On problem of polarization tomography, I

Abstract

The polarization tomography problem consists of recovering a matrix function f from the fundamental matrix of the equation $Dη/dt=π_{\dotγ}fη$ known for every geodesic $γ$ of a given Riemannian metric. Here $π_{\dotγ}$ is the orthogonal projection onto the hyperplan $\dotγ^{\perp}$. The problem arises in optical tomography of slightly anisotropic media. The local uniqueness theorem is proved: a $C^1$- small function f can be recovered from the data uniquely up to a natural obstruction. A partial global result is obtained in the case of the Euclidean metric on $R^3$.