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Background Purpose Methods Results Conclusion The first step in computed tomography (CT) reconstruction is to estimate attenuation pathlength. Usually, this is done with a logarithm transformation, which is the direct solution to the Beer‐Lambert Law. At low signals, however, the logarithm estimator is biased. Bias arises both from the curvature of the logarithm and from the possibility of detecting zero counts, so a data substitution strategy may be employed to avoid the singularity of the logarithm. Recent progress has been made by Li et al. [IEEE Trans Med Img 42:6, 2023] to modify the logarithm estimator to eliminate curvature bias, but the optimal strategy for mitigating bias from the singularity remains unknown.The purpose of this study was to use numerical techniques to construct unbiased attenuation pathlength estimators that are alternatives to the logarithm estimator, and to study the uniqueness and optimality of possible solutions, assuming a photon counting detector.Formally, an attenuation pathlength estimator is a mapping from integer detector counts to real pathlength values. We constrain our focus to only the small signal inputs that are problematic for the logarithm estimator, which we define as inputs of <100 counts, and we consider estimators that use only a single input and that are not informed by adjacent measurements (e.g., adaptive smoothing). The set of all possible pathlength estimators can then be represented as points in a 100‐dimensional vector space. Within this vector space, we use optimization to select the estimator that (1) minimizes mean squared error and (2) is unbiased. We define “unbiased” as satisfying the numerical condition that the maximum bias be less than 0.001 across a continuum of 1000 object thicknesses that span the desired operating range. Because the objective function is convex and the constraints are affine, optimization is tractable and guaranteed to converge to the global minimum. We further examine the nullspace of the constraint matrix to understand the uniqueness of possible solutions, and we compare the results to the Cramér‐Rao bound of the variance.We first show that an unbiased attenuation pathlength estimator does not exist if very low mean detector signals (equivalently, very thick objects) are permitted. It is necessary to select a minimum mean detector signal for which unbiased behavior is desired. If we select two counts, the optimal estimator is similar to Li's estimator. If we select one count, the optimal estimator becomes non‐monotonic. The oscillations cause the unbiased estimator to be noise amplifying. The nullspace of the constraint matrix is high‐dimensional, so that unbiased solutions are not unique. The Cramér‐Rao bound of the variance matches well with the expected I−0.5${{I}^{ - 0.5}}$ scaling law and cannot be attained.If arbitrarily thick objects are permitted, an unbiased attenuation pathlength estimator does not exist. If the maximum thickness is restricted, an unbiased estimator exists but is not unique. An optimal estimator can be selected that minimizes variance, but a bias‐variance tradeoff exists where a larger domain of unbiased behavior requires increased variance. [ABSTRACT FROM AUTHOR]

Background: The detectability performance of a CT scanner is difficult to precisely quantify when nonlinearities are present in reconstruction. An efficient detectability assessment method that is sensitive to small effects of dose and scanner settings is desirable. We previously proposed a method using a search challenge instrument: a phantom is embedded with hundreds of lesions at random locations, and a model observer is used to detect lesions. Preliminary tests in simulation and a prototype showed promising results. Purpose: In this work, we fabricated a full‐size search challenge phantom with design updates, including changes to lesion size, contrast, and number, and studied our implementation by comparing the lesion detectability from a nonprewhitening (NPW) model observer between different reconstructions at different exposure levels, and by estimating the instrument sensitivity to detect changes in dose. Methods: Designed to fit into QRM anthropomorphic phantoms, our search challenge phantom is a cylindrical insert 10 cm wide and 4 cm thick, embedded with 12 000 lesions (nominal width of 0.6 mm, height of 0.8 mm, and contrast of −350 HU), and was fabricated using PixelPrint, a 3D printing technique. The insert was scanned alone at a high dose to assess printing accuracy. To evaluate lesion detectability, the insert was placed in a QRM thorax phantom and scanned from 50 to 625 mAs with increments of 25 mAs, once per exposure level, and the average of all exposure levels was used as high‐dose reference. Scans were reconstructed with three different settings: filtered‐backprojection (FBP) with Br40 and Br59, and Sinogram Affirmed Iterative Reconstruction (SAFIRE) with strength level 5 and Br59 kernel. An NPW model observer was used to search for lesions, and detection performance of different settings were compared using area under the exponential transform of free response ROC curve (AUC). Using propagation of uncertainty, the sensitivity to changes in dose was estimated by the percent change in exposure due to one standard deviation of AUC, measured from 5 repeat scans at 100, 200, 300, and 400 mAs. Results: The printed insert lesions had an average position error of 0.20 mm compared to printing reference. As the exposure level increases from 50 mAs to 625 mAs, the lesion detectability AUCs increase from 0.38 to 0.92, 0.42 to 0.98, and 0.41 to 0.97 for FBP Br40, FBP Br59, and SAFIRE Br59, respectively, with a lower rate of increase at higher exposure level. FBP Br59 performed best with AUC 0.01 higher than SAFIRE Br59 on average and 0.07 higher than FBP Br40 (all P < 0.001). The standard deviation of AUC was less than 0.006, and the sensitivity to detect changes in mAs was within 2% for FBP Br59. Conclusions: Our 3D‐printed search challenge phantom with 12 000 submillimeter lesions, together with an NPW model observer, provide an efficient CT detectability assessment method that is sensitive to subtle effects in reconstruction and is sensitive to small changes in dose. [ABSTRACT FROM AUTHOR]

Background: Photon counting detectors (PCDs) for x‐ray computed tomography (CT) face spectral distortion from pulse pileup and charge sharing. The photon counting scheme used by many PCDs is threshold–subtract (TS) with pulse height analysis (PHA), where each counter counts up‐crossing events when pulses exceed an energy threshold. PCD data are not Poisson‐distributed due to charge sharing and pulse pileup, but the counting statistics have never been studied yet. Purpose: The objectives of this study were (1) to propose a modified photon counting scheme, direct energy binning (DB), that is expected to be robust against pulse pileup; (2) to assess the performance of DB compared to TS; and (3) to evaluate its counting statistics. Methods: With DB scheme, counter k starts a timer upon an up‐crossing event of energy threshold k, and adds a count only if the next higher energy threshold (k+1) was not crossed within a short time window (hence, the pulse peak belongs to the energy bin k). We used Monte Carlo (MC) simulation and assessed count‐rate curves and count‐rate‐dependent spectral imaging task performance for conventional CT imaging as well as water thickness estimation, water–bone material decomposition, and K‐edge imaging with tungsten as the K‐edge material. We also assessed count‐rate‐dependent measurement statistics such as expectation, variance, and covariance of total counts as well as energy bin outputs. The agreement with counting statistics models was also evaluated. Results: The DB scheme improved the count‐rate curve, that is, mean measured counts as a function of input count‐rate, and peaked with 59% higher count‐rate capability than the TS scheme (3.5 × 108 counts per second (cps)/mm2 versus 2.3 × 108 cps/mm2). The Cramér–Rao lower bounds (CRLB) of the variance of basis line integrals estimation for DB was better than those for TS by 2% for the conventional CT imaging, 30% for water–bone material decomposition, and 32% for K‐edge imaging at 1000 mA (at 7.3 × 107 cps/sub‐pixel after charge sharing). When count‐rates were lower, PCD data statistics were dominated by charge sharing: the variance of total counts and lower energy bins was larger than the mean counts; the covariance of bin data was positive and non‐zero. When count‐rates were higher, PCD data statistics were dominated by pulse pileup: the variance of data was lower than the mean; the covariance of bin data was negative. The transition between the two regimes occurred smoothly, and pulse pileup dominated the statistics ≥400 mA (when the count‐rate after charge sharing was 2.9 × 107 cps/sub‐pixel and the probability of count‐loss for DB was 37%). Both DB and TS had good agreement with Yu–Fessler's models of total counts; however, DB had a better agreement with Wang's variance and covariance models for energy bin data than TS did. Conclusions: The proposed DB scheme had several advantages over TS. At low to moderate flux, DB could improve the resilience of PCDs to pulse pileup. Counting statistics deviated from the Poisson distribution due to charge sharing for lower count‐rate conditions and pulse pileup for higher count‐rate conditions. [ABSTRACT FROM AUTHOR]

Deep learning-based image reconstruction and noise reduction (DLIR) methods have been increasingly deployed in clinical CT. Accurate assessment of their data uncertainty properties is essential to understand the stability of DLIR in response to noise. In this work, we aim to evaluate the data uncertainty of a DLIR method using real patient data and a virtual imaging trial framework and compare it with filtered-backprojection (FBP) and iterative reconstruction (IR). The ensemble of noise realizations was generated by using a realistic projection domain noise insertion technique. The impact of varying dose levels and denoising strengths were investigated for a ResNet-based deep convolutional neural network (DCNN) model trained using patient images. On the uncertainty maps, DCNN shows more detailed structures than IR although its bias map has less structural dependency, which implies that DCNN is more sensitive to small changes in the input. Both visual examples and histogram analysis demonstrated that hotspots of uncertainty in DCNN may be associated with a higher chance of distortion from the truth than IR, but it may also correspond to a better detection performance for some of the small structures.