On the problem of local adaptive estimation in tomography

The principle of tomography is to reconstruct a multidimensional function f rom observations of its integrals over hyperplanes. We consider here a mode l of stochastic tomography where we observe the Radon transform Rf of the f unction f with a stochastic error. Then we construct a 'data-driven' estima tor which does not depend on any a priori smoothness assumptions on the fun ction f. Considering pointwise mean-squared error, we prove that it has (up to a log) the same asymptotic properties as an oracle. We give an example of Sobolev classes of functions where our estimator converges to f(x) with the optimal rate of convergence up to a log factor.