Radon-to-Helmholtz mappings and nonlinear diffraction tomography

This paper addresses a number of approximate, analytically invertible solutions of the scalar Helmholtz equation. Primary attention is devoted to the Glauber approximation (GA) derived for the far-field pattern. It is shown that the GA has the form of a nonlinear Radon-to-Helmholtz (RtH) mapping, which transforms a sinogram of the scattering potential into an approximate solution of the Helmholtz equation. A proposal of how to construct a position space counterpart of the GA is formulated. Also, it is established that a paraxial version of the Glauber model coincides, up to an inessential constant factor, with a momentum-space representation of the Mazar-Felsen propagator, which describes forward-scattered waves. For weakly scattering objects, these solutions are reduced to the conventional Born/Rytov approximations, which may, however, preserve the parametrization and sampling formats of the original nonlinear models. Since all RtH mappings are analytically invertible, they can be applied to the (nonlinear) diffraction tomography of penetrable objects. In particular, the Glauber model, which has been largely ignored for years, is shown to provide efficient inversion of synthetic data. The resulting tomograms clearly outperform the Born inversions, even for moderately scattering potentials.