Advertisement

Analysis of Convergence and Smoothness by the Formalism of Laurent Polynomials

  • Nira Dyn
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

In order to design “good” subdivision schemes, tools for analyzing the convergence and smoothness of a scheme, given its mask, are needed.

A Laurent polynomial, encompassing all the available information on a subdivision scheme to be analysed, (a finite set of real numbers), is the basis of the analysis. By simple algebraic operations on such a polynomial, sufficient conditions for convergence of the subdivision scheme, and for the smoothness of the limit curves/surfaces generated by the subdivision scheme, can be checked rather automatically. The chapter concentrates on univariate subdivision schemes, (schemes for curve design) because of the simplicity of this case, and only hints on possible extensions to the bivariate case (schemes for surface design). The analysis is then demonstrated on schemes from the first two chapters of this volume.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. S. Cavaretta, W. Dahmen, and C. A. Micchelli. Stationary subdivision. Memoirs of AMS 93, 1991.Google Scholar
  2. 2.
    I. Daubechies, I. Guskov, and W. Sweldens. Regularity of irregular subdivision. Constr. Approx. 15, 1999, 381–426.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    N. Dyn. Subdivision schemes in Computer-Aided Geometric Design. Advances in Numerical Analysis, Vol II, Wavelets, Subdivision Algorithms and Radial Basis Functions, W. Light (ed.), Clarendon Press, Oxford, 1992, 36–104.Google Scholar
  4. 4.
    N. Dyn, J. A. Gregory, and D. Levin. A four-point interpolatory subdivision scheme for curve design. Comput. Aided Geom. Design 4, 1987, 257–268.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    N. Dyn, J. A. Gregory, and D. Levin. Analysis of linear binary subdivision schemes for curve design. Constr. Approx. 7, 1991, 127–147.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    N. Dyn and D. Levin. Analysis of asymptotically equivalent binary subdivision schemes. J. Math. Anal. Appl. 193, 1995, 594–621.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    N. Dyn and D. Levin. Subdivision schemes in geometric modeling. To appear in Acta Numerica, 2002.Google Scholar
  8. 8.
    N. Dyn, D. Levin, and A. Luzzatto. Non-stationary interpolatory subdivision schemes reproducing spaces of exponential polynomials. Preprint.Google Scholar
  9. 9.
    N. Dyn, D. Levin, and C. A. Micchelli. Using parameters to increase smoothness of curves and surfaces generated by subdivision. Comput. Aided Geom. Design 7, 1990, 129–140.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    N. Dyn. Interpolatory subdivision schemes. This volume.Google Scholar
  11. 11.
    J. A. Gregory. An introduction to bivariate uniform subdivision. Numerical Analysis 1991, D. F. Griffiths and G. A. Watson (eds.), Pitman Research Notes in Mathematics, Longman Scientific and Technical, 1991, 103–117.Google Scholar
  12. 12.
    A. Luzzatto. Multi-Scale Signal Processing, Based on Non-Stationary Subdivision. PhD Thesis, Tel-Aviv University, 2000.Google Scholar
  13. 13.
    C.A. Micchelli and H. Prautzsch. Uniform refinement of curves. Linear Algebra Appl. 114/115, 1989, 841–870.MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Sabin. Subdivision of box-splines. This volume.Google Scholar
  15. 15.
    M. Sabin. Eigenanalysis and artifacts of subdivision curves and surfaces. This volume.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Nira Dyn
    • 1
  1. 1.School of Mathematical SciencesTel-Aviv UniversityIsrael

Personalised recommendations