The discussion of cryo-EM on Monday ranged into the properties of maximum-likelihood estimators. This got me thinking about the possibility of doing something that Leslie Greengard had challenged me with a month or two ago: Could we use diffraction-imaging data in which we have many exposures, each at random (that is, unknown) Euler angles, but in each of which we only have a few photons? Do we need enough photons in every image to align it confidently? My theory (born last night and started in earnest today) is the following: As long as we have a correct probabilistic model for both the orientation distribution (isotropic, perhaps) and as long as our model or representation for the three-dimensional (squared norm of the) Fourier transform contains the true function (a tall order), we should be able to use images of any quality, even single-photon images.
I spent the day writing this up in a document and planning how I would code it. A Christmas project, perhaps. Of course it is almost never that you could have either a correct model of the orientation distribution or a complete representation of the function space. But that just reminds us of the obvious point that all of physical science is impossible; that's not a show-stopper.