  We propose an inverse scattering scheme of recovering a polyhedral obstacle
in $\mathbb{R}^n$, $n=2,3$, by only a few high-frequency acoustic
backscattering measurements. The obstacle could be sound-soft or sound-hard. It
is shown that the modulus of the far-field pattern in the backscattering
aperture possesses a certain local maximum behavior, from which one can
determine the exterior normal directions of the front sides/faces. Then by
using the phaseless backscattering data corresponding to a few incident plane
waves with suitably chosen incident directions, one can determine the exterior
unit normal vector of each side/face of the obstacle. After the determination
of the exterior unit normals, the recovery is reduced to a finite-dimensional
problem of determining a location point of the obstacle and the distance of
each side/face away from the location point. For the latter reconstruction, we
need make use of the far-field data with phases. Numerical experiments are also
presented to illustrate the effectiveness of the proposed scheme.
