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Advances in Computational Mathematics

, Volume 11, Issue 1, pp 41–54 | Cite as

Using Laurent polynomial representation for the analysis of non‐uniform binary subdivision schemes

  • David Levin
Article

Abstract

Non‐uniform binary linear subdivision schemes, with finite masks, over uniform grids, are studied. A Laurent polynomial representation is suggested and the basic operations required for smoothness analysis are presented. As an example it is shown that the interpolatory 4‐point scheme is C 1 with an almost arbitrary non‐uniform choice of the free parameter.

Keywords

Subdivision Scheme Laurent Series Generate Polynomial Tension Parameter Uniform Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 453 (1991).Google Scholar
  2. [2]
    I. Daubechies and J. Lagarias, Two scale difference equations I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991) 1338-1410.MathSciNetGoogle Scholar
  3. [3]
    I. Daubechies and J. Lagarias, Two scale difference equations II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992) 1031-1079.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    C. de Boor, Cutting corners always works, Comput. Aided Geom. Design 4 (1987) 125-131.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    G. Deslauriers and S. Dubuc, Interpolation dyadique, in: Fractals, Dimensions Non Entieres et Applications, ed. G. Cherbit (Masson, Paris, 1989).Google Scholar
  6. [6]
    S. Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl. 114 (1986) 185-204.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    N. Dyn, Subdivision schemes in computer-aided geometric design, in: Advances in Numerical Analysis II, Wavelets, Subdivision Algorithms, and Radial Basis Functions, ed. W.A. Light (Clarendon Press, Oxford, 1992) pp. 36-104.Google Scholar
  8. [8]
    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.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    N. Dyn, J.A. Gregory and D. Levin, Analysis of uniform binary subdivision schemes for curve design, Constr. Approx. 7 (1991) 127-147.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    N. Dyn, J.A. Gregory and D. Levin, Piecewise uniform subdivision schemes, in: Mathematical Methods for Curves and Surfaces, eds. M. Dahlen, T. Lyche and L.L. Schumaker (Vanderbilt University Press, Nashville, TN, 1995) pp. 111-120.Google Scholar
  11. [11]
    N. Dyn, S. Hed and D. Levin, Subdivision schemes for surface interpolation, in: Workshop on Computational Geometry, eds. A. Conte et al. (World Scientific Publications, 1993) pp. 97-118.Google Scholar
  12. [12]
    N. Dyn and D. Levin, Interpolating subdivision schemes for the generation of curves and surfaces, in: Multivariate Interpolation and Approximation, eds. W. Haussmann and K. Jetter (Birkhäuser, Basel, 1990) pp. 91-106.Google Scholar
  13. [13]
    N. Dyn and D. Levin, Analysis of asymptotically equivalent binary subdivision schemes, J. Math. Anal. Appl. 193 (1995) 594-621.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    J.A. Gregory and R. Qu, Non-uniform corner cutting, Brunel University reprint (1988).Google Scholar
  15. [15]
    S. Hed, Analysis of subdivision schemes for surfaces, M.Sc. Thesis, Tel Aviv University (1992).Google Scholar
  16. [16]
    C.A. Micchelli and H. Prautzsch, Computing curves invariant under halving, Comput. Aided Geom. Design 4 (1987) 133-140.zbMATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    C.A. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989) 841-870.MathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • David Levin

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