Efficient estimation of a density in a problem of tomography

The aim of tomography is to reconstruct a multidimensional function from observations of its integrals over hyperplanes. We consider the model that corresponds to the case of positron emission tomography. We have n i.i.d.observations from a probability density proportional to $Rf$, where $Rf$ stands for the Radon transform of the density f.We assume that f is an N-dimensional density such that its Fourier transform is exponentially decreasing. We find an estimator of f which is asymptotically efficient; it achieves the optimal rate of convergence and also the best constant for the minimax risk.