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

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.