On problem of polarization tomography, I

The polarization tomography problem consists of recovering a matrix function f from the fundamental matrix of the equation Dη/dt = πγ̇fη known for every geodesic γ of a given Riemannian metric. Here πγ̇ is the orthogonal projection onto the hyperplane γ̇⊥. The problem arises in optical tomography of slightly anisotropic media. The local uniqueness theorem is proved: a C1-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 R3.