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Numerical Issues When Using Wavelets

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Computational Complexity

Definition of the Subject

Wavelets and related multiscale representations pervade all areas of signal processing. The recent inclusion of wavelet algorithms in JPEG 2000 – the new still‐picture compression standard – testifies to this lasting and significant impact. The reason of thesuccess of the wavelets is due to the fact that wavelet basis represents well a large class of signals, and therefore allows us to detect roughlyisotropic elements occurring at all spatial scales and locations. As the noise in the physical sciences is often not Gaussian, the modeling, in thewavelet space, of many kind of noise (Poisson noise, combination of Gaussian and Poisson noise, long‐memory \( { 1/f } \) noise, non‐stationary noise, …) hasalso been a key step for the use of wavelets in scientific, medical, or industrial applications [41]. Extensive wavelet packages exist now, commercial (see for example [29]) or non commercial (see for example [47,48]), which allows any researcher, doctor, or...

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Notes

  1. 1.

    We only consider here real wavelets. This can be extended to complex wavelets without too muchdifficulty.

  2. 2.

    This integer wavelet transform is based upon a symmetric, bi‐orthogonal wavelet transform built from the interpolating Deslauriers–Dubuc scaling function where both the high-pass filter and its dual has 2 vanishing moments moments [26].

  3. 3.

    In frame theory parlance, we would say that the UWT frame synthesis operator is not injective.

Abbreviations

WT:

Wavelet Transform

CWT:

Continuous Wavelet Transform

DWT:

Discrete (decimated) Wavelet Transform

UWT:

Undecimated Wavelet Transform

IUWT:

Isotropic Undecimated Wavelet Transform

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Author information

Authors and Affiliations

  1. DAPNIA/SEDI-SAP, Service d'Astrophysique, CEA/Saclay, Gif sur Yvette, France

    Jean-Luc Starck

  2. GREYC CNRS UMR 6072, Image Processing Group, Ecole Nationale Supèrieure d'Ingènieurs de Caen, Caen Cedex, France

    Jalal Fadili

Authors
  1. Jean-Luc Starck
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  2. Jalal Fadili
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Starck, JL., Fadili, J. (2012). Numerical Issues When Using Wavelets. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_134

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