Regularizing Orientation Estimation in Cryogenic Electron Microscopy Three-Dimensional Map Refinement through Measure-Based Lifting over Riemannian Manifolds

Motivated by the trade-off between noise robustness and data consistency for joint three-imensional (3D) map reconstruction and rotation estimation in single particle cryogenic-electron microscopy (Cryo-EM), we propose ellipsoidal support lifting (ESL), a measure-based lifting scheme for regu-larizing and approximating the global minimizer of a smooth function over a Riemannian manifold. Under a uniqueness assumption on the minimizer we show several theoretical results, in particular well-posedness of the method and an error bound due to the induced bias with respect to the global minimizer. Additionally, we use the developed theory to integrate the measure-based lifting scheme into an alternating update method for joint homogeneous 3D map reconstruction and rotation es-timation, where typically tens of thousands of manifold-valued minimization problems have to be solved and where regularization is necessary because of the high noise levels in the data. The joint recovery method is used to test both the theoretical predictions and algorithmic performance through numerical experiments with Cryo-EM data. In particular, the induced bias due to the regularizing effect of ESL empirically estimates better rotations, i.e., rotations closer to the ground truth, than global optimization would.