We propose an efficient, deterministic algorithm designed to reconstruct images from real Radon-Transform and Attenuated Radon-Transform data. Its input consists in a small family of recorded signals, each sampling the same composite photon or positron emission scene over a non-Gaussian, noisy channel. The reconstruction is performed by combining a novel numerical implementation of an analytical inversion formula and a novel signal processing technique, inspired by recent work on code reconstruction. Our approach is proven to be optimal under a variety of realistic assumptions. We also indicate several medical imaging applications for which the new technology achieves high fidelity, even when dealing with real data subject to substantial non-Gaussian distortions.