Continuous-Time Multireference Alignment with Applications to Cryo-EM

Abstract

The problem of estimating a signal from noisy cyclically-translated versions of itself is called multireference alignment (MRA). MRA has extensive applications in science and engineering [1], foremost among which is image processing for cryogenic electron microscopy (cryo-EM). In the fi rst part of this paper, we review existing algorithms for MRA in the context of cryo-EM image processing. We then turn to the problem of one-dimensional continuous-time MRA and study the method of invariant features. Whereas most current MRA solutions align and average observations, invariant features avoids alignment by estimating a signal directly from features which are invariant under cyclic translations: the mean, power spectrum, and bispectrum. We study three algorithms for recovering a signal from these invariant features: frequency marching, least-squares optimization, and semidefi nite programming. When observations contain no noise, all three algorithms recover a signal exactly. In the presence of noise, frequency marching fails, while least-squares produces stable estimates when the number of observations grows with the cube of the noise variance. This is the information-theoretic limit at which MRA is possible. We expect semidefi nite programming to achieve similar performance, but this is not studied.