Mojette projections of discrete pixel arrays form good approximations to experimental parallel-beam x-ray intensity absorption profiles. They are discrete sums taken at angles defined by rational fractions. Mojette-like projections form a “half-way house” between a conventional sinogram and fully digital projection data. A new direct and exact image reconstruction technique is proposed here to invert arbitrary but sufficient sets of Mojette data. This new method does not require iterative, statistical solution methods, nor does it use the efficient but noise-sensitive “corner-based” inversion method. It instead exploits the exact invertibility of the prime-sized array Finite Radon Transform (FRT), and the fact that all Mojette projections can be mapped directly into FRT projections. The algorithm uses redundant or “calibrated” areas of an image to expand any asymmetric Mojette set into the smallest symmetric FRT set that contains all of the Mojette data without any re-binning. FRT data will be missing at all angles where Mojette data is not provided, but can be recovered exactly from the “ghost projections” that are generated by back-projecting all the known data across the calibrated regions of the reconstructed image space. Algorithms are presented to enable efficient image reconstruction from any exact Mojette projection set, with a view to extending this approach to invert real x-ray data.