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 gj,k = ( R f)j,k + ηj,k. We show that the reconstruction error N_εʳᵉᶜ= A ηj,k satisfies Nʳᵉᶜ( x;x₀)=limε→0N_εʳᵉᶜ(x₀+ε x), x∈ D. Here x₀ is a fixed point, Dⁿ is a bounded domain, and ηj,k are independent, but not necessarily identically distributed, random variables. Nʳᵉᶜ and N_εʳᵉᶜ 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 ηj,k (and some other not very restrictive conditions on x₀ and A), we prove that Nʳᵉᶜ 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.