Local reconstruction analysis of inverting the Radon transform in the plane from noisy discrete data

In this paper, we investigate the reconstruction error, $N_{\text{e}}^{\text{rec}}(x)$, 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 $N_{\text{e}}^{\text{rec}}(x)$ for $x$ in small, $O(\text{e})$-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: $N^{\text{rec}}(\text{chx};x_0)={\lim }_{\text{e}\to 0}{N}_{\text{e}}^{\text{rec}}(x_0+\text{e}\text{chx})$, for $\check{x}$ in a bounded domain. Here, $N_{\text{e}}^{\text{rec}}$ 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.