Superresolution fluorescence microscopy and
compressed sensing work for sparse point sources. But what if we cannot
control the sources, such as stars or exoplanets, and they are closer to
one another than the diffraction limit? Using a shot-noise model and
parameter-estimation theory, I derive fundamental limits to direct
imaging in this subdiffraction regime, demonstrating that the moments of
the source distribution are the natural parameters to estimate in this
case . I further show that an optical spatial-mode
demultiplexing (SPADE) technique can estimate the second and higher
moments much more accurately than direct imaging can fundamentally do
[1,2]. The recent breakthrough discovery on two-point resolution  can
be regarded as a special case. Realizable with
linear optics and photon counting, SPADE is expected to benefit both
fluorescence microscopy and astronomy.
 M. Tsang, "Subdiffraction incoherent optical imaging via spatial-mode demultiplexing," New Journal of Physics 19, 023054 (2017).
 M. Tsang, "Subdiffraction incoherent optical imaging via spatial-mode demultiplexing: semiclassical treatment," e-print arXiv:1703.08833.
 M. Tsang, R. Nair, and X.-M. Lu, "Quantum Theory of Superresolution for Two Incoherent Optical Point Sources," Physical Review X 6, 031033 (2016).