The goal in Cryo-EM structure determination is to reconstruct 3D macromolecular structures from their noisy projections, taken by an electron microscope at unknown random orientations. Resolving the Cryo-EM problem is of great scientific importance, as the method is applicable to essentially all macromolecules, as opposed to other existing methods such as crystallography. Since almost all large proteins have not yet been crystallized for 3D X-ray crystallography, Cryo-EM seems the most promising alternative, once its associated mathematical challenges are solved. In this talk, we present an extremely efficient and robust algorithm that successfully recovers the projection angles in a globally consistent manner. The key idea of the algorithm is designing a sparse operator defined on the projection data, whose eigenvectors reveal the orientation of each projection. Such an operator is constructed by utilizing the geometry induced on Fourier space by the projection-slice theorem. The presented algorithm is direct (as opposed to iterative refinement schemes), does not require any prior model for the reconstructed object, and shown to have favorable computational and numerical properties. Moreover, our algorithm does not impose any assumption on the distribution of the projection orientations. Physically, this means that the algorithm successfully reconstructs molecules that have unknown spatial preference.

No prior knowledge of tomography, electron microscopy, or spectral graph theory is assumed.

Joint work with Amit Singer, Ronald Coifman and Fred Sigworth.