Novel resolution analysis for the Radon transform in ā„^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 ā„^2, which has a jump across a smooth curve š’® with nonzero curvature. We consider a family of functions f_Ļµ with jumps across a family of curves š’®_Ļµ. Each š’®_Ļµ is an O(Ļµ)-size perturbation of š’®, which scales like O(Ļµ^-1/2) along š’®. Let f_Ļµ^rec be the reconstruction of f_Ļµ from its discrete Radon transform data, where Ļµ is the data sampling rate. A simple asymptotic (as Ļµā†’0) formula to approximate f_Ļµ^rec in any O(Ļµ)-size neighborhood of š’® 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 Hƶlder continuous) š’®_Ļµ. In this paper we provide a full proof of this result, which says that the magnitude of the error between f_Ļµ^rec and its approximation is O(Ļµ^1/2ln(1/Ļµ)). The main assumption is that the level sets of the function H_0(Ā·,Ļµ), which parametrizes the perturbation š’®ā†’š’®_Ļµ, are not too dense.

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