Microlocal inversion of a restricted mixed ray transform for second-order tensor fields in $\mathbb{R}^3$

TL;DR

This paper demonstrates the microlocal invertibility of a restricted mixed ray transform acting on second-order tensor fields in 3D space, enabling the recovery of tensor fields from limited line integrals under specific geometric conditions.

Contribution

It introduces a microlocal inversion method for a restricted mixed ray transform on tensor fields in three dimensions, extending previous results to a more limited geometric setting.

Findings

Proves invertibility of the restricted mixed ray transform using microlocal analysis.

Shows tensor fields can be reconstructed up to a smoothing and known singular term.

Establishes conditions under which the transform is invertible on tensor fields.

Abstract

In this article, we study a restricted mixed ray transform acting on second-order tensor fields in 3-dimensional Euclidean space and prove the invertibility of this integral transform using microlocal techniques. Here, the mixed ray transform is restricted over lines passing through a fixed curve $\gamma$ in $\mathbb{R}^3$ satisfying certain geometric conditions. The main theorem of the article shows that a second-order tensor field can be recovered from its restricted mixed-ray transform up to the kernel of the transform, a smoothing term, and a known singular term.