  This paper is devoted to a novel quantitative imaging scheme of identifying
impenetrable obstacles in time-harmonic acoustic scattering from the associated
far-field data. The proposed method consists of two phases. In the first phase,
we determine the interior eigenvalues of the underlying unknown obstacle from
the far-field data via the indicating behaviour of the linear sampling method.
Then we further determine the associated interior eigenfunctions by solving a
constrained optimization problem, again only involving the far-field data. In
the second phase, we propose a novel iteration scheme of Newton's type to
identify the boundary surface of the obstacle. By using the interior
eigenfunctions determined in the first phase, we can avoid computing any direct
scattering problem at each Newton's iteration. The proposed method is
particularly valuable for recovering a sound-hard obstacle, where the Newton's
formula involves the geometric quantities of the unknown boundary surface in a
natural way. We provide rigorous theoretical justifications of the proposed
method. Numerical experiments in both 2D and 3D are conducted, which confirm
the promising features of the proposed imaging scheme. In particular, it can
produce quantitative reconstructions of high accuracy in a very efficient
manner.
