Single pixel X-ray transform and related inverse problems

TL;DR

This paper investigates a nonlinear single pixel X-ray transform, deriving stability, inversion formulas, and numerical validations, offering new insights into reconstructing functions from minimal, nonlinear measurements in imaging.

Contribution

It introduces a novel nonlinear X-ray transform with a single detector, providing stability estimates, inversion formulas, and extending analysis to geodesic integrations.

Findings

Derived stability estimates for the transform

Established an inversion formula for reconstruction

Numerical experiments support theoretical results

Abstract

In this paper, we analyze the nonlinear single pixel X-ray transform $K$ and study the reconstruction of $f$ from the measurement $Kf$. Different from the well-known X-ray transform, the transform $K$ is a nonlinear operator and uses a single detector that integrates all rays in the space. We derive stability estimates and an inversion formula of $K$. We also consider the case where we integrate along geodesics of a Riemannian metric. Moreover, we conduct several numerical experiments to corroborate the theoretical results.