Browsing by Author "Shahzad, Aamir"

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Citation Count: Khan, Faheem...et al. (2020). "A Computational Method for Subdivision Depth of Ternary Schemes", Mathematics, Vol. 8, No. 5.
A Computational Method for Subdivision Depth of Ternary Schemes
(2020) Khan, Faheem; Mustafa, Ghulam; Shahzad, Aamir; Baleanu, Dumitru; M. Al-Qurashi, Maysaa; 56389
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 atk-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.
Citation Count: Mustafa, Ghulam...et al. (2020). "A New Computational Approach to Estimate the Subdivision Depth of n-Ary Subdivision Scheme", IEEE Access, Vol. 8, pp. 187146-187155.
A New Computational Approach to Estimate the Subdivision Depth of n-Ary Subdivision Scheme
(2020) Mustafa, Ghulam; Shahzad, Aamir; Khan, Faheem; Baleanu, Dumitru; Chu, Yuming; 56389
The n-ary subdivision scheme has traditionally been designed to generate smooth curve and surface from control polygon. In this paper, we propose a new subdivision depth computation technique for n-ary subdivision scheme. The existing techniques do not ensure the computation of subdivision depth unless some strong condition is assumed on the mask of the scheme. But our technique relaxes the effect of strong condition assumed on the mask of the scheme by increasing the number of convolution steps. Consequently, a more precise subdivision depth technique for a given error tolerance is presented in this paper.
Citation Count: Shahzad, Aamir...et al. (2020). "A Novel Numerical Algorithm to Estimate the Subdivision Depth of Binary Subdivision Schemes", Symmetry-Basel, vol. 12, No. 1.
A Novel Numerical Algorithm to Estimate the Subdivision Depth of Binary Subdivision Schemes
(2020) Shahzad, Aamir; Khan, Faheem; Ghaffar, Abdul; Mustafa, Ghulam; Nisar, Kottakkaran Sooppy; Baleanu, Dumitru; 56389
Subdivision schemes are extensively used in scientific and practical applications to produce continuous geometrical shapes in an iterative manner. We construct a numerical algorithm to estimate subdivision depth between the limit curves/surfaces and their control polygons after k-fold subdivisions. In this paper, the proposed numerical algorithm for subdivision depths of binary subdivision curves and surfaces are obtained after some modification of the results given by Mustafa et al in 2006. This algorithm is very useful for implementation of the parametrization.