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Coincidence loss can have detrimental effects on the image quality provided by pixelated counting detectors, especially in dose-sensitive applications like cryoEM where the information extracted from the recorded signal needs to be maximized. In this work, we investigate the impact of coincidence loss phenomena on the recorded statistics in counting detectors producing sparse binary images. First, we derive exact analytical expressions for the mean and the variance of the recorded counts as a function of the incoming event rate. Second, we address the problem of the mean and variance of the recorded events (i.e., pixel clusters identified as individual incoming events), which also acts as a function of the incoming event rate. In this frame, we review previous studies from different disciplines on approximated two-dimensional models, and we critically reinterpret them in our context and evaluate the suitability of their adoption in the present case. The knowledge of the first two momenta of the recorded statistics allows inferring about the signal-to-noise ratio (SNR) and the detective quantum efficiency at zero frequency (DQE_{0}). Analytical results are validated through comparison with numerical data obtained with a custom-made Monte Carlo code. We chose a realistic case study for cryoEM application consisting of a 25-µm-thick MAPS detector featuring a pixel size of 10 µm and illuminated with electrons of 300 keV energy over a wide range of incoming rate.

Coincidence loss in counting detectors is the phenomenon whereby an incoming event fails to be recorded by the system due to its proximity, or overlap, with at least another incoming event. Proximity or overlap can, in general, occur either in the time domain, in the spatial domain, or both. In the time domain, coincidence loss problems belong to the category known as ^{1}

The effects of coincidence loss are therefore particularly detrimental in dose-sensitive applications where the need to maximize the information content of the detected signal is pre-eminent. This is, e.g., the case of electron cryomicroscopy (cryoEM), an electron microscopy technique for life science applications which involves the illumination of a highly radiation-sensitive, vitrified biological sample with a high-energy electron beam (typically in the range 100–300 keV), and the detection of the transmitted signal with a two-dimensional pixelated detector array [

To shed some light on the impact of coincidence loss on the recorded signal statistics and on the noise figures in counting detectors producing sparse binary images, we proceed as follows: first, we investigate the statistics of the recorded counts as a function of the incoming event rate, where counts are intended in their classical meaning of (binary) value stored in the pixel counter. In particular, we derive from basic statistical arguments, and without the need for free parameters, analytical expressions for the mean and the variance of the recorded counts. Second, we investigate the statistics of the recorded

In both counting cases, the knowledge of the first two momenta of the recorded signal statistics allows us to derive the corresponding analytical expressions for the SNR and for the DQE at zero spatial frequency (DQE_{0}) as a function of the incoming event rate. Additional figures of merit, namely, the area occupancy (AO) and the coincidence loss fraction (CLF), are also presented.

Analytical results are validated through comparison with numerical data obtained with a custom-made Monte Carlo simulation suite. As a realistic case study, we simulated a MAPS detector for cryoEM applications featuring a thickness of 25 µm and a pixel size of 10 μm, illuminated with 300-keV electrons up to a flux intensity of 200 el/s/pix (electrons per second per pixel), working at the frame rate of 1 kfps and with a counting threshold of 1 keV. Given the nature of the case study,

Let us assume a two-dimensional pixelated array with infinitely large area and with a total number of pixels _{
pix
}. Incoming events constitute a homogeneous random sequence following Poisson statistics, both in time and in space, with constant average rate per pixel and per unit time _{0}. Let _{0}(_{0} (Δ_{
N
} time frames is _{0}(_{
N
}Δ

In the following sections, quantities are described over the time interval of a frame Δ_{
N
} time frames is straightforward, given the statistical independence between consecutive frames and the homogeneity of the involved processes. For any random variable

Let us define the random variable

The derivation of the counts variance for an individual pixel is straightforward from the binomial distribution and yields the following:^{2}

The analysis of the statistical properties of events that occurs randomly, independently, and with uniform probability in a generic mathematical space belongs to the category of Poisson point processes, and the gathering of such events into clusters is known as Poisson clumping or burst. Often, problems in this field necessitate of heuristic approaches to come to an explicit conclusion [

To ease the reading, we outline the basic reasoning at the basis of the Roach model, as described in [_{0}, whose centers are randomly distributed on a continuous, two-dimensional space. If the centers of two zones are closer than 2_{0}, they are considered overlapping. A series of overlapping zones is called a spot. The recorded number of spots determined with the following scheme. Selected _{0}, zone _{0}. If greater, the pair _{0}. As a result, a spot consisting of

Illustration of Roach’s zone selection procedure. Zones _{0}. The distances between all zone centers in the spot and all zone centers not in the spot, beginning with _{0}. Figure adapted from [

The outcome of the neighbors overlapping test can be modeled with the binomial distribution. For instance, probability _{1} that the distance between a zone center and the center of the nearest neighboring zone is greater than 2_{0} (i.e., the first zone is a singlet) corresponds to the probability that _{0} around the center of the first zone—let us call this coincidence area 4_{0}, where _{0} is the area of an individual zone. Let us remember that the spatial distribution of the events is assumed to follow Poisson statistics. Introducing now our notation and defining _{1} can be written as_{0} is the complementary probability 1 − _{1}. The chain of events that brings to an _{0} and 1 sequence in which the nearest neighbor distance is greater, breaking the spot connection to the remaining zones. If the inter-zone distances are independent from each other—a condition that, as recognized by Roach, is _{
n
} for a zone to be part of an _{0}Δ_{0}Δ_{
n
}. However, because _{
n
} of

Defining now the random variable

At this point, it is suitable to bring to the attention to the reader that the Roach model was developed in a continuous two-dimensional domain and for the “simple” case of circular-shaped zones^{3}
_{0} as a sort of

Let us now focus on the variance of the number of recorded events. Due to the complexity of the task in two dimensions, no model could be found neither by the author nor in the consulted literature. However, an analytical solution for the one-dimensional case exists and it deserves some attention. In the context of chromatography, an exact formulation based on statistical arguments was first achieved by Rowe and Davis [_{0} to the coincidence interval. Few years later, in the context of dead time models for _{0} with the system dead time _{
pix
} and _{0} as half the correlation area _{1} in two dimensions (see Eq.

The knowledge of the first two momenta of the recorded statistics allows us to compute the corresponding SNR over a time frame, defined as

More interesting, however, due to its importance and widespread use in the imaging community (see [_{OUT} is the signal-to-noise ratio referring to the recorded signal statistics and SNR_{IN} is the one referring to the incoming signal statistics, which, under the assumption of obeying Poisson statistics, equals to

In its modern acceptance^{4}
_{0}
^{5}

The DQE_{0} of the recorded counts as a function of the incoming event rate can therefore be retrieved using Eqs

Additionally, derived quantities of particular interest, which we would like to mention, are the area occupancy (AO) and the coincidence loss fraction (CLF). The AO is defined as the ratio between the average number of counting pixels per frame and the total number of pixels, coinciding with the average number of recorded counts in a time frame._{0} in the denominator is scaled by _{0} is the actual ideal event rate recorded in the absence of coincidence loss.

i. The study of the counting statistics of pixels featuring binary counts presented here complements a picture already including at least the classical

ii. The results on the recorded event statistics are of general validity, whether the hit digitization occurs directly in the pixel (with binary or non-binary outcome), in the readout electronics, or at the image processing stage. Differences can arise on the value of the coincidence area, according to the specific algorithm used for the event recognition.

iii. A detector with in-pixel counting electronics might have some advantages compared to one working in the charge integrating mode in terms of a slightly smaller coincidence area. It can indeed happen that a pixel receives, in the same time frame, signals originated by more than one event. If the front-end electronics works in the counting mode, the contribution of every event is individually processed (provided they do not undergo pile-up). If the front-end electronics works in the charge integrating mode, it is the sum of the contributions to be processed, making more probable the recording of a hit and the consequent formation of a “bridge” between neighboring events.

iv. The presented models were derived for systems with frame-based readout, but their validity can be extended to systems with the event-based readout as well. In a system with event-based readout, the detection of an event triggers its own readout, but the time stamp associated with it has finite resolution, and therefore, neighboring events within this time interval cannot be distinguished. The time-stamp discretization corresponds to our frame time^{6}

To validate the analytical models, we used results of numerical simulations carried out with an improved version of the Monte Carlo tool used in [^{7}

We then processed each individual electron track with a custom-developed numerical code mimicking the physics of the charge collection and signal formation at the pixelated electrode. The generated charge distribution of each track segment was thus propagated through the remaining sensor thickness to the pixels, and a Gaussian blurring was added to reproduce the effect of thermal diffusion, with a total width depending on an initial intrinsic contribution and, under the assumption of the constant electric field, to a contribution depending on the total travel length. The charge collected by each pixel was converted into energy and a counting threshold was applied, if the energy is higher, the pixel counts 1; otherwise, it counts 0. A random fluctuation representing the electronic noise, normally distributed and assumed uncorrelated among the pixels, was added to the signal. The response of both the sensor and the readout electronics has been assumed uniform in space. At this point, it was possible to extract the statistical distribution of the event multiplicity. Then, for each value of a series of incoming electron rates, a set of 2000 independent images (frames) was generated. The total number of impinging electrons for each individual frame was chosen randomly according to the suitable Poisson statistics and then they were uniformly and randomly distributed across the sensor surface, which covered an area of 2048 × 2048 pixels. A pixel cluster recognition algorithm was then applied to each frame to isolate single events. In order to be considered isolated, two clusters need to be separated by at least one empty pixel.

We chose a case study realistic for CryoEM applications consisting of a 25-µm-thick MAPS detector, covered with 5 µm of Al accounting for the metal layers and featuring square pixels of size 10 µm. The counting threshold energy was 1 keV, the electronic noise was 200 eV rms, and the thermal diffusion between 1 and 3 µm rms. The frame rate was assumed 1 kfps. The energy of the impinging electrons was 300 keV, with values of incoming rates in the range 1–200 el/s/pix.

In order to get a visual feeling of the events distribution recorded on the pixel matrix, two frame sub-regions obtained with an incoming electron rate of 5 el/s/pix and 30 el/s/pix, respectively, are shown in

Example of frames sub-regions obtained with an incoming electron flux of 5 el/s/pix (left) and 30 el/s/pix (right). Pixels with overlapping events have been highlighted in red for visualization purposes.

The single-event multiplicity probability distribution, computed in a condition of no coincidence loss, is shown in ^{2}

Single-event multiplicity probability distribution. Lines are to guide the eye.

^{2} relative error norm (^{2}REN), defined as

(Top) Comparison between recorded count rate curves obtained with numerical simulations (error bars) and predicted by the analytical model of Eq.

_{0}Δ^{2}REN amounts to 3.97%.

(Top) Comparison between recorded count variances in one time frame obtained with the numerical simulations (error bar) and predicted by the analytical model of Eq.

The level of agreement between simulated and predicted data for both for the mean and the variance is such that we can positively conclude on the validity of the proposed model.

_{0} = 20.51 pixels. For accuracy reasons, the fitting was restricted to a range of incoming electron rate 0–80 el/s/pix (the upper bound corresponds to the location of the maximum recorded electron rate) as above this limit, the model tends to overestimate the number of recorded electrons. Within this range, the ^{2}REN of the Roach model amounts to 0.06%. Incidentally, we note that using the Roach model of Eq. _{0}, we obtain a value of 19.26, which is not far from the correct value obtained through the fitting. The curves labeled with “ana. P_{
n
}” show the breakdown of the Roach model into the first five

(Top) Recorded electron rate curves obtained with the numerical simulations (error bars) and their fitting with the analytical models (solid lines). “ana. 2D” corresponds to the fitting with the Roach model of Eq. _{
n
}” corresponds to the contributions of the _{0} is also shown as the reference. (Bottom) Deviation between the numerical simulations and their fitting with analytical models expressed in percentage.

_{0}Δt is also shown as the reference. We observe that up to the incoming electron rate of ∼30 el/s/pix (corresponding to a coincidence loss of 27.6%, see ^{2}REN of 6.8%. For increasing incoming electron rates, both models underestimate the data down to a factor 1/2, as shown in

(Top) Recorded count variance obtained with the numerical simulations (error bars) and predicted with the analytical models (solid lines). “ana. 1D” corresponds to the result obtained with the one-dimensional model of Eq.

The behavior of the DQE_{0} as a function of the incoming electron rate is reported in _{0M} for low incoming electron rates^{8}
_{0E}, on the other hand, it holds_{0E} shows a non-monotonic behavior, while DQE_{0M} grows indefinitely. We can attempt to explain the increasing trend for both DQE_{0M} and DQE_{0E}—for the latter before the unavoidable collapse at high incoming rates due to counting paralysis—in an analogous way of what is already observed in [_{0} is compensated by a loss in spatial resolving capability due to the increasing size of the merged event.

Simulated (symbols) and analytically modeled (solid lines) DQE_{0} for the recorded counts and for the recorded electrons.

Finally,

Simulated (symbols) and analytically modeled (solid lines) area occupancy and coincidence loss fraction.

We investigated the impact of coincidence loss on the recorded count statistics and on the noise performance in counting detectors featuring sparse binary images. First, we derived exact analytical expressions for the mean and the variance of the recorded counts. Second, we addressed the problem of the mean and variance of the recorded events (i.e., pixel clusters identified as a single incoming event). We reviewed, reinterpreted, and evaluated the suitability of approximated models—as no exact solutions exist in two dimensions—previously obtained in several different disciplines, adopting the “Roach model” for the mean and proposing an extension of the one-dimensional exact solution for the variance to the two-dimensional case. For both cases, we derived expressions for the SNR and the DQE_{0}. Model predictions were qualified against numerical simulation carried out with a custom-developed Monte Carlo code, for the CryoEM-realistic case study of a 25-µm-thick MAPS detector featuring a pixel size of 10 μm, a frame rate of 1 kfps, and working in the binary counting mode. The incoming beam consisted of electrons with energy 300 keV and with flux intensities up to 200 el/s/pix, where coincidence loss phenomena are bringing the system well into paralysis. The matching between simulated data and analytical prediction is perfect for the mean and variance of the recorded counts. For the mean recorded electrons, the Roach model fits excellently simulated data up to an incoming electron rate of ∼80 el/s/pix, corresponding to the location of the maximum of the recorded curve. At higher values of incoming rates, the model tends to slightly overestimate the number of recorded events. For the variance of the recorded events, both the existing one-dimensional and the proposed two-dimensional analytical solutions match the simulated data excellently up to an incoming electron rate of ∼30 el/s/pix. At higher values of incoming rates, both models severely underestimate the data, but the proposed two-dimensional extension follows better functional behavior, supporting its adoption. The resulting DQE_{0} shows an increasing behavior as a function of the incoming rate for the recorded counts, while it shows a non-monotonic behavior for the recorded events. The increase above the low incoming rate limit is due to an allegedly reduced sensitivity of the recorded signals resulting from the union of multiple events to the statistical fluctuations of the individual incoming events, in a sort of noise filtering effect. Only in the second case, it ultimately decreases to zero due to the overwhelming system paralysis. Generalization and limitations to the validity of the models were also discussed.

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

PZ: writing–original draft and writing–review and editing.

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

The author would like to thank J. M. Davis for the kind conversation.

Author PZ was employed by DECTRIS Ltd.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

In the classical

For example, by analyzing the statistical distribution of the cluster sizes generate by single, non-overlapping incoming events. Please note that the knowledge of the real incoming flux is needed since

Considerations on the case of elliptical zones are, however, reported in [

Originally, the DQE was conceived as a “large area” property, what is nowadays called zero-frequency DQE [

The notation DQE_{0} is preferred over DQE(0) not to induce to think that the DQE is an actual function of frequency.

Modern event-based readout chips can feature time-stamp discretization down to ns or sub-ns levels, allowing

v. 4-2.1. The physics was set to multiple Coulomb scattering with the cutoff energy of 1 keV for electrons and 100 eV for photons. Fluorescence was enabled, and no biasing was used.

A result of general validity for counting detectors [

In one spatial dimension the recorded event rate per unit length and unit time _{0}. The term Δ_{0} as a function of _{0} such that

For

For