Inverse obstacle problems with backscattering or generalized backscattering data in one or two directions

This paper is concerned with scattering by a smooth compact obstacle in R^d. The main results are that for certain classes of obstacles knowledge of the scattering amplitude in only one or two pairs of incident and reflected directions suffices to recover the Taylor series of a boundary defining function for the obstacle at a point. Thus certain obstacles with real analytic boundaries can be determined.