An inversion formula for transport equation in 3-dimensions using several complex variable analysis

TL;DR

This paper extends the 3D photon transport equation into complex space, deriving inverse solutions on hyper-spheres and hyper-cylinders, and demonstrating their equivalence to classical X-ray and Radon transforms.

Contribution

It introduces a novel complex analysis approach to inverse transport problems in three dimensions, linking different geometric transforms.

Findings

Inverse solutions on hyper-sphere and hyper-cylinder are obtained.

Results are shown to be equivalent and consistent with classical transforms.

The method bridges complex analysis with transport equation inversion.

Abstract

In this paper, the photon stationary transport equation has been extended from $\mathbb{R}^3$ to $\mathbb{C}^3$. A solution of the inverse problem is obtained on a hyper-sphere and a hyper-cylinder as X-ray and Radon transform, respectively. We show that these results can be transformed into each other and they agree with known results.