The problem of polarization tomography: II

Let f be a matrix function on a bounded domain D subset of R-n furnished with a Riemannian metric. For a unit speed geodesic gamma : [0, l] --> D between boundary points, let Phi[f](gamma) = U(l), where U(t) is the solution to the Cauchy problem DU/dt = (Q(gamma)(t) f (gamma(t))) U, U(0) = E, E being the unit matrix. Here Q(xi) is an orthogonal projection onto the space {h is an element of gl(C-n)vertical bar h xi = h*xi= 0, tr h = 0}. We consider the inverse problem of recovering the function f from the data Phi[f] known on the manifold of all unit speed geodesics between boundary points. The problem arises in optical tomography of weakly 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.