An important problem in electron microscopy is considered here. Assume that we have at our disposal a large number of particles with identical structures. A mathematical way of expressing this is that we can, in principle, attach a coordinate system to each of the particles so that the three-dimensional function (in this coordinate system) describing the particle is the same for all particles. In collecting transmission electron microscopic images of the particles for the purpose of reconstructing their common structure, it is typically the case that neither the orientations of the coordinate systems attached to the particles nor the locations of their origins are known to us; they have to be estimated from the electron micrographs. Without knowledge of these orientations and locations, we cannot possibly reconstruct the common structure of the particles. Hence, it is essential to estimate these viewing parameters.

The authors propose a novel method, which starts from the well-known fact that the two-dimensional Fourier transforms of two projections of a particle will coincide on a single line and that, from the locations of such common lines for pairs of projections from a series of projections, one can determine the orientations of all the projections relative to each other. The essential observation of this paper is that not knowing the location of the origin in the projection image is not a major problem, since this results only in a change of phase in the Fourier transform. The authors show that, regardless of where we assume the origins to be in two projection images, the phase difference of the common lines in their Fourier transforms will depend linearly on the distance from the origin in Fourier space. This fact can be used (provided the noise is not overwhelming) to identify the common lines and, consequently, the locations of the projections of the origin of the assumed coordinate system

The problem with applying this method in electron microscopy is that it works only if the signal-to-noise ratio in the projection images is better than 12 dB. In most electron microscope applications, the signal-to-noise ratio is likely to be worse than 0 dB, so it is not clear that the proposed method is of use in its intended area of application.