Let be the scattering relation on a compact Riemannian manifold M with non-necessarily convex boundary, that maps initial
points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction.
Let . be the length of that geodesic ray. We study the question of whether the metric g is uniquely determined, up to an isometry,
by knowledge of 冃 and . restricted on some subset D. We allow possible conjugate points but we assume that the conormal bundle
of the geodesics issued from D covers T M; and that those geodesics have no conjugate points. Under an additional topological
assumption, we prove that 冃 and . restricted to D uniquely recover an isometric copy of g locally near generic metrics, and in particular, near real analytic ones.