On the problem of local adaptive estimation in tomography

Abstract

The principle of tomography is to reconstruct a multidimensional function from observations of its integrals over hyperplanes. We consider here a model of stochastic tomography where we observe the Radon transform Rf of the function f with a stochastic error. Then we construct a `data-driven' estimator which does not depend on any a priori smoothness assumptions on the function 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.