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 $\mathbb{R}^3$ 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 $\epsilon$, and the density of measured directions on the unit sphere is $O(\epsilon^2)$. Let $f_\epsilon$ denote the result of reconstruction from the discrete data. Pick any generic point $x_0$ (i.e., satisfying some mild conditions), where $f$ has a jump. Our first result is an explicit leading term behavior of $f_\epsilon$ in an $O(\epsilon)$-neighborhood of $x_0$ as $\epsilon\to0$. 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 $x_0\not\in S:=\text{singsupp}(f)$ is generic, then $f_\epsilon(x_0)\to f(x_0)$ as $\epsilon\to0$. 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.