The Mojette transform is a form of discrete Radon transform that maps a 2D image (P×Q pixels) to a set of I 1D projections. Several fast inversion methods exist that require O(PQI) operations but those methods are ill-conditioned. Several robust (or well-conditioned) inversion methods exist, but they are slow, requiring O(P²Q²I) operations. Ideally we require an inversion scheme that is both fast and robust to deal with noisy projections. Noisy projection data can arise from data that is corrupted in storage or by errors in data transmission, quantisation errors in image compression, or through noisy acquisition of physical projections, such as in X-ray computed tomography. This paper presents a robust reconstruction method, performed in the Fourier domain, that requires O(P²QlogP) operations.