  Motivated by the task of 2-D classification in single particle reconstruction
by cryo-electron microscopy (cryo-EM), we consider the problem of heterogeneous
multireference alignment of images. In this problem, the goal is to estimate a
(typically small) set of target images from a (typically large) collection of
observations. Each observation is a rotated, noisy version of one of the target
images. For each individual observation, neither the rotation nor which target
image has been rotated are known. As the noise level in cryo-EM data is high,
clustering the observations and estimating individual rotations is challenging.
We propose a framework to estimate the target images directly from the
observations, completely bypassing the need to cluster or register the images.
The framework consists of two steps. First, we estimate rotation-invariant
features of the images, such as the bispectrum. These features can be estimated
to any desired accuracy, at any noise level, provided sufficiently many
observations are collected. Then, we estimate the images from the invariant
features. Numerical experiments on synthetic cryo-EM datasets demonstrate the
effectiveness of the method. Ultimately, we outline future developments
required to apply this method to experimental data.
