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Shahid S Siddiqi - One of the best experts on this subject based on the ideXlab platform.
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chaikin s perturbation Subdivision Scheme in non stationary forms
alexandria engineering journal, 2016Co-Authors: Wardat Us Salam, Shahid S Siddiqi, Kashif RehanAbstract:Abstract In this paper two non-stationary forms of Chaikin’s perturbation Subdivision Scheme, mentioned in Dyn et al. (2004), have been proposed with tension parameter ω . Comparison among the proposed Subdivision Schemes and the existing non-stationary Subdivision Scheme depicts that the trigonometric form is more efficient in the reproduction of circles and ellipses and the hyperbolic form is more suitable for the construction of many analytical curves.
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a combined ternary 4 point Subdivision Scheme
Applied Mathematics and Computation, 2015Co-Authors: Kashif Rehan, Shahid S SiddiqiAbstract:Combined binary 6-point interpolating and approximating Subdivision Scheme with tension parameters is analyzed. It is shown that the resulting curves are C1, C2 interpolating continuous and C1, C2, C3 approximating continuous for different values of tension parameters. The role of the tension parameters in Subdivision Scheme are illustrated using a few examples.
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a new non stationary binary 6 point Subdivision Scheme
Applied Mathematics and Computation, 2015Co-Authors: Shahid S Siddiqi, Wardat Us Salam, Kashif RehanAbstract:In this paper, the hyperbolic form of binary 6-point interpolating non-stationary Subdivision Scheme has been constructed using the hyperbolic function. Some of the important properties of the proposed Scheme has been discussed. The comparison of the proposed Scheme with the trigonometric form of binary 6-point interpolatory non-stationary Scheme is depicted through examples which indicate that the proposed Scheme not only accommodates and is more consistent with the control polygon, but also generates pleasing curves corresponding to the larger parametric values in and outside the parametric interval as compared to the trigonometric form of binary 6-point Scheme developed by Daniel et al. Comparison with some other interpolating non-stationary Subdivision Schemes has also been demonstrated.
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fractal behavior of ternary 4 point interpolatory Subdivision Scheme with tension parameter
Applied Mathematics and Computation, 2015Co-Authors: Shahid S Siddiqi, Wardat Us Salam, Nadeem Ahmad ButtAbstract:In this paper, the fractal curves and surfaces of ternary 4-point interpolatory Subdivision Scheme are developed and analyzed. The fractal range of the tension parameter has also been determined. The behavior of fractal curves and surfaces corresponding to different values of parameter has been depicted through six examples which offers direct means for an effective generation of Subdivision fractals. The real life applications of the proposed fractal Subdivision Scheme has also been demonstrated.
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generation of fractal curves and surfaces using ternary 4 point interpolatory Subdivision Scheme
Applied Mathematics and Computation, 2014Co-Authors: Shahid S Siddiqi, Usama Idrees, Kashif RehanAbstract:In this paper, the generation of fractal curves and surfaces along with their properties, using ternary 4-point interpolatory Subdivision Scheme with one parameter, are analyzed. The relationship between the tension parameter and the fractal behavior of the limiting curve is demonstrated through different examples. The specific range of the tension parameter has also been depicted, which provides a clear way to generate fractal curves. Since the fractal Scheme introduces, in the paper, have more number of control points therefore it gives more degree of freedom to control the shape of the fractal curve.
Ghulam Mustafa - One of the best experts on this subject based on the ideXlab platform.
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a new computational approach to estimate the Subdivision depth of n ary Subdivision Scheme
IEEE Access, 2020Co-Authors: Ghulam Mustafa, Faheem Khan, Dumitru Baleanu, Aamir Shahzad, Yuming ChuAbstract: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.
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a new approach to increase the flexibility of curves and regular surfaces produced by 4 point ternary Subdivision Scheme
Mathematical Problems in Engineering, 2020Co-Authors: Rabia Hameed, Faheem Khan, Dumitru Baleanu, Ghulam Mustafa, Amina Liaqat, Maysaa M Alqurashi, Yuming ChuAbstract:In this article, we present a new Subdivision Scheme by using an interpolatory Subdivision Scheme and an approximating Subdivision Scheme. The construction of the Subdivision Scheme is based on translation of points of the 4-point interpolatory Subdivision Scheme to the new position according to three displacement vectors containing two shape parameters. We first study the characteristics of the new Subdivision Scheme analytically and then present numerical experiments to justify these analytical characteristics geometrically. We also extend the new derived Scheme into its bivariate/tensor product version. This bivariate Scheme is applicable on quadrilateral meshes to produce smooth limiting surfaces up to continuity.
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the numerical solution of fourth order nonlinear singularly perturbed boundary value problems via 10 point Subdivision Scheme based numerical algorithm
AIP Advances, 2020Co-Authors: Syeda Tehmina Ejaz, Dumitru Baleanu, Ghulam Mustafa, Safia Malik, Yuming ChuAbstract:The Subdivision Scheme is used to illustrate smooth curves and surfaces. It is an algorithmic technique which takes a coarse polygon as an input and produces a refined polygon as an output. In this paper, a 10-point interpolating Subdivision Scheme is used to develop a numerical algorithm for the solution of fourth order nonlinear singularly perturbed boundary value problems (NSPBVPs). The studies of convergence and approximation order of the numerical algorithm are also presented. The solution of NSPBVPs is presented to see the efficiency of the algorithm.
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a 6 point Subdivision Scheme and its applications for the solution of 2nd order nonlinear singularly perturbed boundary value problems
Mathematical Biosciences and Engineering, 2020Co-Authors: Ghulam Mustafa, Syeda Tehmina Ejaz, Dumitru Baleanu, Kaweeta Anjum, Ali Ahmadian, Soheil Salahshour, Massimiliano FerraraAbstract:In this paper, we first present a 6-point binary interpolating Subdivision Scheme (BISS) which produces a C2 continuous curve and 4th order of approximation. Then as an application of the Scheme, we develop an iterative algorithm for the solution of 2nd order nonlinear singularly per-turbed boundary value problems (NSPBVP). The convergence of an iterative algorithm has also been presented. The 2nd order NSPBVP arising from combustion, chemical reactor theory, nuclear engi-neering, control theory, elasticity, and fluid mechanics can be solved by an iterative algorithm with 4th order of approximation.
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the 4 point α ary approximating Subdivision Scheme
Open Journal of Applied Sciences, 2013Co-Authors: Abdul Ghaffar, Ghulam Mustafa, Kaihuai QinAbstract:A general formula for 4-point -Ary approximating Subdivision Scheme for curve designing is introduced for any arity 2 . The new Scheme is extension of B-spline of degree 6. Laurent polynomial method is used to investigate the continuity of the Scheme. The variety of effects can be achieved in correspondence for different values of parameter. The applications of the proposed Scheme are illustrated in comparison with the established Subdivision Schemes.
Faheem Khan - One of the best experts on this subject based on the ideXlab platform.
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a new computational approach to estimate the Subdivision depth of n ary Subdivision Scheme
IEEE Access, 2020Co-Authors: Ghulam Mustafa, Faheem Khan, Dumitru Baleanu, Aamir Shahzad, Yuming ChuAbstract: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.
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a new approach to increase the flexibility of curves and regular surfaces produced by 4 point ternary Subdivision Scheme
Mathematical Problems in Engineering, 2020Co-Authors: Rabia Hameed, Faheem Khan, Dumitru Baleanu, Ghulam Mustafa, Amina Liaqat, Maysaa M Alqurashi, Yuming ChuAbstract:In this article, we present a new Subdivision Scheme by using an interpolatory Subdivision Scheme and an approximating Subdivision Scheme. The construction of the Subdivision Scheme is based on translation of points of the 4-point interpolatory Subdivision Scheme to the new position according to three displacement vectors containing two shape parameters. We first study the characteristics of the new Subdivision Scheme analytically and then present numerical experiments to justify these analytical characteristics geometrically. We also extend the new derived Scheme into its bivariate/tensor product version. This bivariate Scheme is applicable on quadrilateral meshes to produce smooth limiting surfaces up to continuity.
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shape preserving properties of a relaxed four point interpolating Subdivision Scheme
Mathematics, 2020Co-Authors: Pakeeza Ashraf, Dumitru Baleanu, Abdul Ghaffar, Irem Sehar, Kottakkaran Sooppy Nisar, Faheem KhanAbstract:In this paper, we analyze shape-preserving behavior of a relaxed four-point binary interpolating Subdivision Scheme. These shape-preserving properties include positivity-preserving, monotonicity-preserving and convexity-preserving. We establish the conditions on the initial control points that allow the generation of shape-preserving limit curves by the four-point Scheme. Some numerical examples are given to illustrate the graphical representation of shape-preserving properties of the relaxed Scheme.
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a unified interpolating Subdivision Scheme for curves surfaces by using newton interpolating polynomial
Open Journal of Applied Sciences, 2013Co-Authors: Faheem Khan, Irem Mukhtar, N BatoolAbstract:This paper presents a general formula for (2m + 2)-point n-ary interpolating Subdivision Scheme for curves for any integer m ≥ 0 and n ≥ 2 by using Newton interpolating polynomial. As a consequence, the proposed work is extended for surface case, which is equivalent to the tensor product of above proposed curve case. These formulas merge several notorious curve/surface Schemes. Furthermore, visual performance of the Subdivision Schemes is also presented.
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a new 4 point quaternary approximating Subdivision Scheme
Abstract and Applied Analysis, 2009Co-Authors: Ghulam Mustafa, Faheem KhanAbstract:A new 4-point quaternary approximating Subdivision Scheme with one shape parameter is proposed and analyzed. Its smoothness and approximation order are higher but support is smaller in comparison with the existing binary and ternary 4-point Subdivision Schemes.
Kashif Rehan - One of the best experts on this subject based on the ideXlab platform.
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chaikin s perturbation Subdivision Scheme in non stationary forms
alexandria engineering journal, 2016Co-Authors: Wardat Us Salam, Shahid S Siddiqi, Kashif RehanAbstract:Abstract In this paper two non-stationary forms of Chaikin’s perturbation Subdivision Scheme, mentioned in Dyn et al. (2004), have been proposed with tension parameter ω . Comparison among the proposed Subdivision Schemes and the existing non-stationary Subdivision Scheme depicts that the trigonometric form is more efficient in the reproduction of circles and ellipses and the hyperbolic form is more suitable for the construction of many analytical curves.
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a combined ternary 4 point Subdivision Scheme
Applied Mathematics and Computation, 2015Co-Authors: Kashif Rehan, Shahid S SiddiqiAbstract:Combined binary 6-point interpolating and approximating Subdivision Scheme with tension parameters is analyzed. It is shown that the resulting curves are C1, C2 interpolating continuous and C1, C2, C3 approximating continuous for different values of tension parameters. The role of the tension parameters in Subdivision Scheme are illustrated using a few examples.
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a new non stationary binary 6 point Subdivision Scheme
Applied Mathematics and Computation, 2015Co-Authors: Shahid S Siddiqi, Wardat Us Salam, Kashif RehanAbstract:In this paper, the hyperbolic form of binary 6-point interpolating non-stationary Subdivision Scheme has been constructed using the hyperbolic function. Some of the important properties of the proposed Scheme has been discussed. The comparison of the proposed Scheme with the trigonometric form of binary 6-point interpolatory non-stationary Scheme is depicted through examples which indicate that the proposed Scheme not only accommodates and is more consistent with the control polygon, but also generates pleasing curves corresponding to the larger parametric values in and outside the parametric interval as compared to the trigonometric form of binary 6-point Scheme developed by Daniel et al. Comparison with some other interpolating non-stationary Subdivision Schemes has also been demonstrated.
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generation of fractal curves and surfaces using ternary 4 point interpolatory Subdivision Scheme
Applied Mathematics and Computation, 2014Co-Authors: Shahid S Siddiqi, Usama Idrees, Kashif RehanAbstract:In this paper, the generation of fractal curves and surfaces along with their properties, using ternary 4-point interpolatory Subdivision Scheme with one parameter, are analyzed. The relationship between the tension parameter and the fractal behavior of the limiting curve is demonstrated through different examples. The specific range of the tension parameter has also been depicted, which provides a clear way to generate fractal curves. Since the fractal Scheme introduces, in the paper, have more number of control points therefore it gives more degree of freedom to control the shape of the fractal curve.
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symmetric ternary interpolating c 1 Subdivision Scheme
2012Co-Authors: Shahid S Siddiqi, Kashif RehanAbstract:A ternary 4-point interpolating Subdivision Scheme is proposed that generates the limiting curve of C 1 continuity. Performance of the proposed Subdivision Scheme is improved using a tension parameter. The improved Subdivision Scheme generates a family of C 1 limiting curves for certain range of tension parameter. Laurent polynomial method is used to investigate the derivative continuity of the Subdivision Schemes. The role of the tension parameter is exposed in two examples.
Sergio Amat - One of the best experts on this subject based on the ideXlab platform.
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on a nonlinear 4 point ternary and non interpolatory Subdivision Scheme eliminating the gibbs phenomenon
Applied Mathematics and Computation, 2018Co-Authors: Sergio Amat, Amira Choutri, Juan Ruiz, S ZouaouiAbstract:Abstract A nonlinear ternary 4-point non-interpolatory Subdivision Scheme is presented. It is based on a nonlinear perturbation of the 4-point Subdivision Scheme studied in [16]. The convergence of the Scheme and the regularity of the limit function are analyzed. It is shown that the Gibbs phenomenon, that is classical in linear Schemes, is eliminated. We also establish the stability of the Subdivision Scheme, that is not a consequence of its convergence due to its non-linearity. To the best of our knowledge, this is the first ternary non-interpolatory Subdivision Scheme that can be found in the literature.
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On a nonlinear 4-point quaternary approximating Subdivision Scheme eliminating the Gibbs phenomenon
SeMA Journal, 2013Co-Authors: Sergio Amat, Jacques LiandratAbstract:A nonlinear quaternary 4-point approximating Subdivision Scheme is presented. It is based on a nonlinear perturbation of the quaternary Subdivision Scheme studied in Mustafa and Khan (Abstr Appl Anal 2009:14, 2009 ). The convergence of the Scheme and the regularity of the limit function are analyzed. It is shown that the Gibbs phenomenon, classical in linear Schemes, is eliminated. The stability, that in the nonlinear case is not a consequence of the convergence, is also established. Up to our knowledge, this is the first Subdivision Scheme of regularity larger than three, avoiding Gibbs oscillations and for which the stability is rigorously obtained. All these properties are very important for real applications.
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on a nonlinear Subdivision Scheme avoiding gibbs oscillations and converging towards c s functions with s 1
Mathematics of Computation, 2011Co-Authors: Sergio Amat, Karine Dadourian, Jacques LiandratAbstract:This paper presents a new nonlinear dyadic Subdivision Scheme eliminating the Gibbs oscillations close to discontinuities. Its convergence , stability and order of approximation are analyzed. It is proved that this Scheme converges towards limit functions Holder continuous with exponent larger than 1.299. Numerical estimates provide a Holder exponent of 2.438. This Subdivision Scheme is the first one that simultaneously achieves the control of the Gibbs phenomenon and has limit functions with Holder exponent larger than 1.
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analysis of a new nonlinear Subdivision Scheme applications in image processing
Foundations of Computational Mathematics, 2006Co-Authors: Sergio Amat, Jacques Liandrat, Rosa Donat, Carlos J TrilloAbstract:A nonlinear multiresolution Scheme within Harten's framework is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting Subdivision Scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.
