Nyquist Sampling

The rule that governs your pixel size was not devised for microscopes. In 1924 Harry Nyquist, working at Bell Telephone Laboratories, was asked a commercial question: how fast can distinct pulses be pushed down a telegraph line before they smear into one another and the message becomes unreadable? His answer, published as Certain Factors Affecting Telegraph Speed, was that the maximum rate of distinguishable pulses is twice the bandwidth of the line. Push harder and the pulses do not merely degrade; they become genuinely ambiguous.

Claude Shannon gave the result its general form in 1949, and Vladimir Kotelnikov had reached it independently in the Soviet Union in 1933. What began as a question about how many words per minute a wire could carry became the sampling theorem, and it applies to any band-limited channel whatsoever. A microscope objective is such a channel. That is the whole of the connection, and it is exact rather than an analogy.

To record a repeating pattern you need at least two samples per cycle — one to catch a peak, one to catch the trough between. With fewer, the peaks and troughs are not merely measured coarsely; they cannot be distinguished from those of a slower pattern that happens to pass through the same sample points. In imaging the finest pattern present is set by the objective's cutoff, so the requirement becomes: the pixel footprint at the specimen must be no larger than half the resolution limit.

The footprint is the quantity that matters, and it is pixel pitch divided by total magnification. A pixel size in micrometers means nothing on its own — the same sensor behind a 20x and a 60x objective samples the specimen three times more finely in the second case, with no change to the camera at all.

Undersampling does not simply lose fine detail. It converts that detail into a coarser pattern which is entirely plausible and entirely false. The everyday example is the wagon wheel in a film that appears to rotate slowly backwards: the camera samples 24 times a second, the spokes pass more often than that, and the recorded sequence is consistent with a slow reverse rotation. Nothing in the footage reveals the error, because the footage is a faithful record of what was sampled.

In an image the same mechanism turns a fine grating into a coarse one, and closely spaced structures into broad bands at a spacing that exists nowhere in the specimen. This is why the asymmetry between oversampling and undersampling is absolute rather than a matter of degree. Oversampled data can be summed afterwards to recover exactly the coarser version, whenever you want it. Undersampled data contains a false pattern that is mathematically indistinguishable from a true one, and no processing separates them because the information required to do so was discarded at the sensor.

Continue the 40x / 1.25 NA example at 560 nm, where the resolution limit is 273 nm. Nyquist therefore asks for a pixel footprint of 137 nm or finer. A camera with 4.6 µm pixels gives 4600/40 = 115 nm, comfortably inside. A camera with 6.5 µm pixels gives 162.5 nm, slightly outside — already trimming the finest detail the lens delivers.

Now bin either camera 2×2. The first becomes 230 nm and the second 325 nm, and both are severely undersampled. The 325 nm case caps recoverable detail near 650 nm regardless of the objective in front of it: a 1.25 NA lens is delivering 273 nm detail into a grid that cannot represent anything finer than 650, and the difference is not stored anywhere. Binning to gain signal is a reasonable trade in some experiments. Making it without noticing that it discarded more than half the optical performance you paid for is not.