We present algorithms to reconstruct images from minimal sets of discrete Mojette projections using direct back-projection (DBP) with various forms of correction. This paper extends previous work on discrete projection inversion by Servières et al [1, 2, 3]. The number of Mojette projections needed for exact inversion by DBP (EI-DBP) scales as O(N²). A new form of discrete interpolation is developed to expand the point spread function (PSF) of a minimal (Katz-sufficient) set of discrete projections to encompass new directions and thus augment the size of the reconstruction region to which EI-DBP applies. Additionally, we propose a Fourier domain filter for Mojette back-projection that is built from the discrete PSF of the given Mojette angle set and the autocorrelation function of the image domain. These discrete reconstruction methods are targeted for use with noisy sets of real projection data.