IEEE Trans Med Imaging. 2017 Jul;36(7):1418-1426. doi: 10.1109/TMI.2017.2681939. Epub 2017 Mar 13.
Positron emission tomography (PET) relies on accurate timing information to pair two 511-keV photons into a coincidence event. Calibration of time delays between detectors becomes increasingly important as the timing resolution of detector technology improves, as a calibration error can quickly become a dominant source of error. Previous work has shown that the maximum likelihood estimate of these delays can be calculated by least squares estimation, but an approach is not tractable for complex systems and degrades in the presence of randoms. We demonstrate the original problem to be solvable iteratively using the LSMR algorithm. Using the LSMR, we solve for 60 030 delay parameters, including energy-dependent delays, in 4.5 s, using 1 000 000 coincidence events for a two-panel system dedicated to clinical locoregional imaging. We then extend the original least squares problem to be robust to random coincidences and low statistics by implementing l -norm minimization using the alternating direction method of the multipliers (ADMM) algorithm. The ADMM algorithm converges after six iterations, or 20.6 s, and improves the timing resolution from 64.7 ± 0.1s full width at half maximum (FWHM) uncalibrated to 15.63 ± 0.02ns FWHM. We also demonstrate this algorithm's applicability to commercial systems using a GE Discovery 690 PET/CT. We scan a rotating transmission source, and after subtracting the 511-keV photon time-of-flight due to the source position, we calculate 13 824 per-crystal delays using 5 000 000 coincidence events in 3.78 s with three iterations, while showing a timing resolution improvement that is significantly better than previous calibration methods in the literature.
正电子发射断层扫描（PET）依赖于准确的时间信息，将两个 511keV 的光子对配对成符合事件。随着探测器技术的时间分辨率的提高，探测器之间的时间延迟校准变得越来越重要，因为校准误差可能很快成为误差的主要来源。以前的工作表明，可以通过最小二乘法估计来计算这些延迟的最大似然估计，但对于复杂系统来说，这种方法是不可行的，并且在存在随机数的情况下会降级。我们证明了原始问题可以通过 LSMR 算法迭代求解。使用 LSMR，我们在 4.5 秒内求解了 60030 个延迟参数，包括能量相关的延迟，使用了 1000000 个符合事件，用于一个专门用于临床局部成像的双面板系统。然后，我们通过使用交替方向乘子法（ADMM）算法实现 l-范数最小化，将原始最小二乘问题扩展为对随机符合和低统计量具有鲁棒性。ADMM 算法在经过六次迭代或 20.6 秒后收敛，并将时间分辨率从未经校准的 64.7±0.1s 全宽半最大值（FWHM）提高到 15.63±0.02ns FWHM。我们还使用 GE Discovery 690 PET/CT 演示了该算法在商业系统中的适用性。我们扫描一个旋转的传输源，并在扣除源位置的 511keV 光子飞行时间后，使用 5000000 个符合事件在 3.78 秒内计算了 13824 个每个晶体的延迟，使用了三次迭代，同时显示出的时间分辨率改进明显优于文献中的先前校准方法。
