One shot inverse scattering revisited

We develop and discuss a novel reconstruction algorithm for the inverse source problem and the inverse scattering problem for the Helmholtz equation with well-separated compactly supported sources or scatterers in two-dimensional free space from far field observations of a single radiated or scattered wave. We show that a rational approximation of a Laurent polynomial formed by the low order Fourier coefficients of the given far field pattern can be used to determine straight lines connecting the support of the sources or scatterers to the origin. After repeating this procedure for many different choices of the origin, we apply a filtered backprojection algorithm to recover information on the number and the location of the unknown sources or scatterers. We give numerical examples to illustrate the performance and limitations of our reconstruction algorithm.