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

In this paper, we investigate the reconstruction error, N_ʳᵉᶜ(x), when a linear, filtered back-projection (FBP) algorithm is applied to noisy, discrete Radon transform data with sampling step size ε in two-dimensions. Specifically, we analyze N_ʳᵉᶜ(x) for x in small, O()-sized neighborhoods around a generic fixed point, x₀, in the plane, where the measurement noise values, ηk,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 ηk,j, that the following limit exists: Nʳᵉᶜ(;x₀) = lim→0N_ʳᵉᶜ(x₀+), for x in a bounded domain. Here, N_ʳᵉᶜ and Nʳᵉᶜ are viewed as continuous random variables, and the limit is understood in the sense of distributions. Once the limit is established, we prove that Nʳᵉᶜ 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.