Sampling & the Nyquist limit

Recording a continuous signal on a discrete grid sets a hard upper frequency, above which detail is lost and aliasing corrupts the spectrum

The specimen in a microscope is continuous, but a detector has only finitely many pixels. Sampling replaces that continuous density curve with a list of values read off a regular grid. The step looks like mere bookkeeping, yet it quietly draws a hard frequency line across the whole dataset: if the grid is too coarse, the finest structure is not just unrecorded — it disguises itself as something else and contaminates the data. This page locates that line, explains why it cannot be crossed, and shows how it caps the resolution any Cryo-ET image can reach.

When the grid spacing (the pixel or voxel size) is $\Delta$ , the sampling frequency is $f_s = 1/\Delta$ . The Nyquist–Shannon theorem states that a signal can be reconstructed exactly from its samples only if it contains no frequency above the Nyquist frequency

f_{\text{Nyq}} = \frac{1}{2\Delta},

i.e. one cycle every two pixels. Here $\Delta$ is the real-space distance between neighbouring samples, $f_s = 1/\Delta$ is the number of samples per unit length, and $f_{\text{Nyq}} = f_s/2$ is exactly half the sampling rate — the frequency-domain statement of “you must take at least two samples per cycle.” Frequencies above this limit are not merely lost; they are folded back into the sampled spectrum and masquerade as lower frequencies. This corruption is called aliasing, and once present it cannot be undone, because distinct continuous signals map to identical samples.

True sineSamplesAliased sine

Signal frequency f: 5 cycles/intervalSampling rate fₛ: 6 (Nyquist limit fₛ = 2f = 10)

f = 5 → masquerading as f_alias = 1. Below the Nyquist rate fₛ = 2f the same samples also fit a lower-frequency sine — the high frequency folds back and masquerades as a low one.

The folded frequency lands at a definite point between $0$ and $f_{\text{Nyq}}$ : the true frequency $f$ is mirrored about integer multiples of the sampling rate, giving the alias $f_{\text{alias}} = |f - f_s\,\mathrm{round}(f/f_s)|$ . The term $\mathrm{round}(f/f_s)$ is the nearest integer multiple of the sampling rate to $f$ ; subtracting it and taking the absolute value folds $f$ back into the base interval $[0, f_{\text{Nyq}}]$ . Once $f_s$ drops below $2f$ , this low-frequency alias produces samples identical to the original, and the two become indistinguishable.

A concrete case: take a pixel size $\Delta = 1.5\ \text{Å}$ , so $f_s = 1/1.5 \approx 0.667\ \text{Å}^{-1}$ , the Nyquist frequency is $f_{\text{Nyq}} \approx 0.333\ \text{Å}^{-1}$ , and the finest recordable detail is about $2\Delta = 3\ \text{Å}$ . Now suppose the true structure carries a periodic texture at $0.4\ \text{Å}^{-1}$ (roughly $2.5\ \text{Å}$ spacing). It sits above Nyquist, so it folds back to $|0.4 - 0.667| \approx 0.267\ \text{Å}^{-1}$ (about $3.7\ \text{Å}$ ). A genuine $2.5\ \text{Å}$ feature thus shows up in the image as a spurious $3.7\ \text{Å}$ texture that was never there — superimposed on the real signal and impossible to strip out afterward.

Intuition

Two pixels are the minimum needed to represent one full oscillation — one for the crest and one for the trough. Finer structure than that has nowhere to be recorded, so it reappears disguised as a coarser, spurious pattern. The familiar wagon-wheel effect in video is the temporal version of the same folding: spokes turn so fast that between frames each has rotated most of the way to the next, and the camera “reads” this as a small backward step, so the wheel appears to drift slowly in reverse. Spatial sampling does exactly the same thing, with pixels in place of frames.

In Cryo-ET the pixel size at the specimen plane is fixed by the microscope magnification and the detector, and it sets the highest resolution any subsequent processing can reach: structure finer than $2\Delta$ is simply absent from the data. Choosing a pixel size is therefore a trade-off, since a smaller $\Delta$ raises the Nyquist limit and admits finer detail but spreads the same electron dose over more pixels, lowering the per-pixel signal-to-noise ratio. Dose is tightly capped by radiation damage and cannot scale up as the pixel shrinks, so the trade-off is hard — shrinking pixels alone does not buy a sharper image and can instead bury signal under noise.

This frequency line also marks the safe boundaries of several reconstruction operations. Downsampling an image (binning pixels, increasing $\Delta$ ) lowers Nyquist, so anything above the new Nyquist must be low-pass filtered away first, or those frequencies fold in and corrupt the coarse image — this is why pyramidal multi-resolution processing always blurs before it decimates. Conversely, upsampling or sub-pixel interpolation creates no new high-frequency information: everything above Nyquist was empty to begin with, and interpolation only fills in between existing frequency content, never conjuring detail finer than $2\Delta$ . The sub-pixel registration used when aligning a tilt series relies on the same fact — it can move recorded frequencies to their correct positions but cannot recover what sampling never captured.

Depth

Sampling at spacing $\Delta$ convolves the spectrum with a Dirac comb of period $f_s$ , periodically replicating $F(k)$ along the frequency axis. In real space sampling is multiplication by a comb of delta functions, and by the convolution theorem that becomes convolution with a comb in frequency, so the original spectrum is copied and shifted by $f_s, 2f_s, \dots$ and the copies are summed. If the signal’s support is confined to $|k| < f_{\text{Nyq}}$ the replicas do not overlap and the original is recoverable by an ideal low-pass filter; if it extends beyond, the replicas overlap and add, which is aliasing — the high-frequency tail of the neighbouring copy spills into $[0, f_{\text{Nyq}}]$ and mixes with this copy’s low frequencies, never to be separated. The practical defense is to band-limit before sampling — antialiasing — by blurring or filtering away content above Nyquist, since the contrast transfer function and detector envelope already attenuate the highest frequencies but rarely remove them cleanly.

This also explains why the detector’s physics lowers the effective Nyquist still further. Ideal sampling assumes each pixel is a point; a real pixel is a small square of finite area that averages the intensity falling within it, which is equivalent to first convolving with a one-pixel-wide box and then point-sampling. That box has a $\mathrm{sinc}$ envelope in frequency that drops the response to about $0.64$ at Nyquist and keeps falling toward it. In practice, then, “usable frequencies do not reach Nyquist”: the trustworthy high-frequency ceiling typically lands somewhere between half and two-thirds of $f_{\text{Nyq}}$ — worth keeping in mind when designing a frequency filter or judging the resolution of a map.

Placing this back in the full imaging chain: Nyquist fixes the frequency line; the contrast transfer function decides how each frequency inside that line is discounted and where its contrast flips; and the central-slice theorem together with the missing wedge govern which angular directions go unsampled altogether. Together the three bound the information a Cryo-ET reconstruction actually holds in frequency space. Methods like CryoGEN complete a structure not by inventing frequencies but by using a learned prior to infer which full density is most self-consistent beyond what was sampled, discounted, and left out.

← Signal Processing