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On a stable family of four-point nonlinear subdivision schemes eliminating the Gibbs phenomenon. (English) Zbl 1415.65038

Summary: 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 [N. Dyn et al., in: Mathematical methods for curves and surfaces: Tromsø 2004. Sixth international conference on mathematical methods for curves and surfaces. Brentwood, TN: Nashboro Press. 145–156 (2005; Zbl 1080.65526)]. 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.

MSC:

65D17 Computer-aided design (modeling of curves and surfaces)
41A05 Interpolation in approximation theory
65D05 Numerical interpolation

Citations:

Zbl 1080.65526
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References:

[1] Akram, G.; Bibi, K.; Rehan, K.; Siddiqi, S., Shape preservation of 4-point interpolating non-stationary subdivision scheme, J. Comput. Appl. Math., 319, 480-492, (2017) · Zbl 1360.65057
[2] Conti, C.; Gemignani, L.; Romani, L., From approximating to interpolatory non-stationary subdivision schemes with the same generation properties, Adv. Comput. Math., 35, 2-4, 217-241, (2011) · Zbl 1293.65016
[3] Donat, R.; López-Ureña, S.; Santágueda, M., A family of non-oscillatory 6-point interpolatory subdivision schemes, Adv. Comput. Math., 43, 4, 849-883, (2017) · Zbl 1373.65008
[4] Duchamp, T.; Xie, G.; Yu, T., Smoothing nonlinear subdivision schemes by averaging, Numer. Algorithms, 77, 2, 361-379, (2018) · Zbl 1384.41011
[5] Hameed, R.; Mustafa, G., Family of a-point b-ary subdivision schemes with bell-shaped mask, Appl. Math. Comput., 309, 289-302, (2017) · Zbl 1411.65035
[6] Li, X.; Zheng, J., An alternative method for constructing interpolatory subdivision from approximating subdivision, Comput. Aided Geom. Design, 29, 7, 474-484, (2012) · Zbl 1253.65023
[7] Luo, Z.; Qi, W., On interpolatory subdivision from approximating subdivision scheme, Appl. Math. Comput., 220, 339-349, (2013) · Zbl 1329.65043
[8] Merrien, J. L.; Sauer, T., From Hermite to stationary subdivision schemes in one and several variables, Adv. Comput. Math., 36, 4, 547-579, (2012) · Zbl 1251.41014
[9] Kashif, Rehan; Siddiqi, Shahid S., A family of ternary subdivision schemes for curves, Appl. Math. Comput., 270, 114-123, (2015) · Zbl 1410.65050
[10] Si, X.; Baccou, J.; Liandrat, J., On four-point penalized Lagrange subdivision schemes, Appl. Math. Comput., 281, 278-299, (2016) · Zbl 1410.65025
[11] Siddiqi, S.; Rehan, K., Ternary 2N-point lagrange subdivision schemes, Appl. Math. Comput., 249, 444-452, (2014) · Zbl 1338.65052
[12] Tan, J.; Tong, G.; Zhang, L.; Xie, J., Four point interpolatory-corner cutting subdivision, Appl. Math. Comput., 265, 819-825, (2015) · Zbl 1410.65057
[13] Wang, Y.; Li, Z., A family of convexity-preserving subdivision schemes, J. Math. Res. Appl., 37, 4, 489-495, (2017) · Zbl 1399.65082
[14] Li, X.; Chang, Y., Non-uniform interpolatory subdivision surface, Appl. Math. Comput., 324, 239-253, (2018)
[15] Chaikin, G., An algorithm for high speed curve generation, Comput. Gr. Image Process., 3, 346-349, (1974)
[16] Wang, J.; Zheng, H.; Xu, F.; Liu, D., Fractal properties of the generalized chaikin corner-cutting subdivision scheme, Comput. Math. Appl., 61, 8, 2197-2200, (2011) · Zbl 1219.65026
[17] Amat, S.; Dadourian, K.; Liandrat, J., On an nonlinear subdivision scheme avoiding Gibbs oscillations and converging towards \(C^s\) functions with \(s > 1\), Math. Comp., 80, 274, 959-971, (2010) · Zbl 1217.41002
[18] Amat, S.; Dadourian, K.; Liandrat, J., Analysis of a class of nonlinear subdivision scheme and associated multiresolution transforms, Adv. Comput. Math., 34, 253-277, (2011) · Zbl 1252.65027
[19] Amat, S.; Liandrat, J., On the stability of the PPH nonlinear multiresolution, Appl. Comput. Harmon. Anal., 18, 2, 198-206, (2005) · Zbl 1077.65138
[20] Amat, S.; Dadourian, K.; Liandrat, J.; Trillo, J. C., High order nonlinear interpolatory reconstruction operators and associated multiresolution schemes, J. Comput. Appl. Math., 253, 163-180, (2013) · Zbl 1288.65012
[21] Kuijt, F., Convexity Preserving Interpolation: Nonlinear Subdivision and Splines, (1998), University of Twente, (Ph.D. thesis) · Zbl 0918.41002
[22] Dyn, N.; Floater, M. S.; Hormann, K., A \(C^2\) four-point subdivision scheme with fourth order accuracy and its extensions, (Dæhlen, M.; Mørken, K.; Schumaker, L. L., Methods for Curves and Surfaces. Methods for Curves and Surfaces, Modern Methods in Mathematics, (2005), Nashboro Press: Nashboro Press Tromsø), 145-156 · Zbl 1080.65526
[23] Dyn, N., Subdivision schemes in computer-aided geometric design, (Light, W., Advances in Numerical Analysis, II, (1992), Oxford University Press: Oxford University Press New York), 36-104 · Zbl 0760.65012
[24] Dyn, N.; Levin, D., Subdivision schemes in geometric modelling, Acta Numer., 11, 73-144, (2002) · Zbl 1105.65310
[25] Dyn, N.; Hormann, K.; Sabin, M. A.; Shen, Z., Polynomial reproduction by symmetric subdivision schemes, J. Approx. Theory, 155, 28-42, (2008) · Zbl 1159.41003
[26] Gottlieb, D.; Shu, C-W., On the gibbs phenomenon and its resolution, SIAM Rev., 39, 4, 644-668, (1997) · Zbl 0885.42003
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