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Isotropic polyharmonic B-splines: scaling functions and wavelets.

Dimitri Van De Ville1, Thierry Blu, Michael Unser

  • 1Biomedical Imaging Group, Swiss Federal Institute of Technology Lausanne (EPFL), Switzerland. dimitri.vandeville@epfl.ch

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|November 11, 2005
PubMed
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We introduce isotropic polyharmonic B-splines for multidimensional wavelet analysis, improving convergence properties over previous methods. This enables efficient image processing using the quincunx subsampling scheme.

Area of Science:

  • * Applied Mathematics
  • * Signal Processing
  • * Image Analysis

Background:

  • * Nonseparable, multidimensional wavelet bases are crucial for advanced signal and image processing.
  • * Existing polyharmonic B-splines exhibit limitations in Gaussian convergence and isotropy.
  • * The quincunx subsampling scheme offers finer scale progression but lacks optimal filter design.

Purpose of the Study:

  • * To develop novel, isotropic polyharmonic B-splines with improved convergence properties.
  • * To design efficient wavelet filters for the 2D quincunx subsampling scheme.
  • * To provide an efficient implementation of the discrete wavelet transform using these new splines.

Main Methods:

  • * Construction of isotropic polyharmonic B-splines using a novel localization operator.

Related Experiment Videos

  • * Derivation of polyharmonic spline wavelets and associated filters from scaling functions for the quincunx scheme.
  • * Analysis of wavelet convergence and frequency domain behavior.

    • Main Results:

    • * Isotropic polyharmonic B-splines demonstrate guaranteed convergence to a Gaussian.
    • * A family of polyharmonic spline wavelets for semi-orthogonal transforms is derived, with automatically specified filters.
    • * The derived wavelets are nearly isotropic, approximate an iterated Laplacian at low frequencies, and converge to separated Gabor atoms.

    Conclusions:

    • * Isotropic polyharmonic B-splines offer a robust foundation for multidimensional wavelet analysis.
    • * The developed method provides an efficient and effective approach for image processing via the quincunx scheme.
    • * An efficient FFT-based discrete wavelet transform implementation is presented.