A Family of High Continuity Subdivision Schemes Based on Probability Distribution

Subdivision schemes are famous for the generation of smooth curves and surfaces in CAGD (Computer Aided Geometric Design). The continuity is an important property of subdivision schemes. Subdivision schemes having high continuity are always required for geometric modeling. Probability distribution is the branch of statistics which is used to find the probability of an event. We use probability distribution in the field of subdivision schemes. In this paper, a simplest way is introduced to increase the continuity of subdivision schemes. A family of binary approximating subdivision schemes with probability parameter p is constructed by using binomial probability generating function. We have derived some family members and analyzed the important properties such as continuity, Holder regularity, degree of generation, degree of reproduction and limit stencils. It is observed that, when the probability parameter p = 1/2, the family of subdivision schemes have maximum continuity, generation degree and Holder regularity. Comparison shows that our proposed family has high continuity as compare to the existing subdivision schemes. The proposed family also preserves the shape preserving property such as convexity preservation. Subdivision schemes give negatively skewed, normal and positively skewed behavior on convex data due to the probability parameter. Visual performances of the family are also presented.


INTRODUCTION
Initially, De Rham [1] gave the idea of CAGD with geometry. After that Chaikin [2] introduced a procedure to generate curves from limited number of points. This algorithm was one of the first refinement algorithms to generate curves. Dyn et. al. [3] presented a subdivision scheme with tension parameter which is C 1 continuous for a certain range of parameter. Mustafa and Liu [4] presented a new solid parametric subdivision scheme. 1 , 64 Mustafa et. al. [10] presented a proof of 6-point scheme.
They said that the scheme produce C 2 and C 3 limit curve when w [0,0425], and w[0.0139, 0.0143], respectively.
Bari et. al. [11] presented shape preserving subdivision schemes. Siddiqi and Noreen [12] presented the convexity preserving property of 6-point ternary interpolating subdivision scheme with the tension parameter. Tan et.
al. [13] also discussed the convexity preserving property of 5-point binary scheme.
Zheng et. al. [14] presented a technique to increase the continuity of any subdivision scheme. They multiply the symbol of the scheme with (1+z/2) k factor to get C n+k continuous subdivision schemes. But the technique used by them increase the complexity of the scheme. In this paper, we have presented a way to introduce a new family of subdivision schemes. Our technique is based on probability generating function of Binomial probability distribution. By using this technique one can able to increase the continuity, Holder regularity, degree of generation, degree of reproduction of any subdivision scheme without increasing the complexity of the scheme.
Probability is the chance of occurrence. A variable that shows the probabilities as the outcome of an experiment is called random variable. A list of probabilities associated to each value of random variable is called probability distribution and a function that is used for this purpose is called probability generating function. Binomial probability distribution was introduced by Bernoulli [15].
The paper is organized as follows. In Section 2, we construct a general symbol of family of binary approximating subdivision scheme. Complete analysis of some family members of proposed family is presented in Section 3. Convexity preservation is discussed in Section The scheme corresponding to Equation (7) is:

ANALYSIS OF THE SCHEMES
Aim of this section is to present the analysis of proposed family of binary approximating subdivision schemes. Here we only present the analysis of one family member A 2 of binary subdivision schemes. The analysis of rest of the schemes are similar.
We use Laurent polynomial (symbol) method [18] to calculate integer class continuity, degree of generation and degree of reproduction of the A n schemes. Moreover, Holder regularity analysis is done by using Riouls [19] method. Using [18], the subdivision scheme with symbol Polynomial reproduction of degree d requires polynomial generation of degree d.

Remark-2:
At p = 1/2, the degree of generation of the scheme Equation (8)  Proof: By taking the first derivative of Equation (7) and put z = 1, we get A' 2 (1) = -2p 2 -10p + 9. This implies that  = 1/2 (-2p 2 -10p + 9), for different values of p, the scheme corresponding to thesymbol A 2 has dual as well as primal parametrization. We can easily verify that first and second derivative of A 2 at z = -1 are equal to 0. Further we can also verify Equation (9) for k = 0 and 1.
This completes the proof.
In Table 1, we present the complete analysis of some family members of the proposed family of binary approximating subdivision schemes. Here we see that at p = 1/2 the order of continuity and degree of generation have been increased. Moreover, we also present the Holder regularity analysis. Table 1

Limit Stencil
The limit stencil is a way to obtain a point on the limit curve by using initial control points. The procedure for calculating the limit stencils is presented in [18]. In Table   2, we present the limit stencil of some family members of binary approximating subdivision schemes at p = 1/2.

CONVEXITY PRESERVATION
In this section, we show the convexity preservation of the scheme A 2 . It is clear from Fig. 1(a-c) that if initial control points are strictly convex. Then the limit curves generated by the scheme corresponding toA 2 show positively, normal and negatively skewed behavior on convex data for p 1/2, p = 1/2 and prespectively.   (11) By using Equation (11), the scheme Equation (8) We use mathematical induction to prove This implies C 0 for  1  2 .
Hence proposed schemeA 2 preserves convexity. This completes the proof.

COMPARISON AND APPLICATIONS
In this section, we present the comparison and applications of the proposed family of schemes.

Comparison of Continuity Analysis
Here we present the comparison of continuity analysis of the proposed family of schemes with existing parametric subdivision schemes. It is clear from Table 3 that our proposed family gives higher continuity comparative to the existing parametric subdivision schemes. Table 3 shows the comparison of continuity analysis.
Here, E, OC, A n and OC 1/2 denote the existing schemes, order of continuity of existing schemes, proposed family of schemes and continuity of proposed schemes at p = 1/ 2 respectively.

Applications
Here we discuss the visual performance of the proposed family of subdivision schemes. The control polygons are drawn by doted lines and the smooth curves obtained by our proposed schemes by full lines. Fig. 2(a-c) represents the applications of proposed scheme A 2 at p = 1/64, 1/2 and 9/10. Fig. 3(a-f)