polygon obtained from the scheme (13).
Step 3. In this step, we calculate and restrict next calculations to only the points Q k+1 2i and Q k+1 2i+1 denoted by black solid squares. Again, the fifteen points, denoted by black solid squares in Figure 4, are calculated by the refinement rules of the 4-point relaxed binary B-spline scheme of degree-5 which is defined as follows: In this figure, black polygon is the initial polygon made by the above scheme.
Step 4. Now, we calculate the displacement vectors from the points defined in (14) to the points defined in (13). ese vectors are D k+1 Figure 4, the above displacement vectors and the other related vectors are shown by green vectors.
Step 5. In this step, we calculate vectors αD k+1 2i and αD k+1 2i+1 , where α is any real number that can change the magnitude or the direction or both of the vectors D k+1 2i and D k+1 2i+1 . In Figure 4, these vectors are denoted by magenta lines.
Step 6. e last step of our method is to translate the points obtained by the refinement rules in (13) to the new position by using displacement vectors αD k+1 2i and αD k+1 2i+1 . So, we get following refinement rules of the new combined 4-point relaxed scheme In Figure 4, the points of the above proposed scheme are denoted by black solid circles.

Remark 2.
For the positive values of tension parameter α, the curves generated by the proposed subdivision schemes lie outside the initial polygon, whereas for negative values of α, the curves lie inside the initial polygon. For α � 0, the curve generated by the proposed schemes passes through the initial control points.

Analysis of the Proposed Family of Subdivision Schemes
In this section, we prove the characteristics of the proposed subdivision schemes analytically. We prove few theorems and present various algorithms to check these properties.

Support.
e support of a subdivision scheme represents the portion of the limit curve effected by the displacement of a single control point from its initial place. Now, we calculate the support of the proposed family of subdivision schemes.

Theorem 1.
If ϕ 2n+2 is the basic limit function of the family of subdivision schemes (9), then, its support is Proof. To calculate support of the proposed family of subdivision schemes, we define a function which give zero value to all control points except one point, P 0 0 , whose value is one.
Let ϕ 2n+2 : R ⟶ R be the basic limit function such that . erefore, the support of basic limit function ϕ 2n+2 is equal to the support of the subdivision scheme S a 2n+2 .
When we apply a proposed (2n + 2)-point relaxed scheme on the initial data P 0 i , then at first subdivision step, the nonzero vertices are ϕ 2n+2 − 2 0 (2n + 1) At second subdivision step, the nonzero vertices are At third subdivision step, the nonzero vertices are Figure 4: Geometrical construction of the scheme for k � 0 and n � 1.

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Mathematical Problems in Engineering Similarly, at k-th subdivision step, the nonzero vertices are Hence, the support size of basic limit function is the difference between subscripts of the maximum nonzero vertex and the minimum nonzero vertex, i.e.
Applying limit k ⟶ ∞, we get support size of Smoothness Analysis. Now, we calculate the order of parametric continuity of the proposed subdivision schemes by using the Laurent polynomial method. Detailed information about refinement rules, Laurent polynomials, and convergence of a subdivision scheme can be found in [9][10][11].
e continuity of the subdivision schemes can be analyzed by the following theorems.
Theorem 2 (see [10]). A convergent subdivision scheme S a corresponding to the symbol is C n -continuous iff the subdivision scheme S b corresponding to the symbol b(z) is convergent.
where c l i are the coefficients of the scheme S l c with symbol In the following theorem, we prove that the proposed subdivision schemes satisfy the condition which is necessary for the convergence of a subdivision schemes.

Theorem 4.
e family of subdivision schemes with the refinement rules defined in (9) and (10) satisfies the basic sum rule. Proof.
Since Mathematical Problems in Engineering 7 2n j�0 4n + 2 erefore, the family of subdivision schemes with the refinement rules defined in (9) satisfy the basic sum rule which is also the necessary condition for the convergence of a subdivision scheme.
We use eorems 2 and 3 to design Algorithm 1. is algorithm is designed to calculate the level of parametric continuity of the limit curves generated by the proposed subdivision schemes. In Tables 1 and 2, we collect the results obtained from Algorithm 1 for L � 1 and L � 2, respectively. We use the step count method to find out the time complexity of the iterative Algorithm 1. Firstly, we compute the if-case time complexity.
To calculate time complexity of Steps 12-21, we ignore total cost of Steps 18-19. erefore, the else-case time complexity of Algorithm 2 is where C 3 � C 11 + C 12 + 2C 13 + 2C 14 + 2C 15 + 2C 16 + C 17 + C 18 + C 19 + C 20 and C 4 � C 13 + C 14 + C 15 + C 16 . Hence, time complexity of Algorithm 2 is O(n). In the same manner, we can prove that the time complexity of Algorithm 3 is also O(n).
A scheme has the overshoots near to the point ξ of discontinuity if A scheme has the undershoots near to the point ξ of discontinuity if Algorithm 4 is designed to calculate the range of tension parameter α for which the proposed schemes exhibit undershoot Gibbs oscillations. e results obtained from this algorithm are summarized in Table 9. We calculate the time complexity of Algorithm 4 which depends on the input (1) input: e mask a j,2n+2 : j � 0, 1, . . . , 4n + 2 of the family of proposed schemes whose mask symbol is defined in (11), n ∈ Z + (2) set: A j � a j,2n+2 (3) for j � 0 to 4n + 2 do (4) calculate: an interval, say (α j 1 , α j 2 ), for α by solving A j > 0 (5) end for (6) calculate: intersection, say (α 1 , α 2 ), of 4n + 3 intervals (α  value k by using the step count method. e time complexity of this algorithm does not depend on the input value n. Hence, the time complexity of this algorithm is O(k).

Shape Preserving Properties.
Here, we discuss the shape preserving properties, monotonicity and convexity preservation, of the proposed family of subdivision schemes. We prove that the proposed subdivision schemes preserve monotonicity and convexity for a special interval of tension parameter. For this, firstly, we calculate an interval of α for which the proposed subdivision schemes having the bellshaped mask. en, by using results given in [13], we prove that the proposed schemes preserve monotonicity and convexity for a special interval of α. A 2N-degree polynomial 2N j�0 b j x j is said to be a polynomial with bell-shaped coefficients if it satisfied the following conditions:        e mask symbol of the proposed family of schemes is defined in (10) which is a polynomial of degree-(4n + 2) satisfying the following two conditions: e other two conditions depend on value of α which can be calculated by Algorithm 5. e outputs obtained from Algorithm 5 are given in Table 10. Proof. By eorem 4, we see that the proposed subdivision schemes satisfy the basic sum rule. Also from Algorithm 5, we get that the proposed 4-point, 6-point, and 8-point subdivision schemes have bell-shaped mask for − 1.555555556 < α < − 0.6666666667, − 1.200000000 < α < − 0.5882352941, and − 1.149842822 < α < − 0.9523809524, respectively, so it is easy to prove the required result by using ( [13], eorem 3). Proof. By eorem 4, we see that the proposed subdivision schemes satisfy the basic sum rule. Also from Algorithm 5, we get that the proposed 4-point, 6-point, and 8-point subdivision schemes have bell-shaped mask for − 1.555555556 < α < − 0.6666666667, − 1.200000000 < α < − 0.5882352941, and − 1.149842822 < α < − 0.9523809524, respectively. Furthermore from (11), we can also see that the mask symbols of the proposed subdivision schemes have the factor (1 + z) 2 . Hence, the required result is proved by ( [13], eorem 4). e proposed Algorithm 5 needs O(n) time to give output.

Numerical Examples
In this section, we show the numerical performance of the proposed schemes, based on the results which have proved in previous section. Figure 5 displays the support of the subdivision scheme when the initial data is taken from a basic limit function defined in (16). Figure 6 shows the curves fitted by the proposed (2n + 2)-point subdivision schemes. It is clear from this figure that the tension parameter involved in the proposed subdivision schemes provides relaxation in drawing and designing different types of curves by using the same initial points. Figures 7 and 8 show the performance of the proposed subdivision schemes when the initial data is taken from monotonic and convex functions, respectively. Figures 9 and 10 show that the proposed subdivision schemes also carry the characteristics of keeping the shape of the initial polygon and of fitting the noisy data points, respectively, which is an important characteristic of the proposed family. In Literature, it is hard to find a single subdivision scheme which carries both characteristics simultaneously. e performance of the tensor product scheme of the proposed 4-point scheme in surface fitting is shown in Figure 11. Subfigures Figures 11(e), 11(f ), 11(g), 11(h), 11(m), 11(n), 11(o), and 11(p), respectively are the mirror images in ij-plane of Subfigures 11(a), 11(b), 11(c), 11(d), 11(i), 11(j), 11(k), and 11(l).

Concluding Remarks
In this article, we have derived the general refinement rules of the family of (2n + 2)-point primal binary schemes. e proposed schemes are originated from the schemes by Deslauriers and Dubuc [8]. We have also presented certain efficient algorithms with linear time complexity to check the properties of the proposed schemes. e beauty of the proposed family of schemes is that they carry the characteristics of both of its parent schemes. e proposed schemes show optimal continuity and polynomial generation. Proposed schemes reproduce polynomials up to degree one, while for specific value of tension parameters, they can reproduce polynomials up to degree 2n + 1. e proposed schemes eliminate Gibbs phenomenon for negative values of tension parameter. Moreover, for a specific range of tension parameter, the proposed schemes also preserve monotonicity and convexity. e proposed schemes provide flexibility in fitting curves and surfaces, that is, using the same data points, we can variate the 2D and 3D models in between or around the control polygon/mesh. Proposed schemes are a good choice for noisy data. In future, we will derive the nonuniform form of the proposed schemes.
Since from the methods used in our work, some of the most representative computational intelligence algorithms can be used to solve the problems, such as Monarch Butterfly Optimization [14,15], Earthworm Optimization Algorithm [16], Elephant Herding Optimization [17], Moth Search Algorithm [18], Slime Mould Algorithm [19], Hunger Games Search [20], Runge Kutta Optimizer Method [21], Colony Predation Algorithm [22], and Harris Hawks Optimization Method [23]. ese algorithms/methods can be used in the future to solve these type of problems. [24,25].

Data Availability
e data used to support the findings of the study are available within this paper.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.