Resolution analysis of inverting the generalized Radon transform from discrete data in R^3

08/13/2019 ∙ by Alexander Katsevich, et al. ∙ 0

A number of practically important imaging problems involve inverting the generalized Radon transform (GRT) R of a function f in R^3. On the other hand, not much is known about the spatial resolution of the reconstruction from discretized data. In this paper we study how accurately and with what resolution the singularities of f are reconstructed. The GRT integrates over a fairly general family of surfaces S_y in R^3. Here y is the parameter in the data space, which runs over an open set V⊂ R^3. Assume that the data g(y)=( R f)(y) are known on a regular grid y_j with step-sizes O(ϵ) along each axis, and suppose S=singsupp(f) is a piecewise smooth surface. Let f_ϵ denote the result of reconstruction from the descrete data. We obtain explicitly the leading singular behavior of f_ϵ in an O(ϵ)-neighborhood of a generic point x_0∈ S, where f has a jump discontinuity. We also prove that under some generic conditions on S (which include, e.g. a restriction on the order of tangency of S_y and S), the singularities of f do not lead to non-local artifacts. For both computations, a connection with the uniform distribution theory turns out to be important. Finally, we present a numerical experiment, which demonstrates a good match between the theoretically predicted behavior and actual reconstruction.

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