On a stable family of four-point nonlinear subdivision schemes eliminating the Gibbs phenomenon

Abstract The Chaikin’s scheme presents the interesting property of not producing Gibbs phenomenon. The problem arises when we need a scheme that provides a higher order of approximation. In this case linear schemes are not a good option, as they produce Gibbs phenomenon close to discontinuities. In this article a new four-point nonlinear family of subdivision schemes that eliminate the Gibbs phenomenon is presented. It is based on the linear family of four-point subdivision schemes depending on a tension parameter introduced in Dyn et al. (2005). A simple algebraic transformation leads to an easy way of introducing nonlinearity in the original family of schemes. The non-interpolatory characteristic of the nonlinear scheme can be modulated just varying the value of the tension parameter. Results about the stability, convergence and the elimination of the Gibbs phenomenon are presented. Some numerical comparisons of the results obtained in the generation of curves are also shown, leading to the conclusion that the high order nonlinear schemes are more suitable for this purpose.

[1]  Nira Dyn,et al.  Polynomial reproduction by symmetric subdivision schemes , 2008, J. Approx. Theory.

[2]  Jacques Liandrat,et al.  Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms , 2011, Adv. Comput. Math..

[3]  Shahid S. Siddiqi,et al.  Ternary 2N-point Lagrange subdivision schemes , 2014, Appl. Math. Comput..

[4]  Jacques Liandrat,et al.  On four-point penalized Lagrange subdivision schemes , 2016, Appl. Math. Comput..

[5]  Jean-Louis Merrien,et al.  From Hermite to stationary subdivision schemes in one and several variables , 2012, Adv. Comput. Math..

[6]  Xin Li,et al.  Non-uniform interpolatory subdivision surface , 2018, Appl. Math. Comput..

[7]  Zhongxuan Luo,et al.  On interpolatory subdivision from approximating subdivision scheme , 2013, Appl. Math. Comput..

[8]  Jieqing Tan,et al.  Four point interpolatory-corner cutting subdivision , 2015, Appl. Math. Comput..

[9]  Rosa Donat,et al.  A family of non-oscillatory 6-point interpolatory subdivision schemes , 2017, Adv. Comput. Math..

[10]  Jacques Liandrat,et al.  High order nonlinear interpolatory reconstruction operators and associated multiresolution schemes , 2013, J. Comput. Appl. Math..

[11]  Jacques Liandrat,et al.  On the stability of the PPH nonlinear multiresolution , 2005 .

[12]  Rabia Hameed,et al.  Family of a-point b-ary subdivision schemes with bell-shaped mask , 2017, Appl. Math. Comput..

[13]  Shahid S. Siddiqi,et al.  Shape preservation of 4-point interpolating non-stationary subdivision scheme , 2017, J. Comput. Appl. Math..

[14]  Chi-Wang Shu,et al.  On the Gibbs Phenomenon and Its Resolution , 1997, SIAM Rev..

[15]  D. Levin,et al.  Subdivision schemes in geometric modelling , 2002, Acta Numerica.

[16]  Jianmin Zheng,et al.  An alternative method for constructing interpolatory subdivision from approximating subdivision , 2012, Comput. Aided Geom. Des..

[17]  Jacques Liandrat,et al.  On a nonlinear subdivision scheme avoiding Gibbs oscillations and converging towards Cs functions with s>1 , 2011, Math. Comput..

[18]  Shahid S. Siddiqi,et al.  A family of ternary subdivision schemes for curves , 2015, Appl. Math. Comput..

[19]  George Merrill Chaikin,et al.  An algorithm for high-speed curve generation , 1974, Comput. Graph. Image Process..

[20]  Lucia Romani,et al.  From approximating to interpolatory non-stationary subdivision schemes with the same generation properties , 2010, Adv. Comput. Math..

[21]  Feng Xu,et al.  Fractal properties of the generalized Chaikin corner-cutting subdivision scheme , 2011, Comput. Math. Appl..

[22]  Gang Xie,et al.  Smoothing nonlinear subdivision schemes by averaging , 2017, Numerical Algorithms.