---
title: "Cryo Electron Tomography: Spend Dose, Average Copies Interactive Lab | Sandbox Physics"
description: "Can averaging many copies of a structure pull its signal out of the noise in a cryo-electron tomogram? Yes, up to a point."
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last_modified: "2026-10-27"
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E44 · Cryo tomography

# Cryo Electron Tomography: Spend Dose, Average Copies

**Can averaging many copies of a structure pull its signal out of the noise in a cryo-electron tomogram?** Yes, up to a point. Cutting many copies of the same structure out of the tomogram and averaging them lifts the signal above the noise. Averaging cannot create orientations that were never recorded, though. The frozen sample can also take only so much electron dose, and that total has to be split across the whole tilt series.

Choose the thickness of the frozen slab, the order in which tilts are recorded and the dose, then pull repeated copies of a structure out of the tomogram reconstructed from the counts.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

This is a teaching model of scalar absorption along parallel rays, not phase-contrast cryo-EM or a calibrated microscope. The densities are fictional Gaussians. The ice has an assigned mean free path of 350 nm, and the specimen fades with dose on a scale of 200 electrons per square nanometre. Counts come from seeded counting statistics or from expected-count snapshots, which are clearly marked when selected. The negative log of the counts goes through a finite Ram-Lak filter and limited-angle back-projection; the true specimen never enters processing. There is no CTF, diffraction, multiple scattering, motion correction or angular refinement, and no claim about experimental resolution. Hardware and specimen magnifications are schematic. The fictional particles share one orientation and their rough positions are supplied; correlation only shifts each copy to match the first measured crop. The same missing wedge remains, and there is no particle picking, orientation search, classification or FSC.

Physics tutorial

## Cryo Electron Tomography: Spend Dose, Average Copies

**Background** Cryo electron tomography tilts a frozen specimen, images it at many angles and combines the images into a three-dimensional volume. Here you choose the slab thickness, the acquisition order and the dose, reconstruct a tomogram from the counts and extract repeated structures from it.

**Why it matters** Every electron that images the frozen specimen also degrades it, so the dose must be spread thinly across the tilts. Averaging many identical copies is one way to get the signal back.

Start with the essentials

- **Focus question**: Can averaging many copies of a structure pull its signal out of the noise in a cryo-electron tomogram?
- **One-sentence intuition**: Averaging more copies reduces some of the noise, but it cannot supply angles that were never recorded.

### Core mathematical model

#### Projection along parallel rays

$$
\begin{aligned}u&=x\cos\theta+z\sin\theta\\P_\theta(u,y)&=\int\mu(x,y,z)\,ds\end{aligned}
$$

The tilt axis runs along the specimen's vertical direction. In this scalar model, each camera pixel measures the integral along one ray.

#### From counts to optical depth

$$
\begin{aligned}\langle C\rangle&=N_0 e^{-b}\overline{e^{-P_\theta}}\\\widehat P_\theta&=-\ln\!\left(\frac{\max(C,1/2)}{N_0}\right)-b\end{aligned}
$$

The known ice baseline is subtracted, but the counts it removed and its noise stay. Each cryo frame integrates the density over its exposure, and that density changes with dose in the assigned way.

#### Filtered back-projection

$$
\begin{aligned}Q_\theta&=\mathcal F^{-1}\!\left[|\nu|\mathcal F\{P_\theta\}\right]\\\widehat\mu&\approx\sum_j w_j Q_{\theta_j}(x\cos\theta_j+z\sin\theta_j,y)\end{aligned}
$$

Trapezoid weights in angle cover only the recorded angles. An optional three-tap low-pass filter suppresses noise but also softens detail. The missing wedge is never renormalized away.

### Common difficulties

#### What a passing target does not mean

**Typical misconception** A successful target is a calibrated cryo workflow or a resolved experimental structure.

**Better mental model** Passing only shows success inside this teaching model. This is a teaching model of scalar absorption along parallel rays, not phase-contrast cryo-EM or a calibrated microscope. The densities are fictional Gaussians. The ice has an assigned mean free path of 350 nm, and the specimen fades with dose on a scale of 200 electrons per square nanometre. Counts come from seeded counting statistics or from expected-count snapshots, which are clearly marked when selected. The negative log of the counts goes through a finite Ram-Lak filter and limited-angle back-projection; the true specimen never enters processing. There is no CTF, diffraction, multiple scattering, motion correction or angular refinement, and no claim about experimental resolution. Hardware and specimen magnifications are schematic. The fictional particles share one orientation and their rough positions are supplied; correlation only shifts each copy to match the first measured crop. The same missing wedge remains, and there is no particle 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:** All frames share the total dose, so adding angles changes the counts in each frame.

2. 02

   #### Acquire and inspect

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

   **What to observe:** If you change the acquisition settings now, the stored series stays the same until the next acquisition.

3. 03

   #### Align and reconstruct

   Enable marker alignment from the counts, choose a reconstruction filter, then commit the reconstruction.

   **What to observe:** Blue negative lobes, structures stretched in depth and angular gaps all show the limits of reconstructing from incomplete projections.

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 serves as the correlation template, and no known density goes into the average.

Physical conclusion

Keep what was recorded, what the reconstruction computed from it and the known reference apart.

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)