A non-linear inverse scattering algorithm is presented that uses a local shape function (LSF) approximation to parametrize very strong scatterers in the presence of a transient excitation source. The LSF approximation was presented recently in the context of continuous-wave (CW) excitation and was shown to give good reconstructions of strong scatterers such as metallic objects. It is shown that the local (binary) shape function may be implemented as a volumetric boundary condition in a finite-difference time domain (FDTD) forward scattering solver. The inverse scattering problem is then cast as a non-linear optimization problem where the N-dimensional Frechet derivative of the scattered field is computed as a single backpropagation and correlation using the FDTD forward solver. Connection between the new algorithm and a similar method employing the distorted Born approximation is shown. Computer simulations show that the LSF method employing a FDTD forward solver has superior convergence properties over the corresponding distorted-Born algorithm.