4.1.

Calibration Methods

In CMOS architecture, charge-voltage conversion and the first amplification stage are implemented directly at the pixel level. This means that each pixel has its own amplification circuit, likely generating non-uniform conversion gain between pixels. In addition to the Fano and electronic noise, small variations in the pixel gains can broaden the energy line and degrade the energy resolution of the sensor. To overcome this degradation, correction of the pixel-to-pixel gain dispersion can potentially improve the spectral performance. The X-ray calibration process involves illuminating the detector with a source emitting X-ray photons at known energies, ideally with several lines to increase calibration accuracy. The correlation between the observed spectrum (in ADU) and the known spectrum (in keV) can be measured using different approaches. A common method is the so-called “peak fitting.” It consists of fitting every line in the observed ADU spectrum with a Gaussian function to determine the line centroid and find the relation between these centers and the expected incident photon energy. A linear relation usually provides a reliable energy calibration, but a more complex relation (e.g., quadratic) can be employed to correct for non-linear readout electronic effects over a large energy range. Another possible approach is the “energy calibration via correlation” (ECC)^22,23 initially employed for CdTe semiconductor detectors. The method relies on finding the maximum correlation between a synthetic spectrum (template) of the incident source and the observed spectrum. Similar to the peak fitting approach, the ECC method can be used to find linear or more complex relations. ECC offers multiple possible advantages over peak fitting. It allows a more robust and accurate calibration using lines and background as a whole, provides flexible ways of tracking the energy calibration evolution over time using a previously calibrated dataset, and has been shown to outperform peak fitting in the case of a low-statistics spectrum.²² The latter is particularly interesting for energy calibration of CMOS sensors, where a large number of pixel gains need to be calculated, causing significant challenges in generating sufficient statistics for each pixel. Both methods were used in our study, and their results have been compared to determine the strength of each approach (Sec. 4.4).

4.2.

Data Collection for the Energy Calibration

The Sony IMX290 CMOS sensor is composed of 2.1 million pixels, and its precise energy calibration requires the measurement of a gain for each pixel. This requires generating a spectrum for each pixel, where each pixel spectrum is composed of single-pixel events to avoid mixing the gains of adjacent pixels. It also means collecting data under stable conditions to limit the contribution of external effects (e.g., temperature) on the gains measured.

To keep the acquisition time at a reasonable level, we illuminated the sensor with a bright ${}^{55}\mathrm{Fe}$ radioactive source ($\sim 38\mathrm{MBq}$), which produces $\sim 7000\text{events}/\mathrm{s}$ on the detector. The camera was run at 10 fps to limit the pile-up and reduce the dark current to a negligible level for the operating conditions. To limit the data flow and save storage space, we performed real-time event extraction to capture only the relevant information, as described in Sec. 3. The camera temperature was kept within 0.2°C throughout the acquisition using a TEC and a PID loop.

The ${}^{55}\mathrm{Fe}$ radioactive source produces two main energy lines at 5.90 and 6.49 keV. The fraction of single-pixel events obtained at 5.9 keV was measured to be $\sim 30\%$. Previous work on the IMX290 attempted to measure the gain dispersion by dividing the sensor area into $5\times 5$ sub-areas and measured a dispersion of 0.14%.²⁰ Although this method can give a rough idea of the gain dispersion, it can still mix in small-scale gain variations caused by mismatch variations of the pixel elements (e.g., transistors, photodiodes) during the manufacturing processes.

We performed ${}^{55}\mathrm{Fe}$ simulations to determine the optimal number of single-pixel events per pixel to accurately recover a given gain dispersion. We estimated that for a gain dispersion of $\sim 0.1\%$, more than 1000 events per pixel are needed with both calibration methods. A more precise estimate of the optimal number of individual events is discussed further in Sec. 6. Based on these estimates and the count rate produced by the radioactive source, we targeted 14 consecutive days of acquisition to achieve $\sim 1200$ events per pixel.

Our acquisition sequence consists of two main steps. First, the offset map is calculated using 300 frames (30 s of data collection). The setup configuration prevents the removal of the radioactive source during this acquisition time. We thus derived the offset map by applying a sigma clipping to mitigate the effect of X-ray events on the calculation. The second phase consists of collecting 6000 consecutive frames (10 min of acquisition) and performing a real-time X-ray event extraction using the offset map, as described in Sec. 3. We repeat this sequence every 10 min for 14 days to track the evolution of the offsets and to ensure efficient real-time event extraction. When the camera is turned on, the temperature may temporarily fluctuate faster due to the additional heat produced by the readout electronics until a new equilibrium is reached. To have only datasets with a stable camera temperature, the first 1.5 h of datasets was removed from our post-processing analysis.

4.3.

Post-processing and Per-Pixel Spectrum

The data set collected over 14 days generated 1 TB of data. Processing such a large volume of data remains challenging with current laboratory computing resources. Thus, we performed the post-processing analysis on the Massachusetts Institute of Technology (MIT) SuperCloud,²⁴ a supercomputing cluster at MIT that currently provides individual allocation of 384 CPUs. The post-processing analysis was designed and optimized to minimize computational time, resulting in $\sim 1$ day of computation on 384 CPUs. Single-pixel events were first extracted from each 10-min event list file and then redistributed to their respective pixels along the 14 days of acquisition. Figure 2(a) shows the count map of the 2.1 million pixels from the measurement. We determined a median count of 1368 and observed a non-uniform distribution of counts across the sensor area. The lower left corner had a count of more than 2000, whereas the upper right corner had a count of $\sim 500$. This could be due to a misalignment of the ${}^{55}\mathrm{Fe}$ source with the sensor creating an inhomogeneous illumination pattern. The potential impact of this inhomogeneity on our analysis is discussed further in Sec. 4.4. We also observed that a significant number of pixels exhibited a square pattern, where the central pixel has a higher count rate than the adjacent pixels (see the inset zoom region in Fig. 2). Similarly, when we extracted two-pixel events from the datasets, we noticed an opposite pattern trend in the count map, where the central pixel has a lower count rate than the adjacent pixels. These patterns appear to be associated with pixels with higher noise levels, as shown in the noise map in Fig. 2(b). In this respect, pixels with higher noise levels cause the migration of single-pixel events into two-pixel events in adjacent pixels. This is a consequence of using the same split threshold for all pixels during the event extraction process. Implementing a pixel-dependent split threshold based on individual noise levels would mitigate this effect and result in a more homogeneous count map.

Fig. 2

(a) Count map of single-pixel events obtained after 14 days of acquisition with the ${}^{55}\mathrm{Fe}$ source. The upper and right insets represent the sum of counts by column and row, respectively. The upper right inset shows a zoom of a small area of the sensor outlining the square patterns observed. (b) Noise map derived from 300 frames at the beginning of one of the 10-min acquisition sequences. The upper and right insets show the average noise by column and row, respectively. The upper right inset shows a zoom-in on a small region of the sensor, highlighting the pixels with higher noise than average producing the squared patterns observed on the count map.

We ended up with one spectrum for each of the 2.1 million pixels, made up entirely of single-pixel events. Figure 3 shows an example of two spectra from two pixels in the same column (#662) but in different rows (#192 versus #242). The two main emission lines produced by the radioactive source are clearly detected at 5.90 keV ($\mathrm{Mn}\text{-}\mathrm{K}\alpha$) and 6.49 keV ($\mathrm{Mn}\text{-}\mathrm{K}\beta$). In addition, the Si-K escape line of $\mathrm{Mn}\text{-}\mathrm{K}\alpha$ (4.16 keV) is marginally detected. Both spectra were calibrated in energy with the same gain, and a small energy shift between them is noticeable, contributing to the broadening of the lines when combined. We measured a spectral resolution of 122 eV at 5.9 keV using a Gaussian fit, close to the intrinsic Fano limit at this temperature (119 eV) and consistent with a $2e-$ rms noise device.

Fig. 3

${}^{55}\mathrm{Fe}$ spectra of single-pixel events from the same column (#662) but in different rows (#192 versus #242) of the sensor area calibrated with the same gain value. The $\mathrm{Mn}\text{-}\mathrm{K}\alpha$ (5.90 keV) and $\mathrm{Mn}\text{-}\mathrm{K}\beta$ (6.49 keV) lines produced by the ${}^{55}\mathrm{Fe}$ radioactive source and the Si-K escape line of $\mathrm{Mn}\text{-}\mathrm{K}\alpha$ (4.16 keV) are detected in both spectra. A small horizontal shift is visible between the two spectra. The gray dashed line shows the best-fit Gaussian model used to measure the spectral energy resolution for the pixel ID (662, 242).

4.4.

Gain Measurement and Optimal Counts per Spectrum

Two different calibration methods, peak fitting and ECC, were used to determine the gain for each pixel (see Sec. 4.1). We used the enhanced version of ECC with adaptive mesh refinement²³ to discretize the parameter space and reduce the computational time.

For both methods, we assumed a linear calibration in energy, expressed as $E=\text{gain}\times PHA+\text{offset}$, where PHA is the pulse height amplitude. Because the ${}^{55}\mathrm{Fe}$ source produces two emission lines within a limited energy range (0.5 keV), it can be challenging to constrain a small residual offset sometimes required to improve the energy calibration of some detectors. We first performed an energy calibration using the peak fitting method, considering two free parameters (gain and offset). The ratio of the offset to its error returned a median value of 1.2, suggesting that the residual offset, expected to be close to zero, cannot be well-constrained with the current data. This is supported by real-time data processing, in which the offset map subtracted from the frames prior to extracting the X-ray events is updated every 10 min. This ensures that any systematic offset drift is minimized over time and that the residual offset should tend to zero. We therefore assumed an offset of 0 for the energy calibration in the rest of our analysis.

The gain map derived by the peak fitting method considering only one free parameter (gain) is shown in Fig. 4. The gain distribution shows a normal distribution with a mean of 294 ADU/keV and a 1-sigma dispersion of 0.4%. This observed variation exceeds that reported in the literature using $5\times 5$ binning regions (0.14%),²⁰ suggesting potentially significant interdevice variability or discrepancies due to the method used. The upper inset of Fig. 4 represents the average gains for each column. A difference was observed between the average gains of odd and even columns. This difference is constant along the columns ($\sim 1\mathrm{ADU}/\mathrm{keV}$) and larger than the standard deviation of the average column gains. This trend was not observed in the right inset of the 2D gain map (Fig. 4), where even and odd rows have consistent average gains along rows. The feature observed for the average gain columns is thought to be due to the parallelism of the readout specific to CMOS technology, where the signal from an entire row can be read out simultaneously. In this design, each column has its own second-stage signal amplifiers, correlated double sampling, and analog-to-digital converters (ADCs). In addition, these components can be distributed on either side of the sensor area to save space, with even-column components on one side and odd-column components on the other. No significant difference was observed between the gains measured by ECC and peak fitting methods, with ${\widetilde{\text{gain}}}_{\mathrm{ECC}}=293.66\pm 1.18\mathrm{ADU}/\mathrm{keV}$ and ${\widetilde{\text{gain}}}_{\text{peak fitting}}=293.59\pm 1.18\mathrm{ADU}/\mathrm{keV}$.

Fig. 4

Gain map derived from ${}^{55}\mathrm{Fe}$ spectra of single-pixel events using the peak fitting method on the $\mathrm{Mn}\text{-}\mathrm{K}\alpha$ and $\mathrm{Mn}\text{-}\mathrm{K}\beta$ lines. The upper and right insets represent the average gain by column and row, respectively. Average gains for even (odd) columns and rows are indicated by blue (black) lines. The upper right inset shows a zoom of a small area of the sensor outlining the small-scale gain variation.

We investigated the performance and advantages of each method with respect to the number of single-pixel events in the spectrum (sample size). Due to the inhomogeneity observed in the count map in Fig. 2, which could bias the results, we performed this analysis by randomly selecting 10,000 pixels within the 90th to 95th percentile of the count distribution. This criterion yields an initial pixel sample of 104,737 pixels with counts between 1664 and 1720 photons. For each pixel, we generated a set of spectra with different sample sizes (10, 50, 100, 200, 500, …, events) by randomly selecting events from their initial distribution. For each sample size, we applied both ECC and peak fitting to the set of 10,000 spectra and measured their gains. The energy resolution of the $\mathrm{Mn}\text{-}\mathrm{K}\alpha$ line was then determined by combining the 10,000 spectra into a single spectrum after correcting for individual gain variations. The evolution of spectral resolution as a function of sample size is shown in Fig. 5. Both methods converge to the same energy resolution when spectra are composed of more than $\sim 500$ single-pixel events. At lower counts, the ECC method outperforms peak fitting, showing an improvement in spectral performance for statistics greater than 10 photons. In contrast, peak fitting requires a minimum of 100 photons to achieve results similar to using a single gain for all pixels. Given a gain dispersion of 0.4%, this analysis provides guidance on the optimal statistics required to accurately correct for the per-pixel gain dispersion of the sensors. As all pixels in our data set have more than 492 single-pixel events, it is unlikely that the inhomogeneous count pattern observed in the count map (Fig. 2) has affected our per-pixel gain correction.

Fig. 5

Energy resolution of single-pixel events at 5.9 keV ($\mathrm{Mn}\text{-}\mathrm{K}\alpha$) as a function of sample size. The energy resolution is measured using a Gaussian fit on a spectrum combining 10,000 pixels from the high-count region of Fig. 2 and corrected for per-pixel gain variation using the peak fitting (red) and ECC (blue) methods. The dashed gray line represents the energy resolution assuming the same gain for all pixels. The Fano limit expected at room temperature is indicated by a dashed-dotted line.

5.1.

Per-Pixel Gain Correction

Correcting the per-pixel variation of CMOS sensors is expected to improve the spectral performance of the device. To validate our gain measurement, we corrected the gain dispersion using the gain map visible in Fig. 4 on a 10-min data set, corresponding to $\sim 4$ million photons detected. The combined spectrum of single-pixel and all events before and after the gain correction is shown in Fig. 6. As expected, the energy resolution is improved after correcting the per-pixel gain variation from $136.1\pm 0.2\mathrm{eV}$ to $123.6\pm 0.2\mathrm{eV}$ ($\sim 12\mathrm{eV}$) for single-pixel events and from $144.5\pm 0.6\mathrm{eV}$ to $136.1\pm 0.6\mathrm{eV}$ ($\sim 10\mathrm{eV}$) for all events spectra.

Fig. 6

Spectrum of single-pixel events (a) and all events (b) before per-pixel gain correction (in black) and after per-pixel gain correction (in blue or red). The gray, dark blue, and dark red dashed lines show the best Gaussian fit used to derive the energy resolution.

The improvement observed after the per-pixel gain correction confirms our ability to measure and correct the gain variation. The spectral performance of single-pixel events is approaching the theoretical limit of silicon detectors of 119 eV at room temperature (300 K) for an electron–hole pair creation energy of 3.67 eV and a Fano factor of 0.118.²⁵ The difference observed for single-pixel events compared with the Fano limit could be mainly attributed to the electronic noise of $\sim 2e-$ rms measured for this device. In addition, the electronic noise might increase the observed difference between single-pixel and all events to some extent by the fraction of energy loss below the split threshold, known as the charge-sharing effect, which depends on the detector geometry and multiplicities. This effect can be mitigated empirically or by Monte Carlo simulations⁵ but was not corrected for in this analysis. Additional broadening of the lines compared with the Fano limit could result from the use of the same split threshold for the entire sensor area, producing the square patterns observed in Fig. 2 and the overestimation of the energy of some events.

5.2.

Temperature Dependence of Gains

After a pause of 6 days, we acquired a second data set under the same configuration and operating conditions except that the camera temperature was increased from 21°C to 26°C. The objective was to investigate the temperature dependence of the gain, in particular, the per-pixel dispersion. Figure 7(a) shows the comparison of the gain distribution at the two temperatures. The gain dispersion at both temperatures was consistent at 0.4%, but the distribution showed a median shift of 0.15 ADU/keV (0.05%) when the temperature was raised by 5°C. At first order, this suggests that all pixel gains are affected in a similar way as the temperature changes and that the gain dispersion remains constant within the temperature variations observed during data acquisition ($\lesssim 0.2°\mathrm{C}$). In a broader context, measuring the per-pixel gain dispersion at one temperature may offer the possibility of mitigating the gain dispersion at another temperature, at least within a certain temperature range. Additional data sets over a broader temperature range are needed to confirm the temperature dependence of the gains.

Fig. 7

(a) Histogram of the 2.1 million gains at 21°C (blue) and 26°C (orange). (b) Total number of counts per 10-min file from the $\mathrm{Mn}\text{-}\mathrm{K}\alpha$ line ($5.7<E<6.1\mathrm{keV}$) as a function of time at 21°C (blue) and 26°C (orange). The dashed black line represents the expected radioactive decay rate of ${}^{55}\mathrm{Fe}$ during the observation period. The lighter orange circles indicate count rates at 26°C, reduced by a constant offset of 0.5% to obtain a ${}^{55}\mathrm{Fe}$ decay rate consistent with the one measured on the 21°C data set. The exact cause for this offset remains to be explored in future work.

5.3.

Device Stability over Time

Our long acquisition data sets allow us to measure the stability of the device over several days at a high count rate ($\sim 7000$ counts per second). In Fig. 7(b), the total number of counts per 10 min file from the $\mathrm{Mn}\text{-}\mathrm{K}\alpha$ ($5.7<\mathrm{E}<6.1\mathrm{keV}$) is shown for both temperatures. Hot pixels have been removed from each file, and the gain temperature dependence observed in Fig. 7(a) has been corrected. We observed that at both temperatures the count rate in the $\mathrm{Mn}\text{-}\mathrm{K}\alpha$ line decreases progressively with time. The measured decay is consistent with the expected radioactive decay rate of ${}^{55}\mathrm{Fe}$ (half-life decay of 2.737 years) over this period. However, we noticed an additional constant offset of $\sim 0.5\%$ in the counts to the expected trend at 26°C. By removing this offset, we can obtain a decay consistent with the data set acquired at 21°C. The exact reasons for this offset will be the subject of future work.

All pixels with an abnormal count rate (more than 10 counts per 10 min) were considered as hot pixels. The number of hot pixels was very stable over time for both temperatures, with a median number of hot pixels of $39\pm 6$ at 21°C and $44\pm 6$ at 26°C. At 26°C, we also saved the offset and noise map every 10 min to track their evolution over time. We found that the noise was very stable, with an equivalent electronic noise of ${2.10}_{-0.01}^{+0.05}e-\mathrm{rms}$. The offset level was configured to a value of 200 ADU, and we measured a value of ${199.76}_{-0.63}^{+0.60}\mathrm{ADU}$, corresponding to a fluctuation of $\sim 1e-$ over 16 days. Our measurements demonstrated excellent stability in terms of readout noise, hot pixels, and dark current level of the camera over more than 30 days of data acquisition.