  In single-particle cryo-electron microscopy (cryo-EM), the efficient
determination of orientation parameters for 2D projection images poses a
significant challenge yet is crucial for reconstructing 3D structures. This
task is complicated by the high noise levels present in the cryo-EM datasets,
which often include outliers, necessitating several time-consuming 2D clean-up
processes. Recently, solutions based on deep learning have emerged, offering a
more streamlined approach to the traditionally laborious task of orientation
estimation. These solutions often employ amortized inference, eliminating the
need to estimate parameters individually for each image. However, these methods
frequently overlook the presence of outliers and may not adequately concentrate
on the components used within the network. This paper introduces a novel
approach that uses a 10-dimensional feature vector to represent the orientation
and applies a Quadratically-Constrained Quadratic Program to derive the
predicted orientation as a unit quaternion, supplemented by an uncertainty
metric. Furthermore, we propose a unique loss function that considers the
pairwise distances between orientations, thereby enhancing the accuracy of our
method. Finally, we also comprehensively evaluate the design choices involved
in constructing the encoder network, a topic that has not received sufficient
attention in the literature. Our numerical analysis demonstrates that our
methodology effectively recovers orientations from 2D cryo-EM images in an
end-to-end manner. Importantly, the inclusion of uncertainty quantification
allows for direct clean-up of the dataset at the 3D level. Lastly, we package
our proposed methods into a user-friendly software suite named cryo-forum,
designed for easy accessibility by the developers.
