Advertisement

Numerical Algorithms

, Volume 5, Issue 3, pp 165–177 | Cite as

Iterative polynomial interpolation and data compression

  • Morten DÆhlen
  • Michael Floater
Article

Abstract

In this paper we look at some iterative interpolation schemes and investigate how they may be used in data compression. In particular, we use the pointwise polynomial interpolation method to decompose discrete data into a sequence of difference vectors. By compressing these differences, one can store an approximation to the data within a specified tolerance using a fraction of the original storage space (the larger the tolerance, the smaller the fraction).

We review the iterative interpolation scheme, describe the decomposition algorithm and present some numerical examples. The numerical results are that the best compression rate (ratio of non-zero data in the approximation to the data in the original) is often attained by using cubic polynomials and in some cases polynomials of higher degree.

Keywords

Interpolation Method Storage Space Data Compression Interpolation Scheme Polynomial Interpolation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Arge and M. DÆhlen, Data dependent subdivision, preprint (1992).Google Scholar
  2. [2]
    C. Chui,An Introduction to Wavelets (Academic Press, Boston, 1992).Google Scholar
  3. [3]
    G. Deslauriers and S. Dubuc, Symmetric iterative interpolation processes, Constr. Appr. 5 (1989) 49–68.Google Scholar
  4. [4]
    R.A. deVore, B. Jawerth and B. Lucier, Surface compression, preprint (1991).Google Scholar
  5. [5]
    R.A. deVore, B. Jawerth and B. Lucier, Image processing through wavelet transform coding, preprint (1991).Google Scholar
  6. [6]
    S. Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl. 114 (1986) 185–204.Google Scholar
  7. [7]
    N. Dyn, D. Levin and J. Gregory, A 4-point interpolatory subdivision scheme for curve design, Comp. Aided Geom. Des. 4 (1987) 257–268.Google Scholar
  8. [8]
    M. DÆhlen and T. Lyche, Decomposition of splines,Mathematical Methods, CAGD and Image Processing, eds. T. Lyche and L. Schumaker (Academic Press, Boston, 1992) pp. 135–160.Google Scholar
  9. [9]
    M.S. Floater, Pointwise polynomial interpolation, Research Report, SINTEF-SI, Oslo (1992).Google Scholar
  10. [10]
    T.A. Foley, Interpolation and approximation of 3-D and 4-D scattered data, Comp. Math. App. 13 (1987) 711–740.Google Scholar
  11. [11]
    S. Mallat, Multiresolution approximations and wavelet orthonormal bases ofL 2(ℓ), Trans. Amer. Math. Soc. 315 (1989) 69–87.Google Scholar
  12. [12]
    Y. Meyer,Ondelettes et Opérateurs (Hermann, Paris, 1990).Google Scholar
  13. [13]
    M.J.D. Powell,Approximation Theory and Methods (Cambridge University Press, Cambridge, 1981).Google Scholar
  14. [14]
    O. Rioul, Simple regularity criteria for subdivision schemes, SIAM J. Math. Anal. (1992), to appear.Google Scholar

Copyright information

© J.C. Baltzer AG, Science Publishers 1993

Authors and Affiliations

  • Morten DÆhlen
    • 1
  • Michael Floater
    • 1
  1. 1.SINTEF-SIOsloNorway

Personalised recommendations