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

Abstract

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 $f_\epsilon$ with jumps across a family of curves $\mathcal S_\epsilon$. Each $\mathcal S_\epsilon$ is an $O(\epsilon)$-size perturbation of $\mathcal S$, which scales like $O(\epsilon^{-1/2})$ along $\mathcal S$. Let $f_\epsilon^{\text{rec}}$ be the reconstruction of $f_\epsilon$ from its discrete Radon transform data, where $\epsilon$ is the data sampling rate. A simple asymptotic (as $\epsilon\to0$) formula to approximate $f_\epsilon^{\text{rec}}$ in any $O(\epsilon)$-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{\"o}lder continuous) $\mathcal S_\epsilon$. In this paper we provide a full proof of this result, which says that the magnitude of the error between $f_\epsilon^{\text{rec}}$ and its approximation is $O(\epsilon^{1/2}\ln(1/\epsilon))$. The main assumption is that the level sets of the function $H_0(\cdot,\epsilon)$, which parametrizes the perturbation $\mathcal S\to\mathcal S_\epsilon$, are not too dense.