Uniqueness of two-dimensional tomography with unknown projection directions

We consider uniqueness of two-dimensional parallel beam tomography in which both the object being imaged and the projection directions are unknown. This problem occurs in certain practical applications. For example, in magnetic resonance imaging there may be uncertainty in the projection directions due to the involuntary motion of the patient. The three-dimensional version of this problem occurs in cryo electron microscopy of viral particles, where the projection directions may be completely unknown due to the random orientations of the particles being imaged. We show that the problem is related to some algebraic geometric properties of a certain system of homogeneous polynomials. We also show that for sufficiently asymmetric objects, the object is uniquely determined up to an orthogonal transformation by the projection data from unknown directions.