Feasibility of tomography with unknown view angles

In the standard two-dimensional (2-D) parallel beam tomographic formulation, it is generally assumed that the angles at which the projections were acquired are known. We have recently demonstrated, however, that under fairly mild conditions these view angles can be uniquely recovered from the projections themselves. In this paper, we address the question of reliability of such solutions to the angle recovery problem using moments of the projections. We demonstrate that under mild conditions, the angle recovery problem has unique solutions and is stable with respect to perturbations in the data. Furthermore, we determine the Cramér-Rao lower bounds on the variance of the estimates of the angles when the projection are corrupted by additive Gaussian noise. We also treat the case in which each projection is shifted by some unknown amount which must be jointly estimated with the view angles. Motivated by the stability results and relatively small values of the error bounds, we construct a simple algorithm to approximate the ML estimator and demonstrate that the problem can be feasibly solved in the presence of noise. Simulations using this simple estimator on a variety of phantoms show excellent performance at low to moderate noise levels, essentially achieving the Cramér–Rao bounds.