  On a time-oriented Lorentzian manifold $(M,g)$ with non-empty boundary
satisfying a convexity assumption, we show that the topological,
differentiable, and conformal structure of suitable subsets $S\subset M$ of
sources is uniquely determined by measurements of the intersection of future
light cones from points in $S$ with a fixed open subset of the boundary of $M$;
here, light rays are reflected at $\partial M$ according to Snell's law. Our
proof is constructive, and allows for interior conjugate points as well as
multiply reflected and self-intersecting light cones.
