Efficient estimation of a density in a problem of tomography

The aim of tomography is to reconstruct a multidimensional function From ob servations of its integrals over hyperplanes. We consider the model that co rresponds to the case of positron emission tomography. We have n i.i.d. obs ervations from a probability density proportional to Rf, where Rf stands fo r the Radon transform of the density f. We assume that f is an N-dimensiona l density such that its Fourier transform is exponentially decreasing. We f ind an estimator of f which is asymptotically efficient; it achieves the op timal rate of convergence and also the best constant for the minimax risk.