A model for photon counting X-ray event reconstruction uncertainty

Evaluation of detectors for a soft X-ray imaging spectrometer has resulted in the need to understand the effect of charge spreading on apparent detector noise properties, and therefore achievable energy resolution. This paper presents a mathematical model for the processes leading to increased uncertainty within a simplified X-ray reconstruction process. This is a description for additional uncertainty introduced by the process of collecting X-ray generated electrons into a region of noisy pixels and reconstructing the recorded pixels values back into an estimated X-ray energy value. The predictions of the model, and preliminary experimental verification are shown.


Introduction
In soft X-ray photon counting imaging spectrometers utilising direct detection of X-rays, incoming X-rays are converted into charge within a detector.This charge is confined within pixels, the charge within each pixel is readout and used in later analysis to infer the energy of the original X-ray, relying on a direct relationship between X-ray energy and number of charge carriers produced [1] (0.27 e − eV −1 in silicon, for example [2]).This requires X-ray interactions to be sparse (to avoid overlapping) and good noise properties (to accurately measure the charge generated).
Work for a soft X-ray imaging spectrometer has identified predictable behaviours in the effect of X-ray event reconstruction on energy resolution.These behaviours can be used to predict the optimal reconstruction strategies for an X-ray detector and ensure that the minimum amount of energy uncertainty is introduced by event reconstruction.
The following investigation was conducted during detector evaluation and testing for the Auroral X-ray Imaging Spectrometer (AXIS) instrument proposed for study of the Earth's high atmosphere [3].Detectors considered for AXIS have included electron multiplying charge coupled devices (EMCCDS), standard CMOS image sensors (CISs), and the X-ray optimised CIS221-X [4].The AXIS science goals require sensitivity to low flux, sub-1 keV, X-rays while maintaining good energy resolution.The scope of this work did not permit design changes to the physical properties of the X-ray detectors under consideration; therefore, there was additional emphasis on the handling of data to achieve best performance.

The spread of charge
The spread of charge from an X-ray into the pixels of a CCD-like soft X-ray detector beyond the initial, incident, pixel is an expected behaviour and can be used to improve spatial resolution [5].However, it also introduces additional uncertainty in the measurement of charge which is particularly problematic -1 -for observations of low energy X-rays with high energy resolution requirements.This uncertainty increases the noise seen in an X-ray population, reducing energy resolution.The uncertainty arises for two reasons, the first is the introduction of multiple pixels worth of noise into any measurement, the second is due to the loss of charge into charge distribution tails where the amount of charge collected is equivalent to the uncertainty in any pixel measurement, both illustrated in figure 1.These map into specific concerns for low energy X-rays.Firstly, below 700 eV the attenuation length of X-rays in silicon is less than a micron and so, in a modern silicon backside illuminated detector, most X-rays are absorbed far from charge storage regions and therefore have the greatest opportunity for lateral spreading in undepleted, field free, regions prior to confinement in a pixel.Furthermore, the amount of charge generated is not large (approximately 140 e − for a 500 eV X-ray), exacerbating issues relating to the charge distribution tails.
Figure 1.A one dimension illustration of reconstruction thresholding resulting in the exclusion of charge in four archetypal events with the same integral value.The percentage of event charge recovered by summing pixels above the example threshold is indicated in each plot.The reconstruction threshold is generally set with knowledge of pixel noise.Reproduced with permission from [6].
Often there is a large energy resolution advantage to selecting X-rays based on the measured charge spread.X-rays will be graded with higher quality events observed to only occur in a single pixel being distinguished from edge of pixel X-rays or high charge spread X-rays (for example [7]).This permits a trade-off of energy resolution and number of recorded X-rays in later analysis.However, when using grading for optimum energy resolution there are large sensitivity penalties, and it is sometimes not possible for faint observation targets.
While the following analysis relates to current X-ray detectors observing very soft X-rays, charge reconstruction in the detectors of the next generation of flagship X-ray observatories with small pixels and large detector thicknesses is a known area of concern [8].

Reconstruction uncertainty model
A model for the process of pixel inclusion and exclusion during event reconstruction was developed.It aimed to predict and analyse the effect of charge spreading on the additional uncertainty introduced into X-ray interaction event reconstruction in various X-ray detector architectures.This model uses a simplified version of the two threshold reconstruction technique commonly used for space based detectors [7,9].In this process: 1.A stack of calibrated X-ray detector frames is analysed, identifying incident X-rays with an initial detection pixel value threshold.
2. Anti-coincidence processing is performed to prevent adjacent pixels, likely recording the same interaction, from both being counted as X-rays.Only the brightest pixel in any interaction will be counted as a potential X-ray event.
3. Reconstruction of X-ray values by summing all pixels surrounding (and including) a detected event above a second, lower, reconstruction threshold.This can identify X-rays with charge split over multiple pixels, the reconstruction threshold is also known as the split threshold.
This simple process ignores more complex pattern matching behaviours, (for instance figure 1 of [7]).This has been deemed an acceptable trade-off between accuracy and tractability, as these more complex behaviours only affect a small number of X-ray interaction events.In addition, pattern matching is often used as a prelude to X-ray event grading, which is not always an option for low flux applications.The reconstruction and detection thresholds for a detector are often set in relation to the per-pixel expected noise, values three to five times the per-pixel rms noise are common (for example, [9,10]).This approach addresses the consequence of including too many noisy pixels during reconstruction but may ignore the effect of sub-threshold pixels containing charge.Additionally, separating the values of detection and selection threshold allows the detection threshold to be selected for increased noise rejection, where the minimum X-ray energy is known.A more detailed understanding of the reconstruction process was desired when comparing EMCCDs, for which charge spread is expected to be an issue and suggests the use of a low threshold, and the CIS221-X which was expected to have less charge spread.
The aim of the reconstruction model is to predict the additional uncertainty due to the reconstruction process that may be expected in a detector.This is only one source of uncertainty affecting the measurement of X-rays in a detector.Often noise contributions from excessive dark current, CTI, image lag and charge smear are more dominant in the overall noise of the detector.However, in modern detectors these other sources have been suppressed or greatly reduced.

Charge distribution
The first step in the reconstruction model is to produce an expected population of X-ray interaction events, to which the simplified reconstruction process can be applied.
The charge collected in each pixel associated with an event is dependent on the pixel location, the X-ray interaction location and depth, the energy of the photon, and the parameters associated with the X-ray detector's properties.That is, the probability that a pixel contains an amount of charge due to an X-ray may be expressed by equation (2.1).

𝑃(𝑛 𝑒
Where (  − ,X-ray ) is the probability of   − electrons being collected in a pixel due to an X-ray interaction;  pix and  pix are the pixel coordinates (in column and row indices);  ph ,  ph and  ph are the coordinates of the initial X-ray interaction;  ph is the X-ray photon energy; and param detector is an array of device parameters including pixel geometry, semiconductor properties and operating conditions.When evaluating equation (2.1) certain simplifications can be made.In a sufficiently large detector, performance will be dominated by pixels well away from the array edge.In this case values for  pix and  pix can be resolved relative to the pixel in which an X-ray is incident, without considering X-ray events that spill out of the image area.If we are considering interaction and pixel coordinates relative to the incident pixel, values for  ph and  ph are restricted to values representing the area of the incident pixel with a uniform distribution. ph has been measured from the back (illuminated) surface, the distribution of  ph values for a population of X-rays have an exponential probability distribution.For the purposes of this analysis  ph was fixed.The required detector parameters (param detector ) are dependent on the method of charge distribution modelling.These parameters represent the properties of the X-ray detector and are largely specific to a single detector.However, for the purposes of this analysis only pixel geometry parameters have been changed, and the properties of the semiconductor have been selected to be representative of a broad range of X-ray detectors.The equations from [11] were used for modelling charge transport behaviour in field-free and depleted regions of a CCD, as these had previously been identified as a good predictor of event charge distribution [12].
The inclusion of the equations of [11] makes the evaluation of (  − ) non-trivial.Therefore, to probe the reconstruction properties of each detector, a set of Monte-Carlo trials were run using the charge distribution model to convert an expected population of X-rays into an expected population of recorded X-ray events.

Charge exclusion uncertainty
The probability that a pixel is included in an event is defined in equation (2.2).That is, the probability that the X-ray generated signal of a pixel, added to any noise, is greater than the detection threshold: Where   − ,X-ray is the number of X-ray generated charge carriers;   − ,pixel is the number of charge carriers charge contributed, or removed, by noise with a variance of  pix ; and  thresh is the inclusion threshold value, expressed as charge.In this equation both   − ,X-ray and   − ,pixel are random variables, rather than the mean or rms values.If we assume the noise in each pixel of the detector is roughly Gaussian, the probability of inclusion for a given signal level, threshold and root mean square (rms) pixel noise is provided by equation (2.3): Where: The dependence of the probability of a pixel being ignored on the threshold value and pixel value have been illustrated in figure 2.
-4 - The charge lost from a given pixel, for a given a distribution of pixel values, will have a distribution described by (2.4): Where the term  lost is the amount of charge excluded following a pixel being ignored, and (ignored|  − ,X-ray ) is the probability that a pixel is ignored, given it contains   − ,X-ray electrons.This implies that the mean charge lost from any given pixel will be given by equation (2.5).
Where  ph is the maximum charge expected from an X-ray interaction event.The total lost charge can then be found by summing the mean charge lost values across the pixels containing charge from an X-ray event, using the distribution of expected pixel values from the Monte Carlo trials: And more importantly the variance in charge lost can be found by summing the variance in charge lost across all event pixels: This  2 lost represents the uncertainty associated with the exclusion of charge from a reconstructed event.In an ideal X-ray detector this value stays low, either because noise is low, so any X-ray event charge puts a pixel above any reconstruction threshold, or because the amount of charge spread beyond an initial pixel is very low.

Per-pixel noise
There is also a penalty for including pixels when measuring an X-ray, in the form of additional per-pixel noise sources.Over the region of interest around the incident pixel the total variance included from per-pixel sources has been modelled with: Where B( included,,  ) is a Bernoulli trial for the inclusion of pixel , , which will have a probability of inclusion determined by the expected value for pixel , , as per equation (2.3) using the pixel value determined by equation (2.1).The mean value of  2 pix is representative of the expected per-pixel noise across the population of reconstructed events.That is: included,,  × ( pix ) 2 (2.9)

Reconstruction noise
The total rms noise of the reconstruction process is: In practice the additional noise due to the inclusion of many pixels is anti-correlated with the uncertainty associated with excluding charge and limits the range of behaviours the model can describe accurately.This can be observed for high reconstruction thresholds, where single pixel events are largely unaffected, and split pixel events have charge excluded.In such cases the two populations lead to a multi-modal spectrum with a population of single energy X-ray events giving rise to multiple peaks, single pixel events will form a peak at the correct energy, and then multi-pixel events will form peaks of lower energy.This behaviour cannot be captured by attempting to measure the total noise across an X-ray population with equation (2.10).It is not recommended to apply the model to such a situation, however examining the properties of  lost from equation (2.4) may indicate when it is likely to occur.

Model predictions
The reconstruction model was applied to simplified examples of X-ray detectors considered for soft X-ray applications, specifically AXIS, to understand the behaviour of each.The detectors could be represented with three archetypes: a high noise, high charge spread detector (a standard CIS or CCD); high noise, low charge spread detector (CIS221-X); and a low noise, high charge spread detector (e.g., an EMCCD).
A first archetype with small pixels and relatively high per-pixel noise, representative of a standard CIS or CCD detector, indicated that there is a trade-off to be made between the per-pixel noise and excluded charge uncertainty, illustrated in figure 3.This indicates that these X-ray detectors are likely to face a significant energy resolution penalty for including more than single-pixel events, a known behaviour.
-6 - A second archetype, with large pixels and full depletion but high per pixel noise more representative of the CIS221-X.Full depletion eliminates the field free region at the back of the detector [4], which is responsible for much of the lateral spread of charge.Here, the exclusion uncertainty was seen to level off quite quickly meaning there is little additional penalty for a high reconstruction threshold (illustrated in figure 4).This is because according to the charge distribution model, with large pixels, and no field free region, most X-ray interactions are recorded in a single pixel.The population of split pixel events is either quickly subjected to charge exclusion or consists of events that can be well reconstructed even with high reconstruction thresholds, for instance interactions with 50 % of the signal recorded in two pixels.-7 -

JINST 19 P05017
Finally, a third archetype was distinguished by very low per pixel noise, but small pixels and a larger field free region, representative of an EMCCD.This had a relationship opposite to that of the CIS221-X model.Evaluation of the charge distribution with the model of [11] indicated that the majority of identified X-rays would be spread across many pixels.This resulted in the performance illustrated in figure 5, showing that the penalty for including many pixels was low while the noise introduced by excluding pixels with a high reconstruction threshold was predicted to be high.While, qualitatively, the behaviour of the high noise detector looks similar to the first archetype, the optimum point occurs at a threshold value of around 4 e − , and the total noise is much lower than the high noise detector at that point.

Experimental verification
The model's main predictions relate to unobservable variables, measuring an event value combines the charge exclusion noise and per-pixel noises.However, by accounting for Fano charge generation noise the predicted total reconstruction noise can be related to X-ray peak FWHM, an observable parameter in the data using equation (3.1).
Where  is the energy to charge conversion factor in silicon (3.65 eV e − −1 [2]),  ph is the X-ray photon energy,  is the Fano factor (approximately 0.115 in silicon, and 1.115 in the EMCCD due to the EM register [13]), and  event,tot is the output of the reconstruction noise model.For Lab-based tests raw image frames are often available, permitting the overall effect of the reconstruction threshold to be investigated by re-running event reconstruction algorithms for different threshold values on the same dataset.During separate experimental campaigns high quality monochromatic X-ray data was collected with a CIS221-X [14] and multi-peak fluorescence X-ray spectra of comparable energy was collected from an EMCCD [3].These datasets have permitted the observation of some of the behaviours -8 -predicted by the model.X-ray spectrums from these two detectors are displayed in figure 6, shown for various reconstruction thresholds.In these plots the axis limits have been set to highlight the X-ray peak of interest for the verification.The CIS221-X, corresponding to the second archetype, displayed behaviour consistent with the predictions of the reconstruction model.Illustrated in figure 7, the measured FWHM value was relatively stable after an initial decrease with increasing reconstruction threshold.The measurements indicate limitations of the reconstruction model, there is a 5 eV FWHM improvement from 20 e − to 50 e − threshold value.X-ray events identified as being affected by this threshold change (their reconstructed value changing by more than 140 eV) were also more likely to be identified as edge of pixel events.The excess number of edge-of-pixel events reflects that the charge distribution model does not capture all charge transport behaviours in the complex CIS pixel.
Interestingly the results indicate that a simple reconstruction threshold of three to five times the per-pixel rms noise value (10.5 e − to 17 e − ) would result in a reasonable FWHM value.However, during investigation of this detector a threshold of 22 e − was used because there is no energy resolution penalty for a higher threshold value, and this grants a level of robustness to increases in per-pixel noise.
The CCD201-20 EMCCD corresponds to the third modelled archetype.The measured and predicted FWHM values have been shown in figure 8, with a level of qualitative agreement.The 65 eV increase in FWHM at 0 e − reconstruction threshold value can been explained by a 19 e − rms increase in noise (plotted as the normalised trace).This increased FWHM has been recorded in this device in simpler FWHM predictions [3].Even accounting for this initial difference, the energy resolution penalty for higher reconstruction thresholds is much greater than predicted.
One possible explanation is a lack of charge loss modelling and charge smear or CTI effects in the charge distribution model.This is illustrated in figure 9, which records histograms of pixel values within the X-ray events in the underlying X-ray event populations.The central plot of the -9 -  figure, representing the incident pixel values show very different behaviour with fewer pixels having a value corresponding to the X-ray energy in the experimental data.This may indicate that while the charge distribution model captured charge spreading behaviours adequately (indicated by good matches in surrounding pixels), and per-pixel noise was adequately modelled, captured in the FWHM behaviour prediction to for reconstruction thresholds < 5 e − , correlated charge loss behaviours result in behavioural deviations for higher reconstruction thresholds.This is likely to relate to behaviours of the back surface, which has been represented with a perfectly reflecting boundary in this work.This is not truly representative of its properties.
-10 - From a threshold selection perspective, this data indicates that an uninformed threshold selection (three to five times per pixel rms noise, 6 e − to 10 e − would result in worse than optimum energy resolution performance.During analysis of EMCCD X-ray data very low thresholds were used.Despite its discrepancies the charge distribution model still had utility in informing initial threshold selections and providing some explanation for these experimental results.Experimental results, while differing from prediction, were useful in informing further work.
This analysis supported the predictions of the reconstruction model.This work also informed further work with these soft X-ray detectors by recommending the optimum data analysis strategies.The specific investigation in which both detectors have been considered requires high efficiency excluding few X-ray detections, precluding the use of event grading and down selection to improve energy resolution.The predictions and verification of the model of section 2 has suggested aggressive reconstruction thresholds for both detectors, very low for the EMCCD, and relatively high for the CIS221-X.This is expected to have wider relevance than the immediate work.

Discussion and conclusions
A model used for estimating the effect of event reconstruction has been presented.Its Predictions and an experimental verification for different classes of X-ray detector have been shown.Event reconstruction has been demonstrated as a source of additional uncertainty leading to a loss of energy resolution for soft X-ray imaging spectrometers.
Following the findings of this work use of the threshold sweep procedure used during experimental verification is suggested to ensure that the best reconstruction threshold is used during analysis of X-ray data.Application of the reconstruction model to new detectors is expected to be straightforward, because charge distribution process, which is the primary point at which the properties of the new detector are input into the model, is effectively a black box to the rest of the model, and the equations of [11] can easily be changed for a more appropriate, or available, model.The change of charge distribution model could also include the effects of CTI and charge smear, demonstrated in section 3.
In certain classes of soft X-ray detector, the improvement seen in energy resolution from an informed reconstruction threshold selection may be slight.A three to five times per-pixel rms noise threshold has seen as optimal in at least one of the detectors tested.However, even in such cases evaluation of the reconstruction model may permit an explanation of the energy resolution seen, by accounting for emergent uncertainty not necessarily linked to simple noise properties of the detector.Furthermore, investigation into reconstruction uncertainty may permit the selection of reconstruction thresholds more robust to increases in noise expected of the lifetime of a, particularly space-based, X-ray detector.
Historically, the uncertainty due to event reconstruction has been minimised by grading events by quality.Lower event value uncertainty and higher energy resolution is then achieved by down selecting events based on their quality, the criteria for which can be very strict depending on desired resolution improvements.A key outcome from this work was the ability to use ungraded data for both EMCCDs and the CIS221-X while maintaining good energy resolution.In both cases this has resulted in performance not especially divergent from the performance of graded data.

Figure 2 .
Figure 2. Probability of pixel exclusion for various threshold values, using 10 e − rms pixel noise.Reproduced with permission from [6].

Figure 3 .
Figure 3. Modelled reconstruction noise for 1400 eV X-rays in an approximate model of a standard CIS, for reconstruction threshold.Reproduced with permission from [6].

Figure 4 .
Figure 4. Modelled reconstruction noise for 1400 eV X-rays in an approximate model of the CIS221-X, for reconstruction threshold.Reproduced with permission from [6].

Figure 5 .
Figure 5. Modelled reconstruction noise for 1400 eV X-rays in an EMCCD, for reconstruction threshold.Reproduced with permission from [6].

Figure 6 .
Figure 6.(a) 1400 eV X-ray peak observed in spectrum gathered with the CIS221-X and (b) 1487 eV (Al K) X-ray peak observed in spectrum gathered with the CCD201-20 EMCCD.Each spectrum has been plotted for different event reconstruction threshold values to illustrate the change in energy resolution seen.

Figure 7 .
Figure 7. 1400 eV measured and predicted X-ray population properties for reconstruction threshold.The predicted FWHM includes expected reconstruction noise and Fano noise.This data did not include the pixel level gain correction of[14].

Figure 8 .
Figure 8. 1487 eV (Aluminium K) event population properties for reconstruction threshold.Measured in CCD201 data collected for [3].While there is a large quantative discrepancy between predicted and measured values, the experimental device exhibited many of the same expected behaviours.

Figure 9 .
Figure 9.Plot of pixel values in monte carlo (red) and experimental (blue) datasets, centred on incident pixel in each X-ray event.The effect of EM-register noise and charge smear has been added to the monte carlo pixel values.While many properties are well captured by the monte carlo data, the primary X-ray peak is more prominent in the incident pixel.