Resolution analysis of inverting the generalized $N$-dimensional Radon transform in $\mathbb R^n$ from discrete data

TL;DR

This paper analyzes the resolution limits of inverting the generalized N-dimensional Radon transform from discrete data, providing a detailed limit analysis of the reconstructed singularities and their behavior as data sampling becomes finer.

Contribution

It introduces a method to compute the discrete transition behavior of the reconstructed function near singularities, advancing understanding of resolution in Radon transform inversions from discrete data.

Findings

Derived the limit of scaled reconstructions near singularities

Provided a framework to compute resolution of the inversion process

Analyzed the impact of data discretization on reconstruction accuracy

Abstract

Let $\mathcal R$ denote the generalized Radon transform (GRT), which integrates over a family of $N$-dimensional smooth submanifolds $\mathcal S_{\tilde y}\subset\mathcal U$, $1\le N\le n-1$, where an open set $\mathcal U\subset\mathbb R^n$ is the image domain. The submanifolds are parametrized by points $\tilde y\subset\tilde{\mathcal V}$, where an open set $\tilde{\mathcal V}\subset\mathbb R^n$ is the data domain. The continuous data are $g={\mathcal R} f$, and the reconstruction is $\check f=\mathcal R^*\mathcal B g$. Here $\mathcal R^*$ is a weighted adjoint of $\mathcal R$, and $\mathcal B$ is a pseudo-differential operator. We assume that $f$ is a conormal distribution, $\text{supp}(f)\subset\mathcal U$, and its singular support is a smooth hypersurface $\mathcal S\subset\mathcal U$. Discrete data consists of the values of $g$ on a lattice $\tilde y^j$ with the step size $O(\epsilon)$. Let $\check f_\epsilon=\mathcal R^*\mathcal B g_\epsilon$ denote the reconstruction obtained by applying the inversion formula to an interpolated discrete data $g_\epsilon(\tilde y)$. Pick a generic pair $(x_0,\tilde y_0)$, where $x_0\in\mathcal S$, and $\mathcal S_{\tilde y_0}$ is tangent to $\mathcal S$ at $x_0$. The main result of the paper is the computation of the limit $$ f_0(\check x):=\lim_{\epsilon\to0}\epsilon^\kappa \check f_\epsilon(x_0+\epsilon\check x). $$ Here $\kappa\ge 0$ is selected based on the strength of the reconstructed singularity, and $\check x$ is confined to a bounded set. The limiting function $f_0(\check x)$, which we call the discrete transition behavior, allows computing the resolution of reconstruction.