Direct reconstruction of macromolecular structures from cryogenic electron microscopy (Cryo-EM) data has shown to be a challenge both in the homogeneous and heterogeneous setting. In this work we propose a new direct reconstruction method based on a combination of recent developments on protein geometry and orientation estimation. Even though this method is set up for the homogeneous setting, we aim to gain insight into challenges that atomistic methods have been facing for the heterogeneous case. In numerical experiments we observe that the method is able to recover the structure to almost inter-atomic resolution from as few as 100 2D Cryo-EM images due to the strong bias the regularizer gives. We conclude this work with a discussion on how the obtained results indicate possibilities and challenges for the generalization to the heterogeneous case.

We address recovery of the three-dimensional backbone structure of single polypeptide proteins from single-particle cryo--electron microscopy (Cryo-SPA) data. Cryo-SPA produces noisy tomographic projections of electrostatic potentials of macromolecules. From these projections, we use methods from shape analysis to recover the three-dimensional backbone structure. Thus, we view the reconstruction problem as an indirect matching problem, where a point cloud representation of the protein backbone is deformed to match two-dimensional tomography data. The deformations are obtained via the action of a matrix Lie group. By selecting a deformation energy, the optimality conditions are obtained, which lead to computational algorithms for optimal deformations. We showcase our approach on synthetic data, for which we recover the three-dimensional structure of the backbone.