Nyquist Ratio

The Nyquist ratio is how many pixels span one resolution element: ratio = resolution limit ÷ pixel footprint at the specimen, where the footprint is pixel pitch ÷ total magnification. Nyquist's requirement of two samples per cycle is met at a ratio of 2. Below 2 the grid is too coarse for the detail the optics deliver and aliasing follows; above about 3 the same photons are divided among more pixels without recording anything new.

A note on a discrepancy you may meet elsewhere, including in an earlier version of this entry: some write the ratio with a factor of two in the denominator, which shifts the satisfied condition to 1 rather than 2. Both conventions describe the same physics; mixing them produces thresholds that disagree with the formula beside them. The pixels-per-resolution-element form is used throughout this glossary because it can be read directly as a count.

Two is the theoretical floor, and practice sits a little above it. The reason is contrast rather than geometry: a lens does not transmit detail at full contrast right up to its cutoff, it fades toward it, and a pixel integrates over its own area, adding a second attenuation that grows as the footprint approaches the limit. Sampling at 2.3 to 2.5 keeps most of the transferred contrast rather than only the bare information, and it gives deconvolution something to work with.

Beyond about 3 the returns reverse. The same photons spread over more pixels, so each pixel's signal falls while read noise is paid once per pixel regardless — a real cost at very low light, and a reason oversampling is cheap rather than free. The saving grace is that it is reversible: sum the pixels afterwards and you have exactly the coarser image. Undersampling offers no equivalent undo.

A 3.7 µm sensor at 25x, 1.1 NA, 525 nm. Footprint = 3700/25 = 148 nm. Resolution = 0.61 × 525 / 1.1 = 291 nm. Ratio = 291/148 = 1.97 — a hair under the floor of 2, so this pairing is at the boundary: essentially all of the optical detail is captured, with no margin for the contrast loss near the cutoff. Sound in practice, and not a combination to bin.

A 6.5 µm sensor at 40x, 1.25 NA, 560 nm. Footprint = 6500/40 = 162.5 nm. Resolution = 273 nm. Ratio = 273/162.5 = 1.68 — undersampled, losing the finest detail the objective delivers. Moving the same camera to 60x gives a footprint of 108 nm and a ratio of 2.52, comfortably in range. Nothing about the camera changed; the pairing did.