phase retrieval

I came back from #AstroHackWeek and #DSEsummit all fired up to work on Cryo-EM, but then Leslie Greengard, Charlie Epstein (Penn) and Jeremy Magland (SCDA) distracted me on the phase retrieval problem: In one-dimensional problems, if you know the squared amplitude of the Fourier transform of an all-positive function on a bounded domain, you do not know the function: There are true degeneracies. In higher dimensions, there are degeneracies possible, but generic two-d and three-d all-positive scenes in bounded domains are uniquely specified by the norm of the Fourier transform the vast majority of the time. Or so we think. There are arguments, not proofs, I believe. Anyway, Magland has coded up (an improved version of) one of the standard algorithms for retrieving the phases of the Fourier transform and spent the last week or two testing and adjusting it. He seems to find that the solutions are unique, but that they are very hard to find. We spent the day arguing around this. I am trying not to get sucked in!