Reconstructing classes of 3D FRI signals from sampled tomographic projections at unknown angles

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.