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 is analyzed by calculating their time complexity. The graphical representations and graphical properties of the proposed schemes are also analyzed.
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 refinements. Due to its clarity and simplicity, subdivision schemes have been esteemed in many fields 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. The most important interpolatory 4-point binary subdivision scheme was proposed by Dyn et al.  which gives the curves with continuity. This scheme was extended to the 6-point binary interpolatory scheme by Weissman . The 6-point scheme presented by Weissman  gives the limiting curves with continuity. Most of the approximating schemes were developed from splines. Two of the most famous approximating schemes are Chaikin’s subdivision scheme  and cubic B-spline subdivision scheme , which actually generate uniform quadratic and cubic B-spline curves with continuity and continuity, respectively. There 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 specific values assumed by the parameters. Pan et al.  presented a combined ternary 3-point relaxed subdivision scheme with three tension parameters. Novara and Romani  extended their technique to construct a 3-point relaxed ternary combined subdivision scheme with three tension parameters to generate curves up to continuity. Recently, Zhang et al.  also used similar technique to construct a 4-point relaxed ternary scheme with four tension parameters to generate the curves up to smoothness. The 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.
The 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.
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 which maps a polygon to a refined polygon is defined as
The symbol of the above subdivision scheme is given by the Laurent polynomialwhere is called the mask of the subdivision scheme. The necessary condition for the binary subdivision scheme (1) to be convergent is that its mask satisfies the basic sum rulewhich is equivalent to the following relation:
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 , where is the input of the given algorithm. The time complexity indicates the number of times the operations are executed in an algorithm. The 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.
Definition 4. Worst-case time complexity denotes the longest running time of an algorithm given any input of size . It gives an upper bound on time requirements.
Definition 5. Best-case time complexity denotes the shortest running time of an algorithm given any input of size . It gives a lower bound on time requirements.
3. Family of the Combined Primal Schemes
In this section, we discuss the construction of the family of -point binary relaxed schemes. These schemes are the binary schemes, so are the combination of two refinement rules. One of these rules, consists of the affine combination of -point of level to insert a new point at level , is used to update the vertex of the given polygon; hence, it is called the vertex rule. The other refinement rule, consists of the affine combination of -points of level to insert a new point at level , 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.
3.1. 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  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.
The -point interpolatory subdivision schemes defined by Deslauriers and Dubuc  can reproduce polynomials up to degree and generate the limiting curves with . Whereas, the -point relaxed B-spline approximating schemes are -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. The 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. The 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  and the family of approximating B-spline schemes. Therefore, 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 6-point scheme do not give the artifacts as presented by the 6-point scheme  in Figure 3(a). This 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(b).
Construction of the family of schemes is discussed in coming subsections in detail.
3.2. Framework for the Construction of Family of Subdivision Schemes
The family of -point interpolatory subdivision schemes, constructed by Lagrange’s interpolatory polynomials presented by Deslauriers and Dubuc , is defined by two subdivision rules. The vertex rule is , and the edge rule is
The refinement rules of the family of -point relaxed approximating schemes, i.e. degree- binary B-spline subdivision schemes, arewhere . Let us denote the displacement vectors from the degree- B-spline refinement points to the refinement points of -point interpolatory subdivision scheme defined in (5) after one step of refinements by , respectively. Hence, the displacement vectors are
Now, a new family of primal combined -point subdivision schemes can be obtained by translating the points , that are provided by the binary interpolatory subdivision schemes (5) to the new position according to the displacement vectors , respectively.which can be further written as
The rules defined in (9) are the refinement rules of the proposed family of -point combined schemes, which are equivalent to the following general form:
Remark 1. If , the family of proposed -point schemes converges to the family of -point interpolatory schemes proposed in . Whereas if , the family of proposed -point schemes converges to the family of -point relaxed approximating B-spline schemes of degree-.
3.3. The Geometrical and Mathematical Interpretation of the Family of Schemes
The family of proposed schemes (9) is obtained by moving the points and of the family of schemes (5) to the new position according to the displacement vectors and , respectively. In order to make the construction of the family of schemes understandable, we explain the following steps by fixing and .
Step 1. We take the points , , , , , , , and as the initial control points, i.e. at zeroth level of subdivision. The 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 and 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  on the initial pointsAll the points inserted by the above subdivision scheme are shown in Figure 4; however, we restrict our next calculations to only two points and . 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 and 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). These vectors are and . In Figure 4, the above displacement vectors and the other related vectors are shown by green vectors.
Step 5. In this step, we calculate vectors and , where is any real number that can change the magnitude or the direction or both of the vectors and . In Figure 4, these vectors are denoted by magenta lines.
Step 6. The 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 and . So, we get following refinement rules of the new combined 4-point relaxed schemeIn Figure 4, the points of the above proposed scheme are denoted by black solid circles.
Remark 2. For the positive values of tension parameter , the curves generated by the proposed subdivision schemes lie outside the initial polygon, whereas for negative values of , the curves lie inside the initial polygon. For , the curve generated by the proposed schemes passes through the initial control points.
4. 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.
The 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 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, , whose value is one.Let be the basic limit function such that . Moreover, and . Therefore, the support of basic limit function is equal to the support of the subdivision scheme .
When we apply a proposed -point relaxed scheme on the initial data , then at first subdivision step, the nonzero vertices areAt second subdivision step, the nonzero vertices areAt third subdivision step, the nonzero vertices areSimilarly, at -th subdivision step, the nonzero vertices areHence, 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 , we get support size of . Hence, is the support region of .
4.2. 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 [9–11]. The continuity of the subdivision schemes can be analyzed by the following theorems.
Theorem 2 (see ). A convergent subdivision scheme corresponding to the symbolis -continuous iff the subdivision scheme corresponding to the symbol is convergent.
Theorem 3. The scheme corresponding to the symbol is convergent iff its difference scheme corresponding to the symbol is contractive, where . The scheme is contractive ifwhere are the coefficients of the scheme 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.
Proof. The family of subdivision schemes (9) satisfies basic sum rule ifSinceHence,andTherefore, 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 Theorems 2 and 3 to design Algorithm 1. This 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 and , 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.
Time complexity of Steps 2–15 of Algorithm 1 (if-case). Tables 3–6.
Hence, the if-case time complexity of Algorithm 1 iswhere , , , and . Now, we compute the else-case time complexity of Algorithm 1.
Time complexity of Steps 16–25 of Algorithm 1 (else-case).
The else-case time complexity of Algorithm 1 iswhere , , , and . Hence, the time complexity of Algorithm 1 is .
4.3. Degrees of Polynomial Generation and Polynomial Reproduction
Generation and reproduction degrees are used to examine the behaviors of a subdivision scheme when the original data points lie on the curve of a polynomial. We suppose that the original data points are taken from a polynomial of degree . If the control points of the limit curve lie on the curve of the polynomial having same degree (i.e. ), then we say that the subdivision scheme generates polynomials of degree . If the control points of the limit curve lie on curve of the same polynomial, then we say that the subdivision scheme reproduces polynomials of degree . Mathematically, let denote the space of polynomials of degree and , , an approximation operator generates polynomials of degree if , whereas reproduces polynomials of degree if . Furthermore, the generation degree of a subdivision scheme is the maximum degree of polynomials that can theoretically be generated by the scheme, provided that the initial data is taken correctly. Evidently, it is not less than the reproduction degree. The readers may consult [9, 12] to find out more details about polynomial generation and polynomial reproduction of a subdivision scheme. We design Algorithm 2 and Algorithm 3, based on the results given in , to check the degrees of polynomial generation and polynomial reproduction of proposed schemes, respectively. Outputs obtained by implementing these algorithms are summarized in Tables 7 and 8, respectively.
Now, we find the time complexity of Algorithm 2. Firstly, we compute the if-case (Steps 2–11) time complexity and then we find the else-case time complexity (Steps 12–22). Hence, the worst-case time complexity will be the time complexity of Algorithm 2.
Time complexity of Steps 2–11 of Algorithm 2 (if-case).
To calculate the time complexity of Steps 2–11, we either consider the total cost of Steps 6–7 or of Steps 8–9. So, we consider the worst-case which is Steps 6–7. Hence, the if-case time complexity of Algorithm 2 iswhere and . Now, we compute the else-case time complexity (Steps 12–21).
Time complexity of Steps 12–21 of Algorithm 2 (else-case).
To calculate time complexity of Steps 12–21, we ignore total cost of Steps 18–19. Therefore, the else-case time complexity of Algorithm 2 iswhere and . Hence, time complexity of Algorithm 2 is . In the same manner, we can prove that the time complexity of Algorithm 3 is also .
4.4. Gibbs Oscillations
According to , linear subdivision schemes exhibit overshoots or undershoots Gibbs oscillations near to the points of discontinuity of a discontinuous function. The schemes that show undershoots generate curves free from oscillations while the schemes with overshoots give unpleasant oscillations in the limit curves. Let , and a function is said to be discontinuous at a point , ifwith .
A scheme has the overshoots near to the point of discontinuity if
A scheme has the undershoots near to the point of discontinuity if
Algorithm 4 is designed to calculate the range of tension parameter for which the proposed schemes exhibit undershoot Gibbs oscillations. The results obtained from this algorithm are summarized in Table 9. We calculate the time complexity of Algorithm 4 which depends on the input value by using the step count method. The time complexity of this algorithm does not depend on the input value . Hence, the time complexity of this algorithm is .