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 $f_{\epsilon }(x)$, $x\in {\mathbb{R}}^2$, $\epsilon \to 0$, 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 $(f_{\epsilon }^{\text{rec}}-K_{\epsilon }\ast f_{\epsilon })(x_0+\epsilon \check{x})=O({\epsilon }^{1/2}\ln (1/\epsilon ))$, where $x_0\in \mathcal{S}$ and $K_{\epsilon }$ is an easily computable kernel.