Inversion formula for inverse scattering within the Born approximation

The problem of determining a localized scattering potential V(r) from its associated scattering amplitude f(s, s(0)) is addressed within the first Born approximation. The conventional methods, based on Fourier synthesis, for obtaining approximate solutions to the problem are reviewed briefly. A new reconstruction method is proposed that is in the form of an integral transform over the scattering amplitude, assumed to be specified over all observation directions s and over all incident unit wave vectors s(0). The proposed method is shown also to be applicable to the problem of determining the interatomic distance function integrald(3)r'V(r + r')V*(r') from the magnitude square of the scattering amplitude.