Resolution Analysis of Inverting the Generalized Radon Transform from Discrete Data in $\mathbb{R}^3$

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 discrete 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 nonlocal 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.