scholarly journals Error bounds for a class of subdivision schemes based on the two-scale refinement equation

2011 ◽  
Vol 236 (2) ◽  
pp. 265-278 ◽  
Author(s):  
María Moncayo ◽  
Sergio Amat
Keyword(s):  
Error Bounds ◽  
Refinement Equation ◽  
Subdivision Schemes

scholarly journals (2n-1)-Point Ternary Approximating and Interpolating Subdivision Schemes

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Muhammad Aslam ◽  
Ghulam Mustafa ◽  
Abdul Ghaffar
Keyword(s):  
Explicit Formula ◽  
Error Bounds ◽  
Computational Cost ◽  
Subdivision Schemes ◽  
Special Cases ◽  
Better Than

We present an explicit formula which unifies the mask of(2n-1)-point ternary interpolating as well as approximating subdivision schemes. We observe that the odd point ternary interpolating and approximating schemes introduced by Lian (2009), Siddiqi and Rehan (2010, 2009) and Hassan and Dodgson (2003) are special cases of our proposed masks/schemes. Moreover, schemes introduced by Zheng et al. (2009) can easily be generated by our proposed masks. It is also proved from comparison that(2n-1)-point schemes are better than2n-scheme in the sense of computational cost, support and error bounds.

scholarly journals The Mask of Odd Pointsn-Ary Interpolating Subdivision Scheme

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Ghulam Mustafa ◽  
Jiansong Deng ◽  
Pakeeza Ashraf ◽  
Najma Abdul Rehman
Keyword(s):  
Explicit Formula ◽  
Error Bounds ◽  
Subdivision Scheme ◽  
Subdivision Schemes ◽  
Total Absolute Curvature ◽  
Absolute Curvature ◽  
Special Cases ◽  
Preservation Property ◽  
Subdivision Curves ◽  
And Control

We present an explicit formula for the mask of odd pointsn-ary, for any oddn⩾3, interpolating subdivision schemes. This formula provides the mask of lower and higher arity schemes. The 3-point and 5-pointa-ary schemes introduced by Lian, 2008, and (2m+1)-pointa-ary schemes introduced by, Lian, 2009, are special cases of our explicit formula. Moreover, other well-known existing odd pointn-ary schemes including the schemes introduced by Zheng et al., 2009, can easily be generated by our formula. In addition, error bounds between subdivision curves and control polygons of schemes are computed. It has been noticed that error bounds decrease when the complexity of the scheme decreases and vice versa. Also, as we increase arity of the schemes the error bounds decrease. Furthermore, we present brief comparison of total absolute curvature of subdivision schemes having different arity with different complexity. Convexity preservation property of scheme is also presented.

scholarly journals A Novel Numerical Method for Computing Subdivision Depth of Quaternary Schemes

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 809
Author(s):  
Aamir Shahzad ◽  
Faheem Khan ◽  
Abdul Ghaffar ◽  
Shao-Wen Yao ◽  
Mustafa Inc ◽  
...  
Keyword(s):  
Numerical Method ◽  
Error Bound ◽  
Error Bounds ◽  
Computational Technique ◽  
Optimal Error ◽  
Subdivision Schemes ◽  
Subdivision Depth

In this paper, an advanced computational technique has been presented to compute the error bounds and subdivision depth of quaternary subdivision schemes. First, the estimation is computed of the error bound between quaternary subdivision limit curves/surfaces and their polygons after kth-level subdivision by using l0 order of convolution. Secondly, by using the error bounds, the subdivision depth of the quaternary schemes has been computed. Moreover, this technique needs fewer iterations (subdivision depth) to get the optimal error bounds of quaternary subdivision schemes as compared to the existing techniques.

scholarly journals A Computational Method for Subdivision Depth of Ternary Schemes

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 817
Author(s):  
Faheem Khan ◽  
Ghulam Mustafa ◽  
Aamir Shahzad ◽  
Dumitru Baleanu ◽  
Maysaa M. Al-Qurashi
Keyword(s):  
Error Bounds ◽  
Numerical Experiments ◽  
Computational Method ◽  
Subdivision Schemes ◽  
Practical Applications ◽  
Refinement Procedure ◽  
Subdivision Algorithm ◽  
Subdivision Depth ◽  
Error Distance ◽  
New Framework

Subdivision schemes are extensively used in scientific and practical applications to produce continuous shapes in an iterative way. This paper introduces a framework to compute subdivision depths of ternary schemes. We first use subdivision algorithm in terms of convolution to compute the error bounds between two successive polygons produced by refinement procedure of subdivision schemes. Then, a formula for computing bound between the polygon at k-th stage and the limiting polygon is derived. After that, we predict numerically the number of subdivision steps (depths) required for smooth limiting shape based on the demand of user specified error (distance) tolerance. In addition, extensive numerical experiments were carried out to check the numerical outcomes of this new framework. The proposed methods are more efficient than the method proposed by Song et al.

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