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Physics-Informed Sequential Reconstruction of Scanning Probe Microscopy Images with Calibrated Uncertainty

Surya Prakash Reddy, Sergei V. Kalinin

arXiv:2609.10638v1eess.IV

TL;DR

SPM images mix surface structure with artifacts that evolve during sequential acquisition, while common correction approaches provide limited quantitative uncertainty. The paper uses reliability-weighted Bayesian state estimation with Kalman filtering and RTS smoothing; quality weighting and smoothing reduce pooled reconstruction RMSE, though complex variants offer no uniform benefit.

  • Problem

    Conventional correction uses separate post-processing steps, while physics-based approaches address parts of the measurement chain and learned methods face scarce clean experimental ground truth.

    Source: §1 · pp. 2–3

  • Method

    The method treats raster reconstruction as Bayesian sequential state estimation, using an artifact-dependent quality score, a forward Kalman pass, and an RTS backward pass.

    Source: §1 · p. 3, §2.3 · p. 6, §2.4 · p. 7

  • Results

    Quality weighting and RTS smoothing reduce pooled RMSE by 9.5% and 6.6%, respectively, while more complex states and implemented lateral and dual-pass operations provide no uniform benefit.

    Source: §4 · p. 17

  • Takeaways & Limitations

    The framework is most effective for intermittent line failures over a limited predictive range, and reconstructed height should be interpreted alongside diagnostics, relative uncertainty, and the measured physical observable.

    Source: §4 · p. 17

    In the height-only observation model, S6 fast-axis derivative components are not independently observable without additional measurements or coupling assumptions, so S6 is exploratory.

    Source: §2.2 · p. 5

Abstract

from arXiv · show

Scanning probe microscopy (SPM) has become a mainstay of materials science, condensed matter physics, nanotechnology, biology, and semiconductor metrology. An SPM image, however, is acquired sequentially as the probe traverses the surface under closed-loop feedback while thermal drift, feedback dynamics, tip state, mechanical disturbances, and electronic noise evolve during the experiment. Image correction has therefore become an intrinsic part of SPM data analysis. Classical approaches rely on line leveling, filtering, registration, interpolation, and explicit models of tip or scanner distortions, whereas recent work increasingly uses neural networks trained to map corrupted images to corrected ones. Here, we explore a different formulation in which SPM reconstruction is treated as a Bayesian sequential state-estimation problem. The hidden state represents surface height and its low-order evolution along the slow-scan direction; an artifact-dependent quality score continuously modifies the observation likelihood; a forward Kalman pass provides the causal estimate and innovation diagnostics; and a Rauch-Tung-Striebel (RTS) backward pass incorporates information from the complete acquisition. The result is a reconstructed surface, a relative posterior uncertainty map, and line-resolved diagnostic signals. The recursion is computationally lightweight and scales linearly with the number of image pixels, while the physical parameters can in principle be initialized from instrument characterization or from priors learned during previous operation, an extension not exercised here. We organize the method as a multi-axis family of models so that the contributions of state representation, artifact recognition, lateral coupling, and acquisition redundancy can be evaluated independently.

1. Introduction

SPM correction must separate sample structure from evolving acquisition artifacts, yet conventional, physics-based, and learned approaches leave distinct limitations. This work frames raster reconstruction as sequential Bayesian estimation with reliability-weighted observations, causal diagnostics, and RTS smoothing.

1. Introduction

  • Conventional correction separates leveling and filtering from physics-specific artifact models, while learned restorers face scarce clean experimental ground truth and motivate hybrid use where recognition is needed.Source: §1 · pp. 2–3

2.1 SPM acquisition as a state-space process

  • For each fast-axis position, scan lines form an ordered state-space sequence separating deterministic evolution, process variability, measurement mapping, and observation reliability.Source: §2.1 · p. 4, §2 · p. 4

2.2 Low-order state representations

  • S2 propagates height and rate linearly through weak data, but rate conflates surface slope and drift; S6 adds derivative states whose fast-axis components are unobservable from height alone.Source: §2.2 · p. 5

2.3 Observation reliability as a continuous likelihood weight

  • Continuous quality inflates measurement variance so unreliable observations are downweighted rather than rejected; the linewise detector uses four robust statistics but may miss short defects or misclassify structured surfaces.Source: §2.3 · p. 6

2.4 Forward Kalman recursion and adaptive measurement noise

  • Forward innovations provide line-resolved crash, tip-transition, ringing, and mismatch diagnostics; adaptive baseline-noise updates are monitoring safeguards, not demonstrated performance gains.Source: §2.4 · p. 7

2.5 Non-causal reconstruction with the RTS smoother

  • The RTS backward pass uses later observations to constrain gaps from both sides, while reported uncertainty is relative and intervals require empirical calibration because heuristics and model mismatch undermine Gaussian interpretation.Source: §2.5 · pp. 7–8

2.6 Optional lateral and dual-pass information

  • Lateral regularization is heuristic because cross-pixel covariance is not propagated; implemented dual-pass fusion raises benchmark error and may underestimate variance when quality scores differ.Source: §2.6 · p. 9

2.7 Implementation and validation protocol

  • Validation combines 2,880 synthetic runs, disjoint 54-case calibration and test sets, and real-data references from opposite passes, STM peaks, or nominal geometry.Source: §1 · p. 3, §2.7 · p. 9

3. Results and Discussion

Validation progresses from controlled synthetic tests to experimental checks without clean pixel-level ground truth, using reconstruction metrics and physical observables suited to each setting.

3.1 Reconstruction of controlled artifact classes

  • Reconstructions bridge one-line failures and gaps of 8 or 32 lines when the state model remains predictive, while relative uncertainty rises toward the least-constrained part of a gap.The controlled examples also include simultaneous corruption in both passes; drift, oscillation, and tip-change strata are evaluated separately in the factorial analysis.Source: §3.1 · p. 10, Figure 3 · p. 10

3.2 Controlled model comparisons

  • Paired comparisons favor confidence weighting and RTS smoothing over added model complexity, while higher-order states show no resolved average advantage and the implemented pass-fusion rule worsens pooled error.These findings support M07—the low-order, quality-weighted, single-pass model with RTS smoothing—as the primary model for experimental interpretation, without establishing universal superiority.Source: §3.2 · p. 11, Figure 4 · p. 11

3.3 Metrics for physically meaningful reconstruction

  • Raw uncertainty intervals under-cover in the held-out M08 calibration experiment, and empirical calibration brings coverage closer to nominal without reaching it.This demonstrates the procedure, not calibrated uncertainty for M07 or transferability to new instruments.Source: §3.3 · p. 12, Figure 5 · p. 13

3.4 Experimental data and external baselines

  • With failure locations unknown, archived M10 has lower mean failed-region error than the baselines, but its advantage diminishes as gaps widen; blind correction also alters clean regions, and Red3 cannot establish registered accuracy.Supplying the true mask makes classical methods competitive or better and greatly reduces the spectral penalty; the Red3 orientation audit is ambiguous.Source: §3.4 · pp. 12–13, Figure 5 · p. 13

3.5 Physical validation without pixel-level ground truth

  • Across six STM acquisitions, single-pass Kalman variants balance scan-line suppression with lattice-peak retention, whereas stronger smoothing can suppress lattice peaks along with background.These reciprocal-space checks test structural preservation rather than clean pixelwise accuracy.Source: §3.5 · p. 14, Figure 7 · p. 15

3.6 AFM calibration-grating metrology and detector stress test

  • Most calibration-grating variants retain pitch and step height relative to raw scans, but unweighted dual-pass fusion introduces substantial step-height bias and quality weighting improves repeatability in this small set.The detector assigns very low quality to one strongly periodic scan, where legitimate structure resembles artifact statistics; without certified step-height uncertainty, this establishes geometric self-consistency rather than absolute accuracy.Source: §3.6 · p. 15
  • Low-order prediction, quality weighting, and smoothing support recovery across intermittent failures, but long-gap reconstruction increasingly depends on surface assumptions; the current model omits finite-tip, coordinate-error, hysteresis, and detailed feedback dynamics.The ablations also identify limitations in quality weighting, lateral coupling, and the present dual-pass likelihood.Source: §3.6 · pp. 16–17
  • Empirical calibration improves held-out M08 interval coverage but leaves residual undercoverage and cannot guarantee transfer across operating conditions or repair an incorrect surface model.Covariance indicates information loss under the assumed model; it is not by itself evidence that intervals are calibrated.Source: §3.6 · p. 16

4. Conclusion and future work

The paper frames SPM reconstruction as filtering followed by smoothing, with quality-weighted observations and line-based constraints. Its controlled model family identifies strengths and limits, while future extensions remain conditional on identifiable parameters and implementation support.

Conclusion and future work

  • Quality weighting and RTS smoothing reduce pooled RMSE by 9.5% and 6.6%, respectively, while more complex states and implemented lateral and dual-pass operations provide no uniform benefit.The ablations favor M07 as a transparent primary configuration, not as a universally superior model.Source: §4 · p. 17
  • The framework is most effective for intermittent line failures over a limited predictive range, and STM lattice checks and AFM dimensional tests reveal detector sensitivity to physical structure.Reconstructed height should be interpreted alongside acquisition diagnostics, relative uncertainty, and the measured physical observable.Source: §4 · p. 17
  • An unscented-filtering extension with a nonlinear observation model requires its additional parameters to be identifiable, while real-time deployment would require acquisition and computing interfaces.Future models are motivated to distinguish sample evolution from instrument response.Source: §4 · p. 17

Data Availability

The data-availability statement identifies two repositories and a supplementary snapshot, with different conditions for regenerating synthetic and experimental components.

Data Availability

  • The reconstruction framework and a development repository are available at the two stated GitHub locations.The original framework is in Sergei V. Kalinin’s repository; the corresponding author’s development repository is separately identified.Source: Data Availability · p. 18
  • The JAP_supplementary_repository snapshot contains analysis notebooks, figure-generation code, editable PowerPoint assembly, frozen numerical results, and a file manifest.Synthetic figures and quantitative panels can be regenerated from the snapshot.Source: Data Availability · p. 18
  • Regenerating experimental image components requires original acquisitions and stored reconstruction arrays, whose access conditions must be confirmed with the authors.A versioned archival release and DOI have not yet been assigned.Source: Data Availability · p. 18

Supplementary Material

The supplied supplementary-material metadata identify a partial paper title, its authors, and their department and institution.

Supplementary Material

  • The title fragment is “Physics-Informed Sequential Reconstruction of Scanning Probe Microscopy Images with”.The supplied title text ends at “with”.Source: Supplementary Material · p. 23
  • The listed authors are Surya Prakash Reddy and Sergei V. Kalinin.Source: Supplementary Material · p. 23
  • Both authors are affiliated with the Department of Materials Science and Engineering at the University of Tennessee, Knoxville, Tennessee.Source: Supplementary Material · p. 23

S1. Scope, data, and reproducibility

The supplement records the reproducibility materials and defines a controlled model family whose variants differ in state, confidence, lateral coupling, and pass handling.

S1. Scope, data, and reproducibility

  • The reviewer-facing computational snapshot contains frozen results, analysis scripts, figure sources, manifest checks, and a manuscript-to-code map; a public archival release, tag, commit identifier, and DOI are planned before final publication.Experimental image components require original acquisitions and stored reconstruction arrays, with access conditions to be confirmed.Source: S1. Scope, data, and reproducibility · p. 23
  • The primary model is M07_S2_CT_G0_P1, while S6 higher-order state, GC edge-aware lateral regularization, and P2 dual-pass fusion are exploratory ablations.Normalized intensities are used internally, with dimensional calibration restored for physical metrology outputs.Source: S1. Scope, data, and reproducibility · p. 23

S2. Controlled model family

  • The model axes encode S2/S6 state families, C0/CT observation reliability, G0/GC lateral regularization, and P1/P2 acquisition-pass handling; four logical combinations are not implemented.GC is never paired with unweighted confidence C0.Source: S2. Controlled model family · pp. 23–24
  • The observation equation is y_k = Cx_k + v_k, with C selecting the first state component through C = [1, 0, …, 0].Source: S2. Controlled model family · p. 23

S3. Numerical implementation

The implementation runs Kalman and RTS recursions along each fast-scan column, adjusts measurement covariance using observation quality, and applies safeguards that limit how posterior uncertainty can be interpreted.

S3. Numerical implementation

  • Each fast-scan column receives a Kalman and Rauch–Tung–Striebel recursion, with observation quality modifying measurement covariance.The innovation-based covariance update subtracts predicted observation variance before applying a floor and exponential averaging.Source: S3. Numerical implementation · p. 25
  • The measurement-variance update uses a 0.950 memory factor and runs after burn-in only when mean line quality exceeds 0.7.Otherwise, the previous measurement variance is retained.Source: S3. Numerical implementation · pp. 25–26
  • CT variants run the forward pass twice before one RTS pass; line 0 uses the first observation, and wrapper burn-in is max(25, floor(0.10 N_k)) lines.The first CT pass estimates innovation RMS for the tip-change factor; the second uses the updated quality map.Source: S3. Numerical implementation · p. 26
  • Height and selected derivative bounds improve numerical robustness but break an exact linear–Gaussian posterior interpretation, so raw σ is only a relative diagnostic unless independently calibrated.The implemented quality weighting is R/q, not the more aggressive R/q squared alternative.Source: S3. Numerical implementation · p. 27, Figure S1 · p. 28

S4. Line-quality detector and scale sensitivity

The CT detector combines scan-line features into a quality score, but its thresholds remain scale-sensitive and a strongly periodic calibration scan is a documented failure case.

S4. Line-quality detector and scale sensitivity

S5. Synthetic benchmark and factorial effects

The synthetic benchmark compares model variants and component effects, while separate archives assess external baselines and held-out uncertainty calibration. Raw uncertainty under-covers, whereas calibrated intervals approach nominal coverage on held-out cases.

S5. Synthetic benchmark and factorial effects

  • The archive contains 2,880 model-stage records across 120 synthetic cases and 12 variants; four of 16 logical S/C/G/P combinations are unimplemented.GC is paired only with CT; Figure S2 reports unnormalized metrics and runtime, while Figure 5 uses within-regime normalized radar scores.Source: S5. Synthetic benchmark and factorial… · p. 28, Figure S2 · p. 29
  • 6.25%–12.79% and 4.90%–8.40% are clustered-by-case intervals for RMSE reductions from quality weighting and RTS smoothing, respectively.The clustered bootstrap gives wider intervals than the paired-record bootstrap, with the same pooled directions.Source: Table S4 · p. 30, S5. Synthetic benchmark and factorial… · p. 30

S6. External baselines and tuning

S7. Held-out uncertainty calibration

S8. Paired-pass consistency

Paired-pass comparisons are consistency diagnostics rather than accuracy tests because the opposite acquisition direction is not clean ground truth. The section also describes exploratory dual-pass diagnostics and STM lattice analysis without independent clean images.

S8. Paired-pass consistency

  • The Red3 orientation audit is ambiguous, so Figure 7 uses stored coordinates without extra registration or reversal and makes no agreement claim.The registered correlation is approximately 0.33, and the selected stored orientation differs from the best data-driven candidate.Source: S8. Paired-pass consistency · p. 32
  • A symmetric comparison reconstructs both acquisition directions before measuring consistency, avoiding the one-sided comparison that can penalize successful artifact removal.Absolute values depend on registration and border masking; Figure S4 uses neither subpixel registration nor a border mask.Source: S8. Paired-pass consistency · p. 32, Figure S4 · p. 32

S9. Exploratory dual-pass diagnostics

  • Figure S5 uses exploratory M28 trace and retrace reconstructions with independent quality classification; reconstruction-minus-trace and uncertainty panels are diagnostics, not ground-truth error maps.The diagnostic does not overturn the factorial finding that the current dual-pass fusion rule is detrimental on average.Source: S9. Exploratory dual-pass diagnostics · p. 32, Figure S5 · p. 33

S10. STM lattice analysis

  • Figure 8(d) reports raw-image periods for four Si acquisitions and excludes the two Au/Si acquisitions.Source: S10. STM lattice analysis · p. 33
  • STM lattice retention is evaluated by checking dominant reciprocal-space peaks at fixed coordinates before and after reconstruction, separately from line-artifact and high-frequency suppression.With no independent clean image, the Si100-Ge-3 reconstruction and its quality and relative-uncertainty maps are qualitative diagnostics.Source: S10. STM lattice analysis · p. 33, Figure S6 · p. 34

S11. AFM calibration-grating analysis

The grating tests preservation relative to raw readings and nominal geometry, not certificate-traceable accuracy. Most variants retain mean step height within 1.3% of the raw estimate, while periodic structure exposes a quality-detector failure.

S11. AFM calibration-grating analysis

  • Most variants retain mean step height within 1.3% of the raw estimate across three scans, but no certified step-height value or uncertainty is available.The 1.3% is the mean bias over the scans relative to mean raw step height; the largest single-scan departure among non-M02 variants is 2.60%.Source: S11. AFM calibration-grating analysis · p. 35, Figure S7 · p. 36
  • Confidence weighting reduces step-height coefficient of variation from 4.93% to 3.69%, but this three-scan repeatability result does not establish lateral calibration or absolute accuracy.The supplied files give a nominal 5.000 μm pitch and approximately 100 nm step, without a traceable calibration certificate.Source: S11. AFM calibration-grating analysis · p. 35, Figure S7 · p. 36
  • M02 has a +9.9 nm mean step-height bias and +16.5 nm worst-case bias, while MOBO3's periodic structure receives persistently low quality.The GP baseline is omitted from the figure panel because its mean bias is approximately +525 nm.Source: S11. AFM calibration-grating analysis · p. 35, Figure S7 · p. 36

S12. Future extensions

The supplement distinguishes prospective model extensions from validated results, measures runtime scaling, and documents a tiered repository whose experimental recomputation depends on access to original data.

S12. Future extensions

  • The validated model is intentionally low order; coupled tip, scanner, hysteresis, feedback, persistent-instrument-state, and sample-prior extensions remain research directions.Dual-pass data should be assimilated as separate likelihoods rather than through the present approximate average; these extensions are not validated by the main manuscript.Source: S12. Future extensions · p. 36, Figure S8 · p. 37

S13. Runtime and scaling

  • Reconstruction time scales linearly with pixel count in the measured implementation, with 43.97–45.69 μs per pixel across a 28-fold range in image size.Median time ranges from 0.104 s at 48 × 48 pixels to 2.882 s at 256 × 256 pixels.Source: S13. Runtime and scaling · p. 37

S14. Computational repository and review workflow

  • The repository supports audit and synthetic recomputation, but a full experimental rerun requires original files and processed arrays not redistributed in the reviewer snapshot.Qualified reviewers may request access subject to facility, provenance, and data-use restrictions.Source: S14. Computational repository and… · p. 38
  • Raw posterior σ remains a relative diagnostic until empirically calibrated, and the AFM-grating analysis supports preservation relative to raw/reference estimates rather than absolute metrology.The supplement also characterizes P2 and lateral coupling as exploratory and opposite-pass comparisons as consistency checks rather than accuracy measurements.Source: S14. Computational repository and… · p. 38

S15. Detailed numerical record and figure provenance

The detailed record finds that model components have trade-offs, calibrated intervals approach nominal coverage, and structural metrics support artifact suppression while qualifying the scope of metrological claims.

S15. Detailed numerical record and figure provenance

  • The numerical record consists of archived results and frozen tables, not a claim that the full benchmark was rerun for the editorial revision.The CSV files retain full precision, per-case results, model definitions, and metric-specific sample counts.Source: S15. Detailed numerical record and… · p. 38

Paired model effects

  • RTS smoothing reduces RMSE by 6.61% overall, while the confidence detector reduces it by 9.50%; edge-aware lateral regularization increases RMSE by 3.35%, and current P2 fusion increases it by 66.26%.The smoothing and confidence gains are larger for crashes, whereas confidence weighting modestly increases RMSE on clean, drift, oscillation, and tip-change strata.Source: Paired model effects · p. 39

Held-out interval results

  • Calibrated intervals cover 91.91% of all pixels and 90.87% of corrupted pixels on 54 held-out cases, versus 73.62% and 72.23% for nominal 95% raw intervals.Calibrated coverage is close to, but does not exactly attain, nominal coverage and does not establish a fully specified Gaussian posterior.Source: Held-out interval results · p. 39

External-baseline results

  • Blind Kalman+RTS has mean failed-region RMSE 0.166 across 30 synthetic cases, but rises to 0.467 for 32 consecutive failed lines, above raw data at 0.435.With the true failure mask, classical methods score 0.071–0.091 versus 0.086 for Kalman+RTS; the frozen comparator is M10, not primary M07.Source: External-baseline results · p. 39

STM structural metrics

  • Across six acquisitions, Kalman variants suppress line artifacts 2.69–2.77 times and retain 92.1–94.6% of tracked lattice-peak amplitude without measurable peak displacement.Two unambiguous period assignments agree with accepted values within 2%; the other assignments are reported without a metrological claim.Source: STM structural metrics · pp. 39–40

AFM grating metrology

  • Across three acquisitions, Kalman variants except the unweighted P2 control retain mean step height within 1.3% of raw readings; the comparisons do not establish certified accuracy.Raw pitches span 4.928–5.009 µm and Kalman pitches 4.927–4.992 µm.Source: AFM grating metrology · p. 40

Figure regeneration

  • Experimental figure recomputation requires original measurements and stored reconstructions, while quantitative panels can be rebuilt from supplied tables.Without an experimental-root path, the assembly reuses supplied experimental image components.Source: Figure regeneration · p. 40
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