Photon-counting x-ray detectors for CT

To count photons is the most intuitive approach for detecting x-rays, and if it were not for technical challenges, photon-counting detectors (PCDs) would have been standard from the beginning of radiology. Early on, gas detectors were commonplace, and Geiger–Müller devices counted individual interactions of ionizing radiation. The Nobel-prize-awarded multi-wire proportional chamber (Charpak 1997) combined photon counting with spatial resolution. The development was driven by fundamental physics research, but photon-counting gas detectors were briefly used for imaging in a Paris hospital (Dubousset et al 2007). In nuclear imaging, photon counting was used from the very beginning for the Anger camera as well as for the first PET system. In this case, a scintillator, typically NaI or CsI, converted the incident gamma rays into visible light that was detected by photosensitive devices. For x-ray imaging the challenges for photon counting are much harder than for nuclear imaging. The average energy of the photons is only around 70 keV as compared to 140 keV for SPECT and 511 keV for PET. Moreover, the x-ray fluence rate for a computed tomography (CT) scan can be up to $10^9\; \mathrm{mm}^{-2}\,\mathrm{s}^{-1}$, while for nuclear imaging it is as low as 100 $\mathrm{mm}^{-2}\,\mathrm{s}^{-1}$, putting much higher constraints on fast pulse processing for x-ray imaging.

The first photon-counting imaging system approved by the U.S. Food and Drug Administration was the Sectra MicroDose Mammography (Åslund et al 2007) in 2011, with around one thousand installations worldwide for breast-cancer screening and diagnosis. The first full-field photon-counting CT prototype was evaluated in the clinic in 2007 based on a CdZnTe detector, and though limited in the count rate it could accommodate, it still produced material-specific images of high quality (Benjaminov et al 2008). There are currently at least four photon-counting CT systems under evaluation. Three of these, including one mobile head CT system, utilize cadmium-based sensors (Yu et al 2016c, Si-Mohamed et al 2017a, Han-soo 2017), and one uses silicon-based sensors (da Silva et al 2019). In particular, one cadmium-based system has generated a large number of publications (Yu et al 2016a, 2016c, Symons et al 2018a, 2018b, 2017b, Pourmorteza et al 2016, 2017). Furthermore, a CdTe-based photon-counting system limited to breast CT (Kalender et al 2017) has been evaluated on patients. We are now at a crossroads to emerging clinical systems within the next few years, and the different solutions will likely continue to compete based on imaging performance, reliability, and cost.

Despite the considerable improvements that CT technology has undergone in nearly half a century of existence, there are numerous areas where the improvements promised by PCDs can provide solutions to clinical problems (Willemink et al 2018). Their higher spatial resolution can improve visualization of lung nodules (Kopp et al 2018); small bone details, such as in temporal bone imaging; small blood vessels, such as coronary arteries where calcium blooming is a problem; or implants such as stents (Symons et al 2018a, Mannil et al 2018, Sigovan et al 2019). The improved contrast-to-noise ratio (CNR), in particular for contrast-enhanced imaging (Gutjahr et al 2016), can be used, for example, to improve the visualization of blood vessels (Symons et al 2018b) or contrast-enhancing tumors. PCDs have also been demonstrated to improve differentiation between grey and white matter in the brain (Pourmorteza et al 2017). Furthermore, the ability to generate material-selective images has applications in neuroradiology, such as distinguishing intracranial hemorrhage from calcification (Hu et al 2016) and iodinated contrast (Phan et al 2012), for which dual-energy CT has been shown to be effective. The potential for improved material-selective imaging with PCDs may also offer better visualization of atherosclerotic plaques by improving the ability to separate materials such as lipid, calcium, and iodine (Alessio and MacDonald 2013, Boussel et al 2014).

Another promising aspect is improved low-dose imaging. The health hazard posed by low-dose ionizing radiation is a contested question, but pediatric patients undergoing CT examinations with cumulative doses of more than 50–60 mGy have been shown to exhibit an increased risk of brain cancer and leukemia (Pearce et al 2012). In spite of this, CT is often the imaging alternative with best benefit versus risk for the patient, and the dose in x-ray imaging should therefore be kept as low as reasonably achievable. By rejecting electronic noise, PCDs can allow new imaging protocols with lower doses than what is possible today. This fact can have important implications for imaging of children and adolescents, who are more radiation-sensitive than adults, and for lung-cancer screening, where it is particularly important to minimize the dose since the examination is performed on large numbers of healthy individuals (Symons et al 2016).

The energy-resolving capabilities of PCDs can be used to reduce artifacts due to beam hardening or metal implants (Nasirudin et al 2015). They can thus provide improved diagnostic quality when imaging regions with weak attenuation differences surrounded by dense bony structures, such as in the brain (Pourmorteza et al 2017). Another potential application is found in the emerging field of radiomics, where large numbers of quantitative radiomic features are extracted from medical images and analyzed to find patterns indicative of diagnosis. These radiomic features are currently dependent on acquisition parameters, which limits reproducibility (Berenguer et al 2018). The potential of photon-counting CT to generate quantitative material maps can mitigate this variability. Improved tissue characterization also shows potential for improving radiotherapy planning (Simard et al 2019).

Among the more speculative proposed applications of photon-counting CT are the use of new contrast agents. Gadolinium, which is used in MRI, is an attractive candidate for photon-counting CT because of the location of its K-edge in the central part of the diagnostic energy range (Roessl et al 2011). However, it is still unclear whether gadolinium contrast agents can generate sufficient contrast at patient-safe concentrations. Dual- or triple-contrast agent protocols, where iodine, gadolinium, and potentially also bismuth are administered at different time points to show different phases of contrast uptake in a single scan, have also been proposed (Symons et al 2017a, 2017b, Dangelmaier et al 2018). Further, there is ongoing research into the development of new nanoparticle-based contrast agents containing heavy elements such as gold, tantalum, or ytterbium (Cormode et al 2017, Kim et al 2018, Lambert et al 2018).

As an example of the clinical use of a high-resolution spectral PCD, an excised human heart with a calcified coronary artery containing an iodinated gel was imaged using a state-of-the-art dual-energy CT scanner (figure 1) and a prototype silicon-based spectral PCD (figure 2). The imaging task was to visualize the artery and separate calcified regions from iodine. The dual-energy image was decomposed into several material basis pairs: calcium (water), iodine (water), calcium (iodine) and iodine (calcium), where the basis not shown is within the parenthesis. The spectral PCD images (figure 2) were decomposed into three bases: water, iodine, and calcium. An overlay image was formed using the basis images in which calcium and iodine were colored red and green, respectively. The high resolution combined with the simultaneous spectral acquisition facilitated by the multi-energy-bin PCD shows potential to improve the diagnostic quality of the CT images.

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The purpose of this article is to summarize and discuss recent insights into the characteristic features of PCDs and how they relate to the imaging performance of the CT system. Section 2 outlines the theoretical benefits of PCDs over energy-integrating detectors (EIDs). Section 3 outlines the different design choices that must be made when developing a PCD. Section 4 discusses how image quality is affected by the detector properties. Section 5 describes the steps needed to generate images from the measured data. Section 6 describes different metrics of imaging performance that can be used to guide the system design. Finally, in section 7, we discuss what these insights mean for the future of photon-counting CT.

In the photon-detection process and/or image formation, different weight can be given to photons of different energy, affecting both the contrast and the noise of the signal.

The contrast between two projection measurements depends on the energy of the transmitted x-ray photons. Generally, low-energy photons carry more contrast information than high-energy photons. Also, if two projection measurements have different material composition, the dependency of contrast on photon energy is elevated, and increased contrast can be obtained by increasing the weight given to photons that carry more contrast information. Weighting some photons more than others increases the variance relative to the mean value, thus reducing the signal-to-noise ratio (SNR). There is a fundamental trade-off between the increase of contrast and the reduction of the SNR, and there is an optimal set of photon weights that maximizes the CNR for a given imaging task (Schmidt 2009). The weighting of photons will often be performed indirectly via the material-decomposition process (see section 5.2), where it can be seen as the utilization of the contrast and noise of the signal to obtain a material thickness estimation with minimum variance.

An EID weights each detected photon by its energy, thus adding additional weight to high-energy photons, for which the contrast is lower. In addition, the non-uniform weighting of photons results in a reduction of the SNR. The reduction of the SNR² of an EID due to the weighting of the photons is commonly referred to as the Swank factor (Swank 1973), and it is given by

$\begin{equation} \mathrm{I_{EID}} = \frac{\left[ \int_{0}^{\infty} \epsilon S(\epsilon) d\epsilon \right]^2}{\int_{0}^{\infty} \epsilon^2 S(\epsilon) d\epsilon}, \end{equation} \tag{ 1 }$

where is signal amplitude and S() is the normalized distribution of signal amplitudes registered by the detector. The magnitude of the Swank factor depends on the distribution of signal amplitudes; the more the amplitudes are distributed over different energies, the lower the Swank factor. The distribution of signal amplitudes in turn depends on the spectrum from the x-ray tube, the attenuation of the imaged object, and the response of the detector.

Ideal purely PCDs intrinsically weight all photons equally (one photon, one count). Counting photons therefore gives relatively more weight to low-energy photons compared to energy integrating, resulting in a higher contrast, in particular for low absorbing materials. ⁴ PCDs also avoid the negative effect of the Swank factor. In addition, spectrally resolving PCDs allow giving different weight to photons of different energy as a part of the signal processing, and the weighting can be, for example, tailored to maximize the CNR.

As an illustration of the effect of photon weighting, consider the task of separating between two projection measurements with a typical 120 kVp x-ray tube spectrum for which the x-rays pass through (a) 10 cm water and (b) 9 cm water plus 1 cm of 10 mg ml⁻¹ iodine-water solution, respectively. In this example, we simulate an ideal PCD that detects all photons and registers the counts in two energy bins, $s_{\mathrm{low}}$ and $s_{\mathrm{high}}$, separated by a threshold at 50 keV. The results are compared to a simulation of an ideal EID that integrates the energy for all transmitted photons for the same imaging task. In order to optimize the CNR for the PCD, a weighted sum of the two energy bins is formed:

$\begin{equation} s = x s_\text{low} + (1-x) s_\text{high}, \end{equation} \tag{ 2 }$

where 0⩽x⩽1, where x = 0.5 corresponds to the PCD without energy weighting. As figures-of-merit we evaluate the following: the contrast between the two projections, defined as ${C} = (s_A - s_B) / s_A$; the signal-to-noise ratio, defined as $\mathrm{SNR} = s_A / \sigma(s_A)$, where σ is the standard deviation; and finally, the contrast-to-noise ratio, defined as $\mathrm{CNR} = {C}\times \mathrm{SNR} = (s_A - s_B) / \sigma(s_A)$.

The contrast, SNR, and CNR for the PCD are plotted versus x in figure 3, and the performance of the EID is included for reference. The SNR of the PCD obtains its maximum at x = 0.5, i.e. equal weight to both energy bins. The contrast, on the other hand, increases monotonically as more weight is given to the low-energy bin (higher x). The CNR, which is the product of the contrast and the SNR, reaches a maximum of 1.17 for x = 0.76. The corresponding relative CNR², which is proportional to x-ray dose, is 1.37. In other words, for this imaging case, it is motivated to sacrifice SNR in order to gain contrast, and thus achieve a superior CNR.

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For the simulated ideal EID (gray lines in figure 3), on the other hand, the suboptimal energy weighting results in a relative contrast of 0.82 (compared to the PCD without energy weighting), and a relative SNR of 0.96 (in accordance with the Swank factor). In total, the sub-optimal energy weighting of the EID results in a relative CNR of 0.79, or, equivalently, a relative CNR² of 0.63, which is to be compared with 1.37 for the PCD with optimal energy weights. In other words, the ideal spectral PCD with optimal energy weights is, for this particular imaging task, 2.17 times more dose efficient compared to the ideal EID. It can be noted that the performance of the EID is similar to that of the spectral PCD with x = 0.27.

Another benefit of energy-resolving PCDs is that they allow measuring the composition of the imaged object through a process known as material decomposition. This method is based on the fact that the linear attenuation coefficient µ(E) of any material in the human body can be well approximated by a linear combination of a small number of basis functions f_j (E): $\mu(E) = \sum_{j = 1}^{N_m} a_j f_j(E)$, where the coefficients a_i are referred to as basis coefficients (Alvarez and Macovski 1976). Since there are two dominant physical processes contributing to x-ray attenuation in the diagnostic CT energy range, namely photoelectric absorption and Compton scattering, N_m  = 2 basis functions are commonly assumed to be sufficient for substances containing only light elements, such as human tissues. In addition to these two, one additional basis function must be included for each heavy element whose attenuation coefficient contains a K-edge discontinuity within the diagnostic energy range (Roessl and Proksa 2007, Schlomka et al 2008).

This low dimensionality means that it is possible to completely characterize the energy dependence of the linear attenuation coefficient at every point in the imaged volume with a small number of energy bins: at least two bins are needed for non-enhanced imaging, and three bins with a contrast agent. Compared to the dual-energy systems in clinical use today, the capability to measure the amounts of three or more basis materials is one of the benefits of PCDs, along with other advantages such as the reduction of spectral overlap, the absence of spatial mismatch between the energy-bin images, and the ability to obtain spectral information also in the peripheral parts of the image as opposed to dual-source systems, where one detector has a limited field of view due to geometric constraints.

From the energy-resolved measured data, the material composition at each point in the imaged volume can be inferred through material decomposition. This process, which can take place before, during, or after the image reconstruction, (see section 5.2) results in a set of reconstructed basis images. These basis images show the distribution within the imaged object of each of the selected basis materials, e.g. water, calcium, and iodine. Once such a set of basis images has been generated, it is straightforward to obtain distribution maps of other substances (e.g. muscle and fat) through a simple linear transformation. Another possibility is to use the basis material maps to generate images of the x-ray attenuation coefficient at each energy (i.e. virtual monoenergetic images; see Leng et al (2017)), which are free of beam-hardening artifacts (see section 5.4).

A further possibility is to form a linear combination of the basis images to generate an image that is optimal for maximizing detectability for a specific imaging task. It turns out that this methodology gives a detectability for an optimal linear observer that is equal to the detectability obtained from optimal weighting of the original energy-bin images, provided that photon statistics are sufficient (Alvarez 2010, Persson et al 2018b). In other words, as long as the detector is operating far enough above the fluence level that causes photon starvation, the material-decomposition process preserves the information available in the measured data. Since material decomposition also has the theoretical ability to remove beam hardening completely, displaying weighted basis images makes it possible to avoid the trade-off between beam-hardening artifacts and image detectability that is observed when generating weighted sums of the original bin images (Shikhaliev 2005).

Since a larger number of estimated basis components makes the material decomposition process more ill-conditioned, the benefit of adding each new basis material diminishes with an increasing number of basis functions used in the decomposition. At the same time, using a detector with a large number of energy bins makes it possible to choose a number of basis functions that is optimal for a given task. By using four or more energy bins, it is thus possible, in principle, to quantify several contrast agents independently.

Furthermore, it is still an open research question whether the two-basis approximation is exact down to the precision that photon-counting CT scanners are able to measure. Multiple studies indicate that the two-material approximation is insufficient for performing high-precision measurements of the attenuation coefficient (Williamson et al 2006, Bornefalk 2012b, Alvarez 2013). This suggests that it should be possible to separate more than two components (e.g. water, calcium and iron) from each other when imaging unenhanced human tissue. Measuring the concentration of three materials can be desirable when imaging, for example, vulnerable plaques (Wang et al 2011b) or liver iron overload (Luo et al 2015).

However, even if three-material decomposition of unenhanced tissue turns out to be feasible, it is likely that the high sensitivity to noise in three-basis decomposition will limit its application to large-area tasks (Alvarez 2013), unless prior assumptions about mass or volume preservation are included (section 5.2). The two-basis approximation is also inaccurate at sharp edges in the object, which get a unique spectral signature due to the non-linear partial volume effect (Glover and Pelc 1980), a phenomenon that can be used to obtain subpixel spatial information from spectral measurements (Persson et al 2018a).

One of the main advantages of the PCD is the improved spatial resolution compared to conventional detectors. PCDs that have been designed for full-body clinical CT generally have pixel sizes ranging from 0.225 to 0.5 mm at the detector; this is smaller than conventional EIDs today, which have pixel sizes generally on the order of 1 mm (Leng et al 2016, Persson et al 2014, Shefer et al 2013, da Silva et al 2019). Recent developments have, however, enabled improvements of the spatial resolution of EIDs reaching approximately 0.5 mm pixel pitch at the detector (Yanagawa et al 2018).

The spatial resolution of EIDs has been held back by several practical limitations. To mitigate pixel crosstalk, the detectors are made up of scintillator crystals that have to be diced into pixels, and on the edges of each pixel, there is a reflecting material which keeps the secondary light from escaping the pixel (Shefer et al 2013). The finite thickness of the reflectors creates dead area between the pixels, reducing the geometric efficiency. If the pixels are made smaller, the dead area takes up a larger fraction of the detector area. Semiconductor detectors, on the other hand, do not emit any secondary light and therefore require no reflectors. Instead, the detector consists of a continuous piece of semiconductor material, and these direct conversion detectors are pixelated by charge-collecting electrodes. The pixels can therefore be made smaller without losing geometric efficiency. ⁵

Also, due to the nature of the energy-integrating signal formation, there is a small contribution of integrated electronic noise for each detector channel. If the pixels are made smaller, the number of detector channels per detector area increases, as does the total amount of integrated noise. A PCD, on the other hand, can adapt the lowest energy threshold to the noise floor for each channel and avoid counting noise, even for very small pixels.

Several approaches have been suggested for improving the spatial resolution of scintillator detectors, such as using an attenuating grid to obtain a smaller pixel aperture (Flohr et al 2007). However, this comes at the expense of a substantial reduction in dose efficiency. It has also been proposed to use high-resolution flat-panel detectors for CT (Gupta et al 2006). However, there are some remaining issues that have to be addressed in order to make the technology feasible, including the characteristics of the scintillator material (e.g. speed, afterglow, and lag), object scatter corrections due to removal of the anti-scatter grid, and the dynamic range of the detector.

The benefit of the improvement in spatial resolution made possible by PCDs has been demonstrated on multiple occasions (Leng et al 2018, Bartlett et al 2019, Symons et al 2018a, von Spiczak et al 2018). Figure 4 shows an example of the high-resolution performance of a conventional CT system and a silicon-based full-body PCD prototype. Details about the prototype system can be found in da Silva et al (2019). In this example, the PCD can resolve approximately twice as many line pairs per millimeter as the conventional CT system. Most conventional dual-energy CT systems have limited spatial resolution owing to aspects of the system design. For example, if the spectral separation comes from the source, it is common to use a larger focal-spot size to facilitate a sufficient tube output. Spectral PCD systems, on the other hand, will facilitate simultaneous spectral and high-resolution imaging. Figure 5 shows a comparison of the spatial resolution of a state-of-the-art dual-energy system and a silicon-based full-body PCD prototype for imaging the structures of the inner ear.

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In addition to the improved spatial resolution in the transaxial and longitudinal direction caused by the smaller pixel size, PCDs also have an advantage over EIDs in terms of the angular spatial resolution. The reason for this is twofold: First, PCDs do not suffer from scintillator afterglow which can lead to lag between consecutive measurements in EIDs if the sampling rate is high. Second, the electronic noise level in EIDs tends to increase for higher sampling rate, so that increasing the angular resolution leads to a penalty in dose efficiency. PCDs, on the other hand, are able to reject the electronic noise as will be discussed in section 2.4 and can therefore avoid this dose penalty. This allows higher sampling rates, provided that enough bandwidth is available to read out the data (Sjölin and Danielsson 2017). Potentially, the faster sampling rates could also be used to detect rapid motion.

With an increased spatial resolution of the detector comes a risk of introducing aliasing artifacts in the reconstructed image. This can be avoided by maintaining sufficient radial and angular sampling rates. In the radial direction, the sampling rate is automatically increased since the sampling interval (pixel center-to-center distance) is reduced. In the angular direction, on the other hand, the sampling rate (number of views per revolution) must by increased in order to avoid visible angular blur and aliasing artifacts.

As a final remark on spatial resolution, we note that the detector pixel size and the angular sampling rate are not the only factors impacting the spatial resolution of the imaging system. In order to fully utilize the increased spatial resolution of the detector, appropriate adjustments must be made to the focal spot size, image reconstruction grid, and reconstruction algorithms. All of these aspects significantly impact the CT imaging chain.

A further application of photon-counting imaging is low-dose imaging. There are two reasons why PCDs are able to provide superior performance for low-dose imaging tasks compared to energy-integrating detectors.

The first reason is the overall improvement in image quality achievable with PCDs, as described in previous sections. This includes CNR improvement through better utilization of the energy information in the detected x-ray beam and improvement in visualization of small objects. Since the detectability of a feature in the image increases with dose, these potential improvements in image quality at equal dose can be traded for a lower dose by keeping the image detectability fixed and lowering the tube current instead.

Second, PCDs have a particular advantage for imaging with low dose levels, namely their ability to reduce the impact of electronic noise. PCDs use a threshold to separate real counts from noise, and by setting the threshold high enough above the noise floor, the electronic noise can be rejected. In this case, the only degradation caused by electronic noise is a small broadening of the energy-response function due to random fluctuations in the pulse-height measurements. EIDs, on the other hand, measure the total x-ray energy deposited during a certain time interval, and the electronic noise will be included in the measurement as a random additive term. The magnitude of this term is relatively constant, whereas the quantum noise is proportional to the incident fluence rate; therefore the electronic noise can go more or less unnoticed at high doses but becomes prohibitive at low doses or for large patients (Duan et al 2013). On the other hand, the detective quantum efficiency (DQE) (see section 6.3) of a PCD is, theoretically, constant down to the zero-flux limit. The relative advantage of photon-counting CT compared to energy-integrating CT is therefore larger at low dose levels (Yu et al 2016a). PCDs may therefore allow new low-dose imaging protocols at dose levels where electronic noise is prohibitive with current state-of-the-art scanners, with potential applications in pediatric imaging and lung cancer screening (Symons et al 2016).

PCDs for x-ray CT are so-called direct-conversion detectors, wherein the x-ray photons are converted directly into an electric signal as opposed to first being converted to visible light. PCDs for CT generally consist of semiconductor sensors with an applied bias voltage. An interacting photon creates a cloud of charge carriers, creating a signal which is processed and registered by an application-specific integrated circuit (ASIC). Each detected photon interaction results in a count in an energy bin corresponding to the energy deposited in the interaction.

Despite many years of development, PCDs have only recently approached the performance necessary for being used in clinical CT scanners. The main challenges are achieving good performance at high count rate, obtaining good spectral fidelity, and manufacturing a full-field detector with sufficiently low density of imperfections at a competitive cost. In the following, we examine the different detector design considerations that need to be accounted for in order to optimize detector performance.

There are currently two main converter material candidates: cadmium (zinc) telluride (CdTe or CZT) and silicon (Si). Both candidates have pros and cons, some of which we will mention here, and their properties are summarized in table 1. More comprehensive accounts for the properties of the detector material and the functionality of the ASICs can be found in the literature (Ballabriga et al 2016).

3.2.1. Detector physics

One of the main differences between CdTe/CZT and Si detectors is the relative x-ray stopping power. CdTe has a high linear attenuation coefficient, requiring only about 1.7 mm to stop 95% of the x-rays in a 120 kVp spectrum filtered by 30 cm of water. Silicon, on the other hand, has a relatively low atomic number and requires roughly 55 mm to stop the same fraction of x-rays under the same conditions. As it is not feasible to make the semiconductor wafers much thicker than a few millimeters, the silicon wafers are mounted edge-on with respect to the incoming x-rays (Bornefalk and Danielsson 2010). This way, the effective depth of the detector is determined not by the thickness of the wafer, but by the length of the wafer, allowing the detector to have as long an absorption length as necessary. See figure 6 for an illustration of the face-on (CdTe/CZT) and edge-on (Si) geometries.

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Another difference between the detector materials is the prevalence of the different types of x-ray interactions that occur in the material. In materials containing heavy elements (high-Z materials), such as CdTe/CZT, the main interaction mechanism is photoelectric absorption; Compton and Rayleigh scattering accounts for only a few percent of the total absorption. Silicon, on the other hand, has a high probability of Compton interactions, which dominate over the photoelectric effect for photons with energy higher than 48 keV. An example of the distribution of primary interactions in a 50 mm Si detector and a 1.6 mm CZT detector is shown in table 2. The distribution of interactions is computed for a typical 120 kVp x-ray tube spectrum that has been filtered through 30 cm of water (Cranley et al 1997, Hubbell and Seltzer 2004).

Table 2. The distribution of primary interactions in a 50 mm deep Si detector and a 1.6 mm deep CZT detector for a 120 kVp spectrum attenuated by 30 cm of water.

Interaction	Si	CZT
Total	0.94	0.94
Photoelectric	0.28	0.87
Compton	0.56	0.04
Rayleigh	0.10	0.04

However, keep in mind that it is not straightforward to assess the signal quality based on the distribution of primary interactions in the detector material. For example, in the event of a Rayleigh scattering, there is a high likelihood that the photon will scatter with a small angle and interact again within the same detector pixel. In the same way, the charge from a photoelectric interaction can be shared between two pixels, distorting the measured energy and resulting in a double count. A Compton-scattered photon deposits parts of its energy in the first interaction and continues in a new direction. If a Compton-scattered photon is detected at least once, it adds to the detection efficiency of the detector. However, the spectral information is largely lost. If the Compton-scattered photon is detected more than once, then the signal is degraded due to count multiplicity (see section 4.2).

For high-Z detector materials, there is a high probability of K-fluorescence emission, namely, a photoelectric interaction with a high-Z element (e.g. Cd and Te) results in the emission of K-shell characteristic x-rays (Shikhaliev et al 2009). For CZT, approximately 70% of all photoelectric absorptions result in the emission of characteristic x-rays (Krause 1979). The energies of the emitted characteristic x-rays lie between 23 and 27 keV, and the mean free path in the detector material is approximately 120 µm. The emission and reabsorption of characteristic x-rays result in a skewing of the energy spectrum (see section 4.4), a reduction of the spatial resolution (see section 4.3), and count multiplicity, which reduces the quantum efficiency of the detector (see section 4.2).

As illustrated in this section, the detection mechanisms of PCDs are complex. In the end, what matters for the signal quality is (1) the number of photons that are registered at least once (section 4.1), (2) the count multiplicity (section 4.2), (3) the spatial distribution of the counts (section 4.3), and (4) the associated spectral response (section 4.4). To get a full picture of the imaging performance, these effects must be evaluated simultaneously (section 6.3).

3.2.2. Collection of charge carriers

3.2.3. Sensor thickness

3.2.4. Charge sharing

3.2.5. Electronic noise

3.2.6. Depth segmentation

3.2.7. Imperfections in the detector material

3.2.8. Other factors

3.3.1. The analog channel

3.3.2. The digital channel

The ASIC channel on a multi-bin PCD has several pulse-height comparators (thresholds) that are used to identify the arrival of a new photon pulse and to estimate the energy of the photon interaction (see figure 7). Each comparator compares the amplitude of the output signal from the analog channel to a programmable reference voltage supplied by a digital-to-analog converter. The comparator returns a one or a zero depending on whether the signal exceeded the reference voltage or not. The photon counts are categorized into a set of counters (energy bins), generally one for each comparator.

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The number of energy thresholds varies between different photon-counting ASICs (Taguchi and Iwanczyk 2013). In order to perform a two-basis material decomposition, it is necessary to have only two energy thresholds. However, the performance of the material decomposition depends on the position of the thresholds, and the optimal selection of the thresholds is task dependent. Having more thresholds ensures that a close-to-optimal CNR can be obtained for all imaging tasks in a scan (Shikhaliev 2008, Zheng et al 2020), and that close to the minimum possible noise can be achieved in material-specific images (Alvarez 2011, Faby et al 2015). Also, more thresholds can improve the effectiveness of corrections, such as charge-sharing or pileup correction (see section 3.3.2).

The way the digital part of the ASIC analyzes the output from the comparators is commonly referred to as the counting mode, and the exact implementation differs between ASICs. The counting modes are generally categorized as either paralyzable or non-paralyzable, referring to the behavior of the channel under heavy pulse pileup (section 4.5).

For a non-paralyzable detector, when a photon pulse crosses the lowest threshold, a counter is incremented, and a dead time is initiated during which no other photons are counted. After the dead time, the channel can count again. The dead time is selected such that the signal induced by the longest input pulses are below the lowest energy threshold at the end of the dead time. With increasing number of detected photons per second on the channel, the output from the non-paralyzable channel approaches a fixed number (the maximum number of dead times that fit within a single readout interval).

In a paralyzable detector, the dead time is extended until the signal drops below the threshold. At high flux rates, pileup can cause the signal level to exceed the threshold level for longer periods, resulting in increased dead time. Paralyzable behavior occurs, for example, if the detector is configured to count the number of times the input signal crosses each threshold, known as the threshold-crossing frequency. In presence of severe pileup, the input signal does not fall below the highest threshold, and the threshold-crossing frequency therefore drops to zero.

It is also possible to implement other counting modes, such as detection of local maxima (peak sample and hold) using the comparators, and Hsieh and Pelc (2016) have shown that more sophisticated counting modes can substantially improve the performance of the detector.

3.3.3. Charge summing and anti-coincidence logic

The key figure of merit for the dose efficiency of an x-ray detector is the quantum efficiency, which describes how many of the incoming photons are detected. For the quantum efficiency, it does not matter if the photon is detected in a photoelectric event or a Compton-scattering event, or if it is affected by K-fluorescence or charge sharing. As long as the photon is detected at least once, it contributes to the detection efficiency.

Naturally, the thickness of the detector material impacts the quantum efficiency. However, thicker detectors can affect other important aspects of the detector performance. For face-on detectors, a thicker material affects the charge collection time, and therefore the energy response, via an increase of charge trapping/recombination and charge sharing. Edge-on detectors do not suffer from the same downside to increasing the depth of the detector, apart from the increase in detector material and the resulting more stringent demands on alignment precision, since the distance the charge travels remains the same.

The position of the lowest threshold can have a large impact on quantum efficiency by determining how many low-amplitude input pulses are detected. Any process of spectral degradation that causes some of the photon pulses to become lower than the lowest threshold leads to a reduction of the DQE. For Si, this effect can be caused by charge sharing and Compton scattering. For CdTe/CZT, this effect can, in addition, be caused by K-fluorescence emission/reabsorption and the trapping/recombination of charge carriers. Photon-induced pulses dropping below the lowest energy threshold as a consequence of charge sharing can lead to regions of insensitivity at the boundaries between pixels (Tlustos et al 2006). The lowest energy threshold also determines the number of double counts to a large extent by inclusion or exclusion of low-energy charge sharing and K-fluorescence reabsorption. For Si, the lowest threshold is often set at around 5–15 keV in order to include many Compton interactions (Bornefalk and Danielsson 2010, Persson et al 2014), and for CdTe/CZT, the lowest energy threshold is often assumed to lie at 20–25 keV (Xu et al 2011, Shikhaliev 2009, Yu et al 2016a). In practice, it may not be optimal to reject electronic noise completely, but instead to operate the detector with the threshold at a low enough level where a small number of false counts are generated by the noise floor, as long as the extra noise is outweighed by the benefit of detecting more primary detection events (Rajbhandary and Pelc 2018).

The semiconductor wafer is not necessarily sensitive in the entire bulk of the detector. For CdTe/CZT, there can be a dead layer near the back side of the sensor which is insensitive to x-rays (Matsumoto et al 2000, Moralles et al 2007), ⁶ and for Si, there is generally an inactive guard ring around the wafer protecting against leakage current. The dead layers can have a large impact on the performance of the detector, in particular for CdTe/CZT since the detector material is highly attenuating. For example, a 20 µm dead layer on a CdTe/CZT detector reduces the quantum efficiency by approximately 9%, whereas a 200 µm guard ring on a Si detector reduces the quantum efficiency by approximately 1.5% (for a 120 kVp spectrum attenuated by 10 cm of water). A dead layer on CdTe/CZT detectors can also form as a consequence of high x-ray flux (Du et al 2002). Furthermore, there is a dead layer close to the pixels, where an interaction gives rise to charges that only move through part of the weighting field and therefore generate a reduced signal (Boucher 2013).

As a consequence of scatter, K-fluorescence, or charge sharing, a photon can be counted more than once. Count multiplicity gives extra weight to a fraction of the photons, resulting in a reduction of the zero-frequency DQE and noise correlations between neighboring pixels (Michel et al 2006).

As a consequence of count multiplicity, the zero-frequency DQE is reduced by a factor

$\begin{equation} I_\text{multiplicity} = \frac{\left(\sum_{n = 1}^{\infty} n r_n \right)^2 }{\sum_{n = 1}^{\infty} n^2 r_n} , \end{equation} \tag{ 3 }$

where r_n is the fraction of detected photons that are counted n times. Comparing equation (3) to the Swank factor of an EID (1), we can see that the expressions are identical apart from that the signal amplitude () in equation (1) is exchanged for the multiplicity (n) in equation (3).

Counting photons in more than one pixel results in pixel-to-pixel correlations and a non-white noise power spectrum (NPS) (Xu et al 2014). The detector NPS can be obtained by taking the discrete Fourier transform (DFT) of the auto-covariance of the counts measured by the detector array (Cunningham 2000). In the special case for which a fraction p of the λ photons that are counted in a pixel are counted also in the neighboring pixels (p/2 to the left and p/2 to the right), the auto-covariance is given by

$\begin{equation} \mathrm{Cov}\left[ N_{0}, N_{i} \right] = \begin{cases} \lambda(1 + p), & \text{for } i = 0\\ \lambda p, & \text{for } |i| = 1\\ 0, & \text{for } |i| \gt 1,\\ \end{cases} \end{equation} \tag{ 4 }$

where N_i is the number of counts in the ith pixel. To illustrate the effect of the pixel-to-pixel correlations on the noise, consider the NPS normalized by the square of the mean number of counts in each pixel:

$\begin{equation} \mathrm{NPS}_\text{norm}(u) = \frac{\mathrm{NPS}(u)}{\lambda^2(1+p)^2}, \end{equation} \tag{ 5 }$

where $\mathrm{NPS}(u) = \mathrm{DFT}[\mathrm{Cov}\left[N_{0}, N_{i} \right]]$, and u is spatial frequency. Examples of $\mathrm{NPS}_\text{norm}$ for different values of p are shown in figure 8 (the curves have been normalized by 1/λ to obtain unity $\mathrm{NPS}_\text{norm}$ for p = 0, and the spatial frequency axis is normalized by the sampling frequency). The increase of $\mathrm{NPS}_\text{norm}$ at the zero frequency is inversely proportional to the decrease of the zero-frequency DQE given by (3):

$\begin{equation} \mathrm{NPS}_\text{norm}(u = 0) = 1/I_\text{multiplicity} = \frac{(1-p) + 4n}{((1-p) + 2n)^2}. \end{equation} \tag{ 6 }$

An important observation is that the pixel-to-pixel correlations lead to a higher number of events detected in each pixel; each pixel registers λ(1 + p) photons, as opposed to only λ. This can also be seen as the integral of the normalized NPS reduces as p increases. When the pixels are considered together, however, it becomes clear that the quantum efficiency has indeed been reduced, as indicated the increase of the zero-frequency NPS. This property of correlated noise must be considered when, for example, evaluating the detector's CNR performance.

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The spatial resolution of the PCD is determined, first and foremost, by the center-to-center distance between the electrodes on the semiconductor surface. For edge-on detectors (e.g. silicon), the thickness of the semiconductor wafer determines the spatial resolution in one of the dimensions.

Charge sharing, K-fluorescence reabsorption, and Compton scattering degrade the spatial resolution somewhat by causing events to be detected in positions other than where they first interacted with the detector. This effect leads to a blur of the point-spread function (PSF), and therefore a degradation of the modulation transfer function (MTF). These effects can partially be corrected using anti-coincidence logic or similar methods.

Charge sharing, unless corrected, also limits the smallest feasible pixel size indirectly due to the increase in count multiplicity and the degradation of the energy response of the detector. If pixels that are too small are used, the detector will lose much of its spectral capabilities and some of its dose efficiency (Rajbhandary et al 2018).

The spatial resolution of the detector is not constant over the range of detected energies. High-energy interactions are predominantly registered close to the center of the pixel, since charge sharing occurs at the pixel boundaries, giving them a more narrow PSF, whereas low-energy interactions can be registered over the full extent of the pixel (Stierstorfer et al 2019). This implies that the detector MTF has an energy dependence, and the spatial resolution can be enhanced by giving more weight to the high-energy bins at the cost of increased noise in the reconstructed image. In the case that monoenergetic images are formed from material basis maps, the spatial resolution will therefore depend on the choice of monoenergetic energy. In order to have a full characterization of the spatial resolution of the detector, the full energy range should be evaluated.

4.3.1. The spatial resolution and noise trade-off

A common misconception when it comes to high-resolution detectors is that having small pixels comes with a noise penalty. Indeed, the noise in each detector pixel will increase if the pixels are smaller, since fewer photons are detected in each pixel. However, high-resolution image acquisition improves the resolution-noise trade-off in the reconstructed image. That is, for the same image noise, the high-resolution system will have better spatial resolution at equal dose, and conversely, for the same spatial resolution, the high-resolution system will have lower noise. The literature predicts noise variance reductions between 14% and 83% from a 2× increase in native resolution (Baek et al 2013, Kachelrieß and Kalender 2005). The highest noise reductions are observed when reconstructing high-resolution images. For low-resolution images (e.g. for soft-tissue imaging) the benefit is reduced. In addition, the high-resolution system can use reconstruction kernels with higher cut-off frequency without introducing aliasing artifacts than a system with lower native resolution. An example of the benefit of a native high-resolution detector is shown in figure 9, in which the two images to the left are acquired using a conventional CT system reconstructed with a soft kernel (a) and a bone kernel (b), and the right-most image (c) was acquired using a prototype PCD, with roughly half the pixel size, and reconstructed using a high-resolution kernel. The native high-resolution acquisition of the PCD improves the spatial resolution and noise trade-off, and allows reconstructing high-resolution images with less noise and aliasing artifacts.

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In contrast to properties like detection efficiency and spatial resolution, which are important for all detectors, the ability to measure the energy distribution of the incident spectrum is unique to energy-resolving detectors. The precision with which each deposited energy is measured is partly determined by the number of energy thresholds (see section 3.3.2). At the same time, the number of thresholds is not the only factor that affects the spectral imaging performance.

The energy-resolving capability of PCDs is also determined by non-ideal effects in the detector materials and the readout electronics that give a non-ideal energy response, i.e. cause the registered signal to deviate from the expected amplitude. In a hypothetical ideal PCD, each photon gives rise to an electrical signal with amplitude proportional to the incident photon energy. In CdTe-based and CZT-based detectors (figure 10(a)), K-fluorescence gives rise to two secondary clusters of peaks in the detected spectrum distinct from the photopeak: fluorescence peaks at the fluorescence energy (23 and 26 keV for Cd, and 27 and 31 keV for Te), (Thompson et al 2009) and K-escape peaks at the original energy minus the fluorescence energy (Shikhaliev et al 2009, Xu et al 2011). The response of a silicon-based detector (figure 10(b)), on the other hand, exhibits a large fraction of Compton interactions at low energies and reproduces the photoelectric part of the spectrum with diminished magnitude, but it is otherwise not distorted in the way characteristic of the response from a CdTe-/CZT-based detector (Bornefalk and Danielsson 2010). In both CdTe/CZT and Si detectors, charge sharing causes spectral tailing, which is a background of charge-shared counts of relatively constant magnitude (figure 10) (Xu et al 2011, Liu et al 2014). Spectral tailing can also be caused by incomplete charge collection (section 3.2.7).

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In addition, electronic noise in the readout electronics causes further spectral degradation by adding a random error to the registered amplitude of each pulse. In practice, this characteristic has the effect of convolving the detected spectrum with a Gaussian kernel. Typical photopeak widths reported for detectors designed for high-flux x-ray imaging are 3.5–5.4 keV full width at half maximum (FWHM) for Si (Xu et al 2013a) and 5–10 keV FWHM for CdTe/CZT (Iwanczyk et al 2009, Brambilla et al 2013, Barber et al 2015). Spectral response models incorporating physical effects and electronic properties have been published for silicon (Liu et al 2015a, 2015b) and CdTe/CZT (Schlomka et al 2008, Cammin et al 2014, 2016, Taguchi et al 2016, 2018b).

The effect of the spectral distortion is that it degrades the performance of the detector for tasks that have a strong energy-dependent component. While the performance for density-imaging tasks, such as detection of non-enhancing tumors, is only weakly affected by non-ideal energy resolution (Schmidt 2010), imaging tasks involving calcium and iodine are more strongly affected (Rajbhandary et al 2018). Good energy-resolving capability is especially important for material quantification tasks, since these cannot be performed at all with a non-energy-resolving CT. The impact of the detector energy response on material quantification performance has been investigated by several authors (Roessl and Herrmann 2009, Wang and Pelc 2012, Rajbhandary et al 2018).

Apart from the detector energy response, the incident spectrum shape is also an important factor affecting spectral imaging performance, and multiple studies have investigated how performance can be optimized by varying the x-ray tube voltage (kVp) and prepatient filtering, such as with a K-edge filter (Wang and Pelc 2012, Chen et al 2015, Atak and Shikhaliev 2015, 2016). To understand how the incident spectrum shape interacts with the detector response function, note that misregistration of photon energies does not degrade spectral imaging performance if it is possible to correct for the spectral distortion and thereby recover the original spectrum. Instead, degradation of imaging performance occurs only when there is overlap between detected energies originating from different incident energies, so that it is impossible to recover the incident spectrum without ambiguity (Persson et al 2018b). Figure 11 illustrates how K-fluorescence can give rise to spectral overlap, depending on the incident energy spectrum distribution. K-fluorescence and charge sharing give rise to spectral tailing that overlaps with incident energies present in the true spectrum, which exacerbates the harmfulness of these effects on the spectral imaging performance. On the other hand, Compton scatter events in a silicon-based detector do not overlap much with the spectrum of photoelectric events. Instead, their detrimental effect on energy-resolving imaging stems from the fact that one energy of a Compton event can correspond to a range of incident energies. The Compton events are therefore less useful than the photoelectric events for determining the incident energy distribution, but they do not confound the spectral information provided by the photoelectric events.

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Photons that interact within a short time period 'pile' onto each other, resulting in count loss, distortion of the Poisson counting process, and degradation of the energy information. In order for a PCD to be feasible for clinical CT, the detector has to be able to cope with very high fluence rates that can exceed $3.5 \times 10^8\,\mathrm{mm}^{-2}\,\mathrm{s}^{-1}$ at the detector (Persson et al 2016). Figure 12 shows how the photon fluence rate is distributed on the detector for an imaging protocol that is demanding in terms of high-count-rate capability. The effects of pulse pileup can be mitigated by decreasing the pixel size, but the benefit is limited since small pixels are more prone to charge sharing, which causes double counting and spectral degradation (Hsieh et al 2018). The mathematical description of pulse processing by an electronic channel is known as a dead-time problem, and these problems have been studied extensively for several decades and have applications in many scientific fields (Müller 1973, Carloni et al 1970, Wielopolski and Gardner 1976). Still, analysis of the effects of pulse pileup in spectral PCDs, and how to account or correct for this effect, is an active topic of investigation today.

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4.5.1. Statistical degradation

In the presence of pileup, the output count process is no longer Poisson distributed. A simple, but commonly used, model of the photon-counting process is the so-called ideal non-paralyzable (INP) model (Tenney 1984). The INP model predicts the statistics and spectral distortion under two simplifying conditions: (1) there is a fixed dead time after each photon detection during which all additional events are lost, and (2) the registered energy in the case of pileup is the sum of the energy of the associated photon interactions. The mean and variance of the output count process $\mathrm{Y}$ for the INP model is given by

$\begin{equation} \mu_\mathrm{Y} = \frac{\lambda}{1 + \lambda \tau}, \: \: \: \: \: \sigma^2_\mathrm{Y} = \frac{\lambda \tau}{(1 + \lambda \tau)^3}, \end{equation} \tag{ 7 }$

where λ is the input count rate and τ is the dead time. The mean value and the variance of the output count rate for a non-paralyzable detector with non-zero pulse length, the so-called semi-nonparalyzable (SNP) model (Xu et al 2011), have been derived analytically (Grönberg et al 2018) and are given by

$\begin{equation} \mu_\mathrm{Y} = \frac{\lambda}{e^{-\lambda \tau_s} + \lambda \tau}, \: \: \: \: \: \sigma^2_\mathrm{Y} = \frac{\lambda e^{-\lambda \tau_s} (2 - e^{-\lambda \tau_s} ) }{(e^{-\lambda \tau_s} + \lambda \tau)^3}, \end{equation} \tag{ 8 }$

where λ is the input count rate, τ is the dead time, and τ_s is the average time at the end of the dead time during which, if a photon interacts, a new dead time is initiated directly after the first dead time. The length of τ_s depends on the length and shape of the photon-induced pulses and is often a large fraction of τ (0.77%–0.86% in Liu et al (2016)). Examples of the mean value and the variance predicted by the INP and the SNP model with τ_s  = 0.8 as a function of count rate is plotted in figure 13. The input and output count rates have been normalized by the characteristic count rate N₀ = 1/τ. In the ideal case, without pileup and with Poisson statistics, the output count rate is equal to the input count rate, and the variance is equal to the mean value.

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Pileup introduces a non-linear response to variations in the flux rate, and the statistics start to deviate from Poisson behavior. The effect of pileup on image quality is therefore not straightforward to assess, and looking at, for example, count-rate linearity tells only half the story. One way to incorporate both the effect of non-Poisson statistics and the non-linearity of the output count rate is to consider the CNR for the task of differentiating between two similar projection measurements with a small difference δ in photon flux. A flux-dependent $\mathrm{dose efficiency}$ can then be defined as the ratio of the CNR² with and without pileup:

$\begin{equation} \mathrm{DE}(\lambda) = \frac{\text{CNR}_\text{with pileup}^2}{\text{CNR}_\text{without pileup}^2} = \frac{ \left( \mu_\mathrm{Y}(\lambda + \delta) - \mu_\mathrm{Y}(\lambda) \right)^2 / \sigma^2_\mathrm{Y}(\lambda) }{(\lambda + \delta - \lambda)^2 / \lambda } = \frac{\left( \mu_\mathrm{Y}(\lambda + \delta) - \mu_\mathrm{Y}(\lambda) \right)^2 }{\delta^2 \sigma^2_\mathrm{Y}(\lambda) / \lambda }. \end{equation} \tag{ 9 }$

An example of the dose efficiency as a function of count rate is plotted in figure 14(a), and the CNR² versus the thickness of attenuating material (here water) is shown for different maximum count rate (N₀, 2N₀, and 3N₀) in figure 14(b). As an example of a maximum input count rate, consider a case for which the maximum x-ray flux on the detector is $3.5 \times 10^{8} \: \mathrm{mm}^{-2}\,\mathrm{s}^{-1}$ (Persson et al 2016), and the detector has 0.25 × 0.25 mm² pixels and a 100 ns dead time; then the maximum input count rate is 2.2N₀ ($3.5 \times 10^{8}\: \mathrm{mm}^{-2}\mathrm{s}^{-1} \times0.25^2 \:\mathrm{mm}^{2} \times 100\times 10^{-9} \:\mathrm{s} = 2.2$). Even though the dose efficiency is reduced for the higher count rates, the CNR² is still high since pileup mainly affects projections that have high statistics.

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The effect of the statistical degradation from pileup in the final CT image is limited by several factors. First, the central rays, where the x-ray flux is generally low due to patient attenuation, contribute much more to the image than the peripheral rays due to the nature of tomographic image reconstruction. Second, the x-ray flux at the periphery of the patient is lowered by the bowtie filter of the x-ray tube. Also, pileup increases the noise for high-flux projections, for which the SNR is high (see figure 14(b). Combined, these effects help mitigate the impact of statistical degradation from pulse pileup.

4.5.2. Spectral distortion

The most straightforward ways of generating reconstructed images from spectral CT data are either summing the counts in all energy bins before reconstruction, thereby generating a conventional-looking CT image, or reconstructing a CT image from the measured data in each energy bin (Yu et al 2016c). In comparison with material decomposition, reconstructing bin images has the advantage of being easy to implement since it does not require accurate models of the source and detector. On the other hand, this method has the drawback of not being able to correct for the effect of beam hardening. The bin images can be reconstructed with filtered backprojection or with an iterative algorithm similar to those used for single-energy CT (Salehjahromi et al 2018). By summing the images with carefully selected weight factors, the optimal CNR for a specific imaging task can then be attained (Schmidt 2009).

Another option is to display the images individually, which has the benefit of showing the attenuation in a narrow energy range, but leads to high noise since only a subset of the photons are used to form the image. This trade-off between precise spectral information and noise can be mitigated by applying postprocessing algorithms that act jointly on the different bin images and exploit similarities between them to reduce noise. An example is the HYPR-LR algorithm, in which low spatial frequencies from the individual bin images is combined with high spatial frequencies from a full-spectrum image, thereby generating images combining spectral information with low noise (Leng et al 2011). This type of bin image noise reduction can also be incorporated in an iterative reconstruction algorithm, such as the spectral prior image constrained compressed sensing (SPICCS) algorithm, which uses total variation regularization to penalize structural differences between the bin images and a full-spectrum image (Yu et al 2016b).

Other, more sophisticated regularization methods that exploit redundancy between the bin images have also been studied, including total nuclear variation (Rigie and La Rivière 2015), total generalized variation (Niu et al 2018b), and dictionary learning (Zhang et al 2017). Denoising based on non-local means filtering where the broad-spectrum image is used to weight pixels according to similarity was studied by Li et al (2017) and Zhang et al (2016 combined non-local-means-like regularization in the spectral dimension with SPICCS. Block-matching, where patches with similar structure in different parts of the image are identified in the broad-spectrum image and used to denoise each other (Harrison et al 2017) have also been demonstrated. The observation that most of the image pixels tend to be spanned by a small number of basis materials, whereas k-edge elements are often confined to localized regions, can be exploited for noise reduction as shown by Gao et al (2011). There are also algorithms exploiting the low dimensionality of spectral CT images in combination with SPICCS (Zeng et al 2020) or patch-based denoising (Niu et al 2018a). Finally, a new and promising research direction is the application of convolutional neural networks for bin image denoising (Clark and Badea 2020).

A standalone material-decomposition step can be performed before the reconstruction (in projection space) or after the reconstruction (in image space). Another possibility is to perform reconstruction and material decomposition simultaneously, as discussed in section 5.3. Image-space decomposition is straightforward to implement, since it can take the form of a simple linear transformation of the energy-selective reconstructed images (Persson et al 2014, Symons et al 2017a). However, this does not compensate for the non-linearity of the x-ray attenuation and therefore does not remove beam-hardening artifacts completely. Iterative image-space decomposition has been proposed as a way of solving this problem (Heismann and Balda 2009).

A more straightforward way of eliminating beam hardening, thereby obtaining a quantitatively reliable image, is by using projection-space material decomposition, where the counts in the different energy bins are mapped to line integrals of basis coefficients before reconstruction. This method is well suited for photon-counting CT with its perfect geometric alignment between energy bins, but it can be more difficult to implement for systems exhibiting mismatch between the high- and low-energy projection rays, such as dual-source CT. In statistical terminology, the mapping from counts to line integrals is called an estimator. If the number of energy bins is equal to the number of basis materials, this estimator can be a simple mapping from counts to estimated line integrals, given by a polynomial fit, for example (Alvarez and Macovski 1976). If there are more energy bins than basis functions, the choice of estimator is more complicated.

There are a number of statistical properties that are desirable for such an estimator: ideally, it should be unbiased, meaning that the output should be equal to the true projected basis coefficient on average; the output should preferably have low variance; and it should be robust in the sense that it is able to perform well under a variety of conditions without generating image artifacts. For example, streak artifacts may result if the estimated basis sinograms contain anomalous measurements.

The maximum-likelihood estimator is commonly used in photon-counting CT research (Roessl and Proksa 2007, Schlomka et al 2008, Si-Mohamed et al 2018). This estimator builds on the fact that the registered counts m_i in each energy bin i is Poisson distributed and finds the set of projected basis coefficients $\boldsymbol{A}$ such that the x-ray beam is most likely to have passed through, given the measured counts. This estimator has the favorable property of being asymptotically unbiased and efficient for large measured count numbers, meaning that the variance approaches its theoretical minimum given by the Cramér–Rao lower bound (CRLB) (see section 6.2) (Kay 1993).

Mathematically, the maximum-likelihood estimate is calculated by minimizing the negative log-likelihood function

$\begin{equation} l(\boldsymbol{A}) = \sum_{i = 1}^{N_b} \lambda_i(\boldsymbol{A})-m_i\ln\lambda_i(\boldsymbol{A}) \end{equation} \tag{ 10 }$

with an iterative algorithm. This equation builds on a model for the expected number of counts $\lambda_i(\boldsymbol{A})$ in each of the N_b energy bins, as a function of the projected basis coefficients $\boldsymbol{A}$. An example of such a model is

$\begin{equation} \lambda_i(\boldsymbol{A}) = \int S(E)w_i(E)\mathrm{e}^{-\sum_{j = 1}^{N_m} A_j f_j(E)}dE \end{equation} \tag{ 11 }$

with N_m denoting the number of basis materials, S(E) denoting the spectral density as a function of energy E, and w_i (E) denoting a weight function describing the spectral sensitivity of the energy bin to different incident photon energies. Ehn et al (2016) proposed a modification of this forward model that can be fitted well to a limited number of calibration measurements.

One drawback of the maximum-likelihood estimator is that it is calculated with an iterative algorithm, which is time-consuming. Other estimators have therefore been proposed with the aim of increasing speed. Alvarez proposed using a linearized, approximate forward model to obtain a linear estimator, which is then corrected using a look-up table to reduce bias (Alvarez 2011, 2016). As an extension of this method, different linearized forward models can be applied in different regions of the space of possible $\boldsymbol{A}$ values (Rajbhandary et al 2017).

Another estimator was proposed by Lee et al (2017a, 2017b). This estimator is a three-step method which builds on approximating the set of possible transmittance functions that can be expressed as $X(E) = \mathrm{e}^{-\sum_{j = 1}^{N_m} A_j f_j(E)}$ as a low-dimensional space. This allows the decomposition problem to be formulated as follows: (1) least-squares fitting of the transmittance X(E) that best agrees with the registered counts; (2) least-squares fitting of a combinations of basis functions $\sum_j A_j f_j(E)$ to $\mathrm{{log}} \; X(E)$; and (3) post-correcting for bias. In recent years, neural network estimators have also attracted attention. These are based on networks of simple processing units, called neurons, that are trained to map measured bin counts directly to the corresponding basis coefficient line integrals (Zimmerman and Schmidt 2015).

Since basis images tend to be noisier than conventional CT images, significant attention has been paid to methods of reducing basis image noise. In analogy with the SPICCS algorithm for bin image denoising, the prior-knowledge-aware iterative denoising method uses an image reconstructed from the sum of all energy bins as a prior image and penalizes structural differences between this image and basis images, after rescaling (Tao et al 2018). More advanced basis decomposition methods, such as block-matching methods (Wu et al 2019), dictionary learning (Wu et al 2020) or neural-network-based methods (Chen and Li 2019), are able to reduce noise by aggregating information from multiple pixels when forming the basis images.

Another way of reducing basis image noise is by enforcing a constraint, such as volume or mass conservation, in the basis decomposition (Liu et al 2009, Mendonça et al 2010, Ronaldson et al 2012, Malusek et al 2013, Lee et al 2014, Curtis and Roeder 2017, Ren et al 2019). Such a constraint provides an additional equation in addition to the x-ray measurements, allowing three-material decomposition with two energy bins or four-basis decomposition with three energy bins. Even if a sufficient number of energy bins are available, it may be desirable to include constraints in order to reduce noise in the decomposed basis images. A drawback of this approach is that the constraints build on prior assumptions about the material compositions that can occur in the body, and if these assumptions are in disagreement with reality, including the constraints will result in systematic errors in the resulting basis images (Bornefalk and Persson 2014, Yveborg et al 2015a). Whether the benefits of such constraints outweigh the drawbacks is therefore dependent on the specific imaging task.

The process of reconstructing photon-counting images depends on which material-decomposition algorithm is used. An ordinary single-channel reconstruction algorithm such as filtered back-projection can be used to reconstruct one image for each energy bin, allowing image-space material decomposition to be performed subsequently. Projection-space material decomposition, on the other hand, can be implemented either as a one-step or two-step inversion, depending on whether the mapping from measured counts to basis images takes place in a single processing step or whether basis sinograms are generated as an intermediate step.

In two-step inversion, projection-space material decomposition is first used to generate a set of basis sinograms, one for each basis material (Schlomka et al 2008). From these basis sinograms, a set of reconstructed basis images is then generated with a reconstruction algorithm such as filtered back-projection or an iterative method. In the case of an iterative method, there is the possibility of choosing whether the image reconstruction should treat the different bases independently (Schirra et al 2013) or as a joint optimization problem (Sawatzky et al 2014). The noise in the basis images resulting from basis decomposition is anti-correlated between basis materials, since the basis decomposition algorithm is able to measure the total attenuation more accurately than it can measure the exact amount of each basis material in the beam path. By modifying the data weighting to take this anti-correlated noise structure into account, the reconstructed image noise can be reduced. At the same time, the coupling between basis materials introduced in this way increases the complexity of the reconstruction problem and can also lead to artifacts caused by crosstalk between the basis images (Persson and Grönberg 2017).

In one-step inversion, on the other hand, the mapping from registered counts to basis images takes place in one single step, eliminating the ray-wise decomposition step. This step typically takes the form of an iterative optimization, which searches for the set of basis images that best corresponds to the measured counts in each projection line and energy bin. In this case, the reconstruction necessarily treats the basis images as a joint optimization problem. The development of algorithms that optimize image quality while satisfying computational requirements is an active research topic. Several one-step algorithms have been proposed, based on primal-dual optimization (Barber et al 2016) or separable quadratic surrogates (Long and Fessler 2014), which can be accelerated with momentum terms (Mechlem et al 2018). A comparison of the computational cost of five such algorithms can be found in Mory et al (2018).

The result of the image reconstruction algorithm is a set of reconstructed images, one for each basis material. These can either be displayed as is (e.g. as maps of contrast agent concentration) or combined together to form a final image. Basis material decomposition is often claimed to increase image noise, and this is true in the sense that the individual basis images (e.g. a K-edge image) exhibit higher noise levels than a conventional CT image formed from the summed counts in all energy bins. On the other hand, as pointed out in section 2.2, the basis images can be combined in a weighted sum and yield the same optimal CNR as a weighted sum of the bin images (Rajbhandary and Pelc 2016). Another option is to combine the basis images with a conventional gray-scale image, which can always be reconstructed from the raw data in addition to the basis images, to form a color-overlay image combining high resolution and/or low noise with a material distribution map (Pourmorteza et al 2016, Symons et al 2017b). Thus, material decomposition does not cause any of the available information to be lost, but rather enables additional ways of presenting the image data.

One or more basis images can be used to form a virtual non-contrast image (Faby et al 2015). Another display technique currently used in dual-energy CT is to form virtual non-calcium images, which can suppress bone and thereby help in visualizing bone marrow (Kellock et al 2017). The different basis images can also be combined to form virtual monoenergetic images (Leng et al 2017, Symons et al 2018b), as is common in dual-energy scanners today. By varying the display energy, the CNR can be optimized for a given imaging task. More advanced ways of forming virtual monoenergetic images include prior-knowledge-aware iterative denoising (Tao et al 2019b) and neural-network-based methods (Feng et al 2019). Approximate material maps and virtual monoenergetic images can also be generated through image-space material decomposition from reconstructed bin images (Zhou et al 2019), although this approach gives less accurate results since it does not eliminate beam hardening.

Another possibility for image display is to form a synthetic Hounsfield image, which is an image with CT numbers equal to those of an energy-integrating CT image for a specified set of acquisition parameters, but with beam-hardening artifacts eliminated (Bornefalk 2012a). This may facilitate the transition from energy-integrating to photon-counting CT by providing image measurements that can be compared to existing reference values.

A common way to compare the performance of two imaging systems is to evaluate the CNR, or perhaps more commonly the CNR², for a given imaging task. However, when comparing detector systems with different energy response, such as an EID and a PCD, the relative CNR² is highly task dependent. The relative performance depends not only on the choice of materials in the two projections for which the CNR is computed, but also on the density, or thickness, of those materials. As an example, consider figure 15, where we have compared the projection domain CNR² of a Si PCD, a Si spectral PCD with optimal energy weighting, and an EID. The two projections that were used to compute the CNR² comprised 20 cm of soft tissue and 19 cm of soft tissue plus 1 cm of an iodine/water solution with varying concentration up to 10 mg ml⁻¹. The CNR² curves reach a minimum at slightly different iodine concentrations, i.e. there is an offset between the curves. Both the PCD and the EID reach zero CNR² for some concentrations of iodine, whereas the spectral PCD never reaches zero due to the optimal energy weighting (Yveborg et al 2013). The relative performance of the photon-counting systems compared to the energy-integrating system has a strong dependence on the iodine concentration (figure 15(b)). Comparing imaging systems with a different energy response through a single relative CNR² measurement is therefore likely to be misleading. In fact, altering the imaging task can change which of the detector systems obtains the highest CNR. This also applies when comparing different photon-counting systems (e.g. silicon-based and CdTe/CZT-based detectors) since they too have different energy response. A possible solution to this problem is to make sure that the selected imaging tasks are clinically relevant and evaluate the performance for a range of imaging tasks in order to get a picture of the overall performance of the system.

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Whereas CNR indicates how much a certain feature stands out from the background relative to the noise level, another measure is needed for assessing the performance for material-specific imaging. This role is filled by the CRLB (Kay 1993). The CRLB gives a lower bound on the variance for any unbiased estimator, or a lower bound on the covariance matrix when estimating a multivariate parameter. In the context of photon-counting CT, the CRLB gives a lower bound on the covariance matrix of estimated projected basis coefficients. Roessl and Herrmann (2009) derived the CRLB for both photon-counting and EIDs and used this formula to study how the noise in basis images depends on detector design parameters. The noise level in the image domain can be obtained from the projection-space CRLB through back-projection (Roessl et al 2011).

It may seem like a lower bound on image variance is of limited value since it does not provide any information about the upper limit of the variance. In practice, however, the maximum-likelihood estimator comes close to attaining the CRLB (Roessl et al 2011). Figure 16 illustrates the CRLB for a three-basis decomposition task, for a CdTe-based and a silicon-based detector model.

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It is important to note that the lower limit of basis projection variance is a lower limit of the variance of any unbiased estimator. If the estimator is allowed to be biased, the variance can be allowed to be lower than this limit. An extreme example would be a material-decomposition algorithm that predicts the same projected basis coefficients regardless of the measured data, which gives an estimator with zero variance and large bias. In practice, it may be desirable to use inequality constraints, such as non-negativity (Barber et al 2016, Long and Fessler 2014), and thereby reduce variance for certain basis configurations at the expense of a small amount of bias.

In contrast to CNR, which is relevant for detection tasks, the CRLB measures the ability of the imaging system to identify the material composition of the imaged object. The CRLB therefore provides a measure of the energy-resolving capability of a PCD, and it has been applied to investigating the relationship between pixel size and energy resolution in PCDs (Rajbhandary et al 2018). Faby et al used the CRLB to validate the statistical optimality of an image-based decomposition method, which was then used to study the effect of energy-bin correlations and compare the performance of photon-counting systems with other dual-energy CT techniques (Faby et al 2015, 2016). Another application of the CRLB is to estimate the covariance matrix between basis line integrals in ray-wise material decomposition in order to obtain the data weighting for the reconstruction step in a two-step material-decomposition algorithm (Schirra et al 2013, Sawatzky et al 2014).

Characterization of conventional x-ray detectors is typically done using the noise-equivalent quanta (NEQ) and DQE. These metrics are based on linear-systems theory and are typically expressed as functions of spatial frequency $\boldsymbol{u}$. The NEQ can be calculated from the incident number of photons q, the large-area gain G, detector MTF, and NPS as follows (Cunningham 2000):

$\begin{equation} \mathrm{NEQ}(\boldsymbol{u}) = \frac{q^2G^2\mathrm{MTF}^2(\boldsymbol{u})}{\mathrm{NPS}(\boldsymbol{u})}. \end{equation} \tag{ 12 }$

The NEQ of a detector is the number of noise quanta that an ideal detector, which registers every incoming photon and its exact position of incidence, would need to register in order to achieve the same detectability as the detector under consideration. This metric is dependent on the illumination conditions. For example, a detector which has an NEQ of 15 000 $\mathrm{mm}^{-2}$ for a particular spatial frequency band when irradiated by 20 000 photons per $\mathrm{mm}^2$ is 75% as efficient as an ideal detector, since the ideal detector could achieve the same detectability with only 15 000 incident photons. The DQE is obtained by normalizing the NEQ by the number of incident photons,

$\begin{equation} \mathrm{DQE}(\boldsymbol{u}) = \frac{\mathrm{NPS}(\boldsymbol{u})}{q}, \end{equation} \tag{ 13 }$

which can be shown to be equal to the squared ideal-linear-observer detectability relative to an ideal detector, $d^{^{\prime}}(\boldsymbol{u})^2/d^{^{\prime}}_{\mathrm{ideal}}(\boldsymbol{u})^2$ (Cunningham 2000). In the above example, the DQE is equal to 15 000/20 000 = 0.75 for the given spatial frequency band.

DQE is a useful metric of detector performance because it takes both noise and resolution into account. It is therefore a better measure of dose efficiency than CNR, which does not incorporate the effect of spatial resolution. Whereas the CNR increases by low-pass filtration of the image after acquisition, such processing leaves the DQE unchanged, reflecting the fact that low-pass filtering does not add information to the image.

PCDs with a single energy bin (i.e. non-energy-resolving detectors) can be analyzed with the traditional NEQ and DQE metrics. An early such study was made by Durst et al (2007); Acciavatti and Maidment (2010) compared the DQE of photon-counting and EIDs; and Stierstorfer et al modeled the DQE of a CdTe-based detector (Stierstorfer 2018, Stierstorfer et al 2019). A common way of simulating signal and noise transfer characteristics is to model the detection process as a cascade of simple processes, the so-called cascaded-systems analysis (Cunningham 2000). Tanguay et al (2013) developed the theory for cascaded-systems analysis of PCDs and used it to study the impact of non-ideal physical effects on detector performance (Tanguay et al 2015, Tanguay and Cunningham 2018). Furthermore, Xu et al (2014) studied a silicon-strip PCD for mammography with a cascaded model, and Ji et al (2018) used a cascaded model to study anti-coincidence logic in a CdTe-based detector.

To analyze the ability of PCDs to perform energy-resolved imaging, cascaded-systems analysis has also been applied to energy-resolving detectors by Tanguay and Cunningham (2018) and in the work of Taguchi et al (2016, 2018a, 2018b), where such a model was used to study the effect of N × N binning of pixels into macro-pixels in a CdTe-based detector. Energy-resolving silicon-strip detectors also been studied, by Fredenberg et al (2010a) and Chen et al (2016). However, fully understanding how the resulting signal transfer and noise characteristics affect image quality requires an extension of the NEQ and DQE concepts to energy-resolving detectors. Important steps in this direction were taken by Richard and Siewerdsen (2008) in the context of dual-energy radiography.

A natural extension of the NEQ and DQE metrics to energy-resolving PCDs was proposed by Persson et al (2018b) and will be briefly outlined here. An equivalent spatial-domain description has been described by Rajbhandary et al (2019). Although it may be tempting to simply extend the NEQ to be dependent on incident photon energy in addition to spatial frequency, this definition does not work with broad-spectrum illumination since it fails to describe how different incident energies can be confounded during the detection process as described in section 4.4. Instead, NEQ needs to be defined as a function of two energy indices so that $\mathrm{NEQ}(\boldsymbol{u},E_1,E_2)$ describes how much the joint presence of photons with energy E₁ and E₂ in the incident spectrum contributes to image detectability at spatial frequency $\boldsymbol{u}$.

To compute this generalized NEQ, it is necessary to know detector characteristics. The signal transfer is described by the transfer function $H_k(\boldsymbol{u})$, which corresponds to the MTF but without absolute value and without the normalization to 1 at $\boldsymbol{u} = 0$. The noise characteristics of the sampled signal is described by the cross-spectral density $\mathit{W}_{d^{+}}$, essentially a frequency-dependent covariance matrix between energy bins. From these quantities, the generalized NEQ is obtained as

$\begin{equation} \mathrm{NEQ}(\boldsymbol{u},E_1,E_2) = \sum_{k_1}\sum_{k_2}\overline{q}(E_1)H_{k_1}^*(\boldsymbol{u},E_1)\left(W_{d^{+}}^{-1}\right)^*_{k_1,k_2}{(\boldsymbol{u})}H_{k_2}(\boldsymbol{u},E_2)\overline{q}(E_2) \end{equation} \tag{ 14 }$

with $\overline{q}(E)$ denoting the spectral photon density and ^* denoting complex conjugate.

Under the assumption that the object consists of a small set $\mathcal{B}$ of basis materials, it is more convenient to express the generalized NEQ in matrix form through a basis transformation. The elements $\mathrm{NEQ}^{\mathcal{B}}_{l_1,l_2}(\boldsymbol{u})$ of this NEQ matrix describe the contribution to image detectability from the addition of differential amounts of basis materials l₁ and l₂. In addition to allowing the detectability to be calculated for detection tasks, this matrix also describes the performance for quantification tasks in a set of decomposed basis images. Namely, the matrix elements of its inverse $\left(\mathrm{NEQ}^{\mathcal{B}}(\boldsymbol{u})^{-1}\right)_{l_1,l_2}$ give, up to a scale factor, the CRLB for the covariance of basis images l₁ and l₂ as a function of frequency (Persson et al 2018b, equation 25). This allows calculating the squared ideal-linear-observer detectability in basis image l as

$\begin{equation} {d^{^{\prime} 2}_{l} = \int \frac{\left(\Delta\hat{\tilde{A}}_l(\boldsymbol{u})\right)^2}{\mathrm{Var}\left({\hat{\tilde{A}}_l(\boldsymbol{u})}\right)}\mathrm{d}\boldsymbol{u}} \end{equation} \tag{ 15 }$

where $\Delta\tilde{A}_l(\boldsymbol{u})$ is the difference between target and background in the Fourier transform $\tilde{A}_l(\boldsymbol{u})$ of the basis image, and $\mathrm{Var}\left(\hat{\tilde{A}}_l(\boldsymbol{u})\right)$ is the variance of an efficient estimator for $\tilde{A}_l$.

As an illustration of equation (15), the relative squared ideal-linear-observer detectability for detection of features of different sizes in an iodine basis image is shown in figure 17 for simulation models of CdTe-based and silicon-based detectors. As shown in this figure, a system with smaller pixels, such as the simulated CdTe system, has an advantage for imaging of small features. At the same time, a system with high energy resolution, such as the simulated silicon system, has an advantage for imaging larger features in a material-selective basis image, in particular for three-basis decomposition where energy resolution is particularly important in order to separate iodine and calcium. It is also interesting to note that the CdTe detector performance exhibits a relatively weak kVp dependence, whereas the silicon detector benefits strongly from using low kVp, where the fraction of Compton scatter is low. These observations show that the spectral linear-systems framework provides important clues for how to best utilize PCDs, especially in view of the ongoing trend towards using lower kVp to save patient dose.

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In analogy with equation (13), a generalized DQE matrix can also be defined by normalizing the NEQ matrix by the NEQ matrix of a detector with perfect quantum efficiency, energy resolution, and spatial resolution:

$\begin{equation} \mathrm{DQE}^\mathcal{B}_{l_1,l_2}(\boldsymbol{u}) = \frac{\mathrm{NEQ}^\mathcal{B}_{l_1,l_2}(\boldsymbol{u})}{\sqrt{\mathrm{NEQ}^{\mathcal{B},\mathrm{ideal}}_{l_1,l_1}\mathrm{NEQ}^{\mathcal{B},\mathrm{ideal}}_{l_2,l_2}}}. \end{equation} \tag{ 16 }$

In analogy with the conventional DQE, it can be shown that a diagonal element $\mathrm{DQE}^\mathcal{B}_{l,l}$ of this matrix gives the task-specific DQE, i.e. the squared ideal-linear-observer detectability $d^{^{\prime}}_l(\boldsymbol{u})^2/d^{^{\prime}}_{\mathrm{ideal}}(\boldsymbol{u})^2$ relative to an ideal energy-resolving detector, for detecting basis material l. Here, the ideal-observer detectability of a multi-channel image should be interpreted as the highest detectability that can be obtained in any image as a weighted sum of the images, where the weight factors are allowed to vary with spatial frequency (Yveborg et al 2015b). The generalized NEQ matrix thus encodes the performance of an energy-resolving PCD for both detection and quantification tasks. How this framework can be applied to studying PCD performance is described by Persson et al (2020).

CNR, CLRB, and DQE are metrics related to the image noise level. In addition to these noise metrics, the systematic error, or bias, in the average value in regions of the image is also an important performance measure, for example when evaluating the conspicuity of artifacts or the accuracy of quantitative image measurements. For the latter case, a relevant metric is the mean-squared error, which combines the bias with the noise variance as (Kay 1993)

$\begin{equation} \mathrm{MSE}(I) = \left(\mathrm{E}\left[I\right]-I^\mathrm{True}\right)^2 +\mathrm{Var}\left[I\right] \end{equation} \tag{ 17 }$

where $\mathrm{E}[\cdot]$ and $\mathrm{Var}[\cdot]$ denote expected value and variance, respectively, and $I^\mathrm{True}$ is the true value of the random image value I.

Although the standard deviation of image noise in a reconstructed image is typically several Hounsfield units, the averaging that takes place when measuring the mean CT number in a region of interest suppresses noise so that even a small systematic error can dominate the MSE. Similarly, when making quantitative measurements in a basis image, it is more important to reduce bias if the measurement region of interest is large, whereas for small regions it may be favorable to reduce noise at the expense of bias by reducing the number of basis materials (Bornefalk and Persson 2014).

One source of error is the log-normalization step transforming counts into attenuation line integrals. Since this operation is non-linear, it introduces an error that becomes larger as the dose decreases (Zhang et al 2018). Other sources of error are inaccuracies in the model used for material decomposition. In order to make quantitative material composition measurements that are limited by quantum noise rather than systematic errors, it is necessary to have a detailed model of the x-ray source, the detector response, and scattered photons from the object (Bornefalk et al 2015). Estimating the distribution of object scatter from energy-resolved measurements is a non-trivial problem since the energy spectra of scattered and primary photons overlap in general. However, energy-resolved measurements can be helpful for scatter correction and/or measurements of the scatter distribution (Sossin et al 2016, 2017).

Despite the fact that more research is needed to address these challenges, measurements of K-edge contrast agents such as gold or iodine in basis images generated with prototype spectral CT systems have been shown to correlate well with the actual concentrations of these elements (Si-Mohamed et al 2017b, 2018).