Analysis of reconstruction from noisy discrete generalized Radon data

We consider a wide class of generalized Radon transforms R, which act in Rⁿ for any n 2 and integrate over submanifolds of any codimension N, 1 N n-1. Also, we allow for a fairly general reconstruction operator A. The main requirement is that 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 g₉, ₊ = (R f) ₉, ₊ + ₉, ₊. We show that the reconstruction error N_^rec= A ₉, ₊ satisfies N^rec (x;x₀) =₀N_^rec (x₀+ x), x D. Here x₀ is a fixed point, Dⁿ is a bounded domain, and ₉, ₊ are independent, but not necessarily identically distributed, random variables. N^rec and N_^rec are viewed as continuous random functions of the argument x (random fields), and the limit is understood in the sense of probability distributions. Under some conditions on the first three moments of ₉, ₊ (and some other not very restrictive conditions on x₀ and A), we prove that N^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 R³, which shows an excellent match between theoretical predictions and simulated reconstructions.