  Let $\sigma$ 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 $\ell$ 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 $\sigma$ and $\ell$ 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
$\sigma$ and $\ell$ restricted to $D$ uniquely recover an isometric copy of $g$
locally near generic metrics, and in particular, near real analytic ones.
