---
title: "Electron Tomography: Reconstruct the Missing View Interactive Lab | Sandbox Physics"
description: "How much of a specimen's three-dimensional structure can a tilt series recover? As much as the recorded tilt range covers, and no more."
canonical_url: "https://sandboxphysics.com/labs/electron-tomography/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/electron-tomography/"
last_modified: "2026-10-27"
---

E42 · Projection tomography

# Electron Tomography: Reconstruct the Missing View

**How much of a specimen's three-dimensional structure can a tilt series recover?** As much as the recorded tilt range covers, and no more. The specimen cannot be tilted through every angle, so a wedge of viewing directions is never recorded, and features in the reconstruction come out stretched along the beam direction. Averaging more copies of a structure lowers the noise but cannot fill in those angles.

Place a marker, change the specimen depth, and record a tilt series. Compare reconstructed horizontal and depth slices as the marker leaves the camera or viewing directions go missing.

> 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. A known-axis, integer horizontal shift is fitted from the placed marker counts using supplied marker coordinates and tilt angles. Weak, absent or cropped markers can leave frames unresolved; those frames keep their original coordinates. The three Gaussian components have a user-set depth aspect, analytic tails and a finite camera field. There is no CTF, diffraction, multiple scattering, within-exposure motion correction or angular refinement, and no claim about experimental resolution. Hardware and specimen magnifications are schematic.

Physics tutorial

## Electron Tomography: Reconstruct the Missing View

**Background** Tomography rebuilds a three-dimensional object from projections taken at many tilt angles. Here you record projections, align and filter them, and inspect perpendicular slices of the result together with the missing wedge.

**Why it matters** The specimen can only be tilted so far, so some viewing directions are never recorded and the reconstruction has to do without them.

Start with the essentials

- **Focus question**: How much of a specimen's three-dimensional structure can a tilt series recover?
- **One-sentence intuition**: More copies reduce some of the noise, but they 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. The mark coordinates, size, strength and tilt axis are supplied calibration. Alignment compares eleven integer horizontal shifts using only the marker-strip counts; its likelihood-gap and goodness-of-fit limits are teaching heuristics. Absent or unresolved marks leave frames uncorrected. The three Gaussian components have adjustable depth, infinite analytic tails and a finite camera field. There is no CTF, diffraction, multiple scattering, within-exposure motion correction or angular refinement, and no claim about experimental resolution. Hardware and specimen magnifications are schematic.

### Run the experiment

1. 01

   #### Predict before recording

   Place the marker horizontally and in depth, and choose the specimen depth aspect. Then set the tilt range, angular step, dose and drift. The marker stays in a known strip away from the specimen features.

   **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, choose a filter, and reconstruct. Compare removing the mark with placing it near the camera edge. Check how many frames have a resolved shift before interpreting the volume.

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

4. 04

   #### Compare with the known reference

   Compare the orthogonal slices with the known input, shown separately in gold, and check the target.

   **What to observe:** The correlation and error against the reference are teaching checks on the same grid, not an experimental resolution.

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)