A Family of Binary Approximating Subdivision Schemes based on Binomial Distribution

A simplest way is introduced to generate a generalized algorithm of univariate and bivariate subdivision schemes. This generalized algorithm is based on the symbol of uniform B-splines subdivision schemes and probability generating function of Binomial distribution. We present a family of binary approximating subdivision schemes which has maximum continuity and less support size. Our proposed family members P 4 , P 5 , P 6 , and P 7 , have C, C, C and C continuities respectively. In fact, we use Binomial probability distribution to increase the continuity of uniform B-splines subdivision schemes up to more than double. We present the complete analysis of one family member of proposed schemes and give a visual performance to check smoothness graphically. In our analysis, we present continuity, Holder regularity, degree of generation, degree of reproduction and limit stencils analysis of proposed family of subdivision schemes. We also present a survey of high continuity subdivision schemes. Comparison shows that our proposed family of subdivision schemes gives high continuity of subdivision schemes comparative to existing subdivision schemes. We have found that as complexity increases the continuity also increases. In the last, we give the general formula for tensor product surface subdivision schemes and also present the visual performance of proposed tensor product surface subdivision schemes.

In this arena, we follow procedures for the shapes that are usually used in the design, computer graphics for their mathematical study. The important tool of CAGD is subdivision schemes. Subdivision schemes are the iterative formulas which are used for generation of smooth curves and surfaces form initial polygon or initial mesh.
Developing new subdivision schemes for curve and surface design has its own importance. In the field of subdivision schemes de Rham [1] and Chaikin [2] are regarded as the pioneers. They developed the corner cutting schemes which generate C 1 limit curve. Now a day wide variety of approximating and interpolating schemes have been proposed in the literature which possess shape parameters. Dyn et. al. [3] presented a four-point subdivision scheme which has C 2 continuity.
Siddiqi and Ahmad [4][5][6] offered three, five and six points approximating subdivision schemes having C 2 , C 4 and C 6 limit curves respectively when the shape parameter w = 1/4. Mustafa et. al. [7] presented 6-point approximating subdivision schemes which give C 6 continuity when the shape parameter w(-1. 6 al. [9] presented 3-point approximating subdivision scheme which is C 3 continuous at  = 1. Ghaffar and Mustafa [10] presented a family of even point ternary approximating subdivision schemes. One family member of 6-point ternary approximating subdivision scheme gives C 7 continuity at w = 1/12. Tan et. al. [11] presented a new four-point shapepreserving subdivision scheme with two shape parameters which has C 3 continuity at  = 1,  = 32.
Hameed and Mustafa [12] presented a family of univariate and bivariate binary approximating subdivision schemes using Lane-Riesenfeld algorithm.
Zheng et. al. [13] presented designing multi-parameter curve subdivision schemes with high continuity. They offered a technique to increase the continuity of subdivision scheme. The way of increasing the continuity is multiplying the symbol with (1+z/2) k factor, after multiplication they get C n+k continuous subdivision schemes. But the demerits of this technique are: when they multiply symbol with factor then complexity, support size and mathematical computation of subdivision schemes are also increased. After this literature, we offer a technique that gives high continuity subdivision schemes, less complexity and support size.
In this paper, we construct a parametric family of subdivision schemes which have high continuity, less complexity and support size. The construction of proposed family based on probability density function of binomial distribution and symbol of uniform B-splines subdivision schemes. We give the comparison of continuity analysis with uniform B-splines and other existing binary subdivision schemes which have high continuity. This paper is also a survey of high continuity subdivision schemes.
The paper is organized as follows. In Section 2, we present construction of a family of binary approximating subdivision schemes. Analysis of the proposed family is presented in Section 3. Section 4 is for general formulae of a family of tensor product surface subdivision schemes.
Applications and conclusions are drawn in Sections 5 and 6 respectively.

CONSTRUCTION OF A FAMILY OF SCHEMES
In this section, we will present an algorithm for parametric families of subdivision schemes. There are two subsections, one is for a family of Binomial distribution based subdivision schemes and other is for a family of binary approximating subdivision schemes which is based on the symbol of uniform Bsplines and probability generating function of binomial distribution.

Binomial Distribution Based Subdivision Schemes
Here we will construct binary approximating subdivision schemes using the binomial distribution. The Binomial distribution is given by:

A Family of Binary Approximating Subdivision Schemes based on Binomial Distribution
where the variablei with 0 in and the parameter n are any positive integers. The parameter p is a probability of successive events; the range of p is 0  p 1. The distribution describes the probability of exactly successes in n (fixed number) trials if the probability of success in a single trial is p (we sometimes also use 1-p, the probability for a failure). Bernoulli first presented in a work which was published in [14]. After substituting the different values of n,n1 in Equation (1), we get (n + 1)point binary approximating subdivision schemes. The general form of (n+1)-point binary approximating subdivision schemes is: If we put p = 1/2 in Equation (3), then we get a well-known Chaikin [2] corner cutting scheme. Similarly, for n = 2, we We can generate a family of (n+1)-point binary approximating subdivision schemes by using the different values of n and p(0,1).

Family of Binary Approximating Subdivision Schemes with High Continuity
This subsection is for the general symbol of a family of binary approximating subdivision schemes with one shape parameter. Probability density function of binomial distribution and (n + 1) the degree polynomial uniform Bsplines for all n1 are used to construct the general symbol of the family of binary approximating subdivision schemes.
The symbol of uniform B-splines [15] subdivision schemes are given as: The probability generating function of binomial distribution [16] is given as: The symbol defined in Equation (4) has C n-1 continuous subdivision schemes. In order to increase the continuity of uniform B-splines and generate a new family of approximating subdivision schemes, we take a product of symbol of uniform B-splines schemes (4) and probability density function of the binomial distribution (5). The general symbol of a family of approximating subdivision schemes is defined as: The simplest form of the symbol is: 2  3  2  3  2  3   2  3  3  3  2  2  3  3  3   3  3  1  ,  4  9  6  ,  3  3  6  6   ,  12  4  6  3  ,  3  9  1  ,  3  3

A Family of Binary Approximating Subdivision Schemes based on Binomial Distribution
This is the general formula for a family of binary approximating subdivision schemes with one parameter.
Our proposed family of binary approximating subdivision schemes with one parameter has high continuity, less support size and many existing subdivision schemes are particular cases of our proposed subdivision schemes.
We get the family members of approximating subdivision schemes corresponding to each value of n in Equation (6), and for different values of p can generate the different subdivision schemes. Table 1 gives the complexity and mask of proposed family of approximating subdivision schemes.

ANALYSIS OF THE SCHEMES
This section aims to present the analysis of a proposed family of binary approximating subdivision schemes. Here we only present the analysis of one family member P 2 of binary subdivision schemes, the analysis of other family members is similar.

Convergence and Smoothness Analysis
Continuity is an important property of subdivision schemes. Continuity of a subdivision scheme refers to the differentiability of the limit curve/surface produced by subdivision process. A high continuous subdivision scheme gives a more smooth limit curve. We use Laurent polynomial (symbol) method [17] to calculate integer class continuity of the P n -schemes. In general P n (z) in Equation (6) is C continuous but when we substitute the value of P=1/2, P n (z) have maximum continuity. Table 2 gives the comparison of continuity analysis of uniform B-splines schemes A n (z), proposed P n (z) schemes in general and proposed P n (z) schemes at p = 1/2.

Theorem-1:
The 3-point binary scheme corresponding to the symbol P 2 is C 2 continuous.

Proof:
The Laurent polynomial of the 3-point binary scheme is obtained by substituting n=2 in Equation (6), which is:

A Family of Binary Approximating Subdivision Schemes based on Binomial Distribution
This implies Therefore by [13], if   on Holder continuity are calculated by using Rioul's method [18]. Moreover, exact upper bounds on Holder continuity can also be derived by using Floater and Muntingh algorithm [19]. In Table 3

Survey and Comparison with Existing Schemes
Here we will present a survey of high continuity parametric subdivision schemes. We also show that our proposed family of binary approximating subdivision schemes gives the highest continuity as compare to existing parametric subdivision schemes.

TENSOR PRODUCT FAMILY OF APPROXIMATING SUBDIVISION SCHEMES
In this section, we will present the generalized formulae The general symbol of a family of approximating subdivision schemes is defined as:  Proof: From Equation (7) we have b 0 = (1-p) 2 , b 1 = 4p(1-p), b 0 = 2p 2 , k = 3, m = 2 and b = 2 thus q = 0,1,2,B o ,B 1 and B 2 are the matrices with elements

TABLE 5. SHOWS THE COMPARISON OF CONTINUITY ANALYSIS OF EXISTING PARAMETRIC SUBDIVISION SCHEMES AND PROPOSED FAMILY OF SCHEMES E,C AND P n DENOTE THE EXISTING SCHEMES, CONTINUITY OF EXISTING SCHEMES AND CONTINUITY OF PROPOSED SCHEMES At p=1/2 RESPECTIVELY
Where i,j = 1 and 2. Thus B 0 , B 1 and B 2 are given by: By [14], the Holder regularity is given by r = k -log b (), where is the joint spectral radius of the matrices B 0 , B By using Equation (9)

Generation and Reproduction Analysis
The subdivision scheme with symbol P n (z) reproduces polynomials of degree d with respect to the parameterizations with Polynomial reproduction of degree d requires the polynomial generation of degree d [13]. Support of the schemes is calculated by [20]. Table 4 summarizes the results of support, generation degree, reproduction degree, shift parameter and parametrization of proposed schemes corresponding to P n (z) and generation degree at p = 1/2 of proposed schemes corresponding to P n (z).   (7) can be written as: Hence generation degree is 2.
Theorem-4: The 3-point binary scheme corresponding to the symbol P 2 (z) reproduces polynomial of degree 1 concerning the dual parameterizations and also gives the primal parametrization at p = 1/4 Proof: By taking the first derivative of Equation (7) and put z = 1, we get (P 2 )(1) = 4p + 3 This implies that  = 4p+3/ 2, so the scheme corresponding to the symbol P 2 (z) has dual parametrization. This scheme also gives the primal parametrization at p = 1/4. We can easily verify that the first and second derivatives of P 2 (z) if z = -1 are equal to 0.
Further, we can also verify (10) for k = 0 and 1. This completes the proof.   Fig. 2(a-f) present limit curves for close polygons and Fig. 3(a-c) give the comparison of limit curves generated by the schemes corresponding to P 2 , P 3 and P 4 . After comparison, we see that P 4 gives more smooth curves as compare to P 2 + and P 3 at p = 1/2 Figs. 4-5(a-e) are the visual performances of surface subdivision schemes P 3 .

CONCLUSION
In this paper, we have presented a generalized algorithm to introduce a parametric family of binary approximating subdivision schemes with high continuity. In our proposed algorithm, the continuity of uniform B-splines subdivision schemes increased up to more than double.
We also give the complete analysis of one family member of proposed schemes. Visual performance shows the smoothness of a family of schemes graphically. Tensor product surface formula of proposed family of schemes is also presented. This paper also contains a survey of high continuity parametric subdivision schemes.