The Momentum Light Ray Transform

TL;DR

This paper investigates the properties and inversion of the Momentum Light Ray Transform (MLRT) for tensor fields, providing conditions for injectivity and developing algorithms for restricted measurements in tensor tomography.

Contribution

It establishes necessary and sufficient conditions for MLRT injectivity on tensor fields and introduces an inversion algorithm for limited measurement scenarios.

Findings

Identifies conditions for MLRT injectivity on tensor fields.

Develops an inversion algorithm for restricted measurements.

Provides theoretical foundations for tensor tomography using MLRT.

Abstract

In this article, we study Momentum Light Ray Transform (MLRT) on symmetric tensor fields. MLRT is an integral transform in time-space domain ($(t,x)\in \mathbb{R}^{1+n}$), which integrates a scalar function or a tensor field along the light rays with a polynomial type weight. We explore necessary and sufficient conditions for injectivity of MLRT, over general order tensors on space dimension $\geq 2$, from full and restricted measurements. Furthermore, we develop an inversion algorithm for MLRTs in the restricted measurement setting. To prove the results, we use tools from tensor tomography, geometry, and analysis.