Sampling the X-ray transform on simple surfaces

TL;DR

This paper investigates optimal sampling strategies for the geodesic X-ray transform on simple surfaces, providing minimal sampling rates and analyzing aliasing artifacts based on surface geometry and coordinate choices.

Contribution

It introduces a framework for discretizing the X-ray transform on simple surfaces, including minimal sampling rates and artifact prediction methods.

Findings

Derived minimal sampling rates for faithful reconstruction

Quantified sampling quality dependence on surface geometry

Explained artifact prediction when aliasing occurs

Abstract

We study the problem of proper discretizing and sampling issues related to geodesic X-ray transforms on simple surfaces, and illustrate the theory on simple geodesic disks of constant curvature. Given a notion of band limit on a function, we provide the minimal sampling rates of its X-ray transform for a faithful reconstruction. In Cartesian sampling, we quantify the quality of a sampling scheme depending on geometric parameters of the surface (e.g. curvature and boundary curvature), and the coordinate system used to represent the space of geodesics. When aliasing happens, we explain how to predict the location, orientation and frequency of the artifacts.