[论文解读] Near optimal neural network estimator for spectral x-ray photon counting data with pileup

Robert Alvarez|arXiv (Cornell University)|Feb 3, 2017

Advanced X-ray and CT Imaging参考文献 1被引用 3

一句话总结

本文提出了一种针对存在堆积效应的光谱X射线光子计数数据的近似最优神经网络估计器，采用混合训练方法，结合低噪声校准数据与添加的多元高斯噪声。该方法实现的均方误差（MSE）接近费舍尔信息矩阵下限（CRLB），偏差可忽略不计，方差接近理论最小值，优于仅使用纯低噪声或高噪声数据训练的网络。

ABSTRACT

-Purpose: A neural network estimator to process x-ray spectral measurements from photon counting detectors with pileup. The estimator is used with an expansion of the attenuation coefficient as a linear combination of functions of energy multiplied by coefficients that depend on the material composition at points within the object [R.E. Alvarez and A. Macovski, Phys. Med. Biol., 1976, 733-744]. The estimator computes the line integrals of the coefficients from measurements with different spectra. Neural network estimators are trained with measurements of a calibration phantom with the clinical x-ray system. One estimator uses low noise training data and another network is trained with data computed by adding random noise to the low noise data. The performance of the estimators is compared to each other and to the Cramer-Rao lower bound (CRLB). Methods: The estimator performance is measured using a Monte Carlo simulation with an idealized model of a photon counting detector that includes only pileup and quantum noise. Transmitted x-ray spectra are computed for a calibration phantom. The transmitted spectra are used to compute random data for photon counting detectors with pileup. Detectors with small and large dead times are considered. Neural network training data with extremely low noise are computed by averaging the random detected data with pileup for a large numbers of exposures of the phantom. Each exposure is equivalent to a projection image or one projection of a computed tomography scan. Training data with high noise are computed by using data from one exposure. Finally, training data are computed by adding random data to the low noise data. The added random data are multivariate normal with zero mean and covariance equal to the sample covariance of data for an object with properly chosen attenuation. To test the estimators, random data are computed for different thicknesses of three test objects with different compositions. These are used as inputs to the neural network estimators. The mean squared errors (MSE), variance and square of the bias of the neural networks' outputs with the random object data are each compared to the CRLB. Results: The MSE for a network trained with low noise data and added noise is close to the CRLB for both the low and high pileup cases. Networks trained with very low noise data have low bias but large variance for both pileup cases. [e-mail] Networks trained with high noise data have both large bias and large variance. Conclusion: With a properly chosen level of added training data noise, a neural network estimator for photon counting data with pileup can have variance close to the CRLB with negligible bias.

研究动机与目标

• 开发一种神经网络估计器，使其在受堆积效应和量子噪声影响的光谱X射线光子计数系统中实现近似最优性能。
• 研究训练数据噪声水平对估计器偏差、方差及相对于Cramér-Rao下限（CRLB）的均方误差（MSE）的影响。
• 确定添加噪声的最优水平，以在偏差与方差之间取得平衡，从而提升估计器性能。
• 通过蒙特卡洛模拟评估估计器在低堆积和高堆积条件下的性能。

提出的方法

• 估计器采用Alvarez-Macovski基函数分解方法，将衰减系数建模为能量相关基函数的线性组合，其权重为与材料相关的系数。
• 神经网络在来自校准体模的测量数据上进行训练，使用三种数据类型：低噪声（1000次曝光平均）、高噪声（单次曝光）以及低噪声加添加的多元高斯噪声。
• 添加的噪声具有零均值，且协方差与校准体模中代表性内部点的协方差一致，以模拟真实的信号相关噪声。
• 通过蒙特卡洛模拟光子计数探测器的性能，模拟理想化的堆积效应和量子噪声，使用不同厚度和组成的物体。
• 在不同堆积水平（η₀ = 0.1 和 η₀ = 1）下，将估计器的均方误差（MSE）、方差和平方偏差与CRLB进行比较。
• Cramér-Rao下限（CRLB）作为最优估计器性能的理论基准。

实验结果

研究问题

• RQ1 能否通过噪声增强数据训练的神经网络估计器，在存在堆积效应的光谱X射线光子计数系统中实现近似最优性能？
• RQ2 训练数据噪声水平如何影响神经网络估计器的偏差与方差？
• RQ3 结合低噪声与添加噪声数据进行训练，是否能获得优于仅使用低噪声或高噪声数据训练的性能？
• RQ4 堆积效应对CRLB及估计器性能有何影响，特别是在物体厚度增加时的信噪比权衡方面？

主要发现

• 使用低噪声数据与添加噪声训练的神经网络估计器，其均方误差（MSE）非常接近Cramér-Rao下限（CRLB），表明实现了近似最优性能。
• 仅使用低噪声数据训练的估计器表现出较低偏差，但方差显著高于CRLB，尤其是在高堆积条件下。
• 使用高噪声数据（单次曝光）训练的估计器存在较大的偏差与方差，导致MSE明显高于CRLB。
• 在高堆积情况下（η₀ = 1），随着物体厚度增加，CRLB反而下降，这是由于堆积参数减小所致，从而抵消了因透射光子数减少而预期的噪声增加。
• 添加噪声训练的估计器方差低于仅使用低噪声数据训练的网络，表明噪声增强可提升泛化能力并减少对无噪声数据的过拟合。
• 结果证实，经过精心选择的训练噪声水平可使神经网络实现接近最小方差且偏差可忽略，从而逼近理论性能极限。