This thesis develops theory and algorithms for a Riemannian geometric approach to inverse problems in cryogenic electron microscopy (Cryo-EM). It is divided into two parts, motivated by the sub problem of orientation estimation and that of modelling of protein dynamics. The thesis is concluded with a discussion on how to bring these pieces together to solve the so-called continuous heterogeneous reconstruction problem and a reflection on the general implications and new opportunities that a Riemannian geometry-based approach has for and brings to signal processing and recovery problems.

In the first part of the thesis we consider how to use ideas from global optimisation on Riemannian manifolds to regularise the orientation estimation sub problem. Our approach is motivated by the two main challenges in orientation estimation: the high noise levels present in Cryo-EM data and the non-convexity of typical variational problems for estimating orientations. To overcome these challenges jointly, we construct a new regularised global optimisation scheme to solve a variational problem for orientation estimation in a more noise-robust fashion.

In the second (and main) part we focus on constructing a Riemannian manifold for protein conformations. In particular, we are interested in constructing Riemannian geometry for protein conformations such that physically realistic protein conformations live in low-dimensional geodesic subspaces. Before constructing a Riemannian manifold, we first consider how curvature causes a discrepancy between data only looking low-dimensional and data actually being low-dimensional, and also consider how to address such curvature effects. Next, we construct a Riemannian manifold of protein conformations with computationally feasible manifold mappings such that realistic protein dynamics data really are low-dimensional under the proposed Riemannian structure, i.e., not suffering from curvature effects. Thirdly, we argue that it could be beneficial to have additional structure on a Riemannian protein geometry to what we have so far. In particular, we consider pullback geometry as a candidate class of Riemannian geometries that comes with the structure of interest and in addition offers a large amount of geometries to choose from. However, instead of directly trying to approximate the target Riemannian geometry, we take a step back and consider how a pullback Riemannian structure affects downstream data processing first and consider how one should go about constructing proper pullback manifolds given a target geometry.

In the conclusions we see that the constructed individual parts can be combined into two variational problems for solving the continuous heterogeneous reconstruction problem, one of which being a new strategy for solving general inverse problems under a certain type of learned regulariser. Next, in the light of the findings in this thesis, we also advocate more generally for the development of methodology for processing data under data-driven Riemannian geometry. A key overall takeaway from this thesis is then that over time having a proper account of the Riemannian geometry of our data can change the way we go about handling them in any data analysis pipeline with classically Euclidean data.