Unbiased zero-count correction method in low-dose high-resolution photon counting detector CT

Objective. To address the zero-count problem in low-dose, high-spatial-resolution photon counting detector CT (PCD-CT) without introducing statistical biases or degrading spatial resolution. Approach. The classical approach to generate the sinogram projection data for estimating the line integrals of the linear attenuation coefficients of the image object is to take a log transform of detector counts, which requires zero counts to be replaced by positive numbers. Both the log transform and the zero-count replacement introduce biases. After analyzing the statistical properties of the zero-count replaced pre-log and post-log data, a formula for the statistical sinogram bias was derived, based on which a new sinogram estimator was empirically constructed to cancel the statistical biases. Dose- and object-independent free parameters in the proposed estimator were learned from simulated data, and then the estimator was applied to experimental low-dose PCD-CT data of physical phantoms for validation and generalizability testing. Both bias and noise performances of the proposed method were evaluated and compared with those of previous zero-count correction methods, including zero-weighting, zero-replacement, and adaptive filtration-based methods. The impact of these correction methods on spatial resolution was also quantified using line-pair patterns. Main Results. For all objects and reduced-dose levels, the proposed method reduces the statistical CT number biases to be within ± 10 HU, which is significantly lower than the biases given by the classical zero-count correction methods. The Bland-Altman analysis demonstrated that the proposed correction led to negligible sinogram biases at all attenuation levels, whereas the other correction methods did not. Additionally, the proposed method was found to have no discernible impact on image noise and spatial resolution. Significance. The proposed zero-count correction scheme allows the CT numbers of low-dose, high-spatial-resolution PCD-CT images to match those of standard-dose and standard-resolution PCD-CT images.


Introduction
The high rate of clinical CT utilization has culminated in concerns regarding the underlying health risks associated with the exposure of patients to ionizing radiation (McCollough et al 2012). In recent years, promising results from semiconductor-based photon counting CT (PCD-CT) (Leng et al 2019, Rajendran et al 2021,Taguchiet al 2021, Treb et al 2022 has piqued community interest in the potential for PCD-CT to provide a new paradigm for further radiation dose reduction. The underlying design and physical principles governing PCD-CT technology provide inherent advantages compared to conventional scintillatorbased energy-integrating detectors (EIDs). The prominent attributes that enable PCD-CT to reduce radiation dose are the improved detective quantum efficiency and the capacity to reduce or eliminate electronic noise.
Despite the benefits of PCD-CTs, many practical challenges still exist that must be appropriately addressed to realize the full potential of PCD-CT imaging in the low-dose regime. One such challenge is the issue of registering a raw count of zero for a given detector element. Zero counts in the raw data must be corrected to avoid division-by-zero when the logarithmic normalization operation is performed in order to obtain the sinogram data for estimating the line integrals of the linear attenuation coefficients of the image object. The probability of registering a count of zero for a given detector element is significantly higher for PCDs when compared with conventional detectors. This increased probability is partially the result of the smaller PCD pixel size of 0.1-0.3 mm, compared to EIDs with elements on the order of 1 mm, and the allocation of counts to multiple energy bins for spectral CT imaging applications. Figure 1, illustrates this increased probability for PCDs to register a count of zero. The histograms in this figure compare the counts generated in an EID (PaxScan 4030CB, Varex Imaging) and a PCD (XC-Hydra FX50, Varex Imaging) across four exposure levels. With all other imaging parameters matched, the PCD pixels were binned to a 0.2 mm × 0.2 mm size to compare to the native EID pixel size of 0.194 mm × 0.194 mm. Following a detector dark current correction on the EID counts, the PCD had a significantly higher number of non-positive measurements when compared to the EID across four exposure levels, highlighting the concern regarding zero counts in PCD-CT imaging. This concern becomes further exacerbated when considering low-dose, high-spatial-resolution PCD-CT applications, large patient imaging, and metallic or highly attenuating structures in the patient anatomy.
Although the prominence of zero-counts is greater for PCD-CT imaging, the issue of zero-counts and negative measurements resulting from electronic noise, has been well documented for low-dose EID-CT imaging (Lee et al 2016, Ye et al 2019. Correction schemes that have been developed to address non-positive measurements in EID-CT imaging include the replacement of non-positive measurements with artificial positive values (Gilland et al 1999, Wang et al 2006, Lee et al 2016, the filtering of the raw counts (Hsieh 1998, Kachelrie et al 2001, Thibault et al 2006, Chang et al 2016, Lee et al 2016, and iterative reconstruction methods with imposed non-negativity constraints or appropriate statistical modeling (La Riviere et al 2006, Wang et al 2006, Thibault et al 2007, Haase et al 2019. The most straightforward correction to non-positive measurements is to replace these values with an artificial positive constant or replace negative measurements with their absolute Figure 1. Demonstration of the increased probability for registering a zero count in PCD-CT imaging. The PCD pixels were binned to a size of 0.2 × 0.2 mm and compared to an EID with a pixel size of 0.194 mm × 0.194 mm with all other imaging parameters were matched. The PCD had significantly more zero counts when compared to the non-positive counts generated in the EID across all four exposure levels shown. values. However, these replacement-based correction schemes corrupt the raw measurements by altering the true statistical nature of the data and introducing additional bias in the final reconstructed images. This was best exemplified in the work by Lee et al (2016), in which they demonstrated that non-positive replacement methods, including an absolute value method and a replacement algorithm, resulted in the introduction of a positive bias to the local pre-logarithm counts and a negative bias to the post-logarithm distributions. When significant corrections are required, the bias introduced by these replacement corrections may manifest as significant shading artifacts in the final reconstructed image.
To reduce the bias introduced by the replacement of non-positive measurements, spatial filtration of raw detector counts can be used (Hsieh 1998, Thibault et al 2006, Takahashi et al 2014. The drawbacks to the filtration methods are the potential introduction of correlations among neighboring measurements in the sinogram data and the inherent degradation of the spatial resolution that occurs when significant corrections are necessary. The correlations introduced by mean-preserving filters violate the conditional independence assumption utilized in many statistical reconstruction techniques often employed in conjunction with these aforementioned pre-processing filters. In this work, a theoretical analysis was performed to highlight the impact of zero-count replacement on the statistical properties of both the raw PCD counts and the post-logarithm sinogram data. A correction framework was then proposed with the following components: (1) replacement of zero-counts in the raw PCD counts with an artificial positive constant and (2) a novel data-driven Laurent expansion correction method to reduce statistical bias and the bias introduced by the replacement of zero-counts. Experimental validation studies were then conducted using quantitative or anthropomorphic phantoms to demonstrate the efficacy and robustness of the proposed correction framework.

Methods
2.1. Impact of zero-count replacement on the statistical properties of PCD counts Before a zero-count correction, the raw count output, N, of a given PCD pixel is known to follow the Poisson distribution under the assumption of negligible pulse pileup. The probability mass function (PMF) of N is: where λ is the expected value of the random variable N. When N = 0, the log transform cannot be taken to produce the line integrals of the linear attenuation coefficients for image reconstruction. However, if zerocounts are replaced by a positive non-integer constant, N c , to enable the log transform, then the PMF of the zeroreplaced counts, ¢ N , will have to be modified as follows. Using the above modified Poisson PMF, the kth moment of ¢ N for k > 0 can be calculated as follows: Therefore, it becomes evident that the statistical properties of ¢ N deviate from those pertaining to the Poisson distribution, that the un-modified counts N follow.
Furthermore, from equation (3), the first moment of the modified Poisson distribution is N c e −λ + λ, which is statistically biased as shown below: Thus, the replacement of zero-counts by N c introduces a positive bias of N c e −λ . It is also clear that an effort to eliminate statistical bias would automatically result in the consequence of N c being set to 0.
To find a way out, namely, to reduce statistical bias while enabling the needed log-transform, it is illuminating to also look into the noise variance of ¢ N . Using equation (3), the variance of ¢ N can be readily calculated as follows: In comparison to the variance of N, the variance of ¢ N has an additional term of ( ) l -- . Since the noise variance depends on the free parameter N c , one can choose an optimal N c to minimize the variance. This can be easily accomplished by setting the first order derivative of variance s ¢ N 2 with respect to N c : to zero, we can get the N c value that minimizes the variance of ¢ N : This result motivates us to consider another scheme to minimize both noise variance and statistical bias, i.e. to minimize the mean square errors (MSE), i.e. the sum of noise variances and squared bias, as follows: This yields an optimal selection of parameter N c : With the above insight into the statistical properties of zero-count replaced detector counts, we can determine the impact of zero-count replacements in the post-logarithm projection data.
2.2. Impact of zero-count replacement on the statistical properties of post-log sinogram data Following the replacement of zero-counts, the post-logarithm sinogram data, ¢ y , can be calculated as where 〈N 0 〉 denotes the ensemble average of air scan counts. Note that this is a random variable because ¢ N is a random variable. Moreover, the nonlinear nature of the log-transform also leads to a statistical bias in the logtransformed projection data, as shown below. To estimate the associated statistical bias relative to the following ground truth projection data: , 12 0 a Taylor expansion of equation (11) about λ is needed: where ( ) l á ¢ -ñ N k denotes the kth central moments of ¢ N .

Empirical construction of an unbiased sinogram estimator
As presented in appendix A, the central moment of ¢ N is related to the central moment of N by Using the result in equation (15), the bias of ¢ y in equation (14) can be calculated to obtain the following result: We can now discuss the proposed new scheme in this paper to correct the statistical bias caused by the introduction of the zero-count correction. Inspired by our previous work (Chen et al 2022, Li et al 2023, we made the following key observation about ¢ -N k (appendix B): where coefficients B k,j are independent of λ. Compared with equation (16), one can observe that both formulae have similar structures. The difference between the two is the expansion coefficients A k versus B k,j . Therefore, we propose to add a summation of terms ¢ N 1 k to the modified projection data ¢ y in equation (11): with the hope that the statistical biases can be eliminated by choosing the proper expansion coefficients {C k } that are independent of λ and the image object.
The selection of coefficients {C k } can be formulated as a least squared optimization problem as follows: where || · || 2 are the standard least squares. This can be readily solved using the widely available numerical solvers provided that the training data can be obtained with M pairs of training data { } = N p , The training data can be obtained from numerical simulations as described in the following sub-sections. This method is generally referred to as parameter calibration or machine learning. Regardless of the name of the method, one cannot avoid answering the following essential scientific question: once {ˆ} C k are obtained by solving equation (19), how generalizable are they when being applied to PCD data acquired under different conditions? To answer this question, we used simulated PCD data and equation (19) to solve for {ˆ} C k , and then applied the correction to experimental PCD data acquired with different objects and at different dose levels for generalizability testing. The simulation and experimental methods are described as follows.

Simulation methods
In this work, the values of {ˆ} C k were determined by solving equation (19) using numerically simulated ¢ N data: for a given assumed λ level, an ensemble of Poisson-distributed random numbers, {N}, were repeatedly generated (for 1 × 10 6 times) using the Matlab's poissrnd function. Zeros in {N} were replaced by a value of N c = 1/3 to generate { } ¢ N , and then an ensemble averaging was used to estimate á ¢ ñ -N K . The process was repeated at M (M = 37) different λ levels ranging from 1 to 10 with a uniform step size of 0.5 and from λ=15 to 100 with a step size of 5. The set of lambda values was selected for two distinct reasons. First, lambda values from 0 to 10 exhibit significant zero counts. Therefore, parameter learning can be fine-tuned to address the bias associated with the replacement of zero counts, in conjunction with the bias associated with the logarithmic normalization operation. Second, lambda values from 15 to 100 do not demonstrate a significant percentage of zero counts, but the statistical bias associated with the log function is still significant. For lambda values greater than 100, the statistical bias becomes insignificant. As such, this parameter learning strategy aims to have a generalizable correction framework that works across all dose levels, even when zero counts are not a significant issue. To determine the optimal choice of K, coefficients were learned for K = 3 to K = 6, and then applied to the low dose CTP404 module of the Catphan 600 phantom that is described in detail in section 2.5.1. The residual bias was then quantified as the square root of the sum of squares for all material inserts. The K value with the least residual bias was then selected. The choice of N c = 1/3 was guided by the results shown in figure 2. It was found that the performance of the proposed correction scheme was invariant with respect to bias for N c values from 0.1 to 0.65. The correction framework can be implemented with the user's desired selection of N c , and the choice can be chosen to improve performance with respect to bias, noise variance, or MSE as demonstrated in figure 2.
With the simulated data, equation (19) can be written as The above linear equation array can be written with the following matrix-vector notation: The optimization problem in equation (19) is equivalent tô The solution to this least squares problem is given bŷ In this work, {ˆ} C k were calculated up to k = 4 from the simulated data using equation (23). Once calculated, their values were fixed when applied to the experimental PCD-CT data.

Experimental methods
Experimental data were acquired with a benchtop PCD-CT imaging system shown in figure 3. The PCD (XC-Hydra FX50, Varex Imaging) has a 0.75 mm thick cadmium telluride (CdTe) sensor layer, 5120 × 60 pixels (0.1 mm × 0.1 mm each), and two adjustable energy thresholds. The PCD was operated in the anti-coincidence mode. The anti-coincidence mode was implemented by the PCD manufacturer to address the charge-sharing effect, in which a single interacting photon may induce pulses to be counted for more than one detector element. This mode is designed to detect coincident count events within a given detector element, and its adjacent detector elements. For each detected coincident events, the total induced charges within a given pixel block are assigned to a single detector element (Ballabriga et al 2013). In a prior publication, our group demonstrated that PCD counts follow the Poisson distribution when this mode was turned off (Ji et al 2017). Figure 4 is presented to show that PCD counts still follow the Poisson distribution under the anti-coincidence mode. For non-spectral imaging acquisitions, only a single energy threshold controller was activated and set to 15 keV to reject electronic noise. The x-ray source is a rotating-tungsten anode tube (G-1592 with B-180H housing, Varex Imaging). For all imaging studies, the nominal focal spot was fixed at 0.6 mm, and the tube potential was set to 120 kV. A beam collimator and a 0.25 mm copper filter are attached to the tube. A motorized rotary stage was used to rotate the image object: during each 360 degree PCD-CT acquisition, 1200 projection views were acquired. The source-toiso and source-to-detector distances were 57.2 cm and 103.4 cm, respectively. For each image object, high-dose PCD-CT scans were repeatedly acquired at 320 mAs, and 48 mGy weighted CTDI (16 cm). After an ensemble averaging of the high-dose raw PCD counts, a log transform was taken to generate the reference-standard sinogram, namely p in equation (12). Each image object also received repeated  (18) were learned separately. The measured bias was plotted for each value of N c to demonstrate that the bias is invariant with respect to N c choice within the range of 0.1-0.65. low-dose PCD-CT scans at 20 mAs, and 3 mGy weighted CTDI (16 cm phantom). The histograms in figure 5 illustrate typical detector count levels in the low-dose scans of the three phantoms.
The proposed zero-count correction was applied to each set of low-dose PCD counts before a log transform was taken to generate the low-dose sinogram. The standard filtered backprojection algorithm was used to reconstruct PCD-CT images from the sinogram. A ring artifact correction (Prell et al 2009) and a PCD-CT band artifact correction  were applied for each of the reconstructed CT images generated in this work. Apart from the ring correction and zero corrections, no additional low signal corrections or image correction methods were applied. All images were reconstructed to a 512 × 512 image matrix and the reconstruction slice thickness was 2.8 mm for all PCD-CT image results.  The proposed correction method was compared with three other zero-count correction schemes (figure 6): • Zero weighting: the log transform of the raw PCD counts is taken without the replacement of zeros. The detector elements that registered zeros in the raw counts are then zeroed out in the sinogram domain.
• Zero replacement: zeros are replaced with the constant N c without further corrections.
• Adaptive alpha-trimmed mean filter (AATM): this adaptive filtration method has been used for low-count corrections in CT. Details regarding the AATM implementation method are provided in (Hayes et al 2018).

Quantitative CT phantom imaging
To evaluate the performance of the proposed correction method at different local contrast levels, the CTP404 module of a Catphan600 phantom (The Phantom Laboratory) was used. This module consists of 8 cylindrical material inserts consisting of Teflon, Delrin, Acrylic, Polystyrene, low-density Polyethylene (LDPE), polymethylpentene (PMP), and air ( figure 3). A total of 50 low-dose scans and 26 high-dose scans were acquired. Sinogram and CT number biases of the four correction frameworks were measured in the difference images generated by subtracting each low-dose measurement from the ensemble average of the high-dose scans. Bland-Altman analysis was performed for each of the four correction schemes investigated. Each point on a Bland-Altman plot corresponds to a point in the post-log sinogram. The horizontal axis is the mean of the low-dose and reference-standard sinograms; the vertical axis is the difference between the low-dose and reference-standard sinograms. Furthermore, sinogram biases were plotted as a function of λ to investigate the generalizability of each correction scheme. Each point on the sinogram-λ plot corresponds to a point in the post-log sinogram. The corresponding λ values for each point on the sinogram were generated by taking the ensemble average of the raw counts for the 50 low-dose acquisitions.
For each set of reconstructed low-dose PCD-CT images, its difference was generated relative to the standard PCD-CT image. Circular regions of interest (ROIs) were placed at locations of 7 different material inserts and the phantom center. Each ROI mean yields one sample of CT number bias. The average CT number bias was calculated by taking the ensemble average across the 50 low-dose datasets, and its 95% confidence interval was constructed from  s 1.96 50 , where σ denotes the standard deviation of the measured ROI biases across the 50 low-dose datasets.
In addition to a bias analysis, the impact of the four zero correction methods on CT noise was evaluated. A total of 15 circular ROIs were placed at the 7 material insert locations, at 4 uniform regions in the periphery of the phantom, and at 4 uniform regions in the center of the phantom. The variance of each ROI was measured in the PCD-CT images for each of the 50 low-dose datasets.

Anthropomorphic phantom imaging
To evaluate the generalizability of the proposed correction method to anthropomorphic objects, a human head phantom (ACS CT Head, Kyoto Kagaku Co., LTD) shown in figure 3 was used. This phantom contains models of a contrast-enhanced cerebral artery tree in the left hemisphere and a low-contrast intraparenchymal hemorrhage model in the right hemisphere. A total of 50 repeated scans were acquired, with 25 low-dose and 25 high-dose acquisitions. Difference images were generated by subtracting each low-dose PCD-CT acquisition from the ensemble average of the high-dose reference scan for each correction method. In addition to difference images, bias was quantified in 3 ROIs within the head phantom: (1) iodinated cerebral artery model, (2) cerebral spinal fluid model (CSF), and (3) brain tissue model. CT number biases of the four correction frameworks were quantified in the generated difference images with the 95% confidence interval constructed as  s 1.96 25 . Figure 6. The reconstruction pipeline for the zero-weighting, zero-replaced, AATM filtration, and proposed correction methods are presented. In this chart, N indicates the raw PCD counts, ¢ N indicates the PCD counts following the replacement of zeros with an artificial positive constant N c .

Spectral CT Imaging
Zero counts become more pronounced when multiple-energy bins are utilized for spectral CT imaging applications. To highlight the impact of an appropriate zero-count correction for spectral CT imaging, we conducted PCD-CT imaging acquisitions with the low and high energy thresholds set to 15 keV and 63 keV, respectively. The image object was a 16 cm cylindrical phantom with Gammex iodine and cylindrical calcium rods inserted at varying concentrations. The concentrations of iodine were 5, 10, and 20 mg ml -1 , and the calcium concentrations were 50, 100, and 200 mg ml -1 . In total, 30 repeated scans were acquired with 20 lowdose and 10 high-dose reference acquisitions. After applying the proposed correction scheme to low-and highenergy PCD data, low-and high-energy PCD-CT images were reconstructed before an image domain two material decomposition was performed to generate water and iodine basis images. The reconstructed high-and low-energy bin images and the iodine and water basis images were employed to generate difference images from the ensemble average of repeated high-dose acquisitions for each image.
Bias was quantified for each of the 6 material inserts in the material basis difference images for the four correction schemes. The 95% confidence intervals were constructed in the same manner as the previous investigations discussed in this work.

Spatial resolution evaluation
The impact of the proposed zero-count correction framework on the spatial resolution of the reconstructed PCD-CT images was evaluated with the Catphan600 528 module. This module has bar patterns that include 1-21 line pairs per centimeter. The line patterns were compared between the proposed correction framework and the three other correction schemes to evaluate if the proposed correction framework introduced any degradation to the spatial resolution. For this investigation, the data was collected in the PCDs high-resolution mode at 4.5 mGy. Line profiles are plotted and compared for the 8 line pairs per centimeter resolution pattern for each correction scheme employed in this work.  (19)) as a function of λ compared to the measured post-log bias.

Validation of theoretical analysis
In figure 7, the theoretical pre-log bias (A), pre-log variance (B), and post-log bias (D) are compared with the corresponding values from the numerical study. The theoretical values were calculated from the derivations provided in equations (5) and (6). The theoretical values show good agreement with the measured values for prelog data. The theoretical model for the post-log bias demonstrates good agreement with the measured zerocount corrected sinogram bias.

Experimental results
3.3.1. Quantitative phantom results figure 8 plots post-log sinogram bias as a function of λ. The zero weighted method generated strong negative biases at λ < 4. Zero weighting significantly underestimates the attenuation for pixels registering zero counts. As such, the sinogram is expected to be negatively biased when the zero weighted correction is applied. In contrast, the zero replaced method introduced a positive sinogram bias at low λ values. This may seem to contradict the fact that the replacement of zeroes with a positive constant will under estimate the attenuation for a given pixel. However, the replacement of zeroes is not the only source of statistical bias. The statistical bias associated with the log transform also introduces bias. The bias introduced by the logarithmic transform is positive in the sinogram domain. Therefore, the positive bias exhibited in the low counts regime for the zero replaced results arises from the statistical bias, which is of greater magnitude when compared to the negative bias introduced by replacing zeroes. This point is further exemplified when examining the results for the AATM correction. AATM filtration inherently increases the overall counts when applied to regions with low counts. As such, a negative bias is introduced in the low count regime (λ < 10). However, there is also the positive bias associated with the log transform. Based on figure 8 (D), it can be seen that the negative bias introduced by the AATM filter is the component of the bias with a greater magnitude in the range of λ = 10 to λ = 5. The positive bias associated with the log transform then becomes the predominant contributor to the bias when λ < 5. The proposed zero-count correction method generated the lowest mean bias. More importantly, the method removed the correlation between bias and λ: the wider vertical spread of the data points at lower λ values in figure 8 is due to higher noise.
Bland-Altman analysis results of the experimental sinogram data are shown in figure 9. The plots of the Bland-Altman analysis exhibit similar behavior to that of the sinogram bias as a function of λ plots. The zero weighted Bland-Altman shows a negative correlation between sinogram bias and the average attenuation, while the zero replaced correction demonstrated a positive correlation between bias and attenuation. The AATM Bland-Altman exhibits a nonlinear nature between sinogram bias and attenuation, which is consistent with the results for the sinogram bias as a function of λ. In comparison, the proposed correction demonstrated no correlation between sinogram bias and attenuation.
Sinograms and reconstructed PCD-CT images of the Catphan phantom are depicted in figure 10. For all phantom results, the reference dose sinograms and reconstructed PCD-CT images displayed in this work are ensemble averages of the reference dose scans. In contrast, the low dose sinograms and reconstructed images are single measurements and not ensemble averages of the repeated scans. For the Catphan phantom, the percentage of zero counts was 6.45% ± 0.58% for the low-dose acquisitions. Zero-weighting significantly degraded the image quality, exhibiting important shading and streak artifacts. These artifacts result from the underestimation of the line integrals that occurs when zero-weighting is performed. There is a marked improvement in the qualitative features of the zero-count replaced PCD-CT image. However, the appearance of 'ghost inserts' can be seen in the difference image. These ghost inserts show the existence of CT number bias. Following the application of the Laurent expansion correction, the manifestation of 'ghost inserts' can no longer be seen, suggesting the bias has been mitigated. This is further demonstrated by the quantitative results in table 1. Results of the CT noise variance analysis shown in figure 11 demonstrate similar noise performance between the zero replaced, AATM filtered and proposed correction schemes. The zero weighted images exhibit markedly increased CT image noise compared to the other correction schemes. Figure 12 shows PCD-CT images of the anthropomorphic head phantom. The percentage of zero counts was 2.13% ± 0.27% for the low-dose acquisitions. The reference dose sinograms and reconstructed PCD-CT images displayed are ensemble averages of the reference dose scans. In contrast, the low-dose sinograms and Figure 10. Sinograms and reconstructed PCD-CT images of the Catphan 600 CTP404 phantom are displayed for the reference highdose scan and the four zero-count correction schemes investigated in this work. Difference images were generated from the reference high-dose images and shown for the four correction schemes employed in this work. Table 1. Comparison of CT number biases for the material inserts and central portion of the Catphan600 phantom for the three zero-count methods studied in this work. The CT number biases were measured with respect to reference high-dose ensemble CT images.

Insert
Zero weighted Zero replaced AATM filter Proposed correction −90 ± 1 7 9 ± 1 −24 ± 1 −2 ± 1 Center −572 ± 2 6 4 ± 1 −68 ± 5 4 ± 1 reconstructed images are single measurements and not ensemble averages of the repeated scans. The low-dose corrected image demonstrates preserved anatomical features compared to the full-dose reconstructed images. For example, the intraparenchymal hemorrhage models can be distinguished as illustrated by the zoomed-in portion of the figure. Zero weighting results in a significant loss of anatomical features and the obfuscation of many intraparenchymal hemorrhage models. This is expected because zero-weighting is a sub-optimal Figure 11. Bar plots demonstrating the CT noise variance in HU 2 for the four zero correction schemes investigated at various ROIs in the Catphan600 CTP404 phantom. Figure 12. PCD-CT reconstructed images of the anthropomorphic head phantom are shown for the reference high-dose scan and the four zero corrections schemes employed in this work. Difference images as related to the reference high-dose images are shown. Additionally, the dashed boxes demonstrate a zoomed-in region of the head phantom, as indicated in the reference high-dose scan.
correction scheme, but the result further highlights the importance of appropriate zero-count corrections. Quantitative analysis of the iodinated cerebral artery model, CSF model, and brain tissue model is presented in table 2.

Spectral CT imaging results
The spectral PCD-CT imaging results depicted in figure 13 highlight the significance of addressing zero-counts when multiple energy bins are employed. The percentage of zero counts in the high-and low-energy bins was 9.15% ± 0.50% and 10.14% ± 0.51% respectively. In the basis images, it can be seen that the bias associated with the replacement of zero counts manifests as both 'ghost inserts' and streak artifacts. These artifacts are further propagated and exacerbated in the material basis images. In contrast, the proposed correction scheme can mitigate these artifacts to generate noise-only difference images compared to reference standard images. A quantitative analysis of the CT number bias in the material basis images for each material insert is tabulated in table 3. Figure 13. Difference images between the high-dose ensemble reference and low-dose zero corrected spectral PCD-CT images, including the reconstructed high-and low energy bin images and the water and iodine basis images. Colorbars are indicated for each type of reconstructed difference image, and the W/L is consistent between the displayed zero correction methods. 3.3.4. Spatial resolution evaluation Finally, line profiles across the 8 lp cm -1 pattern of the Catphan spatial resolution phantom are shown in figure 14, confirming that, as expected, the proposed correction scheme has no discernible impact on the spatial resolution.

Discussion
The numerical simulation results show that the theoretically derived statistical models of the zero-count replaced pre-log and post-log data are accurate. The developed understanding of the statistical properties of PCD-CT data guided us to construct a new sinogram estimator that reduces the bias introduced by the replacement of zero counts and the nonlinear logarithmic transform in the post-logarithm PCD data while Figure 14. Line profiles taken for each zero-count correction method and compared to the reference standard dose level for the 8 lp mm -1 line pattern in the Catphan600 phantom. preserving the spatial resolution of the final reconstructed images. Free parameters in the proposed sinogram estimator were pre-calibrated using simulation data. The utility and generalizability of the proposed sinogram estimator were shown through a series of experimental studies, including an evaluation of the sinogram biases and CT number biases, a demonstration of the correction frameworks capacity to improve spectral-CT imaging, and an investigation on the impact of the correction framework on spatial resolution, CT noise variance, and anthropomorphic phantom imaging. High spatial resolution CT imaging requiring accurate CT numbers has become increasingly important in the clinical imaging domain. Low-dose CT lung imaging, which has an increasingly prominent role in the modern radiology department, requires both high spatial resolution and quantitative accuracy for several tasks. For example, high spatial resolution is required for the detection of small pulmonary changes for interstitial lung disease The images generated from the zero weighting correction scheme highlight the importance of incorporating an appropriate zero correction into the CT image reconstruction pipeline. Severe artifacts, increased image noise, significant CT number biases, and a loss of qualitative image quality were found when zero counts were inadequately corrected. Although a simple zero replacement correction scheme may restore qualitative image quality, such a method will result in significant CT number bias. An AATM filtration method was also implemented and designed to address the issue of zero counts. The AATM filtration method reduced the CT number bias associated with the zero replacement scheme but could not match the performance of the proposed correction scheme in this work.
Additionally, the AATM filtration method introduces spatial correlations, has the potential to degrade spatial resolution in the resulting PCD-CT images, and corrupts the underlying statistical nature of the PCD-CT sinogram data. The proposed correction scheme demonstrated the capacity to mitigate CT number bias without introducing any spatial operations or significant noise. The proposed correction framework does not have a strigent requirement over the choice of the N c value. As shown in figure 2, the method demonstrated good performance for any N c within the range of [0.1-0.65]. Once a value for N c is chosen, parameters {C k } can be learned from simulated data without any dependence on the image object, x-ray source, detector, or system geometry.
One limitation of this work is that all studies were conducted on a benchtop PCD-CT imaging system such that the proposed correction scheme could only be applied to phantom imaging data and not human subject data. A limitation of the proposed correction scheme is that the Laurent expansion sinogram estimator (equation (18)) only converges when λ 1. In the case where λ < 1, the proposed correction scheme is expected to fail as the behavior of the bias at λ < 1 deviates significantly from the cases in which λ 1. The proposed algorithm still demonstrates utility if the number of pixels where λ < 1 is small, but it is expected to perform poorly in the case of many image pixels registering an expected count of less than 1. Table 4. Expanded form of ( ) l á ¢ -ñ N k for k 8.

Conclusion
This work highlights the importance of incorporating an appropriate zero-count correction scheme for lowdose and high spatial-resolution PCD-CT imaging. A statistical derivation was performed to establish a rationale for constructing a novel sinogram estimator to handle the issue of zero counts. An empirically constructed sinogram estimator was proposed and demonstrated the capability to handle zero detector counts encountered in low-dose, high-resolution PCD-CT imaging without introducing statistical biases, without increasing image noise, and without degrading spatial resolution.

Data availability statement
The data cannot be made publicly available upon publication because no suitable repository exists for hosting data in this field of study. The data that support the findings of this study are available upon reasonable request from the authors.
Appendix A. Central moments of ¢ N The central moment term in equation (14) can be expanded through the binomial theorem: Employing the binomial theorem once again, the above equation can be rewritten as: