Exploring the flexibility of $ m $-point quaternary approximating subdivision schemes with free parameter

Reem K. Alhefthi; Pakeeza Ashraf; Ayesha Abid; Shahram Rezapour; Abdul Ghaffar; Mustafa Inc; Reem K. Alhefthi; Pakeeza Ashraf; Ayesha Abid; Shahram Rezapour; Abdul Ghaffar; Mustafa Inc

doi:10.3934/math.20241584

AIMS Mathematics

2024, Volume 9, Issue 11: 33185-33214. doi: 10.3934/math.20241584

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Exploring the flexibility of $ m $-point quaternary approximating subdivision schemes with free parameter

1.
Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2.
Department of Mathematics, Government Sadiq College Women University, Bahawalpur, Pakistan
3.
Department of Medical Research, China Medical University Hospital, Taichung, Taiwan
4.
Department of Mathematics, Ghazi University, Dera Ghezi Khan, Pakistan
5.
Department of Mathematics, Firat University, Elazig 23119, Turkiye
6.
Department of Computer Engineering, Biruni University, Istanbul 34010, Turkiye

Received: 04 August 2024 Revised: 15 October 2024 Accepted: 23 October 2024 Published: 21 November 2024
MSC : 65D05, 65D07, 65D15, 65D17

In this study, we proposed a family of $ m $-point quaternary approximating subdivision schemes, characterized by an explicit formula involving three parameters. One of these parameters served as a shape control parameter, allowing for flexible curve design, while the other two parameters identify different members of the family and determined the smoothness of the resulting limit curves. We conducted a thorough analysis of the proposed schemes, covering their smoothness properties, polynomial generation, and reproduction capabilities. Additionally, we examined the behavior of the Gibbs phenomenon within the family both theoretically and graphically, highlighting the advantages of the proposed schemes in eliminating undesirable oscillations. A comparative study with existing subdivision schemes demonstrated the effectiveness and versatility of our approach. The results indicated that the proposed family offered enhanced smoothness and control, making it suitable for a wide range of applications in computer graphics and geometric modeling.
- quaternary,
- subdivision scheme,
- shape control parameter,
- Laurent polynomial,
- Gibbs phenomenon
Citation: Reem K. Alhefthi, Pakeeza Ashraf, Ayesha Abid, Shahram Rezapour, Abdul Ghaffar, Mustafa Inc. Exploring the flexibility of $ m $-point quaternary approximating subdivision schemes with free parameter[J]. AIMS Mathematics, 2024, 9(11): 33185-33214. doi: 10.3934/math.20241584

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Abstract

In this study, we proposed a family of $ m $-point quaternary approximating subdivision schemes, characterized by an explicit formula involving three parameters. One of these parameters served as a shape control parameter, allowing for flexible curve design, while the other two parameters identify different members of the family and determined the smoothness of the resulting limit curves. We conducted a thorough analysis of the proposed schemes, covering their smoothness properties, polynomial generation, and reproduction capabilities. Additionally, we examined the behavior of the Gibbs phenomenon within the family both theoretically and graphically, highlighting the advantages of the proposed schemes in eliminating undesirable oscillations. A comparative study with existing subdivision schemes demonstrated the effectiveness and versatility of our approach. The results indicated that the proposed family offered enhanced smoothness and control, making it suitable for a wide range of applications in computer graphics and geometric modeling.

References

[1]	Y. Liu, H. Shou, K. Ji, Review of subdivision schemes and their applications, Recent Pat. Eng., 16 (2022), 50–62. http://doi.org/10.2174/1872212116666211229151825 doi: 10.2174/1872212116666211229151825
[2]	P. Ashraf, G. Mustafa, A. Ghaffar, R. Zahra, K. S. Nisar, E. E. Mahmoud, et al., Unified framework of approximating and interpolatory subdivision schemes for construction of class of binary subdivision schemes, J. Funct. Spaces, 2020 (2020), 6677778. https://doi.org/10.1155/2020/6677778 doi: 10.1155/2020/6677778
[3]	G. Mustafa, I. Ivrissimtzis, Model selection for the Dubuc-Deslauriers family of subdivision schemes, In: 14th IMA conference on mathematics of surfaces, 2013, 1–6.
[4]	D. R. Georges, Un peu de mathématiques à propos d'une courbe plane, Elem. Math., 2 (1947), 73–76.
[5]	G. M. Chaikin, An algorithm for high-speed curve generation, Comput. Graph. Image Process., 3 (1974), 346–349. https://doi.org/10.1016/0146-664X(74)90028-8 doi: 10.1016/0146-664X(74)90028-8
[6]	G. Deslauriers, S. Dubuc, Symmetric iterative interpolation processes, Constr. Approx., 5 (1989), 49–68. https://doi.org/10.1007/BF01889598 doi: 10.1007/BF01889598
[7]	N. Dyn, Analysis of convergence and smoothness by the formalism of Laurent polynomials, In: Tutorials on multiresolution in geometric modelling, 2002, 51–68. https://doi.org/10.1007/978-3-662-04388-2_3
[8]	N. Dyn, D. Levin, Subdivision schemes in geometric modelling, Acta Numer., 11 (2002), 73–144. https://doi.org/10.1017/S0962492902000028 doi: 10.1017/S0962492902000028
[9]	M. F Hassan, I. P. Ivrissimitzis, N. A. Dodgson, M. A. Sabin, An interpolating 4-point $C^{2}$ ternary stationary subdivision scheme, Comput. Aided Geom. Design, 19 (2002), 1–18. https://doi.org/10.1016/S0167-8396(01)00084-X doi: 10.1016/S0167-8396(01)00084-X
[10]	H. Zheng, Z. Ye, Z. Chen, H. Zhao, A controllable ternary interpolatory subdivision scheme, Int. J. CAD/CAM, 5 (2005), 29–38.
[11]	G. Mustafa, J. Deng, P. Ashraf, N. A. Rehman, The mask of odd points $n$-ary interpolating subdivision sheme, J. Appl. Math., 2012 (2012), 205863. https://doi.org/10.1155/2012/205863 doi: 10.1155/2012/205863
[12]	A. Ghaffar, G. Mustafa, K. Qin, The 4-point $a$-ary approximating subdivision scheme, Open J. Appl. Sci., 3 (2013), 106–111. https://doi.org/10.4236/ojapps.2013.31B1022 doi: 10.4236/ojapps.2013.31B1022
[13]	P. Ashraf, M. Sabir, A. Ghaffar, K. S. Nisar, I. Khan, Shape-preservation of the four-point ternary interpolating non-stationary subdivision scheme, Front. Phys., 7 (2020), 241. https://doi.org/10.3389/fphy.2019.00241 doi: 10.3389/fphy.2019.00241
[14]	P. Ashraf, G. Mustafa, H. Khan, D. Baleanu, A. Ghaffar, K. S. Nisar, A shape-preserving variant of Lane-Riesenfeld algorithm, AIMS Mathematics, 6 (2021), 2152–2170. https://doi.org/10.3934/math.2021131 doi: 10.3934/math.2021131
[15]	S. Zouaoui, S. Amat, S. Busquier, M. J. Legaz, Some new $n$-point ternary subdivision schemes without the gibbs phenomenon, Mathematics, 10 (2022), 2674. https://doi.org/10.3390/math10152674 doi: 10.3390/math10152674
[16]	G. Mustafa, F. Khan, A new $4$-point $C^{3}$ quaternary approximating subdivision scheme, Abstr. Appl. Anal., 2009 (2009), 301967. https://doi.org/10.1155/2009/301967 doi: 10.1155/2009/301967
[17]	K. P. Ko, A quaternary approximating $4$-point subdivision scheme, J. Korean Soc. Ind App., 13 (2009), 307–314.
[18]	M. Bari, R. Bashir, G. Mustafa, $3n$-point quaternary shape preserving subdivision schemes, Mehran Univ. Res. J. Eng. Technol., 36 (2017), 489–500.
[19]	K. Pervez, Shape preservation of the stationary $4$-point quaternary subdivision schemes, Commun. Math. Appl., 9 (2018), 249–264. https://doi.org/10.26713/cma.v9i3.719 doi: 10.26713/cma.v9i3.719
[20]	S. M. Hussain, A. U. Rehman, D. Baleanu, K. S. Nisar, A. Ghaffar, S. A. Abdul Karim, Generalized $5$-point approximating subdivision scheme of varying arity, Mathematics, 8 (2020), 474. https://doi.org/10.3390/math8040474 doi: 10.3390/math8040474
[21]	A. Nawaz, A. Ghaffar, F. Khan, S. A. A. Karim, A new $7$-point quaternary approximating subdivision scheme, In: Intelligent systems modeling and simulation Ⅱ, 444 (2022), 545–566. https://doi.org/10.1007/978-3-031-04028-3_35
[22]	S. W. Yao, P. Ashraf, A. Ghaffar, M. Kousar, M. Inc, N. Nigar, Fractal and convexity analysis of the quaternary four-point scheme and its applications, Fractals, 31 (2023), 2340088. https://doi.org/10.1142/S0218348X23400881 doi: 10.1142/S0218348X23400881
[23]	J. Zhou, H. Zheng, B. Zhang, Gibbs phenomenon for $p$-ary subdivision schemes, J. Inequal. Appl., 2019 (2019), 48. https://doi.org/10.1186/s13660-019-1998-6 doi: 10.1186/s13660-019-1998-6
[24]	S. Amat, J. Ruiz, J. C. Trillo, D. F. Yanez, Analysis of the Gibbs phenomenon in stationary subdivision schemes, Appl. Math. Lett., 76 (2018), 157–163. https://doi.org/10.1016/j.aml.2017.08.014 doi: 10.1016/j.aml.2017.08.014
[25]	S. S. Siddiqi, M. Younis, The $m$-point quaternary approximating subdivision schemes, Am. J. Comput. Math., 3 (2013), 6–10. https://doi.org/10.4236/ajcm.2013.31A002 doi: 10.4236/ajcm.2013.31A002
[26]	R. Bashir, G. Mustafaa, P. Agarwalb, A class of shape preserving $5$-point $n$-ary approximating schemes, J. Math. Comput. Sci., 18 (2018), 364–380.
[27]	O. Rioul, Simple regularity criteria for subdivision schemes, SIAM J. Math. Anal., 23 (1992), 1544–1576. https://doi.org/10.1137/0523086 doi: 10.1137/0523086
[28]	H. Yang, K. Kim, J. Yoon, A family of $C^2$ four-point stationary subdivision schemes with fourth-order accuracy and shape-preserving properties, J. Comput. Appl. Math., 446 (2024), 115843. https://doi.org/10.1016/j.cam.2024.115843 doi: 10.1016/j.cam.2024.115843
[29]	N. Dyn, K. Hormann, M. A. Sabin, Z. Shen, Polynomial reproduction by symmetric subdivision schemes, J. Approx. Theory, 155 (2008), 28–42. https://doi.org/10.1016/j.jat.2008.04.008 doi: 10.1016/j.jat.2008.04.008
[30]	C. Conti, K. Hormann, Polynomial reproduction for univariate subdivision schemes of any arity, J. Approx. Theory, 163 (2011), 413–437. https://doi.org/10.1016/j.jat.2010.11.002 doi: 10.1016/j.jat.2010.11.002
[31]	N. Dyn, Interpolatory subdivision scheme, In: Tutorials on multiresolution in geometric modelling, 2002, 25–50. https://doi.org/10.1007/978-3-662-04388-2_2
[32]	R. Hameed, G. Mustafa, T. Latif, S. A. A. Karim, Smooth transition and Gibbs oscillation minimization in a 7-point subdivision scheme with shape-control parameters for high smoothness, Results Appl. Math., 23 (2024), 100485. https://doi.org/10.1016/j.rinam.2024.100485 doi: 10.1016/j.rinam.2024.100485

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