1/4 It may come as a surprise to some that Rayleigh's limit to resolution is just a rule of thumb - not a physical law:
"two point sources are regarded as just resolved when the maximum of the Airy disk (PSF) of one source coincides with the first minimum of the PSF of the other"

2/4 Rayleigh's criterion just says that we can't eyeball two sources if they're too close. But this doesn't mean we can't *detect* 1 vs. 2 or more sources statistically even in the presence of noise. With proper tests, we can find very weak signals fully hidden to the naked eye.

3/4 The punchline: ability to resolve two nearby sources follows a power law. Plot shows the min detectable separation between two equal brightness sources as a function of SNR w/ & w/o aliasing given N uniform samples.
Formula: required SNR = c/(N*d^4)
https://ieeexplore.ieee.org/document/1288193

4/4 This relationship & its generalizations have been (re)-discovered many times. Yet it's surprisingly little known. Our work has relatively few citations (5-6/yr), but I'm proud of their diversity: from black holes & gravitational waves to neuronal spikes, microscopy, math, etc