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A method is proposed for the digital reconstruction of images from their projections based on optimising specified performance criteria. The reconstruction problem is embedded into the framework of constrained optimisation and its solution is shown to lead to a relationship between the image and the one-dimensional Lagrange functions associated with each cost criterion. Two types of geometries (the parallel-beam and fan-beam systems) are considered for the acquisition of projection data and the constrained-optimisation problem is solved for both. The ensuing algorithms allow the reconstruction of multidimentional objects from one-dimensional functions only. For digital data a fast reconstruction algorithm is proposed which exploits the symmetries inherent in both a circular domain of image reconstruction and in projections obtained at equispaced angles. Computational complexity is significantly reduced by the use of fast-Fourier-transform techniques, as the underlying relationship between the available projection data and the associated Lagrange multipliers is shown to possess a block circulant matrix structure.