Confidence regions for images observed under the Radon transform

Recovering a function ff from its integrals over hyperplanes (or line integrals in the two-dimensional case), that is, recovering ff from the Radon transform RfRf of ff, is a basic problem with important applications in medical imaging such as computerized tomography (CT). In the presence of stochastic noise in the observed function RfRf, we shall construct asymptotic uniform confidence regions for the function ff of interest, which allows to draw conclusions regarding global features of ff. Specifically, in a white noise model as well as a fixed-design regression model, we prove a Bickel–Rosenblatt-type theorem for the maximal deviation of a kernel-type estimator from its mean, and give uniform estimates for the bias for ff in a Sobolev smoothness class. The finite sample properties of the proposed methods are investigated in a simulation study.