In this paper, we investigate the reconstruction error, $N_\e^{\text{rec}}(x)$, when a linear, filtered back-projection (FBP) algorithm is applied to noisy, discrete Radon transform data with sampling step size $\epsilon$ in two-dimensions. Specifically, we analyze $N_\e^{\text{rec}}(x)$ for $x$ in small, $O(\e)$-sized neighborhoods around a generic fixed point, $x_0$, in the plane, where the measurement noise values, $\eta_{k,j}$ (i.e., the errors in the sinogram space), are random variables. The latter are independent, but not necessarily identically distributed. We show, under suitable assumptions on the first three moments of the $\eta_{k,j}$, that the following limit exists: $N^{\text{rec}}(\chx;x_0) = \lim_{\e\to0}N_\e^{\text{rec}}(x_0+\e\chx)$, for $\check x$ in a bounded domain. Here, $N_\e^{\text{rec}}$ and $ N^{\text{rec}}$ are viewed as continuous random variables, and the limit is understood in the sense of distributions. Once the limit is established, we prove that $N^{\text{rec}}$ is a zero mean Gaussian random field and compute explicitly its covariance. In addition, we validate our theory using numerical simulations and pseudo random noise.

翻译：本文研究了在二维空间中，当对步长为$\epsilon$的含噪离散Radon变换数据应用线性滤波反投影（FBP）算法时，重构误差$N_\e^{\text{rec}}(x)$的性质。具体地，我们分析了平面中固定点$x_0$周围$O(\e)$量级的小邻域内$x$处的$N_\e^{\text{rec}}(x)$，其中测量噪声$\eta_{k,j}$（即正弦图空间中的误差）为随机变量。这些随机变量相互独立但未必同分布。在关于$\eta_{k,j}$前三阶矩的适当假设下，我们证明以下极限存在：$N^{\text{rec}}(\chx;x_0) = \lim_{\e\to0}N_\e^{\text{rec}}(x_0+\e\chx)$，其中$\check x$位于有界域内。这里$N_\e^{\text{rec}}$和$N^{\text{rec}}$被视为连续随机变量，且极限理解为分布意义下的收敛。在建立极限后，我们证明$N^{\text{rec}}$是均值为零的高斯随机场，并显式计算其协方差。此外，我们通过数值模拟和伪随机噪声验证了理论结果。