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Sandbox Physics

E44 · Cryo tomography

Cryo Electron Tomography: Spend Dose, Average Copies

Balance frozen-slab thickness, acquisition order and dose before extracting repeated structures from the count-derived tomogram.

Interactive modelCryo Electron Tomography: Spend Dose, Average Copies
Recorded projection count—\text{—}
Selected recorded tilt—\text{—}
Recorded total incident dose—\text{—}
Dose before selected exposure—\text{—}
Incident electrons per selected pixel—\text{—}
Selected-frame mean recorded counts—\text{—}
Missing angular extent—\text{—}
Count-derived horizontal shift in pixels—\text{—}
Separate reference correlation—\text{—}
Reference relative RMS error—\text{—}
Recorded frozen slab thickness—\text{—}
Selected-frame ice transmission—\text{—}
Assigned final structural retention—\text{—}
Measured copies averaged—\text{—}
Experiment target—\text{—}

Physics tutorial

Cryo Electron Tomography: Spend Dose, Average Copies

BackgroundBalance frozen-slab thickness, acquisition order and dose before extracting repeated structures from the count-derived tomogram.

Why it mattersCan averaging recover signal without inventing missing orientations?

Start with the essentials

Focus question
Can averaging recover signal without inventing missing orientations?
One-sentence intuition
More copies reduce some noise; they do not supply missing angles.

Core mathematical model

Parallel-ray projection

u=xcos⁡θ+zsin⁡θPθ(u,y)=∫μ(x,y,z) ds\begin{aligned}u&=x\cos\theta+z\sin\theta\\P_\theta(u,y)&=\int\mu(x,y,z)\,ds\end{aligned}

The tilt axis is the specimen vertical coordinate. Each camera pixel measures one ray integral in this scalar model.

From counts to optical depth

⟨C⟩=N0e−be−Pθ‾P^θ=−ln⁡ ⁣(max⁡(C,1/2)N0)−b\begin{aligned}\langle C\rangle&=N_0 e^{-b}\overline{e^{-P_\theta}}\\\widehat P_\theta&=-\ln\!\left(\frac{\max(C,1/2)}{N_0}\right)-b\end{aligned}

The known ice baseline is subtracted, but its loss of counts and noise remain. Cryo frames integrate the assigned dose-dependent density over exposure.

Filtered back-projection

Qθ=F−1 ⁣[∣ν∣F{Pθ}]μ^≈∑jwjQθj(xcos⁡θj+zsin⁡θj,y)\begin{aligned}Q_\theta&=\mathcal F^{-1}\!\left[|\nu|\mathcal F\{P_\theta\}\right]\\\widehat\mu&\approx\sum_j w_j Q_{\theta_j}(x\cos\theta_j+z\sin\theta_j,y)\end{aligned}

Angular trapezoid weights cover only recorded angles. A three-tap optional low-pass filter suppresses noise while softening detail; the missing wedge is never renormalized away.

Common difficulties

Scientific boundary

Typical misconceptionA successful target is a calibrated cryo workflow or a resolved experimental structure.

Better mental modelScalar parallel-ray absorption teaching model, not phase-contrast cryo-EM or a calibrated microscope. Fictional Gaussian densities, assigned ice mean free path of 350 nm and dose decay of 200 electrons per square nanometre. Seeded counting statistics or explicitly selected expected-count snapshots. Negative-log counts feed finite Ram-Lak filtering and limited-angle back-projection; no truth enters processing. No CTF, diffraction, multiple scattering, motion correction, angular refinement or experimental resolution claim. Hardware and specimen magnifications are schematic. Same-orientation fictional particles and supplied coarse picks; translation-only correlation against the first measured crop. The same missing wedge persists; no picking, orientation search, classification or FSC.

Run the experiment

  1. 01

    Predict before recording

    Choose tilt range, angular step, total dose and imposed drift. In the cold experiment, set ice thickness and acquisition order.

    What to observe: Dose is shared across all frames. Adding angles changes per-frame counts.
  2. 02

    Acquire and inspect

    Acquire a new series. Select a recorded frame and inspect its camera counts and physical tilt.

    What to observe: Changing acquisition settings now leaves the stored series unchanged until the next acquisition.
  3. 03

    Align and reconstruct

    Enable count-derived marker alignment, choose a reconstruction filter, then commit reconstruction.

    What to observe: Blue negative lobes, stretched depth structure and angular gaps are evidence of the inverse problem.
  4. 04

    Average measured copies

    Extract and average at least four supplied same-orientation picks. Compare the first measured crop with the measured average.

    What to observe: The first measured crop is the correlation template. No known density is inserted into the average.