Analysis of reconstruction from discrete Radon transform data in $\mathbb R^3$ when the function has jump discontinuities
Provides theoretical foundations for accurate reconstruction of discontinuous functions in 3D Radon inversion, addressing a gap in the literature.
The paper analyzes reconstruction from discrete Radon transform data in 3D for functions with jump discontinuities, deriving the leading term behavior near jumps and proving convergence at generic points via a connection to uniform distribution theory.
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 $ε$, and the density of measured directions on the unit sphere is $O(ε^2)$. Let $f_ε$ 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 $f_ε$ in an $O(ε)$-neighborhood of $x_0$ as $ε\to0$. 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 $f_ε(x_0)\to f(x_0)$ as $ε\to0$. The proof of this result reveals a surprising connection with the theory of uniform distribution (u.d.). 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.