Discrete Data Fourier Deconvolution

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 L ².

In practice, only discrete measurements {g _m} of g will be available. Consequently, the construction of a discrete approximation {f _m} to f reduces to performing a deconvolution using a discrete version {g _m} = {p _m}∗{f _m} 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 {f _m} will depend on the form and smoothness of f, the size of the interval truncation, and the level of discretization of the measurements {g _m}. Excellent accuracy for {f _m} is obtained when {g _m} and {p _m} 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 {g _m} are on a suitably fine grid.

Dedicated to Ian H. Sloan on the occasion of his 80th birthday.