On the symmetries of the synchronization problem in Cryo-EM: Multi-Frequency Vector Diffusion Maps on the Projective Plane

Cryo-Electron Microscopy (Cryo-EM) is an important imaging method which allows high-resolution reconstruction of the 3D structures of biomolecules. It produces highly noisy 2D images by projecting a molecule's 3D density from random viewing directions. Because the projection directions are unknown, estimating the images' poses is necessary to perform the reconstruction. We focus on this task and study it under the group synchronization framework: if the relative poses of pairs of images can be approximated from the data, an estimation of the images' poses is given by the assignment which is most consistent with the relative ones.In particular, by studying the symmetries of cryo-EM, we show that relative poses in the group O(2) provide sufficient constraints to identify the images' poses, up to the molecule's chirality. With this in mind, we improve the existing multi-frequency vector diffusion maps (MFVDM) method: by using O(2) relative poses, our method not only predicts the similarity between the images' viewing directions but also recovers their poses. Hence, we can leverage all input images in a 3D reconstruction algorithm by initializing the poses with our estimation rather than just clustering and averaging the input images. We validate the recovery capabilities and robustness of our method on randomly generated synchronization graphs and a synthetic cryo-EM dataset.