Methods for energy dispersive x-ray spectroscopy with photon-counting and deconvolution techniques

Alessandro Forte; Oliver S. Humphries; Sam M. Vinko; Thomas Campbell; Thomas Gawne; Yuanfeng Shi

REVIEW 3 major objections 5 minor 80 references

Methods for energy dispersive x-ray spectroscopy with photon-counting and deconvolution techniques

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper shows that sequentially combining photon-counting cluster identification with Richardson–Lucy deconvolution reconstructs Bragg x-ray spectra more accurately than simple thresholding across all tested photon densities…

desk verdict A genuinely useful methods paper whose hybrid pipeline works in synthetic tests, but the formal Poisson justification in Appendix B is mathematically wrong and needs fixing before the paper is accepted. read the letter →

arxiv 2411.16581 v2 pith:6RHO4PI3 submitted 2024-11-25 physics.app-ph physics.ins-det

Alessandro Forte , Thomas Gawne , Oliver S. Humphries , Thomas Campbell , Yuanfeng Shi , Sam M. Vinko This is my paper · ORCID

classification physics.app-phphysics.ins-det

keywords energy-dispersivex-rayspectroscopyBraggphotoncountingRichardson-Lucydeconvolutionchargespreadingsignal-to-noiseratiovonHámosspectrometerflatcrystal

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

This paper aims to give a complete, step-by-step procedure for turning raw pixel-detector images into calibrated x-ray spectra in Bragg geometry, and to introduce a processing chain that works across the whole detector even when photon density changes sharply from place to place. The chain first identifies and re-accumulates single-photon charge clusters, then applies Richardson–Lucy deconvolution to the crowded regions that remain; the authors call this combined mode 'hybrid'. Their central claim is that this hybrid mode reconstructs spectra with higher SNR and clearer features than simple thresholding at every tested filling fraction and noise level, with the smallest gains at intermediate values where neither sub-method dominates. The practical payoff is that weak spectral features, which often carry the physics in high-energy-density experiments, can be extracted from single-shot images that would otherwise be discarded as too noisy.

What carries the argument

The load-bearing object is the effective charge-spreading function H̃, a translation-invariant kernel that describes how one photon's charge is distributed across neighboring pixels. Two algorithms consume that kernel: a clustering routine that recognizes the 13 possible single-photon cluster shapes and accumulates their charge onto the central pixel, and the Richardson–Lucy iteration, which deconvolves the ADU map under the assumption that the blurred image is Poisson. The hybrid mode applies the clustering first, since it makes single photons brighter and leaves agglomerated clusters unchanged, then hands the image to Richardson–Lucy, which decomposes those agglomerated clusters into a photon-hit map.

What would settle it

Measure the variance-to-mean ratio of ADU values in a flat-field camera image whose true photon intensity is known and whose charge-spreading kernel has fractional weights; the paper's Poisson model predicts a ratio of 1, while the true fractional-weight mixture predicts overdispersion (ratio > 1). A ratio systematically above 1 would show that the RL deconvolution's input-model assumption is violated in exactly the regime the method targets.

Watch

Extended reading notes

Core claim

The central discovery is that the two image-processing philosophies—counting isolated single-photon clusters and deconvolving crowded regions—are complementary, and that applying them in sequence on the same camera image outperforms either alone or simple thresholding. For sparse regions (filling fraction ⲅ0.1), the clustering algorithm exploits the correlation of the point-spread function to collect a photon's charge onto its brightest pixel, raising it above the second threshold; for dense regions (filling fraction ≳1), the Richardson–Lucy algorithm iteratively finds the most likely underlying photon distribution given a known charge-spreading kernel. On synthetic images and on an experimental MgF2 spectrum, the hybrid reconstruction reduces the L2 distance to the true spectrum across all tested λ and noise levels, and it visibly sharpens the Kα and Kβ line structure.

Load-bearing premise

The derivation in Appendix B assumes that the pixel ADU value, after convolution with a fractional charge-spreading kernel, is still a Poisson random variable; this only holds for integer weights, so the formal justification for applying Richardson–Lucy deconvolution to the ADU image is not valid, and the method's success rests on the empirical behavior observed in the synthetic tests.

Editorial extensions

If this is right

If the hybrid mode is correct, weak features such as Raman-scattering peaks or satellite lines in single-shot HED experiments become measurable instead of being lost under thresholding noise.
The method gives a standardized, reproducible recipe for constructing spectra from flat-crystal and von Hámos spectrometers, including energy calibration and solid-angle correction.
Since the hybrid pipeline works on single-shot images, it enables event-resolved analyses (e.g., correlating spectral features with shot-to-shot source changes) that averaging destroys.
The L2-distance benchmark provides a quantitative way to choose processing parameters (thresholds, kernel width) without hand-tuning.
Extension to other pixel detectors (e.g., at XFELs) is direct if the point-spread function is known or measured.

Reading between the lines

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

A rigorous reformulation of the deconvolution step—replacing the Poisson assumption in Appendix B with a proper model of fractional charge sharing—could turn the empirically successful hybrid pipeline into a formally justified estimator, possibly improving the deconvolution further.
Because the method reconstructs single-shot images, it could be combined with sub-pixel centroiding on the identified single-photon clusters, giving simultaneous SNR gain and super-resolution, which the authors note is beyond their current scope.
Applying the same hybrid logic to detectors with larger point-spread functions, such as thicker sensors or higher-energy photons, is a direct testable extension, provided the kernel is measured rather than assumed.
The L2 benchmark on synthetic data could be reused as a parameter-tuning objective on experimental data, e.g., to choose integration widths along the non-dispersive axis.

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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

The main thing you should know: this is a solid, practically useful methods paper. The new contribution is the sequential combination of photon-counting clustering and Richardson-Lucy deconvolution, applied to different regions of the same Bragg spectrometer image depending on local photon density. That hybrid treatment is not, as far as I can tell, in the existing literature, and it is exactly the kind of thing people in HED spectroscopy will want to try. The paper also does a genuinely good job of consolidating the whole spectrum-construction pipeline, from energy calibration for flat-crystal and von Hámos geometries through solid-angle corrections. That guide by itself has value. The synthetic validation is careful and consistent. Across the sweep of fill fraction, noise, and charge-spreading radius, the hybrid method consistently beats simple thresholding. The improvement is modest, as the reader said, but it is real in the simulation. The experimental example is less quantitative, but the feature enhancement is visible and plausible. Now the soft spots. The one that really matters is Appendix B. Equation (B3) assumes that a weighted sum of independent Poisson variables is Poisson, and that is only true for nonnegative integer weights. The charge-spreading kernel has fractional entries, so the ADU map is not Poisson under the paper's own forward model. This means the Richardson-Lucy likelihood, as written, does not match the data. The authors either need to correct the derivation, or state clearly that RL is being used as a heuristic deconvolution whose empirical performance is what justifies it. The synthetic tests do not repair the flaw, because they use the same fractional-convolution model to generate the data — they only prove the pipeline can invert its own forward model. The other weaknesses are more minor. The experimental validation has no ground-truth spectrum, so the improvement is asserted rather than measured. Error bars in Fig. 14 neglect processing uncertainties. No code or data are provided, which limits reproducibility. None of these individually sink the paper, but together they mean the current version should not appear without revision. Bottom line: read this if you work on Bragg x-ray spectroscopy, especially in HED science. The hybrid idea is worth taking seriously, and the guide is useful. As a referee, I would not desk-reject it — send it to review, but insist on fixing the Appendix B derivation and adding a clear statement about the heuristic status of RL.

Referee Report

3 major / 5 minor

Summary. The manuscript presents a processing pipeline for constructing x-ray spectra from pixel-detector images in Bragg spectroscopy, combining photon-counting cluster identification with Richardson-Lucy (RL) deconvolution. It gives energy-calibration procedures for flat-crystal and von Hámos spectrometers, derives solid-angle corrections, and validates the proposed 'hybrid' pipeline on synthetic images across filling fraction, charge-spreading radius, and noise level, reporting consistently lower L2 distances than simple thresholding. The method is then applied to an experimental MgF2 spectrum to demonstrate improved feature visibility.

Significance. If the statistical derivation were sound, this would be a useful methods paper: it collects in one place the steps of energy calibration, camera-parameter estimation, photon clustering, deconvolution, and solid-angle correction, and it provides a quantitative comparison of processing strategies. The synthetic study covers a sensible parameter space (filling fractions from 0.01 to 10, two spreading radii, several noise levels), and the reported improvement of the hybrid method over thresholding is plausible. The experimental demonstration on a real MgF2 spectrum is a strength of the paper. However, the formal justification for applying RL to the ADU image is based on an incorrect property of Poisson variables in Appendix B, and the synthetic validation is self-consistent rather than independent; as written, the claim that RL computes the 'most likely' photon-hit distribution is not established. These issues are load-bearing for the central claim, so the manuscript needs revision before it can be accepted.

major comments (3)

[Appendix B, Eq. (B3)] The derivation treats a weighted sum of independent Poisson variables as Poisson. This is valid only when each weight is a nonnegative integer, because the Poisson property is preserved by thinning/superposition with integer weights. The charge-spreading kernel H in Eq. (B1) has fractional entries from the Gaussian convolution, so the equality in the last line of Eq. (B3) is false. Consequently the ADU map Aij is not Poisson with mean (I * H)ij, and its variance generally exceeds its mean. Equation (18), the likelihood used by Richardson-Lucy, therefore does not describe the actual data distribution, and the statement that the RL output is the most likely estimate of Iij is unsupported. The authors should either derive a correct model (e.g., a compound-Poisson or thinned-Poisson formulation) or explicitly reframe RL as a heuristic deconvolution that works well in practice but is not a maximum-likelihood estimator under the stated model.
[Section IIIA and Section IIIB] The synthetic validation is self-consistent rather than independent: the images in Figs. 11-13 are generated from exactly the forward model assumed by the inversion (known spectrum, Gaussian charge-spreading kernel, Gaussian read noise), and the same kernel and noise parameters are then used for deconvolution. This demonstrates algorithmic consistency but cannot validate the Poisson assumption for real data. In the experimental section, the camera parameters ADUsp, sigma_N, and lambda are estimated by fitting a synthetic histogram to the same experimental image that is later used to produce Fig. 14, so the experimental comparison has no independent ground truth. The relative improvement over simple thresholding may still be valid, but the stronger claim that the methods provide the 'most likely' or statistically optimal reconstruction is not supported by the presented evidence.
[Section IID2, Eq. (18)-(19)] The RL algorithm is applied to A'_ij, which is the output of nonlinear preprocessing: thresholding at 1.5 sigma_N (Eq. 13), cluster accumulation onto the brightest pixel (Eq. 14), and a second threshold at ADUsp - sigma_N. This processed image is not a raw Poisson realization corrupted by additive Gaussian noise, so the model in Eq. (18) does not describe the actual input to RL. The paper notes in Appendix B that the discussion can be extended to include thresholding and charge accumulation, but no such extension is provided. The authors should either account for the nonlinear preprocessing in the statistical model or qualify the claim that RL computes the most likely photon-hit distribution.

minor comments (5)

[Figure 13] The horizontal axis tick labels appear to be rendered incorrectly as '10-2 10-1 100 1010' instead of powers of ten such as 10^{-2}, 10^{-1}, 10^0, 10^1; this should be fixed for readability.
[Eq. (26)] The interval notation (E_l - Delta E/2, E_l - Delta E/2) is missing a '+' sign; it should be (E_l - Delta E/2, E_l + Delta E/2).
[Appendix B, Eq. (B4)] The summation notation 'i'k' in (i'j')' and 'ik in (ij)' is unclear and likely contains typographical errors; it should be rewritten with explicit subpixel indices k,l and k',l'.
[Section IIIB] The statement that the PIXIS-XF camera parameters were 'provided in the relevant documentation' cites reference [60], which is a general review of x-ray detectors and does not appear to be the camera datasheet; a direct citation to the manufacturer documentation would be more appropriate.
[Section IID1] The sentence 'the shapes of single-photon clusters (i.e. the pixels in which the charged produced by the photon has leaked) are few' contains a grammatical error ('charged' should be 'charge') and is awkwardly phrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the hybrid-processing claim is supported by synthetic forward-model tests independent of the target spectrum, and the experimental demonstration is not a prediction of a fitted quantity.

full rationale

The paper's derivation chain is self-contained. The central quantitative claim that sequential clustering plus Richardson-Lucy deconvolution ('Hybrid') beats simple thresholding is evaluated on synthetic images generated by a stated forward model (known spectrum S(E_l), Poisson photon placement, Gaussian charge spreading with width R_s, Gaussian noise sigma_N), where the reconstruction target S(E_l) is never supplied to the algorithm. Using the same kernel in the RL step as in the generator is standard inverse-problem validation rather than circularity, and the comparison is fair because all methods share the same thresholds and parameters. The experimental demonstration in Sec. III B fits ADU_sp, sigma_N, and lambda to the experimental histogram of the same MgF2 image before processing; this is a parameter-tuning or overfitting concern, not an equation-level circularity, since those four scalars do not algebraically determine the reconstructed spectral shape. The self-citations (Refs. [19], [41], [44]) provide experimental data and instrument-function background, but the method's validity does not rest on a self-cited uniqueness theorem or an imported ansatz. The one substantive weakness is Appendix B Eq. (B3), which treats a fractional-weight sum of Poisson variables as Poisson; that is a false statistical step and undermines the formal justification of RL, but it is not circular, because it does not assume the conclusion it purports to prove. Therefore no step in the derivation reduces by construction to its inputs, and the circularity score is 0.

Assumptions & free parameters 10 free parameters · 8 assumptions · 0 invented entities

The central claim rests on several domain assumptions about the detector (constant charge-spreading radius, constant read noise, constant ADU conversion within regions), on standard Bragg geometry, and on a specific probabilistic model in Appendix B that contains an incorrect use of the Poisson reproductive property. The latter is the most fragile assumption; if it fails, the RL deconvolution step lacks theoretical grounding even though it may still work empirically.

free parameters (10)

ADUsp = 43 ADU (MgF2 experiment)
ADU counts per photon; estimated by fitting a synthetic histogram to the experimental histogram (Section IIIB), used to convert ADU to photon number via Eq. (20).
sigma_N = 4 ADU (MgF2 experiment)
Read noise standard deviation; estimated by the same histogram fit; used in thresholding Eq. (13).
lambda = not reported
Fill fraction; estimated by fitting the synthetic histogram to the experimental histogram; governs whether clustering or deconvolution dominates.
Rs = 1.7 pixel units (MgF2 experiment)
Charge spreading radius; taken from camera documentation and adjusted in the histogram fit; defines the Gaussian kernel for charge spreading and deconvolution.
lower threshold = 1.5 sigma_N
Chosen as a compromise to cut noise and retain signal (Eq. 13); could be optimized via Appendix A but is fixed in this work.
upper threshold = ADUsp - sigma_N
Second threshold after clustering to discard events that are unlikely to be single photons; chosen by hand, roughly 20% of photons discarded.
geometric parameters Lambda (alpha_x, alpha_y, alpha_z, D, theta_B) = (-1.9 deg, 55.8 deg, 0 deg, 79.5 mm, 34.7 deg)
Fitted to the Mg K-alpha emission line by minimizing the loss function Eq. (8); defines the energy map for the flat crystal spectrometer.
von Hamos parameters R and L = not reported
Optimized to match bright calibration lines for the von Hamos geometry (Section IIB2); values for the Fe spectrum are not given.
Richardson-Lucy iteration count = not reported
Number of iterations controls the deconvolution strength and reconstruction quality; not stated in the paper, so the exact reconstruction is not fully specified.
energy bin width Delta E = not reported
Chosen to balance quantization error and spectral smoothness; the actual binning used in Figs. 12-14 is not specified.

assumptions (8)

standard math Bragg's law and the geometric energy map formulas (Eq. 2 and Eq. 10)
Used to assign photon energies to detector positions; the flat crystal formula assumes a point source and a delta rocking curve.
domain assumption The charge spreading radius Rs is constant across the detector region of interest (Section IIA)
The authors state they restrict to a narrow photon energy range so Rs can be treated as constant; if Rs varies, the deconvolution kernel is wrong.
domain assumption The read noise sigma_N is constant across the camera (Section IIC)
Used to compute a single threshold for the whole image; per-pixel noise variation is ignored.
domain assumption ADUsp is constant within each camera region (Section IIC)
Needed to interpret the histogram peaks in Eq. (11) and to convert ADU to photon numbers.
ad hoc to paper The weighted sum of independent Poisson variables is Poisson (Appendix B, Eq. B3)
The derivation uses this property to justify applying Richardson-Lucy deconvolution to the ADU image; the property is false for fractional weights, so the derivation is formally invalid.
domain assumption The emission spectrum is isotropic over the detector solid angle (Section IIE)
Simplifies the solid angle correction in Eq. (26); the authors note the spectrum would otherwise be an average over detected directions.
domain assumption Error bars can be computed by treating N_gamma as direct Poisson counts, neglecting processing errors (Appendix C)
Underestimates uncertainties because thresholding, clustering, and deconvolution introduce additional correlated errors that are not propagated.
domain assumption Dark-image subtraction has removed the bias, so the expected ADU of an unhit pixel is zero (Section IIC)
Assumes facility-provided dark subtraction is accurate; residual bias would shift the histogram and photon counts.

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Cite this review

Pith. "Pith review of Methods for energy dispersive x-ray spectroscopy with photon-counting and deconvolution techniques." pith.science (2026). https://pith.science/paper/6RHO4PI3

@misc{pith2026241116581,
  author       = {Pith},
  title        = {Pith review of: Methods for energy dispersive x-ray spectroscopy with photon-counting and deconvolution techniques},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6RHO4PI3}},
  note         = {Machine review of arXiv:2411.16581}
}

read the original abstract

Spectroscopic techniques are essential for studying material properties, but the small cross-sections of some methods may result in low signal-to-noise ratios (SNRs) in the collected spectra. In this article we present methods, based on combining Bragg spectroscopy with photon counting and deconvolution algorithms, which increase the SNRs, making the spectra better suited to further analysis. We aim to provide a comprehensive guide for constructing spectra from camera images. The efficacy of these methods is validated on synthetic and experimental data, the latter coming from the field of high-energy density (HED) science, where x-ray spectroscopy is essential for the understanding of materials under extreme thermodynamic conditions.

Figures

Figures reproduced from arXiv: 2411.16581 by the authors.

**Figure 2.** Simplified scheme of a pixel detector section illus [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Flat crystal (FC) spectrometer geometry with an illustration of source broadening (inset). The laboratory frame is [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Detector image of MgF2 emission spectrum with the theoretically-computed energy contour (red) fitted to a Mg Kα line. tector, and the detector surface lies along this line (see [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Von Hámos spectrometer setup with an illustration of mosaic focusing (inset). The mosaicity of the crystal increases [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Detector image of Fe emission using a von Hámos [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: (a-d) Synthetic histograms of 700x700 images for [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: The 4 layouts for a single photon cluster. The [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: Illustration of the various steps in the camera image processing. (a) Map of photon hits on the camera for a small [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 10.** Figure 10: Scheme for the solid angles calculations in the flat [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: (a) The normalised spectrum used for producing the synthetic camera images.(b) The charge deposited on the [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 12.** Figure 12: The original spectrum compared, for λ = 0.1 (a) and λ = 1 (b), to the spectra reconstructed with different methods: only the photon counting algorithm (‘Photon counting’), only the deconvolution algorithm (‘Richardson-Lucy’), both the techniques combined (‘Hybrid’), n…

**Figure 13.** Figure 13: L2 distance between the original spectrum and the one reconstructed from the processed and thresholded image as a function of the filling fraction, for two values of Rs. Rs. The smallest difference occurs at intermediate λ values, where neither the photon counting al…

**Figure 14.** Figure 14: Emission spectrum of MgF2 obtained with simple thresholding (red), and processing the images with the photon counting and deconvolution methods (blue). Error bounds are indicated by shaded regions. The peaks in the spectrum correspond to the Kα (from 1250 to 1370 eV) …

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Reference graph

Works this paper leans on

80 extracted references · 78 canonical work pages

[1]

This process involves determining the energy asso- ciated witheachpixel, Eij

Computing the energy map of the camera. This process involves determining the energy asso- ciated witheachpixel, Eij. As we will see, these en- ergies represent averaged values for each pixel, ac- counting for effects such as source broadening and finite pixel size. These effects mix the energies of photons hitting a single pixel, ultimately reducing the ...

work page
[2]

Estimating the photon hits on the detector. For a given camera image,Aij, this steps calculates the most likely distribution of photon hits on the detector, N γ ij, providing a good estimate for the number of photons hitting each pixel (i, j). The methods for performing this operation will be dis- cussed in section IID

work page
[3]

Constructing the spectrum. This final step in- volves placing N γ ij into the appropriate energy bin according to the energy mapEij, with corrections applied for the varying solid angles associated with each energy level. The details of how the image is converted to a spectrum depends on the specific geometry of the spectrometer. While all the spectromete...

work page
[4]

The process begins when photons enter the detector and strike the photo- sensitive surface, usually made of a doped or com- pound semiconductor (e.g

Photon absorption. The process begins when photons enter the detector and strike the photo- sensitive surface, usually made of a doped or com- pound semiconductor (e.g. doped silicon, GaAs). At x-ray energies, photons interact with the semi- conductor atoms via photoionization, generating electron-hole pairs. The number of these pairs is approximately pro...

work page
[5]

dark current

Charge Generation and Collection. The gen- erated electrons are collected in potential wells cre- ated by an array of electrodes on the detector sur- face. Each pixel in the detector corresponds to one potential well. The number of electrons collected in each well is roughly proportional to the energy contained in the light hitting that pixel, making the ...

work page
[6]

For modern hybrid detectors, this is done in situ with the readout electronics connected to each individual pixel

Charge T ransfer.Once the exposure is complete, the collected charge needs to be transferred to the readout electronics. For modern hybrid detectors, this is done in situ with the readout electronics connected to each individual pixel. In the case of CCDs, this is achieved through a process known as charge transfer, where the charges are moved se- quentia...

work page
[7]

dark runs

Signal Readout. In the final stage, the collected charges are read out by converting them into a voltage signal. This is done using a charge ampli- fier that translates the charge into a correspond- ing voltage. The voltage signal is then sent to an Analog-to-Digital Converter (ADC), which con- verts it into digital counts known as Analog-to- Digital unit...

work page
[8]

Under these as- sumptions, using Bragg’s law [36] and referring to Fig

Flat crystal spectrometer We begin by assuming that the rocking curve of the re- flecting crystal [36] is a Dirac delta centred at the Bragg angle, θB, and that the sample acts as a point source, thereby neglecting source broadening. Under these as- sumptions, using Bragg’s law [36] and referring to Fig. 3, one can derive the following relation between Ei...

work page

Show all 80 references

[9]

Detector image of MgF2 emission spectrum with the theoretically-computed energy contour (red) fitted to a Mg Kα line

Von Hámos spectrometer In the von Hámos (VH) geometry [40], an axis (the dispersion axis) is drawn between the source and the de- 6 0 500 1000 1500 2000 x 0 250 500 750 1000 1250 1500 1750 2000 y fit line E = 1253.6 eV 50 25 0 25 50 75 100 125 150 Figure 4. Detector image of M...

2000
[10]

In practice, how- ever, various broadening mechanisms can cause photons of the same nominal energy to be recorded at different locations on the detector

Broadening Effects The energy of each photon should map precisely to a corresponding position on the detector. In practice, how- ever, various broadening mechanisms can cause photons of the same nominal energy to be recorded at different locations on the detector. As the spect...
[11]

These algorithms can iden- tify single photon clusters, exploiting the correla- tion between neighbouring pixels to reduce the un- certainty on the photon detection

Photon counting algorithms: To address re- gions with small fill fractions ( λ ≲ 0.1), where single photon clusters are present, we use photon counting algorithms. These algorithms can iden- tify single photon clusters, exploiting the correla- tion between neighbouring pixels ...
[12]

As we will see, these two operations can be carried out se- quentially

Image deconvolution techniques: To treat the regions with large fill fractions (λ ≳ 1), where ag- glomerated clusters are present (see next section), we process the image with the Richardson-Lucy de- convolution method [59]. As we will see, these two operations can be carried ...
[13]

the pixels in which the charged produced by the photon has leaked) are few, and given in Fig

Clustering Algorithms If we suppose, as is often the case, that the charge pro- duced by a photon does not spread further than a pixel side length, then the shapes of single-photon clusters (i.e. the pixels in which the charged produced by the photon has leaked) are few, and g...
[14]

If the shape of the clus- ter C matches that in Fig

Single-photon clusters. If the shape of the clus- ter C matches that in Fig. 8, then the charge spread is considered to originate from the brightest pixel, and we accumulate all the ADU values of the clus- ter onto its brightest pixel (¯i, ¯j): A′ ¯i,¯j = X ij∈C Ath ij A′ ij =...
[15]

Agglomerated clusters. In this case we do not modify the ADU values: A′ ij = Ath ij (15) After the clustering algorithm is applied, the image is thresholded once more with a higher threshold, assum- ing that any events that fall below the threshold did not originate from a sin...
[16]

Images deconvolution To address charge spreading and noise contributions in regions with large λ, we apply the Richardson-Lucy (RL) deconvolution algorithm [59] to the matrix A′ ij . This method is an iterative procedure, based on Bayesian inference, that computes the most lik...
[17]

For simplicity, the energy contours on the camera correspond to the columns of the arrays

We create a 40x40 array to represent the camera pixels and a finer 4000x4000 subgrid to depict its surface. For simplicity, the energy contours on the camera correspond to the columns of the arrays
[18]

Their energy, which de- termines the pixel column they land on, is sampled from the spectrum S(El) shown in Fig

A number of photons, specified by the selectedλ, are sent onto the subgrid. Their energy, which de- termines the pixel column they land on, is sampled from the spectrum S(El) shown in Fig. 11a and then converted to an ADU value, taking ADUsp = 120
[19]

The ADU values of the subgrid belonging to the same pixel are then summed together to obtain the ac- tual camera image (see Fig 11c)

Charge spreading is applied by convolving the sub- grid image with a gaussian kernel of widthRs. The ADU values of the subgrid belonging to the same pixel are then summed together to obtain the ac- tual camera image (see Fig 11c)
[20]

Finally, gaussian random noise with standard devi- ation σN is added to the image (Fig. 11d). We evaluated the performances of our techniques for different values of λ, Rs and σN by computing the L2 14 Figure 11. (a) The normalised spectrum used for producing the synthetic cam...
[21]

Born and E

M. Born and E. Wolf,Principles of optics: electromag- netic theory of propagation, interference and diffraction of light (Elsevier, 2013)

2013
[22]

Foord, R

R. Foord, R. Jones, C. Oliver, and E. Pike, The use of photomultiplier tubes for photon counting, Applied op- tics 8, 1975 (1969)

1969
[23]

Peterson, How it works: the charged-coupled device, or ccd, Journal of young investigators3 (2001)

C. Peterson, How it works: the charged-coupled device, or ccd, Journal of young investigators3 (2001)

2001
[24]

Svanberg, Atomic and molecular spectroscopy: basic aspects and practical applications, Vol

S. Svanberg, Atomic and molecular spectroscopy: basic aspects and practical applications, Vol. 6 (Springer Sci- ence & Business Media, 2012)

2012
[25]

Willmott, An introduction to synchrotron radiation: techniques and applications(John Wiley & Sons, 2019)

P. Willmott, An introduction to synchrotron radiation: techniques and applications(John Wiley & Sons, 2019)

2019
[26]

Jaroszynski, R

D. Jaroszynski, R. Bingham, E. Brunetti, B. Ersfeld, J. Gallacher, B. van Der Geer, R. Issac, S. Jamison, D. Jones, M. De Loos, et al., Radiation sources based on laser–plasma interactions, Philosophical Transactions of the Royal Society A: Mathematical, Physical and En- ginee...

2006
[27]

Schmüser, M

P. Schmüser, M. Dohlus, J. Rossbach, and C. Behrens, Free-electron lasers in the ultraviolet and x-ray regime, Springer Tracts in Modern Physics258 (2014)

2014
[28]

Yamada, S

J. Yamada, S. Matsuyama, I. Inoue, T. Osaka, T. In- oue, N. Nakamura, Y. Tanaka, Y. Inubushi, T. Yabu- uchi, K. Tono, et al., Extreme focusing of hard x-ray free-electron laser pulses enables 7 nm focus width and 1022 w cm- 2 intensity, Nature Photonics , 1 (2024)

2024
[29]

A.G.Basden, C.Haniff,andC.Mackay,Photoncounting strategies with low-light-level ccds, Monthly notices of the royal astronomical society345, 985 (2003)

2003
[30]

Eggert, D

J. Eggert, D. Hicks, P. Celliers, D. Bradley, R. McWilliams, R. Jeanloz, J. Miller, T. Boehly, and G. Collins, Melting temperature of diamond at ultrahigh pressure, Nature Physics6, 40 (2010)

2010
[31]

A. B. Zylstra, O. A. Hurricane, D. A. Callahan, A. L. Kritcher, J. E. Ralph, H. F. Robey, J. S. Ross, C. V. Young, K. L. Baker, D. T. Casey,et al., Burning plasma achieved in inertial fusion, Nature601, 542 (2022). 19

2022
[32]

Abu-Shawareb, R

H. Abu-Shawareb, R. Acree, P. Adams, J. Adams, B. Ad- dis, R. Aden, P. Adrian, B. B. Afeyan, M. Aggleton, L. Aghaian, et al. (Indirect Drive ICF Collaboration), Lawson Criterion for Ignition Exceeded in an Inertial Fu- sion Experiment, Phys. Rev. Lett.129, 075001 (2022)

2022
[33]

Abu-Shawareb, R

H. Abu-Shawareb, R. Acree, P. Adams, J. Adams, B. Ad- dis, R. Aden, P. Adrian, B. B. Afeyan, M. Aggleton, L. Aghaian, et al. (The Indirect Drive ICF Collabora- tion), Achievement of Target Gain Larger than Unity in an Inertial Fusion Experiment, Phys. Rev. Lett.132, 065102 (2024)

2024
[34]

Frydrych, J

D.Kraus, A.Ravasio, M.Gauthier, D.O.Gericke, J.Vor- berger, S. Frydrych, J. Helfrich, L. B. Fletcher, G. Schau- mann, B. Nagler, B. Barbrel, B. Bachmann, E. J. Gam- boa, S. Göde, E. Granados, G. Gregori, H. J. Lee, P. Neu- mayer, W. Schumaker, T. Döppner, R. W. Falcone, S. H. ...

2016
[35]

M. Miao, Y. Sun, E. Zurek, and H. Lin, Chemistry under high pressure, Nature Reviews Chemistry4, 508 (2020)

2020
[36]

Toleikis, T

S. Toleikis, T. Bornath, T. Döppner, S. Düsterer, R. R. Fäustlin, E. Förster, C. Fortmann, S. H. Glenzer, S. Göde, G. Gregori, R. Irsig, T. Laarmann, H. J. Lee, B. Li, K.-H. Meiwes-Broer, J. Mithen, B. Na- gler, A. Przystawik, P. Radcliffe, H. Redlin, R. Redmer, H. Reinholz, G...

2010
[37]

L. B. Fletcher, H. J. Lee, T. Döppner, E. Galtier, B. Na- gler, P. Heimann, C. Fortmann, S. LePape, T. Ma, M. Millot, A. Pak, D. Turnbull, D. A. Chapman, D. O. Gericke, J. Vorberger, T. White, G. Gregori, M. Wei, B. Barbrel, R. W. Falcone, C. C. Kao, H. Nuhn, J. Welch, U. Zast...

2015
[38]

Ciricosta, S

O. Ciricosta, S. M. Vinko, H.-K. Chung, B.-I. Cho, C. R. D. Brown, T. Burian, J. Chalupský, K. Engelhorn, R. W. Falcone, C. Graves, V. Hájková, A. Higginbotham, L. Juha, J. Krzywinski, H. J. Lee, M. Messerschmidt, C. D. Murphy, Y. Ping, D. S. Rackstraw, A. Scherz, W.Schlotter,...

2012
[39]

Gawne, T

T. Gawne, T. Campbell, A. Forte, P. Hollebon, G. Perez- Callejo, O. S. Humphries, O. Karnbach, M. F. Kasim, T. R. Preston, H. J. Lee, A. Miscampbell, Q. Y. van den Berg, B. Nagler, S. Ren, R. B. Royle, J. S. Wark, and S. M. Vinko, Investigating mechanisms of state localiza- ti...

2023
[40]

Hollebon, O

P. Hollebon, O. Ciricosta, M. P. Desjarlais, C. Cacho, C. Spindloe, E. Springate, I. C. E. Turcu, J. S. Wark, andS.M.Vinko,Abinitiosimulationsandmeasurements of the free-free opacity in aluminum, Phys. Rev. E100, 043207 (2019)

2019
[41]

T. R. Preston, S. M. Vinko, O. Ciricosta, P. Hollebon, H.-K. Chung, G. L. Dakovski, J. Krzywinski, M. Minitti, T. Burian, J. Chalupský, V. Hájková, L. Juha, V. Vozda, U. Zastrau, R. W. Lee, and J. S. Wark, Measurements of the k-shell opacity of a solid-density magnesium plasma...

2017
[42]

S. M. Vinko, V. Vozda, J. Andreasson, S. Bajt, J. Bi- elecki, T. Burian, J. Chalupsky, O. Ciricosta, M. P. Des- jarlais, H. Fleckenstein, J. Hajdu, V. Hajkova, P. Holle- bon, L. Juha, M. F. Kasim, E. E. McBride, K. Muehlig, T. R. Preston, D. S. Rackstraw, S. Roling, S. Toleiki...

2020
[43]

S. M. Vinko, O. Ciricosta, T. R. Preston, D. S. Rack- straw, C. R. D. Brown, T. Burian, J. Chalupský, B. I. Cho, H. K. Chung, K. Engelhorn, R. W. Falcone, R.Fiokovinini, V.Hájková, P.A.Heimann, L.Juha, H.J. Lee, R. W. Lee, M. Messerschmidt, B. Nagler, W. Schlot- ter, J. J. Tur...

2015
[44]

Q. Y. van den Berg, E. V. Fernandez-Tello, T. Burian, J. Chalupský, H.-K. Chung, O. Ciricosta, G. L. Dakovski, V. Hájková, P. Hollebon, L. Juha, J. Krzywinski, R. W. Lee, M. P. Minitti, T. R. Preston, A. G. de la Varga, V. Vozda, U. Zastrau, J. S. Wark, P. Velarde, and S. M. V...

2018
[45]

Vinko, O

S. Vinko, O. Ciricosta, B. Cho, K. Engelhorn, H.-K. Chung, C. Brown, T. Burian, J. Chalupsk` y, R. Falcone, C.Graves, et al.,Creationanddiagnosisofasolid-density plasma with an x-ray free-electron laser, Nature482, 59 (2012)

2012
[46]

Excellence in x-ray detection: Advanced x-ray cameras for scientific and industrial applications., Princeton In- struments
[47]

C. H. Lewenkopf, E. R. Mucciolo, and A. Castro Neto, Numerical studies of conductivity and fano factor in dis- ordered graphene, Physical Review B—Condensed Mat- ter and Materials Physics77, 081410 (2008)

2008
[48]

M. V. Newberry, Signal-to-noise considerations for sky- subtracted ccd data, Publications of the Astronomical Society of the Pacific103, 122 (1991)

1991
[49]

Du and K

H. Du and K. J. Voss, Effects of point-spread function on calibration and radiometric accuracy of ccd camera, Applied optics 43, 665 (2004)

2004
[50]

Abboud, S

A. Abboud, S. Send, N. Pashniak, W. Leitenberger, S. Ihle, M. Huth, R. Hartmann, L. Strüder, and U.Pietsch,Sub-pixelresolutionofapnccdforx-raywhite beam applications, Journal of Instrumentation 8 (05), P05005
[51]

Hagino, K

K. Hagino, K. Oono, K. Negishi, K. Yarita, T. Kohmura, T.G.Tsuru, T.Tanaka, H.Uchida, S.Harada, T.Okuno, et al., Measurement of charge cloud size in x-ray soi pixel sensors, IEEE Transactions on Nuclear Science66, 1897 (2019)

2019
[52]

Gatti, A

E. Gatti, A. Longoni, P. Rehak, and M. Sampietro, Dy- namics of electrons in drift detectors, Nuclear Instru- mentsandMethodsinPhysicsResearchSectionA:Accel- erators, Spectrometers, Detectors and Associated Equip- ment 253, 393 (1987). 20

1987
[53]

J. C. Mullikin, L. J. van Vliet, H. Netten, F. R. Boddeke, G. Van der Feltz, and I. T. Young, Methods for ccd cam- era characterization, inImage Acquisition and Scientific Imaging Systems, Vol. 2173 (Spie, 1994) pp. 73–84

1994
[54]

A. Shor, Y. Eisen, and I. Mardor, Edge effects in pix- elated cdznte gamma detectors, IEEE Transactions on Nuclear Science 51, 2412 (2004)

2004
[55]

Morton, Photon counting, Applied Optics7, 1 (1968)

G. Morton, Photon counting, Applied Optics7, 1 (1968)

1968
[56]

Zachariasen, A general theory of x-ray diffraction in crystals, Acta Crystallographica23, 558 (1967)

W. Zachariasen, A general theory of x-ray diffraction in crystals, Acta Crystallographica23, 558 (1967)

1967
[57]

Humphries,Isochoric generation and spectroscopic di- agnosis of high energy-density systems, Ph.D

O. Humphries,Isochoric generation and spectroscopic di- agnosis of high energy-density systems, Ph.D. thesis, Uni- versity of Oxford (2020)

2020
[58]

Goldstein, Classical Mechanics (Addison-Wesley, 1980)

H. Goldstein, Classical Mechanics (Addison-Wesley, 1980)

1980
[59]

Gao and L

F. Gao and L. Han, Implementing the nelder-mead sim- plex algorithm with adaptive parameters, Computational Optimization and Applications51, 259 (2012)

2012
[60]

L. v. Hámos, Röntgenspektroskopie und abbildung mit- tels gekrümmter kristallreflektoren, Naturwissenschaften 20, 705 (1932)

1932
[61]

Forte, T

A. Forte, T. Gawne, K. K. Alaa El-Din, O. S. Humphries, T. R. Preston, C. Crépisson, T. Campbell, P. Svens- son, S. Azadi, P. Heighway,et al., Resonant inelastic x- ray scattering in warm-dense fe compounds beyond the sase fel resolution limit, Communications Physics7, 266 (2024)

2024
[62]

Taupin, Théorie dynamique de la diffraction des rayons x par les cristaux déformés, Bulletin de Minéralo- gie 87, 469 (1964)

D. Taupin, Théorie dynamique de la diffraction des rayons x par les cristaux déformés, Bulletin de Minéralo- gie 87, 469 (1964)

1964
[63]

I.Uschmann, E.Förster, K.Gäbel, G.Hölzer,andM.En- sslen, X-ray reflection properties of elastically bent per- fect crystals in bragg geometry, Journal of applied crys- tallography 26, 405 (1993)

1993
[64]

Gawne, H

T. Gawne, H. Bellenbaum, L. B. Fletcher, K. Appel, C. Baehtz, V. Bouffetier, E. Brambrink, D. Brown, A. Cangi, A. Descamps, S. Göde, N. J. Hartley, M.-L. Herbert, P. Hesselbach, H. Höppner, O. S. Humphries, Z. Konôpková, A. Laso, B. Lindqvist, J. Lütgert, M. J. MacDonald, M. M...

2024 arXiv
[65]

T. P. et al., Design and performance characterisation of the hapg von hámos spectrometer at the high energy den- sity instrument of the european xfel, Journal of instru- mentation 13, 0.1088/1748-0221/15/11/P11033 (2020)

2020
[66]

Zastrau, C

U. Zastrau, C. R. D. Brown, T. Döppner, S. H. Glen- zer, G. Gregori, H. J. Lee, H. Marschner, S. Toleikis, O. Wehrhan, and E. Förster, Focal aberrations of large- apertureHOPG von-Hàmos x-ray spectrometers, Journal of Instrumentation 7 (09), P09015
[67]

Zastrau, A

U. Zastrau, A. Woldegeorgis, E. Förster, R. Loetzsch, H. Marschner, and I. Uschmann, Characterization of strongly-bent hapg crystals for von-hámos x-ray spectro- graphs, Journal of Instrumentation8 (10), P10006
[68]

Glenzer, L

S. Glenzer, L. Fletcher, E. Galtier, B. Nagler, R. Alonso- Mori, B. Barbrel, S. Brown, D. Chapman, Z. Chen, C. Curry, et al., Matter under extreme conditions ex- periments at the linac coherent light source, Journal of Physics B: Atomic, Molecular and Optical Physics49, 092001 (2016)

2016
[69]

MacDonald, A

M. MacDonald, A. Saunders, B. Bachmann, M. Bethken- hagen, L. Divol, M. Doyle, L. Fletcher, S. Glenzer, D. Kraus, O. Landen,et al., Demonstration of a laser- driven, narrow spectral bandwidth x-ray source for col- lective x-ray scattering experiments, Physics of Plasmas 28 (2021)

2021
[70]

Mozzanica, M

A. Mozzanica, M. Andrä, R. Barten, A. Bergamaschi, S. Chiriotti, M. Brückner, R. Dinapoli, E. Fröjdh, D. Greiffenberg, F. Leonarski, et al., The jungfrau de- tector for applications at synchrotron light sources and xfels, Synchrotron Radiation News31, 16 (2018)

2018
[71]

Chalupský, J

J. Chalupský, J. Krzywinski, L. Juha, V. Hájková, J. Ci- helka, T. Burian, L. Vyšín, J. Gaudin, A. Gleeson, M. Ju- rek, A. R. Khorsand, D. Klinger, H. Wabnitz, R. So- bierajski, M. Störmer, K. Tiedtke, and S. Toleikis, Spot size characterization of focused non-Gaussian X-ray l...

2010
[72]

Schlesiger, L

C. Schlesiger, L. Anklamm, W. Malzer, R. Gnewkow, and B. Kanngießer, A new model for the description of X-ray diffraction from mosaic crystals for ray-tracing cal- culations, Journal of Applied Crystallography50, 1490 (2017)

2017
[73]

W. Dusi, N. Auricchio, L. Brigliadori, A. Donati, G. Lan- dini, D. Mengoni, E. Perillo, and G. Ventura, A study of the spectroscopic response of planar cdte detectors when irradiated at various angles of incidence, Nuclear Instru- mentsandMethodsinPhysicsResearchSectionA:Accel...

2003
[74]

Gerlach, L

M. Gerlach, L. Anklamm, A. Antonov, I. Grigorieva, I. Holfelder, B. Kanngießer, H. Legall, W. Malzer, C. Schlesiger, and B. Beckhoff, Characterization of HAPGmosaiccrystalsusingsynchrotronradiation,Jour- nal of Applied Crystallography48, 1381 (2015)

2015
[75]

Feldhaus, E

J. Feldhaus, E. Saldin, J. Schneider, E. Schneidmiller, and M. Yurkov, Possible application of x-ray optical el- ements for reducing the spectral bandwidth of an x-ray sase fel, Optics Communications140, 341 (1997)

1997
[76]

Wollenweber, T

L. Wollenweber, T. Preston, A. Descamps, V. Cerantola, A. Comley, J. Eggert, L. Fletcher, G. Geloni, D. Gericke, S. Glenzer,et al., High-resolution inelastic x-ray scatter- ing at the high energy density scientific instrument at the european x-ray free-electron laser, Review o...

2021
[77]

Voigt, M

K. Voigt, M. Zhang, K. Ramakrishna, A. Amouretti, K. Appel, E. Brambrink, V. Cerantola, D. Chekrygina, T. Döppner, R. W. Falcone, K. Falk, L. B. Fletcher, D. O. Gericke, S. Göde, M. Harmand, N. J. Hartley, S. P. Hau-Riege, L. G. Huang, O. S. Humphries, M. Lokamani, M. Makita, ...

2021
[78]

Hölzer, M

G. Hölzer, M. Fritsch, M. Deutsch, J. Härtwig, and E. Förster, kα1,2 and kβ1,3 x-ray emission lines of the 3d transition metals, Phys. Rev. A56, 4554 (1997)

1997
[79]

W. H. Richardson, Bayesian-based iterative method of image restoration, JoSA62, 55 (1972). 21

1972
[80]

Yaffe and J

M. Yaffe and J. Rowlands, X-ray detectors for digital radiography, Physics in Medicine & Biology42, 1 (1997)

1997

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

This process involves determining the energy asso- ciated witheachpixel, Eij

Computing the energy map of the camera. This process involves determining the energy asso- ciated witheachpixel, Eij. As we will see, these en- ergies represent averaged values for each pixel, ac- counting for effects such as source broadening and finite pixel size. These effects mix the energies of photons hitting a single pixel, ultimately reducing the ...

work page

[2] [2]

Estimating the photon hits on the detector. For a given camera image,Aij, this steps calculates the most likely distribution of photon hits on the detector, N γ ij, providing a good estimate for the number of photons hitting each pixel (i, j). The methods for performing this operation will be dis- cussed in section IID

work page

[3] [3]

Constructing the spectrum. This final step in- volves placing N γ ij into the appropriate energy bin according to the energy mapEij, with corrections applied for the varying solid angles associated with each energy level. The details of how the image is converted to a spectrum depends on the specific geometry of the spectrometer. While all the spectromete...

work page

[4] [4]

The process begins when photons enter the detector and strike the photo- sensitive surface, usually made of a doped or com- pound semiconductor (e.g

Photon absorption. The process begins when photons enter the detector and strike the photo- sensitive surface, usually made of a doped or com- pound semiconductor (e.g. doped silicon, GaAs). At x-ray energies, photons interact with the semi- conductor atoms via photoionization, generating electron-hole pairs. The number of these pairs is approximately pro...

work page

[5] [5]

dark current

Charge Generation and Collection. The gen- erated electrons are collected in potential wells cre- ated by an array of electrodes on the detector sur- face. Each pixel in the detector corresponds to one potential well. The number of electrons collected in each well is roughly proportional to the energy contained in the light hitting that pixel, making the ...

work page

[6] [6]

For modern hybrid detectors, this is done in situ with the readout electronics connected to each individual pixel

Charge T ransfer.Once the exposure is complete, the collected charge needs to be transferred to the readout electronics. For modern hybrid detectors, this is done in situ with the readout electronics connected to each individual pixel. In the case of CCDs, this is achieved through a process known as charge transfer, where the charges are moved se- quentia...

work page

[7] [7]

dark runs

Signal Readout. In the final stage, the collected charges are read out by converting them into a voltage signal. This is done using a charge ampli- fier that translates the charge into a correspond- ing voltage. The voltage signal is then sent to an Analog-to-Digital Converter (ADC), which con- verts it into digital counts known as Analog-to- Digital unit...

work page

[8] [8]

Under these as- sumptions, using Bragg’s law [36] and referring to Fig

Flat crystal spectrometer We begin by assuming that the rocking curve of the re- flecting crystal [36] is a Dirac delta centred at the Bragg angle, θB, and that the sample acts as a point source, thereby neglecting source broadening. Under these as- sumptions, using Bragg’s law [36] and referring to Fig. 3, one can derive the following relation between Ei...

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