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.