Perfect Reconstruction of Classes of 3D Non-Bandlimited Signals from Tomographic Projections at Unknown Angles

In this paper, we consider the problem of reconstructing 3D objects from sampled 2D tomographic projections at unknown angles. We consider 3D polyhedrons and we provide a constructive solution to recover the 3D structure of the polyhedron and the projection directions. The proposed method is based on the observation that the 2D projections are actually signals with finite rate of innovation, and this allows us to retrieve the locations of the projected vertices using Prony's method. We are then able to unveil the 3D geometric information of the projection directions, as well as the structure of the polyhedron with an algebraic approach. The reconstruction method can also be applied to 3D point source models and is resilient to noise.