Share This Article:

The m-Point Quaternary Approximating Subdivision Schemes

Full-Text HTML XML Download Download as PDF (Size:139KB) PP. 6-10
DOI: 10.4236/ajcm.2013.31A002    3,635 Downloads   6,292 Views   Citations

ABSTRACT

In this article, the objective is to introduce an algorithm to produce the quaternary m-point (for any integer m>1) approximating subdivision schemes, which have smaller support and higher smoothness, comparing to binary and ternary schemes. The proposed algorithm has been derived from uniform B-spline basis function using the Cox-de Boor recursion formula. In order to determine the convergence and smoothness of the proposed schemes, the Laurent polynomial method has been used.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

S. Siddiqi and M. Younis, "The m-Point Quaternary Approximating Subdivision Schemes," American Journal of Computational Mathematics, Vol. 3 No. 1A, 2013, pp. 6-10. doi: 10.4236/ajcm.2013.31A002.

References

[1] G. M. Chaikin, “An Algorithm for High Speed Curve Generation,” Graph cuts in Computer Vision, Vol. 3, No. 4, 1974, pp. 346-349.
[2] N. Dyn, J. A. Gregory and D. Levin, “A 4-Points Interpolatory Subdivision Scheme for Curve Design,” Computer Aided Geometric Design, Vol. 4, No. 4, 1987, pp. 257-268. doi:10.1016/0167-8396(87)90001-X
[3] N. Dyn, “Tutorials on Multresolution in Geometric Modelling,” In: A. Iske, E. Quak and M. S. Floater, Eds., Summer School Lectures Notes Series: Mathematics and Visualization, Springer, 1995, ISBN: 3-540-43639-1.
[4] S. S. Siddiqi and M. Younis, “Construction of m-Point Approximating Subdivision Schemes,” Applied Mathematics Letters, Vol. 26, No. 3, 2013, pp. 337-343. doi:10.1016/j.aml.2012.09.016
[5] C. Beccari, G. Casciola and L. Romani, “A Non-Stationary Uniform Tension Controlled Interpolating 4-Point scheme Reproducing Conics,” Computer Aided Geometric Design, Vol. 24, No. 1, 2007, pp. 1-9. doi:10.1016/j.cagd.2006.10.003
[6] M. F. Hassan and N. A. Dodgson, “Ternary and Three Point Univariate Subdivision Schemes,” In: A. Cohen, J.-L. Merrien and L. L. Schumaker, Eds., Curve and Surface Fitting: Sant-Malo, Nashboro Press, Brentwood, 2003, pp. 199-208.
[7] M. F. Hassan, I. P. Ivrissimtzis, N. A. Dodgson and M. A. Sabin, “An Interpolating 4-Point Ternary Stationary Subdivision Scheme,” Computer Aided Geometric Design, Vol. 19, No. 1, 2002, pp. 1-18. doi:10.1016/S0167-8396(01)00084-X
[8] K. P. Ko, B.-G. Lee and G. J. Yoon, “A Ternary 4-Point Approximating Subdivision Scheme,” Applied Mathematics and Computation, Vol. 190, No. 2, 2007, pp. 1563-1573. doi:10.1016/j.amc.2007.02.032
[9] G. Mustafa, A. Ghaffar and F. Khan, “The Odd-Point Ternary Approximating Schemes,” American Journal of Computational Mathematics, Vol. 1, No. 2, 2011, pp. 111-118.
[10] S. R. Buss, “3-D Computer Graphics A Mathematical Introduction with OpenGL,” 1st Edition, Cambridge University Press, New York, 2003. doi:10.1017/CBO9780511804991
[11] Y. Tang, K. P. Ko and B. G. Lee, “A New Proof of the Smoothness of 4-Point Deslauriers-Dubuc Scheme,” Journal of Applied Mathematics and Computing Vol. 18, No. 1-2, 2005, pp. 553-562.
[12] C. Conti and K. Hormann, “Polynomial Reproduction for Univariate Subdivision Schemes of any Arity,” Journal of Approximation Theory, Vol. 163, No. 4, 2011, pp. 413-437. doi:10.1016/j.jat.2010.11.002

  
comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.