A multiscale Bayesian approach to quantification and denoising of energy-dispersive x-ray data

Abderrahim Halimi; C\'ecile H\'ebert; C\'eline Lichtensteiger; Duncan T. L. Alexander; Ludovica Tovaglieri; Pau Torruella

REVIEW 3 major objections 5 minor 47 references

A multiscale Bayesian approach to quantification and denoising of energy-dispersive x-ray data

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A Poisson-aware multiscale Bayesian estimator recovers accurate, denoised elemental maps from sparse STEM-EDX spectra, resolving ferroelectric domains at unit-cell scale in PbTiO3.

desk verdict A well-executed and reproducible method paper for sparse STEM-EDX quantification; the main caveat is that quantitative accuracy is only demonstrated on simulations that use the same physics library the method is built on, though the experimental section stays on the spatial-resolution side. read the letter →

arxiv 2502.07473 v1 pith:2JIBQGAR submitted 2025-02-11 cond-mat.mtrl-sci cond-mat.mes-hallphysics.data-an

Pau Torruella , Abderrahim Halimi , Ludovica Tovaglieri , Céline Lichtensteiger , Duncan T. L. Alexander , Cécile Hébert This is my paper · ORCID

classification cond-mat.mtrl-scicond-mat.mes-hallphysics.data-an

keywords BayesianestimationEDXspectrumimagingSTEMchemicalquantificationdenoisingPoissonstatisticsmultiscaleferroelectricdomains

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Energy-dispersive X-ray (EDX) spectrum imaging in a scanning transmission electron microscope is a powerful but photon-starved technique: each pixel often records only a few counts, and standard quantification then produces noisy, unreliable maps. The authors introduce a robust multiscale Bayesian (RMB) estimator that models the detected counts as Poisson, uses simulated elemental and Bremsstrahlung spectra as endmembers, and borrows statistical strength from neighbouring pixels at several spatial scales. They claim that on simulated data this yields more accurate chemical maps at higher spatial resolution than standard line-integration and Cliff-Lorimer quantification, while also returning per-pixel uncertainty. On an experimental atomic-resolution PbTiO3 dataset, the RMB maps resolve ferroelectric domains with unit-cell resolution, something the standard analysis cannot do at the same acquisition time. The broader payoff is a general, training-free pipeline that extracts near-limit chemical information from beam-sensitive or short-acquisition EDX measurements.

What carries the argument

The central object is a hierarchical Bayesian model with a Poisson-Gamma observation model: after a Jensen-inequality approximation, the per-pixel likelihood becomes a product of Gamma distributions over abundances, with rate terms $S_k$ from simulated endmembers. Spatial correlation enters through a latent variable $m_{n,k}$ with a Gaussian Markov-random-field prior whose weights are exponential functions of abundance differences at multiple scales; inverse-Gamma priors on the variances $\psi_{n,k}$ yield uncertainty estimates. Point estimates come from maximum a posteriori optimisation by coordinate descent, with analytical updates for the abundance maps $r_{n,k}^{(\ell)}$, the latent map $m_{n,k}$, and the variance $\psi_{n,k}$. Initial abundances are obtained from Sunsal unmixing, and the endmembers themselves come from the ESPM library, which accounts for acceleration voltage, take-off angle, and detector efficiency.

What would settle it

Acquire an EDX spectrum image from a wedge-shaped standard specimen of known composition whose thickness varies from about 10 nm to 100 nm, run the RMB algorithm, and check whether the recovered atomic percentages drift systematically with thickness or deviate from known values in thicker regions; any such drift would show that the simulated endmembers do not capture absorption effects.

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Extended reading notes

Core claim

The central claim is that a fully Bayesian treatment of EDX spectrum images, in which each pixel's spectrum is generated by a Poisson mixture of simulated atomic-emission and Bremsstrahlung endmembers and abundances are spatially regularised through a latent Gaussian field evaluated at multiple resolutions, outperforms standard quantification. Specifically, the paper argues that the recovered elemental maps are more accurate and preserve higher spatial resolution than conventional kernel-smoothed line-integration/Cliff-Lorimer maps at equal signal levels. The authors demonstrate this on simulated PbTiO3/SrRuO3 spectrum images with 5, 50, and 500 counts per pixel, and then on a 28 nm PbTiO3 film where the RMB maps of Pb and Ti show sharper atomic columns and additional Fourier spots, allowing atomic columns to be located like HAADF and unit-cell c/a ratios to be measured across a ferroelectric domain boundary.

Load-bearing premise

The whole pipeline depends on the simulated endmember spectra correctly matching the real X-ray emission and background of the specimen, and the paper itself limits this to thin samples where absorption is negligible while noting that electron channelling makes atomic-level concentrations not directly meaningful.

Editorial extensions

If this is right

RMB quantification stays accurate down to roughly 5 X-ray counts per pixel spectrum, a regime where matrix-decomposition methods cannot operate.
For a fixed kernel size, RMB maps show higher signal-to-noise ratio and less feature broadening than standard kernel-smoothed Cliff-Lorimer maps on simulated data.
Per-pixel uncertainty estimates come directly from the Bayesian posterior, without propagating k-factor, background, and counting errors separately.
Experimental PbTiO3 maps from only 350 s of acquisition locate all atomic columns correctly relative to HAADF, enabling unit-cell c/a ratio maps that match HAADF-derived values.
Because the posterior can be updated with new data, the algorithm is compatible with on-the-fly abundance estimation during live acquisition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension: the same Poisson-multiscale-Bayesian machinery could be reused for other counting detectors, such as EELS, time-of-flight, or X-ray ptychography, wherever known spectral responses and spatial smoothness hold.
Extension: the uncertainty maps suggest an adaptive version of the algorithm that chooses per-pixel kernel sizes by minimising predicted variance, since large kernels degrade interface localisation.
Extension: if electron-channelling and absorption were incorporated into the endmembers, the method could go beyond thin specimens and make atomic-resolution concentrations physically meaningful, a step the paper does not take.
Extension: a thickness series on a known specimen would directly test the method's boundary, because simulated endmembers ignore absorption.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Pau, this is a solid methods paper and a fair one. The core idea—marrying the multiscale Bayesian Poisson reconstruction that Halimi and others developed for single-photon LIDAR with ESPM-simulated EDX endmembers—is genuinely new and well implemented. The paper does what a good methods paper should: it gives a clear statistical model, a coordinate-descent algorithm with analytical updates, code on GitHub, and benchmarks against synthetic data with known ground truth at 5, 50, and 500 counts per pixel. The comparison to standard Cliff-Lorimer quantification is fair, since both methods get the same spatial smoothing. The experimental PbTiO3 demo uses HAADF images as an independent reference for atomic column positions and for the c/a ratio maps, which is the right way to ground the spatial-resolution claim.

The stress-test worry about the ESPM self-consistency loop is real but not as damaging as it sounds. The simulated data are generated with ESPM and then inverted with ESPM endmembers, so the quantitative accuracy test is internal-consistency. That is standard practice in this community, but it does mean the claim of 'more accurate chemical maps' shouldn't be read as externally validated absolute concentrations—the paper's own experimental section wisely says atomic-resolution EDX concentrations are not always meaningful due to channelling. The abstract and conclusions lean a bit harder on 'more accurate' than the evidence supports; the experimental evidence is about spatial resolution and structural features, not unbiased abundances. That's a mild overclaim, not a fatal one.

The more specific technical soft spot is the surrogate likelihood in Eq. (2), which fixes r_bar using Sunsal on the same dataset. The data therefore enter twice, and the reported per-pixel uncertainties don't include the error in those initial estimates. I'd like to see a sensitivity analysis or a simulation-based calibration check for the uncertainty maps. The paper also lacks repeated-simulation error bars on the SNR/broadening curves in Fig. 4; each point appears to come from a single synthetic dataset. That's easy to fix and would make the comparison much more convincing.

Overall, this deserves a serious referee. The method is reproducible, the evaluation is thoughtful, and the experimental demonstration is honest about its limitations. I'd recommend conditional acceptance: ask for the Sunsal sensitivity check, error bars on the benchmark plots, and a slightly softer phrasing of the accuracy claim in the abstract.

Referee Report

3 major / 5 minor

Summary. The paper introduces a robust multiscale Bayesian (RMB) method for quantitative analysis and denoising of energy-dispersive X-ray (EDX) spectrum images. The method models X-ray counts with a Poisson likelihood, approximates it via a gamma surrogate that uses initial abundance estimates from the SUNSAL unmixing algorithm, and combines this with a multiscale spatial-smoothness prior and an inverse-gamma variance prior. Inference is performed by coordinate-descent MAP estimation. The method is tested on simulated 90x90 spectrum images of PbTiO3/SrRuO3 layers at 5, 50, and 500 counts per pixel, where it is compared with standard Cliff-Lorimer quantification after kernel smoothing, and on an experimental atomic-resolution PbTiO3 spectrum image. In simulations, RMB yields narrower composition histograms, better SNR versus feature broadening, and fewer Otsu-threshold misclassifications. In the experimental data, RMB produces sharper elemental maps, shows additional FFT spots, improves atomic-column localization relative to HAADF positions, and allows unit-cell c/a ratio maps to be extracted.

Significance. If the claims hold, the RMB approach is a valuable contribution to sparse STEM-EDX analysis: it provides denoised quantitative maps with per-pixel uncertainty, is applicable at extremely low counts where matrix-decomposition methods struggle, and is backed by publicly available code on GitHub. The use of physics-based ESPM endmembers and the multiscale spatial prior is a sensible combination, and the independent HAADF-based column-location comparison provides a strong spatial-resolution benchmark for the experimental data. However, the absolute accuracy claim rests on simulations generated with the same ESPM forward model used for quantification, which tests inversion consistency rather than physical fidelity, and the surrogate likelihood fixes initial abundances from the same data, so the reported uncertainties do not account for that estimation step. These limitations need to be addressed or clearly bounded before the accuracy claims can be fully accepted.

major comments (3)

[Methods, Algorithm description; SI, Derivation of eq. (2)] Equation (2) replaces the Poisson likelihood with a gamma surrogate whose shape parameter is 1 + r̅_{n,k}, where r̅ is the SUNSAL estimate obtained from the same spectrum image. The data therefore enter twice: once to set the effective photon count in the likelihood and once in the observed counts themselves. This is load-bearing because the reported posterior variances ψ_{n,k} are computed conditional on this fixed r̅ and do not propagate the uncertainty in the SUNSAL initialization or the approximation error of Jensen's inequality. I recommend calibrating the uncertainty output on the simulated data: check whether the 68% or 95% credible intervals derived from ψ contain the known ground-truth abundances at the expected rates, and test sensitivity of the maps and uncertainties to perturbed or alternative initial abundance estimates. Without such a check, the uncertainty maps should be described as conditional rather than fully calibrated posterior uncertainties.
[Data simulation; Results, Simulated Data; Results, Experimental Data] The central accuracy claim—that the chemical maps are 'more accurate' than standard methods—is supported on simulated data by inverting the same ESPM forward model used to generate the spectra. This is an internal-consistency test: it demonstrates that the algorithm can recover known abundances from its own forward model, but it does not validate the physical fidelity of the ESPM cross-sections, Bremsstrahlung background, or detector response for the experimental geometry. The experimental section itself states that atomic-resolution EDX concentrations are 'not always directly meaningful, owing to electron channelling', and the simulated sample is spatially uniform rather than atomic-resolution. I recommend either providing an external accuracy anchor (for example, comparison with EELS quantification, a stoichiometric standard, or a thickness series where absorption can be estimated) or explicitly limiting the accuracy claim to 'precision and spatial resolution' while stating that absolute concentrations inherit the validity of the ESPM model.
[Methods, Algorithm description; Experimental data acquisition] The Methods state that the ESPM-based endmember approach is valid 'for relatively thin samples, where absorption does not represent a relevant contribution', but the experimental PbTiO3 film is 28 nm thick and is imaged at atomic resolution, where electron channelling is acknowledged to complicate quantification. The magnitude of X-ray absorption and channelling effects at this thickness and for the Pb L and Ti K lines is not estimated or discussed. This is load-bearing for the quantitative elemental maps, although the unit-cell c/a demonstration is less affected because it relies on relative column positions. Please quantify or justify the thin-sample approximation for the specific experimental conditions, or soften the quantitative claims accordingly.

minor comments (5)

[Methods, Algorithm description] The SUNSAL algorithm (Ref. 24) is typeset inconsistently as 'Sunsal' in the main text and supplementary information; use 'SUNSAL' consistently.
[Methods, Algorithm description, Eq. (5)] The inverse-gamma hyperparameters α_l and β_l are said to be 'set to zero', which would make the prior improper; please clarify whether this is intended as a limiting non-informative prior and how the implementation handles it.
[Methods, Algorithm description, Eq. (3) and Results, Simulated Data] The notation for multiscale kernel sizes is ambiguous: the text refers to '2l-1 by 2l-1' windows and later to downsampling sets {1, 3, 5, …, 2l+1}; please define the relationship between the level index l and the kernel size unambiguously.
[Results, Simulated Data, Fig. 4] The quantity labeled 'SNR ≡ σ/I' is actually a noise-to-signal ratio (lower is better); the label and the discussion should be inverted or renamed to avoid confusion.
[Results, Experimental Data] In the paragraph describing Figure 6, the sentence 'all the atomic columns are correctly identified in the RMB map correctly' contains a duplicated adverb; please revise.

Circularity Check

2 steps flagged · score 4.0 of 10

The Bayesian estimator's likelihood is built from a Sunsal estimate of the same data, and the simulated accuracy benchmark is generated and quantified with the same ESPM model, making the quantitative-accuracy claim partly self-referential.

self definitional [Methods, Eq. (2) and surrounding text; supplementary Algorithm 1, step 5.]
"𝑃(𝑦𝑛|𝑟𝑛) ∝ ∏ [𝒢(𝑟𝑛,𝑘; 1 + 𝑟ത𝑛,𝑘, 𝑆𝑘)𝑄ത(𝑦𝑛,𝑡)] ... 𝑟ത𝑛,𝑘 denotes an initial estimate of the abundance at the nth pixel location for the kth element, for which we used the Sunsal unmixing algorithm."

Eq. (2) is presented as the observation likelihood for the abundances r, but its Gamma shape parameter is 1 + r̄, where r̄ is obtained by running Sunsal on the same spectrum image Y that this likelihood is meant to explain. The conditional mode of the Gamma factor for r is proportional to r̄, so the MAP update in Eq. (9) returns a spatially smoothed version of the Sunsal initialization rather than an estimate from an independent generative model. Because r̄ is held fixed in the posterior and variance updates, the reported uncertainties omit the error of this initial unmixing. The data thus enter the likelihood twice, making the posterior a self-referential construction.
other [Methods ('Algorithm description' and 'Data simulation'); Results ('Simulated Data').]
"For relatively thin samples, where absorption does not represent a relevant contribution, the X-ray emission probability of each element in the periodic table is simulated through the ESPM library. ... To test our algorithm, a 90 x 90-pixel spectrum image ... was simulated using the ESPM package."

The simulated datasets used to support the claim of 'more accurate' chemical maps are generated with the same ESPM forward model that supplies the endmembers for quantification. The benchmark therefore verifies only that the estimator can invert the ESPM generator; it does not test the physical fidelity of the ESPM emission probabilities and Bremsstrahlung background for the experimental geometry. The experimental section further states that atomic-resolution EDX concentrations are 'not always directly meaningful, owing to electron channelling', so no independent quantitative anchor is provided. The central quantitative-accuracy claim thus rests on a closed self-consistency loop.

full rationale

The paper's spatial-resolution claim has genuinely independent content: the experimental Pb map is benchmarked against co-acquired HAADF atomic positions, and the c/a ratio maps derived from the RMB EDX map agree with those from HAADF. There is no target-result fitting, and the algorithm is not a mere renaming of a known result. However, two self-referential elements compromise the quantitative-accuracy and uncertainty claims. First, the surrogate likelihood in Eq. (2) uses Sunsal estimates computed from the same spectrum image as fixed shape parameters, so the data are used twice and the posterior mode is effectively a smoothed version of the initialization; the reported variances condition on that initialization and exclude its error. Second, the simulated-data validation is an internal-consistency test: ESPM both generates the synthetic spectrum images and provides the endmembers used for estimation, and the experimental data cannot supply an external quantitative check because the paper explicitly acknowledges channelling effects. These steps make the quantitative-accuracy claim partially circular, while leaving the spatial-resolution demonstration substantially independent. Accordingly, the overall circularity score is 4.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The method rests on a standard Poisson observation model, ESPM endmember fidelity, a smoothness prior, and a variational approximation with data-derived initial abundances. No new physical entities are introduced. The main costs are the user-set kernel/eta parameters and the data reuse in r_bar.

free parameters (4)

Scale-dependent correlation constant eta^(l) = not specified
Defines the width of the exponential weights in Eq (4); controls spatial smoothing at each scale and must be chosen by the user. The paper does not state how it was set for the reported results.
Multiscale downsampling sets / kernel sizes = {1,3,5} for experimental, {3} for simulated, other values in Fig 4
The number of scales L and the kernel sizes are selected by hand; performance depends on this choice.
Initial abundance estimates r_bar from Sunsal = data-dependent (per-pixel)
Used as the fixed shape parameter in the approximate likelihood (2); the same data are used to produce r_bar and then to update R, so the likelihood is not fully independent of the data.
Inverse-gamma hyperparameters alpha_l, beta_l = 0 (default)
Set to zero for non-informative priors; not fitted, but the model allows them to be changed.

assumptions (5)

domain assumption Detected X-ray counts follow independent Poisson distributions (Eq 1)
Standard photon-counting model for EDX; the entire likelihood is built on it.
domain assumption ESPM-simulated endmembers, including Bremsstrahlung background, accurately represent the true spectral response under the experimental conditions
The method quantifies by fitting these endmembers; absorption is neglected (Methods: 'For relatively thin samples...'), and electron channelling is acknowledged to affect atomic-resolution EDX concentrations.
domain assumption Abundances vary smoothly in space, encoded by the latent M with Gaussian prior (Eq 4)
This spatial smoothness prior is what enables denoising; it can bias results at sharp interfaces and atomic-scale features, as acknowledged in the uncertainty discussion.
ad hoc to paper The Jensen-based surrogate likelihood (Eq 2), with r_bar fixed to Sunsal estimates, is an adequate approximation for MAP estimation
The true likelihood is replaced by a lower-bound form; the paper shows this maximizes the bound but does not provide a global optimality or coverage guarantee.
domain assumption Independence between downsampled spectra is assumed when forming the multiscale likelihoods (Eq 3)
Used to build L separate likelihoods; ignores correlations introduced by overlapping low-pass filters.

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Cite this review

Pith. "Pith review of A multiscale Bayesian approach to quantification and denoising of energy-dispersive x-ray data." pith.science (2026). https://pith.science/paper/2JIBQGAR

@misc{pith2026250207473,
  author       = {Pith},
  title        = {Pith review of: A multiscale Bayesian approach to quantification and denoising of energy-dispersive x-ray data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2JIBQGAR}},
  note         = {Machine review of arXiv:2502.07473}
}

read the original abstract

Energy dispersive X-ray (EDX) spectrum imaging yields compositional information with a spatial resolution down to the atomic level. However, experimental limitations often produce extremely sparse and noisy EDX spectra. Under such conditions, every detected X-ray must be leveraged to obtain the maximum possible amount of information about the sample. To this end, we introduce a robust multiscale Bayesian approach that accounts for the Poisson statistics in the EDX data and leverages their underlying spatial correlations. This is combined with EDX spectral simulation (elemental contributions and Bremsstrahlung background) into a Bayesian estimation strategy. When tested using simulated datasets, the chemical maps obtained with this approach are more accurate and preserve a higher spatial resolution than those obtained by standard methods. These properties translate to experimental datasets, where the method enhances the atomic resolution chemical maps of a canonical tetragonal ferroelectric PbTiO3 sample, such that ferroelectric domains are mapped with unit-cell resolution.

Figures

Figures reproduced from arXiv: 2502.07473 by the authors.

**Figure 2.** Quantification accuracy. In the first column, Sr at.% maps calculated through the standard quantification method of spectral line integration followed by CL quantification are shown. The second column of panels shows the Sr at.% map calculated through RMB. The third column shows the pixel histogram of the obtained Sr maps. Columns four to six show the equivalent results for Ti. Each row corresponds to the results fo… view at source ↗

**Figure 3.** shows the results of these comparisons for kernel sizes 3, 7 and 11, this time comparing Pb maps as opposed to Sr and Ti for figure 2. The results for all elements can be found in the supplementary information, as are the full results for kernel sizes 5 and 9. Since our simulated map contains only two distinct, uniform phases, it is reasonable to assume that the classic Otsu thresholding algorithm36,37 applied to th… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: SNR vs resolution evaluation. The measured SNR plotted against feature broadening ratio when applying kernels of diƯerent sizes (K) on the 5 X-ray counts/pixel SI, for simple kernel averaging and for the RMB algorithm. Experimental Data In this section, we analyse an a…

**Figure 5.** Figure 5: Experimental EDX SI analysis. The Pb (red) and Ti (green) maps have been obtained with standard approach (using Velox software) and the RMB approach, for diƯerent eƯective acquisition times. The scale bar corresponds to 1 nm. The FFT of maps at 70 s and 350 s where cal…

**Figure 6.** Figure 6: Atomic column location measurements. HAADF image (a) and Pb maps obtained through standard (b) and RMB quantifications (c). The atomic column positions obtained with atom map are plotted with red dots in each image. d) Mean error in the atomic column position relative …

**Figure 7.** Figure 7: Ferroelectric domain boundary measurement. [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

Discussion (0). Continue with ORCID to comment.

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