# Resolution analysis of inverting the generalized Radon transform from discrete data in $\mathbb R^3$

## TL;DR

This paper analyzes the spatial resolution and accuracy of reconstructing singularities of functions in 3D from discretized generalized Radon transform data, providing explicit singular behavior and conditions for artifact-free recovery.

## Contribution

It offers a detailed analysis of the resolution limits and singularity reconstruction accuracy for the GRT in 3D from discrete data, including explicit formulas and conditions for artifact suppression.

## Key findings

- Explicit leading singular behavior of reconstructed functions near discontinuities.
- Conditions under which singularities do not produce non-local artifacts.
- Numerical experiments confirm theoretical predictions.

## Abstract

A number of practically important imaging problems involve inverting the generalized Radon transform (GRT) $\mathcal R$ of a function $f$ in $\mathbb R^3$. On the other hand, not much is known about the spatial resolution of the reconstruction from discretized data. In this paper we study how accurately and with what resolution the singularities of $f$ are reconstructed. The GRT integrates over a fairly general family of surfaces $\mathcal S_y$ in $\mathbb R^3$. Here $y$ is the parameter in the data space, which runs over an open set $\mathcal V\subset\mathbb R^3$. Assume that the data $g(y)=(\mathcal R f)(y)$ are known on a regular grid $y_j$ with step-sizes $O(\epsilon)$ along each axis, and suppose $\mathcal S=\text{singsupp}(f)$ is a piecewise smooth surface. Let $f_\epsilon$ denote the result of reconstruction from the descrete data. We obtain explicitly the leading singular behavior of $f_\epsilon$ in an $O(\epsilon)$-neighborhood of a generic point $x_0\in\mathcal S$, where $f$ has a jump discontinuity. We also prove that under some generic conditions on $\mathcal S$ (which include, e.g. a restriction on the order of tangency of $\mathcal S_y$ and $\mathcal S$), the singularities of $f$ do not lead to non-local artifacts. For both computations, a connection with the uniform distribution theory turns out to be important. Finally, we present a numerical experiment, which demonstrates a good match between the theoretically predicted behavior and actual reconstruction.