Analysis of reconstruction of functions with rough edges from discrete Radon data in $\mathbb R^2$

Abstract

We study the accuracy of reconstruction of a family of functions $f_\epsilon(x)$, $x\in\mathbb R^2$, $\epsilon\to0$, from their discrete Radon transform data sampled with step size $O(\epsilon)$. For each $\epsilon>0$ sufficiently small, the function $f_\epsilon$ has a jump across a rough boundary $\mathcal S_\epsilon$, which is modeled by an $O(\epsilon)$-size perturbation of a smooth boundary $\mathcal S$. The function $H_0$, which describes the perturbation, is assumed to be of bounded variation. Let $f_\epsilon^{\text{rec}}$ denote the reconstruction, which is computed by interpolating discrete data and substituting it into a continuous inversion formula. We prove that $(f_\epsilon^{\text{rec}}-K_\epsilon*f_\epsilon)(x_0+\epsilon\check x)=O(\epsilon^{1/2}\ln(1/\epsilon))$, where $x_0\in\mathcal S$ and $K_\epsilon$ is an easily computable kernel.