Single Pixel X-ray Transform and Related Inverse Problems

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 of the linearization at zero. 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.