The authors study a two-dimensional reconstruction algorithm due to D.C. Barber and B.H. Brown, applied to a linearized electrostatic inverse problem. First, the authors demonstrate how this algorithm fits within the framework of inverses of generalized Radon transforms studied by G. Beylkin. Second, an iterative improvement of the Barber-Brown algorithm is constructed based on the conjugate residual method. Several numerical results obtained with this iterative algorithm are presented.