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

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.