Novel resolution analysis for the Radon transform in ℝ^2 for functions with rough edges

06/09/2022

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by   Alexander Katsevich, et al.

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Let f be a function in ℝ^2, which has a jump across a smooth curve 𝒮 with nonzero curvature. We consider a family of functions f_ϵ with jumps across a family of curves 𝒮_ϵ. Each 𝒮_ϵ is an O(ϵ)-size perturbation of 𝒮, which scales like O(ϵ^-1/2) along 𝒮. Let f_ϵ^rec be the reconstruction of f_ϵ from its discrete Radon transform data, where ϵ is the data sampling rate. A simple asymptotic (as ϵ→0) formula to approximate f_ϵ^rec in any O(ϵ)-size neighborhood of 𝒮 was derived heuristically in an earlier paper of the author. Numerical experiments revealed that the formula is highly accurate even for nonsmooth (i.e., only Hölder continuous) 𝒮_ϵ. In this paper we provide a full proof of this result, which says that the magnitude of the error between f_ϵ^rec and its approximation is O(ϵ^1/2ln(1/ϵ)). The main assumption is that the level sets of the function H_0(·,ϵ), which parametrizes the perturbation 𝒮→𝒮_ϵ, are not too dense.