Skip to main content
Sandbox Physics

E42 · Projection tomography

Electron Tomography: Reconstruct the Missing View

Acquire, align and filter recorded projections; inspect orthogonal reslices and the missing wedge.

Interactive modelElectron Tomography: Reconstruct the Missing View
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{—}
Experiment target—\text{—}

Physics tutorial

Electron Tomography: Reconstruct the Missing View

BackgroundAcquire, align and filter recorded projections; inspect orthogonal reslices and the missing wedge.

Why it mattersHow much three-dimensional structure can your tilt series support?

Start with the essentials

Focus question
How much three-dimensional structure can your tilt series support?
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.

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

    Check the reference audit

    Compare orthogonal reslices with the separately gold-colored input and check the target.

    What to observe: Reference correlation and error are teaching audits on the same grid, not experimental resolution.