We consider a wide class of generalized Radon transforms $\mathcal R$, which act in $\mathbb{R}^n$ for any $n\ge 2$ and integrate over submanifolds of any codimension $N$, $1\le N\le n-1$. Also, we allow for a fairly general reconstruction operator $\mathcal A$. The main requirement is that $\mathcal A$ be a Fourier integral operator with a phase function, which is linear in the phase variable. We consider the task of image reconstruction from discrete data $g_{j,k} = (\mathcal R f)_{j,k} + \eta_{j,k}$. We show that the reconstruction error $N_\epsilon^{\text{rec}}=\mathcal A \eta_{j,k}$ satisfies $N^{\text{rec}}(\check x;x_0)=\lim_{\epsilon\to0}N_\epsilon^{\text{rec}}(x_0+\epsilon\check x)$, $\check x\in D$. Here $x_0$ is a fixed point, $D\subset\mathbb{R}^n$ is a bounded domain, and $\eta_{j,k}$ are independent, but not necessarily identically distributed, random variables. $N^{\text{rec}}$ and $N_\epsilon^{\text{rec}}$ are viewed as continuous random functions of the argument $\check x$ (random fields), and the limit is understood in the sense of probability distributions. Under some conditions on the first three moments of $\eta_{j,k}$ (and some other not very restrictive conditions on $x_0$ and $\mathcal A$), we prove that $N^{\text{rec}}$ is a zero mean Gaussian random field and explicitly compute its covariance. We also present a numerical experiment with a cone beam transform in $\mathbb{R}^3$, which shows an excellent match between theoretical predictions and simulated reconstructions.

翻译：我们研究一大类广义Radon变换$\mathcal R$，其作用于任意维度$n\ge 2$的$\mathbb{R}^n$空间，并在任意余维$N$（$1\le N\le n-1$）的子流形上进行积分。同时，我们允许相当一般的重建算子$\mathcal A$。主要要求是$\mathcal A$须为具有相位函数的傅里叶积分算子，且该相位函数关于相位变量是线性的。我们考虑从离散数据$g_{j,k} = (\mathcal R f)_{j,k} + \eta_{j,k}$进行图像重建的任务。我们证明重建误差$N_\epsilon^{\text{rec}}=\mathcal A \eta_{j,k}$满足$N^{\text{rec}}(\check x;x_0)=\lim_{\epsilon\to0}N_\epsilon^{\text{rec}}(x_0+\epsilon\check x)$，其中$\check x\in D$。此处$x_0$为固定点，$D\subset\mathbb{R}^n$为有界域，$\eta_{j,k}$为独立但不一定同分布的随机变量。$N^{\text{rec}}$与$N_\epsilon^{\text{rec}}$被视为关于自变量$\check x$的连续随机函数（随机场），且极限按概率分布的意义理解。在$\eta_{j,k}$的前三阶矩满足一定条件（以及对$x_0$和$\mathcal A$的其他非严格限制条件）的情况下，我们证明$N^{\text{rec}}$是零均值高斯随机场，并显式计算其协方差。我们还展示了$\mathbb{R}^3$中锥束变换的数值实验，该实验表明理论预测与模拟重建结果高度吻合。