Estimating alignment errors in sets of 2-D images

We describe a robust and accurate method for the estimation of alignment errors for a set of two-dimensional images, in the case where the true pattern is unknown. The intended application of the proposed method is cryo-electron microscopy, where two-dimensional views of individual proteins in random orientations are observed in the electron microscope at low signal-to-noise ratio. By representing images in the basis of Fourier-harmonic coordinates and constructing averages and average intensities, we demonstrate that the variances of translation and rotational errors as well as of the Gaussian noise can be recovered. This machinery therefore allows one to isolate the various categories of errors that impede the quality of results in single particle reconstructions into constituent parts: translational errors, rotational errors, and additive noise.