We propose a framework to jointly determine the deformation parameters and reconstruct the unknown volume in electron cryotomography (CryoET). CryoET aims to reconstruct three-dimensional biological samples from two-dimensional projections. A major challenge is that we can only acquire projections for a limited range of tilts, and that each projection undergoes an unknown deformation during acquisition. Not accounting for these deformations results in poor reconstruction. The existing CryoET software packages attempt to align the projections, often in a workflow which uses manual feedback. Our proposed method sidesteps this inconvenience by automatically computing a set of undeformed projections while simultaneously reconstructing the unknown volume. We achieve this by learning a continuous representation of the undeformed measurements and deformation parameters. We show that our approach enables the recovery of high-frequency details that are destroyed without accounting for deformations.

Motivated by data analysis for protein conformations, we construct a smooth quotient manifold of point clouds and equip it with a non-trivial metric tensor field, that models which point clouds are close together and which are far apart. We analyse properties of the Riemannian manifold and obtain cheap to compute expressions for important manifold mappings. Furthermore, we investigate potential numerical advantages of using the Riemannian manifold structure in several data processing tasks such as interpolation, computing means and principal component analysis of simulated molecular dynamics (MD) data sets. For the latter, we observe that MD trajectories live in a low-dimensional sub-manifold in the proposed metric.

We consider the problem of recovering the three-dimensional atomic structure of a flexible macromolecule from a heterogeneous single-particle cryo-EM dataset. Our method combines prior biological knowledge about the macromolecule of interest with the cryo-EM images. The goal is to determine the deformation of the atomic structure in each image with respect to a specific conformation, which is assumed to be known. The prior biological knowledge is used to parametrize the space of possible atomic structures. The parameters corresponding to each conformation are then estimated as a linear combination of the leading eigenvectors of a graph Laplacian, constructed by means of the cryo-EM dataset, which approximates the spectral properties of the manifold of conformations of the underlying macromolecule.