  Traditional sampling schemes often assume that the sampling locations are
known. Motivated by the recent bioimaging technique known as cryogenic electron
microscopy (cryoEM), we consider the problem of reconstructing an unknown 3D
structure from samples of its 2D tomographic projections at unknown angles. We
focus on 3D convex bilevel polyhedra and 3D point sources and show that the
exact estimation of these 3D structures and of the projection angles can be
achieved up to an orthogonal transformation. Moreover, we are able to show that
the minimum number of projections needed to achieve perfect reconstruction is
independent of the complexity of the signal model. By using the divergence
theorem, we are able to retrieve the projected vertices of the polyhedron from
the sampled tomographic projections, and then we show how to retrieve the 3D
object and the projection angles from this information. The proof of our
theorem is constructive and leads to a robust reconstruction algorithm, which
we validate under various conditions. Finally, we apply aspects of the proposed
framework to calibration of X-ray computed tomography (CT) data.
