Synthesized monoenergetic images, generated using linear weighted combination of basis material images, portray the anatomy at a selected effective energy. Images at both high and low effective energies have been proposed as clinically useful. This paper studies the dependence of signal-to-noise ratio (SNR) and detective quantum efficiency (DQE) on the selected energy for CdTe PCDs, and for other spectral CT that uses scintillator detectors. DQE is estimated as the squared of SNR for the system being evaluated divided by that of an ideal PCD. Signal is the unbiased line integral of a material of interest and noise is estimated using propagation of the Cramer-Rao Lower Bound through the weighted sum. SNR and DQE are unimodal with the optimal energy dependent on the mean and width of the measured spectrum, on the spectral response, and system, and weakly on the material of interest. For the CdTe detectors simulated, DQE(0) at the optimal energy is relatively tolerant of spectral degradation (85-92% depending on pixel size), but is highly dependent on effective energy, with maximum variation (in 250 μm pixels) of 22-85% for effective energies between 30 to 120 keV. Study of effect of spectral distribution on DQE shows that a wider spectrum shifts the optimum to lower energy and weakens the energy dependence. In comparison to dual kV and dual layer spectral CT, PCDs have lower optimal effective energy and show higher DQE at low effective energies than energy integrating detectors with dual kV spectra.
Photon counting detectors (PCD) are widely credited with having minimum degradation from electronic noise compared to energy integrating detectors. However, they are not immune. We characterized the effect of electronic noise in simulated CdTe PCDs (0.25-1mm pixels) for spectral and effective monoenergetic tasks. Electronic noise was modeled as two separable effects - spectral blurring modeled as convolution with a Gaussian kernel with standard deviation of 7 keV, and false triggering of the lowest energy bin (depending on the threshold). To model false triggering, noise was created by filtering white Gaussian noise with a Gaussian pulse shaping kernel of 40 ns peaking time and, scaled to have a standard deviation of 7 keV, and analyzed numerically to obtain the mean and variance of false triggers at thresholds from 3 to 45 keV with ±3.5 keV hysteresis. PCDs had 5 energy bins, were operated at maximum of 20 % of characteristic count rate unless otherwise specified, and pulse pileup was not modeled. We assume the expected number of false triggers can be predicted and subtracted but that the noise from those events remains. Quantum and false triggering noise were propagated into basis material images using the Cramer-Rao Lower Bound. In basis material images, at the optimal threshold (balancing false triggers and lost true events) there was an 18-24% variance penalty compared to a detector with no electronic noise. For effective monoenergetic imaging, capturing low energy pulses performs asymptotically as well as a detector without electronic noise, with the penalty increasing with increasing energy threshold.
Charge sharing and migration of scattered and fluorescence photons in photon counting detector (PCD) degrade the detector's energy response and cause a single photon to be potentially counted as multiple events in neighboring pixels, leading to correlations of signal and noise. Signal and noise correlations in conventional linear, space-invariant imaging can be usefully characterized by the frequency dependent detective quantum efficiency, DQE(f). The situation is complicated in the PCDs by the spectral dimension. We analyze DQE(f) of CdTe PCDs using a spatial domain method starting from a previously described computation of spatio-energetic cross talk. DQE(f) is estimated as the squared signal-to-noise ratio of the estimate of the amplitude of a small-signal sinusoidal modulation in the object at a frequency f by a given system compared to that with an ideal detector. DQE(f) for spectral and effective monoenergetic imaging are estimated using a multi-pixel Cramer-Rao lower bound for CdTe detectors of different pixel pitch. For a 120 kVp incident spectrum, DQE(0) for a spectral task was ~18%, 25% and 34% for 250 μm, 500 μm and 1 mm pixels, respectively. Positive correlation between same basis material estimates in neighboring pixels from the spatio-energetic cross-talk causes this effect to have least impact at the detector's Nyquist frequency. For effective monoenergetic imaging, DQE(0) at the optimal energy is relatively tolerant of spectral degradation (85-92% depending on pixel size), but is highly dependent on effective energy, with maximum variation (in 250 μm pixels) of 25-85% for effective energies between 30 to 120 keV.
The development of energy-resolving photon-counting detectors for medical x-ray imaging is attracting considerable attention. Since the image quality can be degraded by different nonidealities such as charge sharing, Compton scatter and fluorescence, there is a need for developing performance metrics in order to compare and optimize detector designs. For conventional, non-energy-resolving detectors, this is commonly done using the linear-systems-theory framework, in which the detector performance is described by noise-equivalent quanta (NEQ) and detective quantum efficiency (DQE) as functions of spatial frequency. However, these metrics do not take the energy-resolving capabilities of multibin photon-counting detectors into account. In this work, we present a unified mathematical framework for quantifying the performance of energy-resolving detectors. We show that the NEQ and DQE can be generalized into matrix-valued quantities, which describe the detector performance for detection tasks with both spatial and energy dependence. With this framework, a small number of simple measurements or simulations are sufficient to compute the dose efficiency of a detector design for any imaging task, taking the effects of detector nonidealities on spatial and energy resolution into account. We further demonstrate that the same framework also can be used for assessing material quantification performance, thereby extending the commonly used performance metrics based on the Cramér-Rao lower bound to spatial-frequency-dependent tasks. The usefulness of the proposed framework is demonstrated using simulations of charge sharing and fluorescence in a CdTe detector.
We present a fast, noise-efficient, and accurate estimator for material separation using photon-counting x-ray detectors (PCXDs) with multiple energy bin capability. The proposed targeted least squares estimator (TLSE) is an improvement of a previously described A-table method by incorporating dynamic weighting that allows the variance to be closer to the Cramér–Rao lower bound (CRLB) throughout the operating range. We explore Cartesian and average-energy segmentation of the basis material space for TLSE and show that, compared with Cartesian segmentation, the average-energy method requires fewer segments to achieve similar performance. We compare the average-energy TLSE to other proposed estimators—including the gold standard maximum likelihood estimator (MLE) and the A-table—in terms of variance, bias, and computational efficiency. The variance and bias were simulated in the range of 0 to 6 cm of aluminum and 0 to 50 cm of water with Monte Carlo methods. The Average-energy TLSE achieves an average variance within 2% of the CRLB and mean absolute error of 3.68±0.06×10−6 cm. Using the same protocol, the MLE showed variance within 1.9% of the CRLB ratio and average absolute error of 3.10±0.06×10−6 cm but was 50 times slower in our implementations. Compared with the A-table method, TLSE gives a more homogenously optimal variance-to-CRLB ratio in the operating region. We show that variance in basis material estimates for TLSE is lower than that of the A-table method by as much as ∼36% in the peripheral region of operating range (thin or thick objects). The TLSE is a computationally efficient and fast method for material separation with PCXDs, with accuracy and precision comparable to the MLE.
Charge sharing, scatter and fluorescence events in a photon counting detector (PCD) can result in multiple counting of a single incident photon in neighboring pixels. This causes energy distortion and correlation of data across energy bins in neighboring pixels (spatio-energy correlation). If a “macro-pixel” is formed by combining multiple small pixels, it will exhibit correlations across its energy bins. Charge sharing and fluorescence escape are dependent on pixel size and detector material. Accurately modeling these effects can be crucial for detector design and for model based imaging applications. This study derives a correlation model for the multi-counting events and investigates the effect in virtual non-contrast and effective monoenergetic imaging. Three versions of 1 mm2 square CdTe macro-pixel were compared: a 4×4 grid, 2×2 grid, or 1×1 composed of pixels with side length 250 μm, 500 μm, or 1 mm, respectively. The same flux was applied to each pixel, and pulse pile-up was ignored. The mean and covariance matrix of measured photon counts is derived analytically using pre-computed spatio-energy response functions (SERF) estimated from Monte Carlo simulations. Based on the Cramer-Rao Lower Bound, a macro-pixel with 250×250 μm2 sub-pixels shows ~2.2 times worse variance than a single 1 mm2 pixel for spectral imaging, while its penalty for effective monoenergetic imaging is <10% compared to a single 1 mm2 pixel.
Spectral imaging systems need to be able to produce "conventional" images, and it's been shown that systems with
energy discriminating detectors can achieve higher CNR than conventional systems by optimal weighting.
Combining measured data in energy bins (EBs) and also combining basis material images have previously been
proposed, but there are no studies systematically comparing the two methods. In this paper, we analytically
evaluate the two methods for systems with ideal photon counting detectors using CNR and beam hardening (BH)
artifact as metrics. For a 120-kVp polychromatic simulations of a water phantom with low contrast inserts, the
difference of the optimal CNR between the two methods for the studied phantom is within 2%. For a
polychromatic spectrum, beam-hardening artifacts are noticeable in EB weighted images (BH artifact of 3.8% for 8
EB and 6.9% for 2 EB), while weighted basis material images are free of such artifacts.
We show that in material decomposition, statistical bias exists in the low photon regime due to non-linearity including but not limited to the log operation and polychromatic measurements. As new scan methods divide the total number of photons into an increasing number of measurements (e.g., energy bins, projection paths) and as developers seek to reduce radiation dose, the number of photons per measurement will decrease and estimators should be robust against bias at low photon counts. We study bias as a function of total flux and spectral spread, which provides insight when parameters like material thicknesses, number of energy bins, and number of projection views change. We find that the bias increases with lower photon counts, wide spectrum, with more number of energy bins and more projection views. Our simulation, with ideal photon counting detectors, show biases up to 2.4 % in basis material images. We propose a bias correction method in projection space that uses a multi dimensional look up table. With the correction, the relative bias in CT images is within 0.5 ± 0.17%.
We present a fast, noise-efficient, and accurate estimator for material separation using photon-counting x-ray detectors
(PCXDs) with multiple energy bin capability. The proposed targeted least squares estimator (TLSE) improves a
previously proposed A-Table method by incorporating dynamic weighting that allows noise to be closer to the Cramér-
Rao Lower Bound (CRLB) throughout the operating range. We explore Cartesian and average-energy segmentation of
the basis material space for TLSE, and show that iso-average-energy contours require fewer segments compared to
Cartesian segmentation to achieve similar performance. We compare the iso-average-energy TLSE to other proposed
estimators - including the gold standard maximum likelihood estimator (MLE) and the A-Table1 - in terms of variance,
bias and computational efficiency. The variance and bias of this estimator between 0 to 6 cm of aluminum and 0 to 50
cm of water is simulated with Monte Carlo methods. Iso-average energy TLSE achieves an average variance within 2%
of CRLB, and mean of absolute error of (3.68 ± 0.06) x 10-6 cm. Using the same protocol, MLE showed variance-to-
CRLB ratio and average bias of 1.0186 ± 0.0002 and (3.10 ± 0.06) x 10-6 cm, respectively, but was 50 times slower in
our simulation. Compared to the A-Table method, TLSE gives a more homogenous variance-to-CRLB profile in the
operating region. We show that variance-to-CRLB for TLSE is lower by as much as ~36% than A-Table method in the
peripheral region of operation (thin or thick objects). The TLSE is a computationally efficient and fast method for
implementing material separation technique in PCXDs, with performance parameters comparable to the MLE.