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Nonlinear Subdivision Schemes: Applications to Image Processing

  • Albert Cohen
  • Basarab Matei
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

In classical subdivision schemes, some initial discrete data set v 0 is refined iteratively, following a prescribed linear rule which is summarized by
$$ {v^j} = S{v^{j - 1}} = \cdots = {S^j}{v^0} $$
where v j are the numerical data at resolution 2-j and S the subdivision operator. One is usually interested in the convergence properties of this process to some limit function f = S v 0. In the simplest setting the data v j belongs to the uniform grid 2-j ℤ and convergence means that sup k
$$ \left| {f({2^{ - j}}k) - v_k^j} \right| $$
goes to zero as j tends to +. The analysis of convergence can be performed by various methods, including Fourier analysis by Laurent polynomials [4], when the scheme is uniform.

Keywords

Subdivision Scheme Multiresolution Analysis Laurent Polynomial Subdivision Algorithm Multiscale Representation 
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.
    S. Amat, F. Arandiga, A. Cohen, and R. Donat. Tensor product multiresolution analysis with error control. To appear in Signal Processing, 2001.Google Scholar
  2. 2.
    A. Cohen and R. Ryan. Wavelets and multiscale signal processing. Chapman and Hall, London, 1995.zbMATHGoogle Scholar
  3. 3.
    A. Cohen, N. Dyn, and B. Matei. Quasilinear subdivision schemes with applications to ENO interpolation. Preprint, submitted to Appl. Comput. Harmonic Anal., 2001.Google Scholar
  4. 4.
    N. Dyn. Analysis of convergence and smoothness by the formalism of Laurent polynomials. This volume.Google Scholar
  5. 5.
    B. Matei and A. Cohen. Compact representation of images by edge adapted multiscale transforms. Proceedings of IEEE International Conference on Image Processing, Tessaloniki, October 2001.Google Scholar
  6. 6.
    I. Daubechies. Ten lectures on wavelets. SIAM, Philadelphia, 1992.zbMATHCrossRefGoogle Scholar
  7. 7.
    A. Harten. Discrete multiresolution analysis and generalized wavelets. J. Appl. Num. Math. 12, 1993, 153–193.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    B. Matei. Méthodes multirésolutions non-linéaires — Applications au traitement d’image. PhD thesis, Université P. & M. Curie, Paris, 2002.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Albert Cohen
    • 1
  • Basarab Matei
    • 1
  1. 1.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie CurieParisFrance

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