Abstract: The goal in Cryo-EM structure determination is to reconstruct the 3D structure of molecules 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. A central challenge is devising a reconstruction algorithm that uses only the acquired data and makes no assumptions on the underlying structure.

In this talk, we present an extremely efficient and robust algorithm that successfully recovers the acquisition direction of each projection image. Once all acquisition directions are known, the molecule is reconstructed using standard tomography techniques. The key idea of the algorithm is designing an 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 does not require any prior model for the reconstructed molecule, shown to have favorable computational and numerical properties, and does not impose any assumption on the distribution of the projection orientations, thus applicable to molecules that have unknown spatial preference.

Although the presented algorithm was originally derived in the context of Cryo-EM, it turns out that its underlying ideas are applicable to a large number of problems. This lays the foundations to a new paradigm in machine learning, in which relations between pairs of data points are used to construct a problem-dependent operator, whose eigenvectors encode the coordinates of each data point..

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

Coffee will be served at 12:00 before the lecture
at Schreiber building 006