Weighted angle Radon transform: Convergence rates and efficient estimation

In the statistics literature, recovering a signal observed under the Radon transform is considered a mildly ill-posed inverse problem. In this paper, we argue that several statistical models that involve the Radon transform lead to an observational design which strongly influences its degree of ill-posedness, and that the Radon transform can actually become severely ill-posed. The main ingredient here is a weight function λ on the angle. Extending results for the limited angle situation, we compute the singular value decomposition of the Radon transform as an operator between suitably weighted L2-spaces, and show how the singular values relate to the eigenvalues of the sequence of Toeplitz matrices of λ. Further, in the associated white noise sequence model, we give upper and lower bounds on the rate of convergence, and in several special situations even obtain optimal rates with precise minimax constants. For the severely ill-posed limited angle problem, a simple projection estimator is adaptive in the exact minimax sense.