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.

Selected-frame mean recorded counts $\text{—}$

Missing angular extent $\text{—}$

Separate reference correlation $\text{—}$

Frames with resolved marker shifts $\text{—}$

More measurements and target

Recorded projection count $\text{—}$

Selected recorded tilt $\text{—}$

Recorded total incident dose $\text{—}$

Dose before selected exposure $\text{—}$

Incident electrons per selected pixel $\text{—}$

Count-derived horizontal shift in pixels $\text{—}$

Reference relative RMS error $\text{—}$

Experiment target $\text{—}$

Simulated examples; each plot identifies its data source.

Requested maximum tilt magnitude

\theta_{\max}

45\,\mathrm{{}^\circ}

Total dose for next series

D

100\,\mathrm{\frac{electrons}{nm^2}}

Marker horizontal position for next series

x_m

0\,\mathrm{nm}

Specimen depth aspect for next series

a_z

1

Processing: count-derived marker alignment

0

Advanced settings

Tilt step; endpoints rounded inward

\Delta\theta

6\,\mathrm{{}^\circ}

Assigned maximum horizontal drift in pixels

\delta

3

Counts: 0 expectation, 1 seeded statistics

1

Placed mark: 0 absent, 1 present

1

Marker depth for next series

z_m

0\,\mathrm{nm}

Processing: 0 back-project, 1 ramp, 2 smoothed ramp

2

Selected recorded frame position

f

50\,\mathrm{\%}

Selected reslice index

k

12

Data color gain only

G

1

Learning goal and model limits

Predict · acquire · reconstruct

Make the missing direction visible

Target: cover at least 80 percent of the angles with a step of at most 4 degrees, and use a filtered reconstruction. The reference correlation must be at least 0.65, the relative RMS error at most 80 percent and the mean counts at least 40. If there is any drift, align it.

Acquisition settings take effect on the next new dataset. Processing settings take effect when you run the reconstruction or averaging. Frame, slice and color gain only change the display. Presets load finished examples.

The marker lies in a known horizontal strip; its horizontal position and depth are supplied calibration. Dragging it records and reconstructs a new series. Sliders plan the next operation. Unresolved frames keep their original coordinates.

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.

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

Better mental modelPassing 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

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.
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.
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.
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.

Core mathematical model

Projection along parallel rays

From counts to optical depth

Filtered back-projection

Common difficulties

What a passing target does not mean

Run the experiment

Predict before recording

Acquire and inspect

Align and reconstruct

Compare with the known reference