Even/odd splitting

Cut the data into two copies with independent noise but identical signal — score resolution with FSC, and power self-supervised denoising

Intuition

Split one acquisition into two piles so that the two piles carry identical signal but independent noise. “Signal” is what you actually want to see — the true density of a molecule, a membrane, a complex; “noise” is the random fluctuation layered on top, mostly from the randomness of electron counting (shot noise), which comes out different every time you image the same sample. There are two common ways to cut. By frame: even-numbered frames into one half, odd into the other (see dose fractionation). Or by tilt: even tilt angles into one half, odd into the other. Both halves see the same molecule and the same underlying structure, but the noise on top of each is a separate, independent draw. Once you have this pair of “same-signal, different-noise” copies, things that were impossible become easy — measuring resolution and denoising both rest on it.

What this step does: in, out, with what

Input: raw, not-yet-averaged data. It can be the fractionated movie or tilt-series after alignment but before reconstruction; or two tomograms that were already reconstructed, each from its own independent chain.
What it does: cut the data into two non-overlapping parts by some “even/odd” rule. The most common rules are by frame (frames 1, 3, 5… vs 2, 4, 6… of the movie) or by tilt (projections 1, 3, 5… vs 2, 4, 6…).
Output: two files — usually called half1 / half2 or even / odd. They share a coordinate system and correspond to the same object; they differ only in noise.
With what: the split itself is just selecting frames/tilts by a rule — IMOD’s newstack, the corresponding options in Warp/RELION, or a few lines of your own script all work. What actually matters is where in the pipeline you cut (see below).

Why this step exists

The two big things downstream both need two copies that are “identical except for noise”: one is scoring the resolution of a reconstruction, the other is training a denoiser/restorer with no clean ground truth. Even/odd splitting is the only thing in the pipeline that reliably manufactures such copies, which is why it is the data prerequisite for self-supervised training.

Why independence is the whole point

Whether you are scoring resolution or training a denoiser, it comes down to one rule: what is shared survives a comparison, what is independent does not.

The true signal $s$ is the same in both halves; the noise $n_1$ , $n_2$ is independent and zero-mean. When you correlate or average the two halves:

the signal is in phase across the halves — it adds when summed, and is preserved when correlated;
the noise, being independent and uncorrelated, contributes near zero to a correlation, and cancels at rate $1/\sqrt{2}$ when averaged.

The catch is that this independence cannot be faked. If the two halves share any processing step before the split — say you average and denoise the whole dataset first, then cut it in two — the noise becomes artificially correlated, and both uses break: the correlation inflates, and the denoiser learns the noise itself. So the split must be made at the rawest end of the data, with the two processing chains never touching each other. One slogan: split early, keep the two chains apart.

Use one: FSC for resolution

The gold-standard resolution in subtomogram averaging comes from the Fourier Shell Correlation (FSC). Randomly split the particles into two groups, align and average each group fully independently to get two half-maps, then go into Fourier space and bin the two half-maps into concentric shells by spatial frequency. For each shell, compute the complex correlation:

\mathrm{FSC}(k) = \frac{\sum_{k} F_1(k)\,\overline{F_2(k)}}{\sqrt{\sum_{k}|F_1(k)|^2 \cdot \sum_{k}|F_2(k)|^2}}

Here $k$ is spatial frequency (the shell radius — larger $k$ means finer detail), and $F_1$ , $F_2$ are the Fourier transforms of the two half-maps. At low frequency the halves agree and FSC is near 1. As frequency rises the signal weakens and noise dominates, so the halves stop matching and FSC falls toward 0. The frequency where the curve drops below a chosen threshold, converted back to a real-space scale (in ångströms), is the reported resolution — it answers “how fine can this map be trusted to resolve.”

Depth

Whether you read the threshold at 0.143 or 0.5 depends on whether the half-maps are “gold-standard independent.” The 0.143 criterion assumes two halves processed independently end to end — exactly what even/odd splitting guarantees — and reports a resolution you can trust. If the halves ever peek at each other during alignment (for example by sharing one reference), independence is broken, the high-frequency correlation is artificially raised, and FSC returns an over-optimistic number — the familiar overfitting / noise amplification. Strict half-map independence is the precondition for the FSC value to mean anything.

Use two: self-supervised denoising

The same pair of halves can directly train a denoiser, with no clean ground truth at all — which matters especially in Cryo-ET, where there simply is no “clean” tomogram to use as an answer key.

The idea comes from Noise2Noise: have the network use one half to predict the other. The clean signal is identical across halves, so simply outputting that shared signal brings the network close to both targets at once. The noise, however, is independent, so the network cannot predict one half’s noise from the other’s — trying only pulls it in random directions for no gain. Under an L2-style loss the optimum is the conditional expectation of the two halves, which is precisely the shared clean structure left after the independent noise averages out. The network learns to denoise without ever seeing a clean image.

Depth

This is the data setup behind learned restoration like cryoCARE and DeepDeWedge: reconstruct two independent tomograms from even/odd (by frame or by tilt), feed one as input and the other as target. It is also exactly what the run training step feeds the network — the “paired data” it asks for is the half1 / half2 made here. Push this independent-noise pairing up to the level of generative models and you reach our own reconstruction methods — CryoWGEN likewise learns from paired “same-signal, different-noise” observations, handing both the missing wedge and the noise to a learned prior to infer. In short, even/odd splitting is both the ruler for resolution and the fuel for this family of self-supervised restoration.

Practical trade-offs

Splitting by frame is the cleanest — the two halves come from different electrons of the same exposure, with identical geometry, just half the dose and a lower per-half SNR (see SNR). Splitting by tilt leaves each half missing half the angular sampling, so the missing wedge is wider and each half reconstructs worse, but it is simple to implement and pairs naturally at the tomogram level. Either way the iron rule holds: split early, keep the two chains apart — that is the shared precondition for every use of even/odd.

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