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

Let $f$ be a function in $\mathbb R^2$, which has a jump across a smooth curve $\mathcal S$ with nonzero curvature. We consider a family of functions $f_ε$ with jumps across a family of curves $\mathcal S_ε$. Each $\mathcal S_ε$ is an $O(ε)$-size perturbation of $\mathcal S$, which scales like $O(ε^{-1/2})$ along $\mathcal S$. Let $f_ε^{\text{rec}}$ be the reconstruction of $f_ε$ from its discrete Radon transform data, where $ε$ is the data sampling rate. A simple asymptotic (as $ε\to0$) formula to approximate $f_ε^{\text{rec}}$ in any $O(ε)$-size neighborhood of $\mathcal S$ 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 H{ö}lder continuous) $\mathcal S_ε$. In this paper we provide a full proof of this result, which says that the magnitude of the error between $f_ε^{\text{rec}}$ and its approximation is $O(ε^{1/2}\ln(1/ε))$. The main assumption is that the level sets of the function $H_0(\cdot,ε)$, which parametrizes the perturbation $\mathcal S\to\mathcal S_ε$, are not too dense.

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