E21 · Serial volume electron microscopy

Block-face SEM: Between the Cuts

Cut a resin block with an in-chamber diamond knife, retract it and image the new face. Compare section spacing, counting noise, drift and knife damage before reconstructing the volume.

Overlap with known sampled input $\text{—}$

Volume error against known input $\text{—}$

Count-derived segmented volume $\text{—}$

Known sampled input volume $\text{—}$

Mean central pixel counts $\text{—}$

Recorded depth coverage $\text{—}$

Section to lateral pitch ratio $\text{—}$

Selected exposed-face depth $\text{—}$

Total section positions $\text{—}$

Selected image status $\text{—}$

Summed incident dose over new faces $\text{—}$

Experiment target $\text{—}$

Predict · cut · record · reconstruct

Can a sharp face hide structure between cuts?

Physical section thickness

h

60\,\mathrm{nm}

Dwell per pixel

\tau

2\,\mathrm{\mu s}

Horizontal drift per cut

d_x

3\,\mathrm{\frac{nm}{\mathrm{cut}}}

Prescribed knife-damage band fraction

a

0.15

Lose every nth image: 0 none

n_{\rm gap}

0

r

0

Incident-normalized response threshold

q

0.4

Actual depth spacing: 1 on; 0 assumes 25 nm

c

1

Stored image index: clamps to last available position

k

10

Presentation phase: 0 image, 1 cut

s

0

Target: overlap at least 80 percent, volume error at most 12 percent, recorded depth coverage at least 95 percent, and at least 12 mean central pixel counts. Use actual depth spacing; sections at most 50 nm. The known input is an independent audit, never an inverse input.

The resin block face is perpendicular to the SEM axis. A lateral diamond knife cuts while the SEM is blanked; it retracts before imaging. The stage advances to keep the face at a fixed working position. The assumed BSE information length is 30 nm; section thickness is 50 to 160 nm. Knife damage is a prescribed low-response horizontal band whose position changes between cuts, not a mechanical fracture or chatter solver. Contrast represents a synthetic stained region, not a cell, membrane or staining protocol. No tissue preparation, knife force, charging, environmental-pressure, volume-throughput or real-material calibration. Fixed 50 pA SEM current, 48 by 48 pixels at 25 nm face pitch, 1200 nm field and 960 nm total depth. The synthetic input is a sphere plus a curved tube, with three persistent boundary fiducials. Selected BSE response is an assumed linear mixture: 0.18 background plus 0.44 occupied-region contrast and 0.65 fiducial contrast; detector efficiency is 8 percent. No material identification or calibrated biological structure. The forward model averages four normalized exponential depth bins, applies horizontal drift and cutting artefacts, then stores seeded electron counts: Poisson below 40 expected counts and a normal approximation above. Fixed linear colors clip only the display. The inverse receives stored counts and calibration metadata, never the known specimen or imposed drift. Integer horizontal cross-correlation uses acquired boundary fiducials against the first acquired image, over an eight-pixel search range. No vertical, affine or subpixel correction. A threshold segments the central 700 nm square; fiducials are excluded. Unknown layers and uncovered pixels remain unknown. Each mask forms a slab with its recorded thickness; no interpolated detail or tomography solver. The separately shown known input is used only for overlap and volume-error checks on the same sampling grid. Cumulative incident dose sums exposures on successive faces; it is not dose delivered to one surviving voxel. Presentation phase and camera motion never alter the stored dataset.

Physics tutorial

Block-face SEM: Between the Cuts

BackgroundCut a resin block with an in-chamber diamond knife, retract it and image the new face. Compare section spacing, counting noise, drift and knife damage before reconstructing the volume.

Why it mattersTarget: at least 80 percent overlap with the known sampled input, at most 12 percent volume error, at least 95 percent recorded depth coverage and 12 mean counts per central pixel. Keep actual depth spacing enabled. Keep sections at most 50 nm thick.

Start with the essentials

Focus question: Can a sharp face hide structure between cuts?
One-sentence intuition: A crisp image proves lateral contrast, not complete depth sampling. Registration can correct measured drift; it cannot restore a destroyed or missing section. Correct voxel spacing matters as much as segmentation.

Core mathematical model

Electron counting budget

N_0=\eta\frac{I\tau}{e}

Incident current and dwell set the count scale; the selected BSE response multiplies this budget.

Depth sampling

\begin{aligned}\mathcal T&=\sum_{j=0}^{3}w_j\rho_j\\\rho_j&=\rho(x,y,z_j)\\z_j&=z+(j+\tfrac12)\ell\\w_j&=\frac{e^{-j}-e^{-j-1}}{1-e^{-4}}\end{aligned}

A prescribed finite quadrature of a one-sided exponential information-depth kernel, not electron transport.

Count-derived volume

\begin{aligned}\widehat V&=p^2\sum_k h_k\sum_{x,y}\widehat M_k(x,y)\\\mathrm{IoU}&=\frac{|\widehat M\cap M_*|}{|\widehat M\cup M_*|}\end{aligned}

The threshold mask uses acquired counts. Slabs use actual thickness, including the final partial section. Unknown voxels contribute no invented object volume; known-input overlap is a separate audit.

Common difficulties

A reconstruction is not the known input

Typical misconceptionThe reference shape proves the algorithm recovered it.

Better mental modelThe gold reference is supplied only to an independent audit. The teal reconstruction comes from thresholded acquired counts; noise, damage and gaps remain.

Smaller voxels do not create resolution

Typical misconceptionSetting all voxel edges to 25 nm restores isotropic information.

Better mental modelChanging metadata can distort the shape and volume without adding any measured detail. Use the real section spacing.

Run the experiment

01
Predict the hidden error
Compare coarse sections with the resolved preset. Inspect the face image, orthogonal reslices and known input.
What to observe: Lateral sharpness does not recover structure between cuts.
02
Process the same acquisition
Use the drifting preset, then enable counted-fiducial registration. Change the threshold and voxel calibration.
What to observe: These processing controls reuse the stored count arrays; a changed reconstruction does not imply a new acquisition.
03
Inspect an actual missing layer
Choose missing sections and select the fifth stored position. Observe the dark image tile and unknown reslice row.
What to observe: A missing layer is not an empty part of the specimen.
04
Reach the target
Reduce section thickness and artefacts, use enough dwell for counts, register the measured fiducials and enable actual spacing. Check the target.
What to observe: The target tests overlap, volume error, depth coverage and counting budget.

Core mathematical model

Electron counting budget

Depth sampling

Count-derived volume

Common difficulties

A reconstruction is not the known input

Smaller voxels do not create resolution

Run the experiment

Predict the hidden error

Process the same acquisition

Inspect an actual missing layer

Reach the target