On problem of polarization tomography, I

TL;DR

This paper addresses the polarization tomography problem, proving local uniqueness for recovering a matrix function in anisotropic media and providing partial global results in Euclidean space.

Contribution

It establishes local uniqueness theorems for polarization tomography and offers partial global results in Euclidean space, advancing understanding of inverse problems in optical media.

Findings

Unique recovery of small matrix functions in anisotropic media

Local uniqueness theorem proved for polarization tomography

Partial global results in Euclidean space R^3

Abstract

The polarization tomography problem consists of recovering a matrix function f from the fundamental matrix of the equation $D\eta/dt=\pi_{\dot\gamma}f\eta$ known for every geodesic $\gamma$ of a given Riemannian metric. Here $\pi_{\dot\gamma}$ is the orthogonal projection onto the hyperplan $\dot\gamma^{\perp}$. 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 $R^3$.