Skip to main content
SpringerLink
Log in

Smoothing nonlinear subdivision schemes by averaging

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In the theory of linear subdivision algorithms, it is well-known that the regularity of a linear subdivision scheme can be elevated by one order (say, from C k to C k+1) by composing it with an averaging step (equivalently, by multiplying to the subdivision mask a(z) a (1 + z) factor. In this paper, we show that the same can be done to nonlinear subdivision schemes: by composing with it any nonlinear, smooth, 2-point averaging step, the lifted nonlinear subdivision scheme has an extra order of regularity than the original scheme. A notable application of this result shows that the classical Lane-Riesenfeld algorithm for uniform B-Spline, when extended to Riemannian manifolds based on geodesic midpoint, produces curves with the same regularity as their linear counterparts. (In particular, curvature does not obstruct the nonlinear Lane-Riesenfeld algorithm to inherit regularity from the linear algorithm.) Our main result uses the recently developed technique of differential proximity conditions.

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. Duchamp, T., Xie, G., Yu, T.: A necessary and sufficient proximity condition for smoothness equivalence of nonlinear subdivision schemes. Foundations of Computational Mathematics, pp. 1–46 (2015)

  2. Duchamp, T., Xie, G., Yu, T.P.-Y.: Single basepoint subdivision schemes for manifold-valued data time-symmetry without space-symmetry. Found. Comput. Math. 13(5), 693–728 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Duchamp, T., Xie, G., Yu, T.P.-Y.: On a new proximition condition for manifold-valued subdivision schemes. In: Fasshauer, G.E., Schumaker, L.L. (eds.) Approximation Theory XIV: San Antonio 2013, volume 83 of Springer Proceedings in Mathematics & Statistics, pp 65–79. Springer, Cham (2014)

    Google Scholar 

  4. Dyn, N., Goldman, R.: Convergence and smoothness of nonlinear Lane-Riesenfeld algorithms in the functional setting. Found. Comput. Math. 11, 79–94 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Grohs, P.: A general proximity analysis of nonlinear subdivision schemes. SIAM J. Math. Anal. 42(2), 729–750 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jia, R.Q., Micchelli, C.A.: Using the refinement equations for the construction of pre-wavelets. II. Powers of two Curves and Surfaces (Chamonix-Mont-Blanc, 1990), pp 209–246. Academic Press, Boston, MA (1991)

    Chapter  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Noakes, L.: Non-linear corner cutting. Adv. Comput. Math. 8, 165–177 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ur Rahman, I., Drori, I., Stodden, V.C., Donoho, D.L., Schröder, P.: Multiscale representations for manifold-valued data. Multiscale Model. Simul. 4(4), 1201–1232 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Xie, G., Yu, T. P.-Y.: Smoothness equivalence properties of general manifold-valued data subdivision schemes. Multiscale Model. Simul. 7(3), 1073–1100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Gang Xie’s research was supported by the Fundamental Research Funds for the Central Universities and the National, Natural Science Foundation of China (No.11101146).

Thomas Yu’s research was partially supported by the National Science Foundation grants DMS 1115915 and DMS 1522337.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195-4350, USA

    Tom Duchamp

  2. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

    Gang Xie

  3. Department of Mathematics, Drexel University, 3141 Chestnut Street, 206 Korman Center, Philadelphia, PA, 19104, USA

    Thomas Yu

Authors
  1. Tom Duchamp
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gang Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gang Xie.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duchamp, T., Xie, G. & Yu, T. Smoothing nonlinear subdivision schemes by averaging. Numer Algor 77, 361–379 (2018). https://doi.org/10.1007/s11075-017-0319-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0319-8

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation