An accelerated expectation-maximization for multi-reference alignment.

The multi-reference alignment (MRA) problem entails estimating an image from multiple noisy and rotated copies of itself. If the noise level is low, one can reconstruct the image by estimating the missing rotations, aligning the images, and averaging out the noise. While accurate rotation estimation is impossible if the noise level is high, the rotations can still be approximated, and thus can provide indispensable information. In particular, learning the approximation error can be harnessed for efficient image estimation. In this paper, we propose a new computational framework, called Synch-EM, that consists of angular synchronization followed by expectation-maximization (EM). The synchronization step results in a concentrated distribution of rotations; this distribution is learned and then incorporated into the EM as a Bayesian prior. The learned distribution also dramatically reduces the search space, and thus the computational load, of the EM iterations. We show by extensive numerical experiments that the proposed framework can significantly accelerate EM for MRA in high noise levels, occasionally by a few orders of magnitude, without degrading the reconstruction quality.