Scientific tutorial · Computational Atomic-Resolution STEM: Concepts and Methods · Scientific Computing ·
What Can an Electron Microscope Image Tell Us About Atomic Structure?
A practical introduction to joint structure–nuisance inference, heteroscedastic likelihoods, adequacy gates, protected validation, and scientifically necessary abstention.
Choose your reading depth
Core explanation: read the visible story, example, findings, and limitations for the main argument. Open Methods for how the comparison works, or Technical depth for equations and assumptions. These are choices about detail, not labels for readers. No other tutorial is required.
Start with the image
HAADF-STEM (high-angle annular dark-field scanning transmission electron microscopy) scans a focused electron probe across a specimen and records electrons scattered to large angles. The resulting image is a two-dimensional intensity map of a three-dimensional object. Bright features can carry information about atoms, but the image also depends on the beam, detector, atomic arrangement, and measurement conditions.
Inferring structure reverses image formation: we observe an image and ask which atomic arrangements could have produced it. Different three-dimensional structures or viewing directions can produce similar projections. Identifiability asks whether the available observation can distinguish the alternatives.
A candidate is one proposed structure; orientation is its direction relative to the beam. Nuisance variables, such as brightness scale, background, and image translation, affect the observation without being the structural answer we want. They must still contribute to uncertainty.
The inverse problem
Given one HAADF-STEM image, which atom count, structural family, candidate structure, and orientation are supported—and when should the inference refuse to decide?
Methods: Structure competes with nuisance
Image intensity and background, translation, observation noise, projection degeneracy, and forward-model mismatch can imitate structural differences. The method therefore evaluates structure and nuisance variables jointly instead of fixing nuisance values in advance.
A worked example
Illustrative ambiguity, not a reported measurement: one candidate has a brighter simulated image than another. If detector scale is uncertain, the dimmer candidate with a larger scale may explain the same observation. Fixing scale prematurely can create artificial structural certainty. Allowing plausible scales and backgrounds gives both candidates a fair comparison.
Marginalization combines support over those plausible settings, weighted by the model. It does not guarantee truth: if the actual structure is absent from the candidate set, the method may confidently select the closest wrong alternative. Qualification checks ask whether a structural answer is justified at all.
Technical depth: Posterior over structure and nuisance
Here c is a structural candidate, omega its orientation, eta nuisance parameters, and y the observed image.
Technical depth: Marginalize nuisance before ranking structure
Structural confidence should reflect plausible orientations and nuisance settings rather than a single convenient fit.
High confidence can still be wrong
Controlled synthetic development improved joint structure–orientation recovery when nuisance was modelled, but closed-set out-of-distribution inputs and forward-model mismatch could remain confidently misclassified. Posterior concentration is therefore not sufficient evidence of adequacy.
Technical depth: Empirical heteroscedastic likelihood
Low-gradient residual diagnostics on prospectively matched development acquisitions supported an empirical intensity-dependent variance law. It is not a claim of Poisson counting statistics or calibrated detector gain.
Methods: The frozen Stage-8 safeguards
Table 1
| Safeguard | Purpose |
|---|---|
| Exact-N candidate population across frozen atom counts and structural families | Prevents post-hoc candidate construction after seeing protected outcomes. |
| Deterministic orientation sampling | Makes the orientation search reproducible. |
| Structure-blind preprocessing | Stops candidate identity from influencing image preparation. |
| Intensity/background marginalization | Propagates photometric nuisance into structural uncertainty. |
| Explicit translation nuisance | Tests whether candidate ranking is stable to registration. |
| Acquisition and ROI qualification gates | Blocks inference when the observation does not meet the frozen regime. |
The qualification found a blocker
Development-only checks resolved the orientation and intensity-grid approximations, but translation support remained boundary-limited through the widest declared ladder. The correct scientific action was to block otherwise eligible cases rather than report forced structural labels.
What happened in the write-once run
The single blind-known Phase-A run decoded and accounted for every held-out source. Cases that passed acquisition checks but not the unresolved translation-support gate were blocked. This is qualification evidence, not a successful experimental structure-identification result. No protected-unidentified source was opened.
Technical depth: An abstention rule is part of the estimator
A scientifically honest inverse method returns no structure when its assumptions are not qualified.
The central lesson
Uncertainty quantification is not only a posterior probability. It also requires candidate-population discipline, nuisance propagation, model-adequacy diagnostics, protected validation, and explicit abstention when a qualification gate fails.
Sources and further reading
Uncertainty-Aware Atomistic Inference from HAADF-STEM Images: Identifiability, Nuisance Propagation, and Prospective Experimental Qualification. Manuscript in preparation. This explanation distinguishes illustrative examples from the frozen results; its numerical findings apply to the configurations described above.