How spectroscopic x-ray imaging benefits from inter-pixel communication

Spectroscopic x-ray imaging based on pixellated semiconductor detectors can be sensitive to charge sharing and K-fluorescence, depending on the sensor material used, its thickness and the pixel pitch employed. As a consequence, spectroscopic resolution is partially lost. In this paper, we study a new detector ASIC, the Medipix3RX, that offers a novel feature called charge summing, which is established by making adjacent pixels communicate with each other. Consequently, single photon interactions resulting in multiple hits are almost completely avoided. We investigate this charge summing mode with respect to those of its imaging properties that are of interest in medical physics and benchmark them against the case without charge summing. In particular, we review its influence on spectroscopic resolution and find that the low energy bias normally present when recording energy spectra is dramatically reduced. Furthermore, we show that charge summing provides a modulation transfer function which is almost independent of the energy threshold setting, which is in contrast to approaches common so far. We demonstrate that this property is directly linked to the detective quantum efficiency, which is found to increase by a factor of three or more when the energy threshold approaches the photon energy and when using charge summing. As a consequence, the contrast-to-noise ratio is found to double at elevated threshold levels and the dynamic range increases for a given counter depth. All these effects are shown to lead to an improved ability to perform material discrimination in spectroscopic CT, using iodine and gadolinium contrast agents. Hence, when compared to conventional photon counting detectors, these benefits carry the potential of substantially reducing the imaging dose a patient is exposed to during diagnostic CT examinations.


Introduction
Today, most x-ray imaging modalities rely on detectors that make use of the indirect detection of photons. Based on scintillators, which first convert the absorbed x-ray flux into visible light, a digital signal is obtained that is proportional to the total amount of energy deposited during exposure. These indirect conversion devices are a proven and robust technology and are widely used in fields such as materials research as well as preclinical and medical imaging.
Complementary to this are direct conversion techniques using an opaque semiconductor sensor, in which the absorption of a photon leads to the generation of electron-hole-pairs. By means of a high voltage applied to the sensor, the resulting photocurrent can be measured in some form of readout electronics attached to one of the sensor sides, which can, for instance, be made of amorphous selenium (Pang et al 2001). Integrating this current over time then again gives the total amount of energy deposited in a pixel, but does not require the intermediate step of a scintillator.
An evolution of these direct conversion devices can be seen in photon counting detectors, which are based on readout electronics that are fast enough to resolve the individual current pulses generated by the absorption of single photons. Quantifying the amplitude of these pulses then enables to obtain information about the energy released by a photon interaction. This can be done by either feeding these pulses into multi-channel analysers or at least one adjustable discriminator. The latter commonly acts by discarding every pulse below the voltage applied to it. Calibrated to reference photon energies, these devices can then be used to perform energy selective measurements, in particular spectral computed tomography (CT). This not only allows an increase in contrast by optimal energy weighting (Giersch et al 2004, Shikhaliev 2009, Schmidt 2010, Bornefalk 2011, Yveborg et al 2013, but also enables to perform material discrimination and K-edge imaging using contrast agents (Firsching et al 2008, Schlomka et al 2008, Wang et al 2011, Ronaldson et al 2012, Taguchi and Iwanczyk 2013. As a useful side effect, a chip's electronic noise-which appears as a background of low energy events-can be cut off by setting appropriate energy thresholds. By doing so, one prevents the usual dark current to appear in the resulting images and detector noise then only affects energy resolution and the lowest possible energy at which imaging can still be performed. A prerequisite for this principle to work is that the sensor material offers charge transport properties good enough to prevent a significant amount of charge trapping to occur around defects (Xu et al 2011), which would otherwise lead to a signal amplitude depending on the depth of interaction. For this reason, photon counting-or spectroscopic-x-ray detectors are usually operated with single-crystal sensors or a sensor showing a quality close to this requirement.
Another important consideration is that, when pixel pitches much smaller than the sensor thickness are used, the detector benefits from the small pixel effect (Barrett et al 1995), making the energy measurement sensitive to only one of the two charge carrier types (electrons or holes).
In the last years, cadmium telluride (CdTe) has emerged as a popular choice as such a sensor material. Being available in detector-grade quality and offering relatively high atomic numbers of 48 (Cd) and 52 (Te), it appears as the material of choice for hard x-ray imaging applications such as medical diagnostics. Associated with these high atomic numbers are K-edges at energies of 27 keV and 32 keV, which result in the emission of characteristic x-rays around 23 keV and 27 keV. In contrast to sensor materials with lower atomic numbers, such as silicon  or gallium arsenide , these fluoresence photons can travel comparably large distances, with mean free path lengths of roughly 0.12 mm in the case of the Cd fluorescence (excluding coherent scattering). Triggering additional events in adjacent pixels, the original interaction is then split up into at least two events with separate energies , Xu et al 2011. Those of the fluorescence photons are usually out of the energy range of interest in medical imaging and can therefore be cut-off easily by setting appropriate energy thresholds in a spectroscopic detector. However, the energy of the remaining photoelectron corresponds to the energy released by the original photon interaction minus the energy of the fluorescence photon that escaped. Depending on the pixel pitch, these so-called escape peaks deteriorate and bias the recorded x-ray spectra by adding a continuous background when broadband sources such as x-ray tubes are employed.
In addition to this, an effect called charge sharing also becomes increasingly dominant when pixel sizes are reduced. Since the cloud of free charge carriers has a certain intial extent due to the range of the photoelectron creating it and because it is subject to diffusion on its way through the sensor, there is a chance that it spreads across multiple pixels once it reaches the readout ASIC (application specific integrated circuit). Again, this can lead to a single interaction splitting up into multiple events, each of which is assigned the wrong energy.
Both fluorescence and charge sharing can lead to a dramatic distortion of the x-ray spectrum incident on the detector for small pixels, which results in a reduced ability to discriminate contrast agents in spectroscopic CT . Besides this effect, which directly depends on energy resolution, the following consequences of fluorescence and charge sharing are somewhat more subtle: • The contents of spectral channels are mutually correlated due to photons partially being assigned the wrong energy. Cross-talk between pixels thus leads to a cross-talk between spectral channels.
• The sensitive pixel volume shrinks with an increasing energy threshold, leading to insensitive areas at pixel edges and corners (Tlustos et al 2006). Photons hitting the detector in these regions may not be counted at all and thus remain undetected, which decreases a system's detective quantum efficiency (DQE). Due to this loss of events, channels corresponding to high photon energies may offer a poor signal-to-noise ratio (SNR) and can potentially become completely unusable unless exposure times are increased, which implies both higher radiation doses and acquisition times.
In total, these effects can be deleterious and could so far only be reduced by using larger pixels. However, this strategy implies a higher photon flux per pixel and is therefore more prone to pulse pile-up within the ASIC. This effect is caused by overlapping pulses due to events which are not separated enough in time for an ASIC to perceive them as separate events (Taguchi et al 2010). In these high flux cases, the expected benefits in energy resolution are counteracted by the fusion of events, which again gives wrong photon numbers and energies, preventing the proper separation of contrast agents (Rink et al 2013).
A way to tackle this problem is to let pixels communicate with each other in order to detect events that are spread among neighbouring pixels. To our knowledge, the Medipix3RX ASIC (Ballabriga et al 2013, Frojdh et al 2014, developed by the Medipix3 collaboration centered at CERN, is the first detector to offer this feature in its so-called charge summing mode (CSM). As the name suggests, the chip offers a circuitry whose purpose is to reconstruct the charge distributed among pixels due to fluorescence and charge sharing on an event-by-event basis and to assign it to the pixel which received most of the charge. To this end, the signal deposited in overlapping clusters of 2 × 2 pixels is reconstructed by analogue summing of the preamplifier outputs. The hit is then allocated by the digital circuitry. Exhibiting a pixel pitch of only 55 µm, this feature is a most welcome addition to the chip's architecture in order to maintain a high-level performance in spectroscopic imaging tasks. Recent results have also indicated that, in CSM, a net benefit exists when considering the trade-off between high flux capability and energy response in small pixels. This was explained by the vast reduction of hits occurring at the boundary of the (adaptive) collection volume . As a consequence, the energy response obtained with charge summing does not have an equivalent at a larger pixel pitch operated without this feature.
Since the features and working principle of this chip were described previously (Ballabriga et al 2013), we just recall a number of its aspects that are important to understand what follows: • The CSM can be deactivated to operate the chip in single pixel mode (SPM). In this mode of operation, a pixel works independently from those adjacent to it. This not only allows to assess the benefits of charge summing, but also makes it possible to study in detail the adverse effects of fluorescence and charge sharing.
• While establishing a larger charge collection volume through inter-pixel communication, the spatial resolution imposed by the pixel pitch remains the same in both SPM and CSM. Hence, CSM offers an improved energy response at constant spatial resolution.
• The chip can be operated in a mode that offers an even larger collection volume, for which it can be connected to 110 µm sensors. Here, only one in four pixels is bonded to the sensor and thus receives a signal from it 5 . The charge summing circuitry then spans an area of 220 μm × 220 µm, again by maintining a spatial resolution of 110 µm. Also, in this mode groups of four pixels, three of which are not connected to the sensor, share their discriminators and counters, which allows operating the chip in CSM using four effective energy thresholds (each counting photons with energies equal to or above them) and eight thresholds in SPM. Therefore, this is called the spectroscopic mode.
Although another system offering a charge sharing correction has been presented and characterized recently (Ullberg et al 2013), we are unaware of any reports on how the position of an energy threshold influences imaging properties in the presence of charge summing. More importantly, the effects of charge summing on the image contrast and the discrimination of materials in spectroscopic CT have not yet been investigated. The motivation for this article was therefore to provide a comprehensive study of how a detector benefits from this new feature, which will be done by comparing our results to those obtained in SPM, i.e. with the charge summing circuitry turned off. In particular, the aspects we will investigate include the chip's capability to reconstruct x-ray spectra as well as its DQEs, modulation transfer functions (MTFs) and the achievable contrast-to-noise ratios (CNRs) as a function of the energy threshold and mode of operation. We will also present results from spectroscopic CT, using a PMMA phantom which contains both iodine and gadolinium contrast agents. Using a linear regression approach, we will study how well the two modes compete in a material discrimination task.
The details of all these procedures are given in the next section, while we present our results in section 3 and conclude our findings in section 4. We would like to point out that this study applies to pixellated semiconductor detectors. Our results are therefore not directly transferrable to other approaches, for instance geometries that make use of silicon strip detectors placed 'edge-on' to the incoming radiation field (Bornefalk et al 2010).

Detector configuration
The CdTe sensor studied in this work exhibits a thickness of 2 mm. This represents a choice which offers a high absorption efficiency at photon energies typical of medical applications (nearly 100% at 60 keV and 86% at 100 keV 6 ). Sensors even thicker are possible from a technological point of view, but are more and more prone to charge trapping, which worsens the achievable spectroscopic resolution.
This sensor was bonded to a Medipix3RX detector using a low temperature soldering process. The pixel pitch chosen was 110 µm to better account for the mean free path lengths of the CdTe fluorescences, resulting in a total number of 128 × 128 active pixels, spread over an area of 1.4 cm × 1.4 cm. All measurements were conducted at room temperature, using a bias voltage of −600 V in both SPM and CSM.

Spectroscopic resolution
A Viscom X9160-D ED microfocus x-ray tube was operated with a tungsten target and a 1 mm thick aluminium filter to remove photons of very low energies from the beam. X-ray energy spectra were obtained at 120 kVp in SPM and CSM by scanning the respective energy thresholds. Since the discriminator's architecture implies counting photons above a threshold, it was necessary to differentiate these 'integral scans' to obtain actual photon energy spectra, which we refer to as 'differential'. In order to compare these spectra to the actual one, we also carried out a Monte Carlo simulation of our x-ray tube, based on the EGSnrc code system (Kawrakow 2000) and the BEAMnrc user code (Kawrakow et al 2004) in version V4 2.3.2.

Detective quantum efficiency
For the DQE measurements, we made use of the 60 keV and 88 keV photons emitted by 241 Am and 109 Cd radioisotopes, choosing an appropriate amount of aluminium and copper filters to suppress unwanted low energy photons additionally emitted by these sources. DQEs were calculated for a spatial frequency of zero, which we denote as DQE(0). As the DQE represents the squared ratio of the output to the input SNR at a given spatial frequency, the DQE(0) measures how much the average output count rate of the detector fluctuates in relation to the input Poisson process 7 . In contrast to non-zero spatial frequencies and in the absence of pulse pile-up, this quantity can be derived very easily for a photon counting detector (Michel et al 2006). By measuring the multiplicity m, defined as the number of pixels responding to a single photon interaction and by knowing the sensor's absorption efficiency ϵ, the DQE(0) is given by: Here, represents mean values obtained from a multitude of events. For an energy of 88 keV, the absorption efficiency is around 0.92 and approximately 1 for 60 keV photons. These values therefore represent an upper boundary for the DQE (0)  . The values of m 2 and m 2 were determined using very short exposure times to ensure that only a very small number of pixels per frame would exhibit non-zero count rates. By doing so, pixels reporting simultaneous events within a certain radius are related to fluorescence and charge sharing in almost 100% of cases. We then performed threshold scans to assess the deterioration of the DQE at high threshold levels due to events lost to fluorescence and charge sharing, starting at a very low threshold setting above the noise level and ending a couple of keV above the photon energy. 10 000 frames were recorded per threshold setting to minimize errors and the remaining procedure, including the algorithm to compute multiplicities, was then equivalent to what was reported earlier . The maximum distance for two events to be considered belonging to the same primary photon was set to 550 µm.
In order to account for the decreasing number of hits when the threshold is raised (especially in SPM), the average number of clusters determined for the lowest threshold position was taken to accurately reflect the incoming photon flux. The reduction of this cluster number at higher threshold settings was then included into the effective absorption efficiency ϵ, which therefore decreases with increasing threshold values.
We would like to stress that, as photon counting detectors are comparably novel devices, this method has not yet entered any standards dealing with DQE measurements. However, it represents an appropriate method to obtain the DQE(0) in the absence of pile-up, since, unlike conventional approaches, it does not require measuring a noise power spectrum. In other words, the DQE(0) of a photon counting detector is independent of the exposure time, which is expressed by (1).

Spatial resolution
A detector's spatial resolution is usually given in terms of its MTF, which represents the modulus of the Fourier transform of the line spread function (LSF). It is common to report the so-called presampling MTF, which gives the spatial resolution of the sensor in a manner independent of the pixel pitch, i.e. prior to spatial sampling. As LSFs are difficult to obtain in the x-ray regime, one usually measures the edge spread function (ESF), of which the LSF is the derivative. The ESF can be measured comparably easy by tilting a strongly attenuating edge, in our case made of tungsten, against a pixel row or column. The different degrees by which individual pixels are shadowed then results in an effective oversampling of the ESF and hence the ESF can be measured even for cases where pixels are much larger than the characteristic length scale at which it changes from zero to its maximum.
While it is generally possible to bin, differentiate and Fourier transform the empirical ESF, this binning corresponds to an implicit low-pass filtering, which introduces a systematic error into estimating the MTF. Therefore, we chose to fit an analytical model to the ESF based on assuming a shape for the LSF that can be described as a weighted sum of a normal and a Phys. Med. Biol. 59 (2014) 6195 Laplace distribution. From this, analytical expressions can be obtained for both the ESF (for which the fit is performed) and the MTF (Boone 1994).
The data we used to derive the MTFs were recorded at the TopoTomo beamline of the ANKA synchrotron at an x-ray energy of 35 keV. This energy is above both of the CdTe K-edges and therefore stimulates the emission of characteristic x-rays. It also somewhat corresponds to a worst-case scenario, as in this case the majority of the interaction energy is carried away by the fluorescence, leaving only a minor amount for the photoelectron that more or less stays in place. As a consequence, one can expect an additional broadening of the LSF, since for a given interaction position, the CSM will reconstruct events at different pixels, depending on where exactly the fluorescence went.
Cutting off these fluorescence photons using an appropriate energy threshold also gives additional insights into how well a charge summing circuitry operates. Ideally, one would observe no change of the MTF when varying the energy threshold, as in this case the total charge released would have been reconstructed correctly (i.e. including the fluorescence photon). Therefore, we investigated the MTFs using energy thresholds of 8.5, 20 and 30 keV.
All measurements were performed at six different regions of interest, distributed across the sensor, in order to average over inhomogeneities and to allow the error analysis we describe in section 2.6.

Spectroscopic CT and material discrimination
The main interest in spectroscopic x-ray detectors comes from the field of CT and here it is the discrimination of contrast agents that receives constant attention. One of the main goals of this study was therefore to assess whether a benefit would arise from using charge summing and if yes, how large this would be. The experiments that we describe in the following made use of a contrast agent phantom filled with two different concentrations of iodine and gadolinium contrast agents (50 µmol ml −1 and 250 µmol ml −1 each, figure 1). We would like to point out that, with a phantom this thin (ø 8 mm), the influence of x-ray scatter, which would be present under clinical conditions, cannot be assessed.  The microfocus x-ray tube was operated in the same fashion as before, using a voltage of 120 kVp and a current of 0.6 mA. The detector was placed 1.35 m from the source and 0.37 m from the object, resulting in a magnification of 1.27. The acquisition time was 15 s per projection and a total of 720 projections were recorded over an angular interval of 360°. CT reconstruction was done using Octopus 8.6 (inCT, Aast, Belgium), with the reconstruction settings given in the appendix. A Hounsfield calibration was performed on the PMMA phantom material seperately for each spectral channel and all further analyses were carried out based on these CT numbers. In general, we did not compute difference images to form energy windows (i.e. bins) prior to CT reconstruction, since this procedure is very sensitive to noise. As our results will demonstrate, this is a viable approach.
CT acquisitions were obtained using equidistant thresholds as given in table 1. These aquisitions were performed using the full number of available thresholds in both modes, i.e. eight in SPM and four in CSM. In addition to this, using only every second threshold in SPM allows a direct comparison of the two modes when an equal number of thresholds is used. On the contrary, studying CSM with eight thresholds would have required two separate acquisitions and different levels of noise covariance would have been present among the resulting images (Schmidt et al 2014). For this reason, we refrained from investigating this particular case to ensure an unbiased comparison. Hence, the analysis to follow in section 3.4 will be based on both four and eight thresholds in SPM and four thresholds in CSM. Throughout this article, the images pertaining to a specific threshold setting will be referred to as spectral channels.
In order to find differences in the image quality produced by the two modes of operation, we calculated the contrast-to-noise (CNR) ratios between the contrast agent capillaries and the phantom material according to the following relation (Fast et Here, μ denotes the mean CT number of a region of interest and σ 2 represents the corresponding variance. Material discrimination can be performed in the reconstruction domain using a linear regression model, which has to be calibrated to a specific CT system and the acquisition settings employed (Ronaldson et al 2012). The approach that we introduce below differs from existing methods by modelling the material-decomposed images as a direct linear combination of spectroscopic CT images. Making use of the variables defined in table 2, this problem can be described as follows: In words, for each material i we seek to estimate the vector of coefficients β i , whose entries correspond to the weights of the individual spectral channels represented by the columns of X. Both these columns as well as the y i are simply images re-sorted to form N × 1 vectors, in the latter case corresponding to the known material concentrations or densities in a calibration phantom. Estimating β i is done separately for each material i, using a least squares approach. Solving the inverse problem (3) then gives (Hastie et al 2008): Determining β i (4) is what we call the training process, which was carried out individually for the two modes of operation (SPM and CSM). This training process can, in principle, be performed for as many materials K as desired. In particular, it also works for cases in which the number of materials to be discriminated is higher than the number of spectroscopic channels employed (K > p). This can be seen easily from (3) and (4), as the regressions for individual materials i decouple completely, i.e. they do not 'know' of each other. By pointing this out, we want to rectify the often-heard misconception that the number of spectroscopic channels always needs to be equal to or greater than the number of materials to discriminate. However, in this work, we do not exploit this property and use a total of three materials: iodine, gadolinium and PMMA/water (i.e. K = 3, p = 4 or 8). The contrast agents were quantified in terms of their concentrations in water, while the PMMA/water component was taken as a proxy for soft tissue using the respective densities (1.18 g cm −3 and 1 g cm −3 ). During the training of each β i , the materials unwanted in this particular component were set to 0 in y i . Also, the same number of voxels was used for each material in X to avoid biasing the regression towards the more abundant soft tissue voxels. The columns of X as well as the y i were standardised to zero mean and unit variance. For every regression performed, the trained model was applied to a test set according to (3) as described in the next section and the mean squared error, was obtained as a quality measure. We also calculated CNRs on these material-decomposed images as follows: Here, μ correct represents the mean of all voxels that should be in a given material-decomposed image, μ false denotes those which should not and the σ 2 are again the corresponding variances.
In detail, for the contrast agents this amounted to picking a capillary with a particular concentration and calculating (6) with respect to the remaining image, with the exception of the other capillary of the same sort of contrast agent (having a different concentration). The value for PMMA was calculated with respect to the surrounding air.

Error analysis
For the DQE, MTF and CNR measurements, it is important to know whether differences among measurements are of statistical or systematic nature, such as altering threshold levels or switching between SPM and CSM. Since we are unaware of the probability density functions influencing the errors of our measurements, we resorted to using a non-parametric bootstrap to estimate confidence intervals (Hastie et al 2008). Bootstrapping can be considered a Monte Carlo simulation based on empirical data. Given a number of samples, in our case distributions of multiplicities, MTFs obtained across the sensor surface or CNRs through a multitude of slices, we draw with replacement the same number of samples as contained in the original distribution, yielding a bootstrapping distribution. From this, we derive the corresponding medians. Repeating this a large number of times (1000 in our case) then allows computing 95% confidence intervals for these medians. Where applicable, these intervals are plotted as error bars or coloured areas around the 50% quantiles estimated in the same fashion. If these confidence intervals do not overlap, the corresponding measurements can be considered significantly different at the 5% level. On the other hand, if they do overlap, a significant difference cannot be found at this level, which does not necessarily mean that there is none.
To assess the accuracy of our linear regression approach described in section 2.5, we used a form of leave-one-out cross-validation. This means, given a number of n = 17 slices, we trained the regression on the average of n − 1 slices and tested it on the remaining one. This process was repeated n times until each slice had been used for testing exactly once. The resulting distribution of fit errors (on the test set) was then again bootstrapped as explained before.

Spectroscopic resolution
The 120 kVp x-ray spectra recorded in both SPM and CSM are depicted in figure 2. Both measurements show the Cd fluorescences at a photon energy of around 23 keV. As expected, this complex is a lot more pronounced for the SPM case, and not visible as a distinct peak, but rather as a shoulder due to an excessive charge sharing background. Another very clear advantage of charge summing can be found in the visibility of the tungsten K α fluorescences and even their associated escape peaks are slightly indicated. These occur because the tungsten K α lines at 58 keV and 59 keV, emitted from the source, can stimulate the emission of Cd or Te characteristic photons in the sensor, which in turn leave behind a photoelectron with an energy contributing to this peak 8 . This implies that the pixel pitch of 110 µm, yielding an effective collection area of 220 μm × 220 µm, is still not sufficient to suppress fluorescence events down to a negligible level. Figure 2 additionally shows a spectrum which was obtained from the Monte Carlo simulation by lowpass-filtering with a Gaussian kernel using a FWHM of 4.9 keV. This value was chosen to achieve a close (visual) similarity to the measurement data. Above 35 keV, there is a remarkable agreement between this low-pass filtered Monte Carlo simulation and the CSM measurements. At high energies, the CSM curve eventually falls below the simulated spectra, which is due to events lost to fluorescence and an absorption efficiency that is no longer close to 100% . In contrast to this, the number of photon interactions reported in SPM is strongly overestimated at low energies and underestimated at high energies. This implies a reduction of the DQE at high threshold settings, a behaviour we will study in the next section. Additionally, the higher number of counts at low threshold positions leads to a reduction of dynamic range, since a counter will be more prone to overflowing in these cases.

Detective quantum efficiency
The DQE(0) obtained for the two different photon energies and modes of operation are given in figure 3, where an improved performance is found for CSM. In particular, it stays close to the theoretical maxima of around 1 (60 keV) and 0.92 (88 keV), which are obtained when setting = m m / 1 2 2 in (1). Especially for the 88 keV measurement, the DQE increases slightly when the threshold exceeds the energies of the CdTe characteristic x-rays. In contrast to this, the curves pertaining to SPM rapidly decrease when the threshold is raised above half the photon energy. Again, this is due to fluorescence and especially charge sharing, i.e. more and more events are not counted at higher threshold levels, resulting in a decrease of the detector's active area (see also next section). A similar deterioration can be observed in CSM only for much higher thresholds, when the escape peaks are cut off. Eventually, the threshold enters the photo peak and the DQE(0) drops more and more due to the finite energy resolution.
The results discussed above were obtained for monoenergetic photons. For broad spectra, like those of an x-ray tube, the DQE(0) will be influenced by all individual photon energies.

Spatial resolution
Discussing the presampling MTF of a photon counting detector is conceptually a little more difficult than in the case of energy integrating detectors. The reason for this is the presence of energy thresholds, which can never be set to an effective energy of 0 keV, as the detector's counters would be flooded by noise in this case. A critical point to understanding the results presented in this section is that applying an energy threshold to a charge shared event can amount to cutting off parts of the charge cloud, since those below a pixel's threshold are simply not counted. Hence, it may occur that only one pixel will increase its counter at a high threshold setting, when actually three would respond at a low one. As a consequence, the average size of the measured charge cloud decreases with an increasing energy threshold, a detector's effective LSF gets narrower and thus the effective MTF of a photon counting detector improves. Figure 4 shows the results obtained for the SPM case. As expected, the measured MTF improves when choosing higher thresholds. This effect is especially pronounced for the highest setting, which cuts off both of the CdTe fluorescence energies. While this effect may be helpful on some occasions, it goes along with a decrease of a detector's DQE, as discussed in the last section.
CSM behaves differently, which can be seen in figure 5. When raising the threshold from 8.5 keV to 20 keV, we observe no significant change in the MTF, which indicates that the charge sharing portion of the signal spread is nicely captured. Only when the threshold exceeds the CdTe fluorescence energies, we find a significant, but moderate increase of the MTF at low frequencies, which is due to the pixel pitch still being a little too small for CdTe sensors (also see figure 2). Hence, these results generally discourage choosing pixels much smaller than 110 µm on CdTe sensors, despite the charge summing functionality. In summary, the MTFs vary considerably less when charge summing is activated.

Spectroscopic CT and material discrimination
In figure 6, we show the CT images that represented the basis for material discrimination. Both contrast and noise found in these images clearly favour CSM over SPM. Especially at high threshold levels, the SPM images are affected by noise to much a higher degree than those obtained using CSM, which is a consequence of the inferior DQE observed before. Profiles through the high concentration capillaries are given in figure 7 for a threshold of 48 keV, demonstrating the contrast increase obtained with charge summing. This is further quantified in figure 8. Especially for elevated threshold settings, the higher DQE and contrast measured in CSM achieve a CNR that is at least twice as high as with SPM. Finally, we turn to discussing the material-decomposed images, which are shown in figure 9. Visually, they demonstrate a clear advantage of charge summing in terms of image noise, no matter the amount of thresholds used in SPM. Figure 10 gives the average material decomposition error (MSE) on the test set obtained for the three modes and the differences are both significant as well as substantial. The lack of spectroscopic resolution in SPM leads to a behaviour that hardly profits from adding more energy thresholds. In contrast to this, switching on charge summing produces a decomposition error which is four times lower than without it and three times lower when eight thresholds are allowed in SPM.  figure 11, where we show the CNRs obtained on the material-decomposed images according to (6). Again, we find an increase in CNR by about a factor of two and, in this experiment, only CSM provides a CNR > 1 for the low gadolinium concentration.

Conclusions
We have investigated a method called charge summing with the aim to improve both regular as well as material contrast in spectroscopic CT. This feature was implemented into a pixellated photon counting detector bonded to a CdTe sensor. Studying its effects on various imaging quantities, we found substantial improvements compared to conventional approaches, in which detector pixels act independently. While all the benefits of charge summing that we reported are due to the improved spectroscopic performance, some have implications that go beyond energy response. Firstly, we found that charge summing produces MTF curves which are almost independent of energy thresholds, with residual variations being due to fluorescence not completely captured by the pixel pitch chosen. Secondly, while the absence of charge summing enables to increase a detector's MTF, this comes at the cost of a dramatically reduced DQE. Hence, charge summing represents an improvement over conventional technologies with regard to avoiding the loss of events due to charge sharing and sensor fluorescence. Thirdly, charge summing also improves the dynamic range at low threshold settings: counters will be less prone to overflowing, since a lower number of pixels will respond to a single photon interaction.
In addition to maintaining a high detection efficiency, charge summing offers a higher image quality in regular as well as material-decomposed CT images. In both cases, we demonstrated a notable increase in CNR. In the case of material discrimination, we also found the regression error to reduce by 60-80%, even when using twice the amount of energy thresholds in single pixel mode. The design considerations that, for this small a pixel pitch, forced a reduction of the number of usable energy thresholds in charge summing mode, was therefore a trade-off worth making.
All these results encourage the use of charge summing in upcoming spectroscopic micro-CT systems, where a small pixel pitch is mandatory. Not only is spectroscopic information recovered, the use of this novel feature also enables quicker acquisitions due to the improved DQE. When extrapolating these results to a clinical environment, in which the reduction of imaging dose is crucial, we consider the use of charge summing to be essential, unless very low threshold settings are used. Unfortunately, to our knowledge, current industrial prototypes of clinical spectroscopic CT systems do not exhibit such a feature yet.