Analysis of Reconstruction from Discrete Radon Transform Data in R^3 When the Function Has Jump Discontinuities

In this paper we study reconstruction of a function f from its discrete Radon transform data in R³ when f has jump discontinuities. Consider a conventional parametrization of the Radon data in terms of the affine and angular variables. The step size along the affine variable is ε, and the density of measured directions on the unit sphere is O(ε²). Let f_ε denote the result of reconstruction from the discrete data. Pick any generic point x₀ (i.e., satisfying some mild conditions), where f has a jump. Our first result is an explicit leading term behavior of f_ε in an O(ε)-neighborhood of x₀ as ε→0. A closely related question is why can we accurately reconstruct functions with discontinuities at all? This is a fundamental question, which has not been studied in the literature in dimensions three and higher. We prove that the discrete inversion formula “works,” i.e., if x₀∈ S:=singsupp(f) is generic, then f_ε(x₀)→ f(x₀) as ε→0. The proof of this result reveals a surprising connection with the theory of uniform distribution. This is a new phenomenon that has not been known previously. We also present some numerical experiments, which confirm the validity of the developed theory.