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.

翻译：我们研究了在步长为$O(\epsilon)$的采样下，从离散Radon变换数据重建函数族$f_\epsilon(x)$（$x\in\mathbb R^2$，$\epsilon\to0$）的精确性。对于每个充分小的$\epsilon>0$，函数$f_\epsilon$在粗糙边界$\mathcal S_\epsilon$上存在跳跃，该边界由光滑边界$\mathcal S$的$O(\epsilon)$量级扰动建模。描述扰动的函数$H_0$被假设为有界变差。令$f_\epsilon^{\text{rec}}$表示重建结果，它通过插值离散数据并代入连续反演公式计算得到。我们证明$(f_\epsilon^{\text{rec}}-K_\epsilon*f_\epsilon)(x_0+\epsilon\check x)=O(\epsilon^{1/2}\ln(1/\epsilon))$，其中$x_0\in\mathcal S$，而$K_\epsilon$是一个易于计算的核函数。