Shape reconstruction of acoustic obstacles from the modulus of the far field pattern

We consider the inverse problem of time-harmonic acoustic wave scattering where the shape of an obstacle is reconstructed from a given incident field and the modulus of the far field pattern of the scattered field. Our approach is based on a pair of nonlinear and ill-posed integral equations to be solved for the shape of the unknown boundary. This approach is an extension of the method suggested by Kress and Rundell [5] for an inverse boundary value problem for the Laplace equation. Since the modulus of far field pattern is invariant under translations [4] we can reconstruct the shape of the obstacle but not the location.
    The numerical implementation of the method is described and it is illustrated by numerical examples that the method yields satisfactory reconstructions both for sound-soft and sound-hard obstacles, also in the case when the modulus is given in a limited aperture.