Scattering of directed waves as an invertible Radon-to-Helmholtz mapping

A concept of invertible Radon-to-Helmholtz mapping is proposed as a model of directed (i.e., paraxial) wave scattering. It is shown that not only the Mazar-Felsen (MF) solution, but also Born and Rytov approximations can be expressed in terms of a propagation operator that transforms (the complex exponential of) a linogram of the illuminated object into a set of its diffraction patterns. Since the propagation operator is easily invertible, a unified approach, based on either of these approximations, may be used to recover the scattering potential. For a purely refractive phantom, the superiority of the MF propagator as compared to the classical perturbative solutions is demonstrated.