Unique recovery of unknown projection orientations in three-dimensional tomography

We consider uniqueness of three-dimensional parallel beam tomography in which both the object being imaged and the projection orientations are unknown. This problem occurs in certain practical applications, for example in cryo electron microscopy of viral particles, where the projection orientations may be completely unknown due to the random orientations of the particles being imaged. We show that only three projections are needed to guarantee unique recovery of the unknown projection orientations (up to a common orthogonal transformation), if the object belongs to a certain generic set of objects. In particular, the uniqueness holds for almost all objects. We also show that, if the object belongs to that generic set, $k+1$ projections at unknown orientations suffice to determine uniquely also the geometric moments of the object of order less or equal to $k$. As a consequence, the object belonging to that generic set, is uniquely determined (up to an orthogonal transformation) by almost any infinitely many projections at unknown orientations. We show that the uniqueness problem is related to some properties of certain homogeneous polynomials that depend on the projection orientations and the geometric moments of objects. Here certain theorems of algebraic geometry turn out to be useful. We also provide a system of equations, that is uniquely solvable and gives the desired projection orientations and object's geometric moments.