Generalized and Unified Families of Interpolating Subdivision Schemes

Mustafa, Ghulam

doi:10.4208/nmtma.2014.1313nm

Cited by 12 publications

(7 citation statements)

References 13 publications

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“…ese schemes have been introduced by [2][3][4][5][6]. e other types of subdivision schemes produce shapes which do not pass through the initial data.…”

Section: Introductionmentioning

confidence: 99%

A New Paradigm to Design a Class of Combined Ternary Subdivision Schemes

Tariq

Hameed

Mustafa

2021

Journal of Mathematics

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Subdivision schemes play a vital role in curve modeling. The curves produced by the class of 2 n + 2 -point ternary scheme (Deslauriers and Dubuc (1989)) interpolate the given data while the curves produced by a class of 2 n + 2 -point ternary B-spline schemes approximate the given data. In this research, we merge these two classes to introduce a consolidated and unified class of combined subdivision schemes with two shape control parameters in order to grow versatility for overseeing valuable necessities. However, the proposed class of subdivision schemes gives optimal smoothness in the final shapes, yet we can increase its smoothness by using a proposed general formula in form of its Laurent polynomial. The theoretical analysis of the class of subdivision schemes is done by using various mathematical tools and using their coding in the Maple environment. The graphical analysis of the class of schemes is done in the Maple environment by writing the codes based on the recursive mathematical expressions of the class of subdivision schemes.

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“…ese schemes have been introduced by [2][3][4][5][6]. e other types of subdivision schemes produce shapes which do not pass through the initial data.…”

Section: Introductionmentioning

confidence: 99%

A New Paradigm to Design a Class of Combined Ternary Subdivision Schemes

Tariq

Hameed

Mustafa

2021

Journal of Mathematics

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“…They also worked on subdivision regularization, in which they showed that unified frame work can work well for both curve fitting and noise removal. They generalized unified families of interpolating subdivision schemes of 2n-point and (2n − 1)-point p-ary which generate Lagrange's polynomial for n ≥ 2 and p ≥ 3, presented in [21]. In 2013, Younus and Siddiqi [22] established an algorithm based on Quaternary-point for (m > 1) approximating subdivision scheme which has high smoothness and small support.…”

Section: Introductionmentioning

confidence: 99%

Generalized 5-Point Approximating Subdivision Scheme of Varying Arity

et al. 2020

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The Subdivision Schemes (SSs) have been the heart of Computer Aided Geometric Design (CAGD) almost from its origin, and various analyses of SSs have been conducted. SSs are commonly used in CAGD and several methods have been invented to design curves/surfaces produced by SSs to applied geometry. In this article, we consider an algorithm that generates the 5-point approximating subdivision scheme with varying arity. By applying the algorithm, we further discuss several properties: continuity, Hölder regularity, limit stencils, error bound, and shape of limit curves. The efficiency of the scheme is also depicted with assuming different values of shape parameter along with its application.

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“…Zheng et al [11] introduced a scheme with multi-parameters in 2014. Mustafa et al [12] introduced the families of interpolating schemes with parameters in 2014. In 2017, Feng et al [13] presented a family of non-uniform schemes with variable parameters.…”

Section: Introductionmentioning

confidence: 99%

A Class of Refinement Schemes With Two Shape Control Parameters

et al. 2020

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A subdivision scheme defines a smooth curve or surface as the limit of a sequence of successive refinements of given polygon or mesh. These schemes take polygons or meshes as inputs and produce smooth curves or surfaces as outputs. In this paper, a class of combine refinement schemes with two shape control parameters is presented. These even and odd rules of these schemes have complexity three and four respectively. The even rule is designed to modify the vertices of the given polygon, whereas the odd rule is designed to insert a new point between every edge of the given polygon. These schemes can produce high order of continuous shapes than existing combine binary and ternary family of schemes. It has been observed that the schemes have interpolating and approximating behaviors depending on the values of parameters. These schemes have an interproximate behavior in the case of non-uniform setting of the parameters. These schemes can be considered as the generalized version of some of the interpolating and B-spline schemes. The theoretical as well as the numerical and graphical analysis of the shapes produced by these schemes are also presented.

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Generalized and Unified Families of Interpolating Subdivision Schemes

Cited by 12 publications

References 13 publications

A New Paradigm to Design a Class of Combined Ternary Subdivision Schemes

A New Paradigm to Design a Class of Combined Ternary Subdivision Schemes

Generalized 5-Point Approximating Subdivision Scheme of Varying Arity

A Class of Refinement Schemes With Two Shape Control Parameters

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