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On the Coupling of Decimation Operator with Subdivision Schemes for Multi-scale Analysis

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Mathematical Methods for Curves and Surfaces (MMCS 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10521))

Abstract

Subdivision schemes [5, 11] are powerful tools for the fast generation of refined sequences ultimately representing curves or surfaces. Coupled with decimation operators, they generate multi-scale transforms largely used in signal/image processing [1, 3] that generalize the multi-resolution analysis/wavelet framework [8]. The flexibility of subdivision schemes (a subdivision scheme can be non-stationary, non-homogeneous, position-dependent, interpolating, approximating, non-linear...) (e.g. [3]) is balanced, as a counterpart, by the fact that the construction of suitable consistent decimation operators is not direct and easy.

In this paper, we first propose a generic approach for the construction of decimation operators consistent with a given linear subdivision. A study of the so-called prediction error within the multi-scale framework is then performed and a condition on the subdivision mask to ensure a fast decay of this error is established. Finally, the cases of homogeneous Lagrange interpolatory subdivision, spline subdivision, subdivision related to Daubechies scaling functions (and wavelets) and some recently developed non stationary non interpolating schemes are revisited.

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Notes

  1. 1.

    a scheme U is said to reproduce constant if \(\forall k, f_k=C \implies \forall k, (Uf)_k=C\).

References

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Authors and Affiliations

  1. Centrale Marseille, I2M, UMR 7353, CNRS, Aix-Marseille University, 13451, Marseille, France

    Zhiqing Kui & Jacques Liandrat

  2. Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PSN-RES/SEMIA/LIMAR, CE Cadarache, 13115, Saint Paul Les Durance, France

    Jean Baccou

Authors
  1. Zhiqing Kui
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  2. Jean Baccou
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  3. Jacques Liandrat
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Corresponding author

Correspondence to Zhiqing Kui .

Editor information

Editors and Affiliations

  1. University of Oslo, Oslo, Norway

    Michael Floater

  2. University of Oslo, Oslo, Norway

    Tom Lyche

  3. Université Joseph Fourier, Grenoble, France

    Marie-Laurence Mazure

  4. University of Oslo, Oslo, Norway

    Knut Mørken

  5. Vanderbilt University, Nashville, Tennessee, USA

    Larry L. Schumaker

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Kui, Z., Baccou, J., Liandrat, J. (2017). On the Coupling of Decimation Operator with Subdivision Schemes for Multi-scale Analysis. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2016. Lecture Notes in Computer Science(), vol 10521. Springer, Cham. https://doi.org/10.1007/978-3-319-67885-6_9

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  • DOI: https://doi.org/10.1007/978-3-319-67885-6_9

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67884-9

  • Online ISBN: 978-3-319-67885-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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