Skip to main content
SpringerLink
Log in

Families of univariate and bivariate subdivision schemes originated from quartic B-spline

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

Families of parameter dependent univariate and bivariate subdivision schemes are presented in this paper. These families are new variants of the Lane-Riesenfeld algorithm. So the subdivision algorithms consist of both refining and smoothing steps. In refining step, we use the quartic B-spline based subdivision schemes. In smoothing step, we average the adjacent points. The bivariate schemes are the non-tensor product version of our univariate schemes. Moreover, for odd and even number of smoothing steps, we get the primal and dual schemes respectively. Higher regularity of the schemes can be achieved by increasing the number of smoothing steps. These schemes can be nicely generalized to contain local shape parameters that allow the user to adjust locally the shape of the limit curve/surface.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ashraf, P., Mustafa, G., Deng, J.: A six-point variant on the Lane-Riesenfeld algorithm, Journal of Applied Mathematics, 2014, 7 pages (2014)

  2. Cashman, T.J., Hormann, K., Reif, U.: Generalized Lane-Riesenfeld algorithms. Comput. Aided Geom. Des. 30(4), 398–409 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Comput. Aided Des. 10, 350–355 (1978)

    Article  Google Scholar 

  4. Chaikin, G.M.: An algorithm for high speed curve generation. Comput. Graphics Image Process. 3(4), 346–349 (1974)

    Article  Google Scholar 

  5. Charina, M., Conti, C.: Polynomial reproduction of multivariate scalar subdivision schemes. J. Comput. Appl. Math. 240, 51–61 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Conti, C., Gemignani, L., Romani, L.: From symmetric subdivision masks of Hurwitz type to interpolatory subdivision masks. Linear Algebra Appl. 431, 1971–1987 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Conti, C., Gemignani, L., Romani, L.: A constructive algebraic strategy for interpolatory subdivision schemes induced by bivariate box splines. Adv. Comput. Math. 39, 395–424 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Conti, C., Hormann, k.: Polynomial reproduction for univariate subdivision schemes of any arity. J. Approximation Theory 163, 413–437 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Conti, C., Romani, L.: Dual univariate m-ary subdivision schemes of de Rham-type. Aust. J. Math. Anal. Appl. 407, 443–456 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. Constr. Approx. 5(1), 49–68 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Doo, D., Sabin, M.: Analysis of the behaviour of recursive subdivision surfaces near extraordinary points. Comput. Aided Des. 10(6), 356–360 (1978)

    Article  Google Scholar 

  12. Dyn, N., Heard, A., Hormann, K., Sharon, N.: Univariate subdivision schemes for noisy data with geometric applications. Comput. Aided Geom. Des. 37, 85–104 (2015)

    Article  MathSciNet  Google Scholar 

  13. Dyn, N.: Interpolatory subdivision schemes and analysis of convergence and smoothness by the formalism of Laurent polynomials. In: In: Iske, A., Quak, E., Floater, M.S. (eds.) Tutorials on Multiresolution in Geometric Modeling, Springer, pp 51–68 (2002)

    Chapter  Google Scholar 

  14. Dyn, N., Levin, D., Gregory, J.A.: A 4-point interpolatory subdivision scheme for curve design. Comput. Aided Geom. Des. 4(4), 257–268 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dyn, N., Levin, D.: Subdivision schemes in geometric modelling. Acta Numerica 11, 73–144 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Khan, F., Mustafa, G.: A new non-tensor product C 1 subdivision scheme for regular quad meshes. World Appl. Sci. J. 24(12), 1635–1641 (2013)

    Google Scholar 

  17. Hormann, K., Sabin, M.A.: A family of subdivision schemes with cubic precision. Comput. Aided Geom. Des. 25(1), 41–52 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ivrissimtzis, I.P., Sabin, M.A., Dodgson, N.A.: On the support of recursive subdivision. ACM Trans. Graph. 23(4), 1043–1060 (2004)

    Article  MATH  Google Scholar 

  19. Lane, J.M., Riesenfeld, R.F.: A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Trans. Pattern Anal. Mach. Intell. 2(1), 35–46 (1980)

    Article  MATH  Google Scholar 

  20. Mustafa, G., Ashraf, P., Aslam, M.: Binary univariate dual and primal subdivision schemes. SEMA J. 65(1), 23–35 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  21. Romani, L.: A Chaikin-based variant of Lane-Riesenfeld algorithm and its non-tensor product extension. Comput. Aided Geom. Des. 32, 22–49 (2015)

    Article  MathSciNet  Google Scholar 

  22. Stam, J.: On subdivision schemes generalizing uniform B-spline surfaces of arbitrary degree. Comput. Aided Geom. Des. 18, 383–396 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Zorin, D., Schröder, P.: A unified framework for primal/dual quadrilateral subdivision schemes. Comput. Aided Geom. Des. 18, 429–454 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions. This work is supported by NRPU Project. No. 3183, Pakistan.

Author information

Authors and Affiliations

  1. Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan

    Ghulam Mustafa & Rabia Hameed

Authors
  1. Ghulam Mustafa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rabia Hameed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ghulam Mustafa.

Additional information

Communicated by: Tom Lyche

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mustafa, G., Hameed, R. Families of univariate and bivariate subdivision schemes originated from quartic B-spline. Adv Comput Math 43, 1131–1161 (2017). https://doi.org/10.1007/s10444-017-9519-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-017-9519-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation