Generalized attenuated ray transforms and their integral angular moments

TL;DR

This paper introduces generalized attenuated ray transforms (ART) and integral angular moments, establishing their mathematical properties, connections to differential equations, and implications for inverse problems in tomography and physics.

Contribution

It extends ART operators to complex absorption and polynomial/exponential weights, deriving differential equations and proving uniqueness theorems for boundary value problems.

Findings

Derived inhomogeneous differential equations for ART of order k

Proved uniqueness theorems for boundary-value problems

Established connections between moments of different orders

Abstract

In this article generalized attenuated ray transforms (ART) and integral angular moments are investigated. Starting from the Radon transform, the attenuated ray transform and the longitudinal ray transform, we derive the concept of ART-operators of order $k$ over functions defined on the phase space and depending on time. The ART-operators are generalized for complex-valued absorption coefficient as well as weight functions of polynomial and exponential type. Connections between ART operators of various orders are established by means of the application of the linear part of a transport equation. These connections lead to inhomogeneous differential equations of order $(k+1)$ for the ART of order $k$. Uniqueness theorems for the corresponding boundary-value and initial boundary-value problems are proved. Properties of integral angular moments of order $p$ are considered and connections between the moments of different orders are deduced. A close connection of the considered operators with mathematical models for tomography, physical optics and integral geometry allows to treat the inversion of ART of order $k$ as an inverse problem of determining the right-hand side of a corresponding differential equation.