A projection operator for reconstruction images from sparse, ungridded data

In image reconstruction by the method of convex projections an image vector is successively projected on to subsets defined by various items of prior information and by the measured data. Some of these operators are quite complex and can exhibit numerical instability caused by partial redundancy of the information. We illustrate this, and the application of inverse theory to the method of convex projections, by deriving an operator for projecting on to the subset defined by the measured data and analysing its stability. We show that, with pixel basis functions, nearest-neighbour interpolation with averaging leads to a simple but exact implementation of the stabilized operator. We use this to derive an approximation for some other basis functions.