After I asked for more, Greengard sent me two classic papers (here and here) on diffraction imaging (microscopy). These are beautifully written and very clear. So I now understand the problem well, and the standard solution (which is “oversample by a factor of two!”). One interesting issue is that in real experiments a beam stop kills your low-k modes, so you don't get to see the zero (or near-zero) part of the Fourier Transform. Most of the heavy lifting in standard approaches is done by setting a zero or null region around the object and requiring that the function go to zero there. That strikes me as odd, and only applicable in some circumstances. So I became inspired to try some complementary approaches.
The day also included a conversation with So Hattori, who is going to re-start our search for very long-period planets in the Kepler data. This is an important project: It is Jupiter, not Earth, that is the driver of the present-day Solar System configuration.