Signal-to-Noise Ratio

Independent noise sources combine in quadrature: total variance is the sum of the individual variances, so the standard deviations add as the square root of the sum of squares. Three of them matter here.

Shot noise is Poisson and irreducible. Collect S photoelectrons and they fluctuate by √S. This is a property of light, not of the camera, and no engineering removes it. Background B — autofluorescence, out-of-focus light, stray excitation — carries its own shot noise √B and adds nothing to the signal. Read noise R is a fixed Gaussian penalty charged once per pixel read, independent of how much light arrived, typically 0.3 to 1.6 electrons on a modern sCMOS.

SNR = S / √(S + B + R²)

The asymmetry in that expression is worth staring at: S appears once on top and once underneath. Doubling the signal does not double the SNR; it multiplies it by √2 once you are shot-noise-limited. Four times the exposure for twice the certainty is the central bargain of every photon-starved measurement.

An smFISH spot delivers roughly 300 photons to the camera in one exposure.

The sensor catches about 90% of them, so 300 photons become 270 photoelectrons. Their shot noise is √270 = 16.4 electrons. The cytoplasm around the spot contributes perhaps 50 electrons of autofluorescence, carrying √50 = 7.1 electrons of its own. The readout adds 1.5 electrons.

SNR = 270 / √(270 + 50 + 2.25) = 270 / 17.95 = 15.0. Comfortable: the Rose criterion asks for about 5.

Look at where that noise came from. Shot noise on the signal supplies 84% of the variance, background 15%, and the readout electronics 0.7%. At 300 photons per spot the camera has almost nothing to do with the answer — which is exactly why camera arguments about bright samples go nowhere, and why they become decisive the moment the spot dims.

Everything above is per pixel, and a spot is not one pixel. It is a diffraction pattern spread over several, and detecting it means combining them — which changes the arithmetic in one specific place.

Signal and background are summed, so they are unchanged in total. Read noise is charged per pixel read, so it enters k times for k pixels:

SNR_element = S / √(S + B + k·R²)

At 40× / 1.15 NA the resolution element is 297 nm across, which a 6.5 µm pixel spans with k = 3.34. Our spot's SNR goes from 15.04 per pixel to 14.92 per element — a 0.8% difference, because read noise is only 2.3% of the variance here.

That near-identity is the whole point, and it inverts as the light fails. Read noise equals the shot-noise terms when S + B falls to k·R² = 7.5 electrons. Above that, k barely matters and every competent camera converges. Below it, k multiplies your dominant noise source, and a sensor that reads at 0.3 electrons instead of 1.5 is worth twenty-five times less read variance per pixel.

The Rose criterion says reliable visual detection needs SNR of about 5. It is an empirical threshold from human observers, not a law, but it has held up well enough to plan experiments around.

Run it backwards for the spot above. Holding background at 50 electrons and read noise at 1.5 across 3.34 pixels, SNR = 5 needs S = 52.4 electrons, which at 90% quantum efficiency is 58 photons per spot. Our 300-photon spot has five times the margin it needs.

Two traps. First, SNR of a lone spot on a clean field is not the same as detectability against structured background — a gradient across the field is not averaged away by anything, and it eats the margin the noise calculation says you have. Second, SNR is necessary and not sufficient for counting: two spots 200 nm apart at SNR 50 are still counted as one, because resolution is a separate failure from detection and improving one does nothing for the other.