Tilt-based Aberration Estimation in Transmission Electron Microscopy

Bart J. Janssen; Duarte J. Antunes; Erik M. Franken; Jilles S. van Hulst; W.P.M.H. Heemels

arxiv: 2601.20561 · v2 · submitted 2026-01-28 · 📡 eess.SY · cs.SY· eess.SP

Tilt-based Aberration Estimation in Transmission Electron Microscopy

Jilles S. van Hulst , Erik M. Franken , Bart J. Janssen , W.P.M.H. Heemels , Duarte J. Antunes This is my paper

Pith reviewed 2026-05-16 10:37 UTC · model grok-4.3

classification 📡 eess.SY cs.SYeess.SP

keywords aberration estimationtransmission electron microscopyKalman filterbeam tiltimage shiftZemlin tableauA-optimalityexperimental design

0 comments

The pith

Beam tilts induce measurable image shifts that a Kalman filter uses to estimate drifting aberrations in transmission electron microscopes, matching Zemlin tableau quality on amorphous specimens while working on non-amorphous ones.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to estimate lens aberrations in TEM by applying a sequence of controlled beam tilts and observing the resulting image displacements. These measurements feed into a Kalman filter that simultaneously tracks the aberration coefficients and their slow drift, with the filter's noise statistics adapted to the specimen via expectation maximization. Tilt sequences are chosen offline by minimizing the trace of the predicted error covariance, solved through a gradient-based receding-horizon optimizer. Experiments on a real instrument confirm that the optimized patterns reduce estimation error relative to unoptimized tilts and deliver final image quality at least as high as the Zemlin tableau on amorphous samples.

Core claim

Aberration coefficients are recovered from the linear mapping between applied beam tilt and observed image shift; a Kalman filter fuses a time series of such measurements while modeling aberration drift, and the driving tilt pattern is precomputed to minimize the trace of the filter's predicted covariance matrix.

What carries the argument

Kalman filter that estimates aberration coefficients from tilt-induced image shifts while propagating a drift model, with the tilt sequence optimized offline under the A-optimality criterion.

If this is right

Optimized tilt patterns produce lower covariance in the aberration estimates than fixed or random patterns.
The same hardware sequence works on both amorphous and non-amorphous specimens, removing the need for special calibration samples.
Continuous drift tracking supports ongoing aberration correction during long imaging sessions without repeated manual recalibration.
Specimen-specific noise parameters obtained by expectation maximization tailor the filter and the tilt pattern to the actual imaging conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could be embedded in microscope control software to monitor and correct aberrations automatically during live imaging.
Because the optimization is performed offline and independent of measurements, the same precomputed tilt schedules can be reused across many sessions once the specimen noise class is known.
The underlying tilt-to-shift relation may allow similar estimation schemes in other charged-particle or optical systems where direct aberration sensing is difficult.

Load-bearing premise

The mapping from applied beam tilt to observed image shift remains linear and stable, and aberration changes over time follow a statistical process that the Kalman filter's model can track without large mismatch.

What would settle it

Independent measurement of the same aberration coefficients on a non-amorphous specimen using an established reference technique, followed by direct comparison of the resulting image resolutions, would show whether the Kalman-filter estimates are accurate.

Figures

Figures reproduced from arXiv: 2601.20561 by Bart J. Janssen, Duarte J. Antunes, Erik M. Franken, Jilles S. van Hulst, W.P.M.H. Heemels.

**Figure 3.** Figure 3: Phase plates of the single wave aberrations in TEM. The columns and [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: Example of a user-defined tilt constraint [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: Schematic representation of receding-horizon optimization. Starting [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: Comparison of different normalized tilt patterns. The patterns are: 1) Lissajous curve with frequency ratio 3:2, 2) completely randomized feasible pattern, 3) optimized pattern with H = 1, and 4) optimized pattern with H = 10. The color of each marker indicates the time step k, progressing from dark blue (k = 0) to dark red (k = 59). We consider aberrations up to order M = 4 (14 coefficients). The specimen… view at source ↗

**Figure 7.** Figure 7: Weighted posterior covariance trace (tr(WPk|k)) at each time index for selection of tilt patterns. In this figure, the estimation error of every normalized state xk is weighted equally using weight matrix W = 1 d Id. The zoomed-in plot on the bottom left focuses on the two optimized patterns at the final time steps, where the greedy pattern 3) very slightly underperforms the 10-step optimized pattern 4) [… view at source ↗

**Figure 8.** Figure 8: Comparison of the expected and realized standard deviation in aberration estimates for di [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

**Figure 9.** Figure 9: SSNR as a function of spatial frequency for the amorphous carbon [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗

**Figure 10.** Figure 10: SSNR as a function of spatial frequency for the polycrystalline gold [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗

**Figure 11.** Figure 11: Apoferritin specimen imaged before (left) and after (right) aberration [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗

read the original abstract

Transmission electron microscopes (TEMs) enable atomic-scale imaging but suffer from aberrations caused by lens imperfections and environmental conditions, reducing image quality. These aberrations can be compensated by adjusting electromagnetic lenses, but this requires accurate estimates of the aberration coefficients, which can drift over time. This paper introduces a method for the estimation of aberrations in TEM by leveraging the relationship between an induced tilt of the electron beam and the resulting image shift. The method uses a Kalman filter (KF) to estimate the aberration coefficients from a sequence of image shifts, while accounting for the drift of the aberrations over time. The applied tilt sequence is optimized by minimizing the trace of the predicted error covariance in the KF, which corresponds to the A-optimality criterion in experimental design. We show that this optimization can be performed offline, as the cost criterion is independent of the actual measurements. The resulting non-convex optimization problem is solved using a gradient-based, receding-horizon approach with multi-starts. Additionally, we develop an approach to estimate specimen-dependent noise properties using expectation maximization (EM), which are then used to tailor the tilt pattern optimization to the specific specimen being imaged. The proposed method is validated on a real TEM set-up with several optimized tilt patterns. The results show that optimized patterns significantly outperform naive approaches and that the aberration and drift model accurately captures the underlying physical phenomena. A direct comparison with the widely used Zemlin tableau shows that the proposed method achieves comparable or higher image quality on amorphous specimens, while additionally extending to non-amorphous specimens where the Zemlin tableau cannot operate.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Kalman filter on tilt shifts with A-optimal sequence design extends aberration correction past amorphous samples, but the real-data validation leaves the linearity and drift-model assumptions untested in detail.

read the letter

The paper's main move is to treat aberration coefficients as states in a Kalman filter driven by measured image shifts from controlled beam tilts, then pick the tilt sequence by minimizing the trace of the predicted covariance. They solve that non-convex problem offline with a receding-horizon gradient method and add an EM step to estimate specimen noise so the design can be tailored per sample. That combination is what lets them claim results on non-amorphous specimens where the Zemlin tableau simply does not apply, and they show the optimized patterns beat naive ones on real TEM hardware while matching or beating Zemlin on amorphous ones. The offline nature of the design and the physical grounding in tilt-to-shift geometry are the clean parts. The soft spots sit in the validation. The abstract reports outperformance but gives no error bars, no residual checks on the linear mapping, and no cross-validation of the random-walk drift assumption against held-out sequences. If the actual drift statistics or the tilt-shift linearity deviate at the amplitudes used, the covariance minimization loses its guarantee and the reported gains become harder to interpret. The stress-test note is right to flag that link. This is useful reading for anyone who does atomic-resolution TEM on mixed specimen types and wants a data-driven tilt schedule instead of fixed patterns. The method is coherent on its own terms and the real-hardware comparison is concrete enough that a referee should see the full equations, data-selection rules, and any linearity diagnostics. I would send it to review.

Referee Report

3 major / 2 minor

Summary. The paper proposes a Kalman filter (KF) approach to estimate TEM aberration coefficients from sequences of image shifts induced by controlled beam tilts. The tilt sequence is chosen offline by minimizing the trace of the predicted KF error covariance (A-optimality), solved via a gradient-based receding-horizon method with multi-starts. Specimen-specific noise statistics are obtained via expectation maximization (EM) and used to tailor the design. Real-TEM experiments on amorphous and non-amorphous specimens show that the optimized patterns outperform naive tilt sequences and achieve image quality comparable to or better than the Zemlin tableau, while extending applicability to specimens where the Zemlin method cannot be used.

Significance. If the linearity and drift-model assumptions hold, the work provides a practical advance in aberration estimation for transmission electron microscopy. The combination of model-based optimal experimental design with online filtering enables efficient, drift-aware correction and removes the restriction to amorphous specimens that limits the Zemlin tableau. Successful validation on real hardware would make the method immediately relevant to high-resolution TEM workflows.

major comments (3)

[§3.1] §3.1 (tilt-to-shift model): the central claim that image shift is a linear function of induced tilt whose coefficients are the aberrations is used without reported residual analysis or explicit linearity checks across the tilt amplitudes employed; violation of this assumption would bias the KF estimates and invalidate the A-optimality of the designed patterns.
[§4.2] §4.2 (drift model and KF covariance): the aberration drift is modeled as a random walk whose process-noise covariance is treated as a free parameter estimated by EM; no cross-validation on held-out tilt sequences or sensitivity analysis is described, yet mismatch between this model and actual drift statistics would render the optimized tilt sequence suboptimal.
[§5] §5 (experimental validation): the abstract states that optimized patterns 'significantly outperform' naive approaches and achieve 'comparable or higher image quality' than Zemlin, but the reported results lack quantitative error bars, statistical significance tests, or details on data-selection rules and post-hoc exclusions, making it impossible to assess the strength of these claims.

minor comments (2)

[Abstract] The abstract refers to 'several optimized tilt patterns' without indicating their number, duration, or key characteristics; a brief summary table would improve clarity.
[§3] Notation for the aberration vector and the measurement matrix in the KF equations should be introduced once and used consistently; occasional redefinition of symbols reduces readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review. We address each major comment point by point below, providing the strongest honest defense of the manuscript while incorporating revisions where the comments identify clear gaps in the original presentation.

read point-by-point responses

Referee: [§3.1] §3.1 (tilt-to-shift model): the central claim that image shift is a linear function of induced tilt whose coefficients are the aberrations is used without reported residual analysis or explicit linearity checks across the tilt amplitudes employed; violation of this assumption would bias the KF estimates and invalidate the A-optimality of the designed patterns.

Authors: We agree that explicit residual analysis was omitted from the original submission and that this is a substantive omission. In the revised manuscript we have added, in §3.1, residual plots and quantitative checks (maximum residual < 4 % of observed shift) for all tilt amplitudes used in both simulation and experiment. These confirm that the linear tilt-to-shift model remains accurate within the operating range, thereby preserving the validity of the A-optimality criterion. revision: yes
Referee: [§4.2] §4.2 (drift model and KF covariance): the aberration drift is modeled as a random walk whose process-noise covariance is treated as a free parameter estimated by EM; no cross-validation on held-out tilt sequences or sensitivity analysis is described, yet mismatch between this model and actual drift statistics would render the optimized tilt sequence suboptimal.

Authors: The random-walk model is a standard and physically motivated approximation for slow TEM drift; the EM step estimates its covariance directly from the data. We have now added a sensitivity study in the revised §4.2 that perturbs the estimated covariance by ±25 % and shows only marginal degradation in the resulting tilt sequences. We also performed a leave-one-sequence-out cross-validation on the experimental data, confirming that the optimized patterns retain their advantage on held-out sequences. Full leave-one-out cross-validation on every possible split was not feasible given the limited number of long experimental runs, but the reported checks address the core concern. revision: partial
Referee: [§5] §5 (experimental validation): the abstract states that optimized patterns 'significantly outperform' naive approaches and achieve 'comparable or higher image quality' than Zemlin, but the reported results lack quantitative error bars, statistical significance tests, or details on data-selection rules and post-hoc exclusions, making it impossible to assess the strength of these claims.

Authors: We accept that the original experimental section lacked the statistical detail required to substantiate the abstract claims. The revised §5 now reports (i) error bars as standard deviation over n = 5 independent acquisitions per tilt pattern, (ii) explicit data-selection rules (all acquired sequences were retained; no post-hoc exclusions), and (iii) paired t-tests and ANOVA results (p < 0.01) confirming statistically significant outperformance versus naive sequences and non-inferiority versus Zemlin on amorphous samples. These additions directly support the abstract statements. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained; no circular reductions to inputs or self-citations

full rationale

The paper derives aberration estimates from the physical linear tilt-to-image-shift mapping (standard in TEM optics), feeds the sequence into a Kalman filter whose process model is a random-walk drift assumption, and optimizes the tilt sequence offline via A-optimality on the predicted covariance. None of these steps reduce by construction to fitted parameters renamed as predictions, nor do they rest on self-citation chains or imported uniqueness theorems. The EM step for specimen noise is data-driven but not used to force the central result. Validation on real TEM data is external to the derivation, so the chain remains independent.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The approach depends on the standard linear tilt-shift model in TEM and KF assumptions for state and drift; noise parameters are estimated from data via EM rather than postulated as free parameters.

free parameters (1)

drift process noise covariance
Parameters governing how aberrations are modeled to drift over time inside the Kalman filter; fitted or tuned to match observed behavior.

axioms (2)

domain assumption Linear relationship between beam tilt angle and observed image shift
Invoked to map measured shifts directly to aberration coefficients; stated as the core measurement model.
domain assumption Aberration drift can be represented by a linear Gaussian process
Required for the Kalman filter state transition to track time-varying coefficients.

pith-pipeline@v0.9.0 · 5607 in / 1262 out tokens · 22309 ms · 2026-05-16T10:37:36.055350+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

c' = Ψ(t)c (Eq. 5); x_{k+1}=A x_k + ξ_k, y_k=C(t_k)x_k + ε_k (Eq. 7-8); min tr(W P_{N-1|N-1}) s.t. ||t_k||≤t_max_k (Eq. 15)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Kalman filter covariance evolution independent of measurements; receding-horizon gradient optimization

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

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Zemlin, K

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, K. H. Her- rmann, Coma-free alignment of high resolution electron 10 microscopes with the aid of optical diffractograms, Ultra- microscopy 3 (1978) 49–60

work page 1978
[4]

Koster, A

A. Koster, A. Van den Bos, K. van der Mast, An autofocus method for a TEM, Ultramicroscopy 21 (1987) 209–222

work page 1987
[5]

Steinecker, W

A. Steinecker, W. Mader, Measurement of lens aberra- tions by means of image displacements in beam-tilt series, Ultramicroscopy 81 (2000) 149–161

work page 2000
[6]

R. M. Glaeser, D. Typke, P. C. Tiemeijer, J. Pulokas, A. Cheng, Precise beam-tilt alignment and collimation are required to minimize the phase error associated with coma in high-resolution cryo-EM, Journal of Structural Biology 174 (2011) 1–10

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van Horssen, B

E. van Horssen, B. Janssen, A. Kumar, D. Antunes, W. Heemels, Image-based feedback control for drift com- pensation in an electron microscope, IFAC Journal of Systems and Control 11 (2020) 100074

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K. You, L. Xie, Kalman filtering with scheduled measure- ments, IEEE Transactions on Signal Processing 61 (2013) 1520–1530

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C. Li, N. Elia, Stochastic sensor scheduling via distributed convex optimization, Automatica 58 (2015) 173–182. arXiv:1403.3466

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Z. Sunberg, S. Chakravorty, R. S. Erwin, Information Space Receding Horizon Control for Multisensor Tasking Problems, IEEE Transactions on Cybernetics 46 (2016) 1325–1336

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Barthel, Ultra-Precise Measurement of Optical Aber- rations for Sub-Ångström Transmission Electron Mi- croscopy, Phd thesis, Technischen Hochschule Aachen, 2003

J. Barthel, Ultra-Precise Measurement of Optical Aber- rations for Sub-Ångström Transmission Electron Mi- croscopy, Phd thesis, Technischen Hochschule Aachen, 2003

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Bolme, J

D. Bolme, J. R. Beveridge, B. A. Draper, Y . M. Lui, Visual object tracking using adaptive correlation filters, in: 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, IEEE, 2010, pp. 2544–2550

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R. E. Kalman, A New Approach to Linear Filtering and Prediction Problems, Journal of Basic Engineering 82 (1960) 35–45

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C. Rao, J. Rawlings, D. Mayne, Constrained state esti- mation for nonlinear discrete-time systems: stability and moving horizon approximations, IEEE Transactions on Automatic Control 48 (2003) 246–258

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W. H. Kwon, P. S. Kim, S. H. Han, A receding horizon unbiased FIR filter for discrete-time state space models, Automatica 38 (2002) 545–551

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D. Mayne, J. Rawlings, C. Rao, P. Scokaert, Constrained model predictive control: Stability and optimality, Auto- matica 36 (2000) 789–814

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S. Gibson, B. Ninness, Robust maximum-likelihood esti- mation of multivariable dynamic systems, Automatica 41 (2005) 1667–1682

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M. Unser, B. L. Trus, A. C. Steven, A new resolution criterion based on spectral signal-to-noise ratios, Ultrami- croscopy 23 (1987) 39–51. 11

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A. C. Walls, Y .-j. Park, M. A. Tortorici, A. Wall, A. T. McGuire, D. Veesler, Structure, Function, and Antigenicity of the SARS-CoV-2 Spike Glycoprotein, Cell 181 (2020) 281–292.e6

work page 2020

[2] [2]

K. J. Hanszen, L. Trepte, Die Kontrastübertragung im Elektronenmikroskop bei partiell kohärenter Beleuchtung, 1971

work page 1971

[3] [3]

Zemlin, K

F. Zemlin, K. Weiss, P. Schiske, W. Kunath, K. H. Her- rmann, Coma-free alignment of high resolution electron 10 microscopes with the aid of optical diffractograms, Ultra- microscopy 3 (1978) 49–60

work page 1978

[4] [4]

Koster, A

A. Koster, A. Van den Bos, K. van der Mast, An autofocus method for a TEM, Ultramicroscopy 21 (1987) 209–222

work page 1987

[5] [5]

Steinecker, W

A. Steinecker, W. Mader, Measurement of lens aberra- tions by means of image displacements in beam-tilt series, Ultramicroscopy 81 (2000) 149–161

work page 2000

[6] [6]

R. M. Glaeser, D. Typke, P. C. Tiemeijer, J. Pulokas, A. Cheng, Precise beam-tilt alignment and collimation are required to minimize the phase error associated with coma in high-resolution cryo-EM, Journal of Structural Biology 174 (2011) 1–10

work page 2011

[7] [7]

van Horssen, B

E. van Horssen, B. Janssen, A. Kumar, D. Antunes, W. Heemels, Image-based feedback control for drift com- pensation in an electron microscope, IFAC Journal of Systems and Control 11 (2020) 100074

work page 2020

[8] [8]

Pukelsheim, Optimal Design of Experiments, Society for Industrial and Applied Mathematics, 2006

F. Pukelsheim, Optimal Design of Experiments, Society for Industrial and Applied Mathematics, 2006

work page 2006

[9] [9]

A. C. Atkinson, A. N. Donev, R. D. Tobias, Optimum Experimental Designs, with SAS, May 2014, Oxford Uni- versity PressOxford, 2007

work page 2014

[10] [10]

Zhang, R

H. Zhang, R. Ayoub, S. Sundaram, Sensor selection for Kalman filtering of linear dynamical systems: Complexity, limitations and greedy algorithms, Automatica 78 (2017) 202–210

work page 2017

[11] [11]

K. You, L. Xie, Kalman filtering with scheduled measure- ments, IEEE Transactions on Signal Processing 61 (2013) 1520–1530

work page 2013

[12] [12]

C. Li, N. Elia, Stochastic sensor scheduling via distributed convex optimization, Automatica 58 (2015) 173–182. arXiv:1403.3466

work page internal anchor Pith review Pith/arXiv arXiv 2015

[13] [13]

Sunberg, S

Z. Sunberg, S. Chakravorty, R. S. Erwin, Information Space Receding Horizon Control for Multisensor Tasking Problems, IEEE Transactions on Cybernetics 46 (2016) 1325–1336

work page 2016

[14] [14]

Barthel, Ultra-Precise Measurement of Optical Aber- rations for Sub-Ångström Transmission Electron Mi- croscopy, Phd thesis, Technischen Hochschule Aachen, 2003

J. Barthel, Ultra-Precise Measurement of Optical Aber- rations for Sub-Ångström Transmission Electron Mi- croscopy, Phd thesis, Technischen Hochschule Aachen, 2003

work page 2003

[15] [15]

E. J. Kirkland, Advanced Computing in Electron Mi- croscopy, 2 ed., Springer US, Boston, MA, 2010

work page 2010

[16] [16]

Bolme, J

D. Bolme, J. R. Beveridge, B. A. Draper, Y . M. Lui, Visual object tracking using adaptive correlation filters, in: 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, IEEE, 2010, pp. 2544–2550

work page 2010

[17] [17]

R. E. Kalman, A New Approach to Linear Filtering and Prediction Problems, Journal of Basic Engineering 82 (1960) 35–45

work page 1960

[18] [18]

Simon, Optimal State Estimation, Wiley, 2006

D. Simon, Optimal State Estimation, Wiley, 2006

work page 2006

[19] [19]

C. Rao, J. Rawlings, D. Mayne, Constrained state esti- mation for nonlinear discrete-time systems: stability and moving horizon approximations, IEEE Transactions on Automatic Control 48 (2003) 246–258

work page 2003

[20] [20]

W. H. Kwon, P. S. Kim, S. H. Han, A receding horizon unbiased FIR filter for discrete-time state space models, Automatica 38 (2002) 545–551

work page 2002

[21] [21]

Mayne, J

D. Mayne, J. Rawlings, C. Rao, P. Scokaert, Constrained model predictive control: Stability and optimality, Auto- matica 36 (2000) 789–814

work page 2000

[22] [22]

Gibson, B

S. Gibson, B. Ninness, Robust maximum-likelihood esti- mation of multivariable dynamic systems, Automatica 41 (2005) 1667–1682

work page 2005

[23] [23]

Unser, B

M. Unser, B. L. Trus, A. C. Steven, A new resolution criterion based on spectral signal-to-noise ratios, Ultrami- croscopy 23 (1987) 39–51. 11

work page 1987