Uniqueness of tomography with unknown view angles.

In the standard two-dimensional (2-D) parallel beam tomographic formulation, it is assumed that the angles at which the projections were acquired are known. In certain situations, however, these angles are known only approximately (as in the case of magnetic resonance imaging (MRI) of a moving patient), or are completely unknown. The latter occurs in a three-dimensional (3-D) version of the problem in the electron microscopy-based imaging of viral particles. We address the problem of determining the view angles directly from the projection data itself in the 2-D parallel beam case. We prove the surprising result that under some fairly mild conditions, the view angles are uniquely determined by the projection data. We present conditions for the unique recovery of these view angles based on the Helgasson-Ludwig consistency conditions for the Radon transform, we also show that when the projections are shifted by some random amount which must be jointly estimated with the view angles, unique recovery of both the shifts and view angles is possible.