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.An important intermediate stage in the 3-D reconstruction procedure of cryo-EM is called 2-D classification. The goal of this stage is to produce K 2-D images-called class averages-with higher SNR. Each one of the K images should represent a subset of the projections taken from a similar viewing direction. The 2-D class averages can be used as templates for particle picking, to construct ab initio 3-D structures [3][4][5][6], to provide a quick assessment of the particles, to remove picked particles which are associated with non-informative classes, and for symmetry detection [7][8][9].Different solutions were proposed for the 2-D classification problem. One approach is the referencefree alignment (RFA) technique [10]. RFA tries to align all projection images globally by estimating all individual rotation parameters. However, when the images arise from many different viewing directions, RFA tends to produce large errors as no assignment of in-plane rotational angles can align all images simultaneously. Methods based on expectation-maximization (EM)-an iterative algorithm that aims to find the marginalized maximum likelihood-are also popular. In the context of cryo-EM, the method is usually referred to as Maximum Likelihood 2-D classification (ML2D). The method was first proposed in [11], and is implemented in the popular software package RELION [12,13]. Nevertheless, the EM framework lacks theoretical analysis and may be computationally expensive. In addition, EM suffers from an intrinsic resolution-computational load trade-off, since the sampling of the in-plane rotation angles is Nyquist sampling. In Section 4 we present some numerical results of EM and discuss more of its properties.A different 2-D classification technique is based on multireference...