Identifying convex obstacles from backscattering far field data

The recovery of anomalies from backscattering far field data is a long-standing open problem in inverse scattering theory. We make a first step in this direction by establishing the unique identifiability of convex impenetrable obstacles from backscattering far field measurements. Specifically, we prove that both the boundary and the boundary conditions of the convex obstacle are uniquely determined by the far field pattern measured in backscattering directions for all frequencies. The key tool is Majda's asymptotic estimate of the far field patterns in the high-frequency regime. Furthermore, we introduce a fast and stable numerical algorithm for reconstructing the boundary and computing the boundary condition. A key feature of the algorithm is that the boundary condition can be computed even if the boundary is not known, and vice versa. Numerical experiments demonstrate the validity and robustness of the proposed algorithm.