# Analysis of reconstruction from discrete Radon transform data in $\mathbb R^3$ when the function has jump discontinuities

## TL;DR

This paper analyzes how well discrete Radon transform data can reconstruct functions with jump discontinuities in three dimensions, providing explicit asymptotic behavior and proving accurate reconstruction at generic points.

## Contribution

It provides the first explicit asymptotic analysis of reconstruction near discontinuities in 3D and establishes conditions under which the discrete inversion accurately recovers functions.

## Key findings

- Explicit leading term behavior of reconstructed function near discontinuities
- Discrete inversion formula converges at generic points with jumps as data density increases
- Connection between reconstruction accuracy and uniform distribution theory

## Abstract

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 (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.