scholarly journals Four-Pointn-Ary Interpolating Subdivision Schemes

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Ghulam Mustafa ◽  
Robina Bashir
Keyword(s):  
Simple Algorithm ◽  
Approximation Order ◽  
Divided Differences ◽  
Hölder Regularity ◽  
Wavelet Theory ◽  
Subdivision Schemes ◽  
Holder Regularity

We present an efficient and simple algorithm to generate 4-pointn-ary interpolating schemes. Our algorithm is based on three simple steps: second divided differences, determination of position of vertices by using second divided differences, and computation of new vertices. It is observed that 4-pointn-ary interpolating schemes generated by completely different frameworks (i.e., Lagrange interpolant and wavelet theory) can also be generated by the proposed algorithm. Furthermore, we have discussed continuity, Hölder regularity, degree of polynomial generation, polynomial reproduction, and approximation order of the schemes.

scholarly journals A Family of High Continuity Subdivision Schemes Based on Probability Distribution

2019 ◽  
Vol 38 (2) ◽  
pp. 389-398 ◽  
Author(s):  
Muhammad Asghar ◽  
Muhammad Javed Iqbal ◽  
Ghulam Mustafa
Keyword(s):  
Probability Distribution ◽  
Geometric Modeling ◽  
Probability Generating Function ◽  
Hölder Regularity ◽  
Binomial Probability ◽  
Shape Preserving ◽  
Subdivision Schemes ◽  
Smooth Curves ◽  
The Family ◽  
Holder Regularity

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.

scholarly journals A Six-Point Variant on the Lane-Riesenfeld Algorithm

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Pakeeza Ashraf ◽  
Ghulam Mustafa ◽  
Jiansong Deng
Keyword(s):  
Smoothing Parameter ◽  
Hölder Regularity ◽  
Subdivision Schemes ◽  
Control Polygon ◽  
New Family ◽  
Shrinkage Effect ◽  
The Family ◽  
Holder Regularity

We apply six-point variant on the Lane-Riesenfeld algorithm to obtain a new family of subdivision schemes. We also determine the support, smoothness, Hölder regularity, magnitude of the artifact, and the shrinkage effect due to the change of integer smoothing parameter that characterizes the members of the family. The degree of polynomial reproduction also has been discussed. It is observed that the proposed schemes have less shrinkage effect and as a result better preserve the shape of control polygon.

d-Refinable (dual) pseudo-splines and their regularities

2017 ◽  
Vol 15 (01) ◽  
pp. 1750002
Author(s):  
Yanfeng Shen ◽  
Shouzhi Yang ◽  
Dehui Yuan
Keyword(s):  
Linear Independence ◽  
New Method ◽  
Hölder Regularity ◽  
Refinable Functions ◽  
Subdivision Schemes ◽  
Holder Regularity

In this paper, we introduce the concept of [Formula: see text]-dilation (dual) pseudo-splines in [Formula: see text] with dilation [Formula: see text], characterize the linear independence of the integer shifts, the subdivision schemes of polynomial reproduction and Hölder regularity for (dual) pseudo-splines with general dilation. We present a new method to determine the Hölder regularity of refinable functions with general dilation [Formula: see text]. Furthermore, we compare the regularities between pseudo-splines and dual ones.

scholarly journals Generalized 5-Point Approximating Subdivision Scheme of Varying Arity

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 474 ◽  
Author(s):  
Sardar Muhammad Hussain ◽  
Aziz Ur Rehman ◽  
Dumitru Baleanu ◽  
Kottakkaran Sooppy Nisar ◽  
Abdul Ghaffar ◽  
...  
Keyword(s):  
Error Bound ◽  
Shape Parameter ◽  
Geometric Design ◽  
Hölder Regularity ◽  
Subdivision Scheme ◽  
Subdivision Schemes ◽  
Computer Aided Geometric Design ◽  
Approximating Subdivision Scheme ◽  
Computer Aided ◽  
Holder Regularity

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.

scholarly journals Hölder Regularity of Geometric Subdivision Schemes

2015 ◽  
Vol 42 (3) ◽  
pp. 425-458 ◽  
Author(s):  
T. Ewald ◽  
U. Reif ◽  
M. Sabin
Keyword(s):  
Hölder Regularity ◽  
Subdivision Schemes ◽  
Holder Regularity
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