Aprend Technology Mountain View, California 94043, USA.
Med Phys. 2011 May;38(5):2324-34. doi: 10.1118/1.3570658.
This paper describes a noniterative estimator for the energy dependent information from photon counting detectors with multibin pulse height analysis (PHA). The estimator uses the two function decomposition of the attenuation coefficient [R. E. Alvarez and A. Macovski, Phys. Med. Biol. 21, 733-744 (1976)] and its output is the line integrals of the basis set coefficients. The output noise variance and bias is compared to other noniterative estimators and to the Cramèr-Rao lower bound (CRLB).
The estimator first computes an initial estimate from a linearized maximum likelihood estimator. The errors in the initial estimates are determined at a set of points from measurements on a calibration phantom. The errors at these known points are interpolated to create two-dimensional look up tables of corrections to the initial estimates. During image acquisition, the linearized maximum likelihood estimate for each data point is used as an input to the correction look up tables, and the final output is the sum of the estimate and the correction. The performance of the estimator is compared to generalizations of the polynomial and rational polynomial estimators for multibin data. The estimators are compared by the mean square error (MSE) and its components, the bias, and the variance of the output. The variance is also compared to the CRLB. The performance is simulated with two to five bins PHA data. The CRLB at a fixed object thickness is also computed as a function of the number of bins.
For two bin data, all the estimators' variances are equal to the CRLB. With three or more bins, only the proposed estimator achieves the CRLB while the others, which were not optimized for noise performance, have much larger output variance. The bias of the proposed estimator is equal to the polynomial estimator for calibration phantoms with 40 or more steps, that is, 1600 combinations of basis materials, but is larger than the rational polynomial bias. In all cases at the photon counts tested, the MSE is essentially equal to the variance, indicating that the bias errors are negligible compared to the variance.
The estimator provides a noniterative method to compute the energy dependent information from multibin PHA data that achieves the CRLB over a wide range of operating conditions and has low output bias. The estimator can be calibrated based on the measurements of a calibration phantom; so, it does not require measurements of the x-ray energy spectrum or the detector response functions.
本文介绍了一种用于多-bin 脉冲高度分析 (PHA) 光子计数探测器的能量相关信息的非迭代估计器。该估计器使用衰减系数的两函数分解[R. E. Alvarez 和 A. Macovski, Phys. Med. Biol. 21, 733-744 (1976)]，其输出是基集系数的线积分。将输出噪声方差和偏差与其他非迭代估计器以及克拉美-罗下限 (CRLB) 进行了比较。
该估计器首先从线性化最大似然估计器计算初始估计值。在一组校准体模的测量值上确定初始估计值的误差。在这些已知点插值误差，以创建初始估计值的二维校正查找表。在图像采集期间，每个数据点的线性化最大似然估计值用作校正查找表的输入，最终输出是估计值和校正值的和。通过均方误差 (MSE) 及其分量、偏差和输出的方差来比较估计器的性能。还比较了方差与 CRLB。使用两到五个 bin PHA 数据模拟性能。还计算了在固定物体厚度下的 CRLB 作为 bin 数的函数。
对于两-bin 数据，所有估计器的方差均等于 CRLB。使用三个或更多 bin 时，只有所提出的估计器达到 CRLB，而其他未针对噪声性能进行优化的估计器的输出方差要大得多。对于具有 40 个或更多步骤（即 1600 种基础材料组合）的校准体模，所提出的估计器的偏差等于多项式估计器，但大于有理多项式偏差。在所有测试的光子计数情况下，MSE 实质上等于方差，表明与方差相比，偏差误差可以忽略不计。
该估计器提供了一种从多-bin PHA 数据计算能量相关信息的非迭代方法，在广泛的工作条件下实现了 CRLB，并具有较低的输出偏差。该估计器可以基于校准体模的测量值进行校准；因此，它不需要测量 X 射线能谱或探测器响应函数。
