Binary Approximating Subdivision Scheme Research Articles

A shape-preserving variant of Lane-Riesenfeld algorithm

<abstract> <p>This paper introduces a family of shape-preserving binary approximating subdivision schemes by applying a shape-preserving variant on the Lane-Riesenfeld algorithm. Using the symbols of subdivision schemes, we determine convergence and smoothness, Hölder continuity, and support size of the limit curves. Furthermore, these schemes produce monotonic and convex curves under the certain conditions imposed on the initial data.</p> </abstract>

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 P4, P5, P6, and P7, have C7, C9, C11 and C13 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.

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.

Univariate approximating schemes and their non-tensor product generalization

AbstractThis article deals with univariate binary approximating subdivision schemes and their generalization to non-tensor product bivariate subdivision schemes. The two algorithms are presented with one tension and two integer parameters which generate families of univariate and bivariate schemes. The tension parameter controls the shape of the limit curve and surface while integer parameters identify the members of the family. It is demonstrated that the proposed schemes preserve monotonicity of initial data. Moreover, continuity, polynomial reproduction and generation of the schemes are also discussed. Comparison with existing schemes is also given.

Fractal Generation by 7-Point Binary Approximating Subdivision Scheme

Fractal Generation by 7-Point Binary Approximating Subdivision Scheme

A Five-Point Subdivision Scheme with Two Parameters and a Four-Point Shape-Preserving Scheme

In order to improve the flexibility of curves, a new five-point binary approximating subdivision scheme with two parameters is presented. The generating polynomial method is used to investigate the uniform convergence and C k -continuity of this scheme. In a special case, the five-point scheme changes into a four-point scheme, which can generate C 3 limit curves. The shape-preserving properties of the four-point scheme are analyzed, and a few examples are given to illustrate the efficiency and the shape-preserving effect of this special case.

A NEW CLASS OF BINARY APPROXIMATING SUBDIVISION SCHEMES

In this article, we present a general algorithm to generate a new class of binary approximating subdivision schemes and give derivation of some family members. We discuss important properties of derived schemes such as: convergence, continuity, Hlder regularity, degree of polynomial generation and reproduction, support, limit stencils and artifacts. Furthermore, visual performance of proposed schemes has also been presentedIn this article, we present a general algorithm to generate a new class of binary approximating subdivision schemes and give derivation of some family members. We discuss important properties of derived schemes such as: convergence, continuity, Hlder regularity, degree of polynomial generation and reproduction, support, limit stencils and artifacts. Furthermore, visual performance of proposed schemes has also been presented.

Integer-point Binary Approximating Subdivision Schemes

A general formula to generate a family of integer-point binary approximating subdivision schemes with a parameter is proposed in this paper. It is shown that some existing approximating subdivision schemes are special cases of this kind of subdivision. The proposed subdivision schemes have higher smoothness and approximating order compared with those of the existing subdivision schemes.

Construction of m-point binary approximating subdivision schemes

In this paper, an algorithm to construct m -point (for any integer m > 1 ) binary approximating subdivision schemes has been developed using the Cox–de Boor recursion formula. Some properties like symmetry of basis functions and polynomial reproduction are also discussed. It can be observed that most of the existing binary approximating schemes (corner cutting scheme developed by Chaikin, and 3-point, 4-point, 5-point and 6-point schemes introduced by Siddiqi and Ahmad) are either special cases or can be reproduced by the proposed algorithm.

Analysis of a 6-point binary subdivision scheme

In this paper, a linear 6-point binary approximating subdivision scheme with support [−6, 5] is fully investigated. It is shown that the scheme is simple and has elegant properties. We prove that the scheme can have high order continuity, polynomial reproduction and convexity preserving properties simultaneously. Furthermore, we explore the trade-offs among these properties. Examples are given to demonstrate the efficiency and flexibility of the scheme.