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Discrete Data Fourier Deconvolution

  • Frank de Hoog
  • Russell Davies
  • Richard Loy
  • Robert AnderssenEmail author
Chapter

Abstract

In many practical situations, the recovery of information about some phenomenon of interest f reduces to performing Fourier deconvolution on indirect measurements g = p ∗ f, corresponding to the Fourier convolution of f with a known kernel (point spread function) p. An iterative procedure is proposed for performing the deconvolution of g = p ∗ f, which generates the partial sums of a Neumann series. However, the standard convergence analysis for the Neumann series is not applicable for such deconvolutions so a proof is given which is based on using Fourier properties in L2.

In practice, only discrete measurements {gm} of g will be available. Consequently, the construction of a discrete approximation {fm} to f reduces to performing a deconvolution using a discrete version {gm} = {pm}∗{fm} of g = p ∗ f. For p(x) = sech(x)∕π, it is shown computationally, using the discrete version of the proposed iteration, that the resulting accuracy of {fm} will depend on the form and smoothness of f, the size of the interval truncation, and the level of discretization of the measurements {gm}. Excellent accuracy for {fm} is obtained when {gm} and {pm} accurately approximate the essential structure in g and p, respectively, the support of p is much smaller than that for g, and the discrete measurements of {gm} are on a suitably fine grid.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Frank de Hoog
    • 1
  • Russell Davies
    • 3
  • Richard Loy
    • 2
  • Robert Anderssen
    • 1
    Email author
  1. 1.CSIRO Data 61CanberraAustralia
  2. 2.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  3. 3.School of MathematicsCardiff UniversityCardiff, WalesUK

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