BACKGROUND Attenuation correction in positron emission tomography remains challenging in the absence of measured transmission data. Scattered emission data may contribute missing information, but quantitative scatter-to-attenuation (S2A) reconstruction needs to input the reconstructed activity image. Here, we study S2A reconstruction as a building block for joint estimation of activity and attenuation. METHODS We study two S2A reconstruction algorithms, maximum-likelihood expectation maximization (MLEM) with one-step-late attenuation (MLEM-OSL) and a maximum-likelihood gradient ascent (MLGA). We study theoretical properties of these algorithms with a focus on convergence and convergence speed and compare convergence speeds and the impact of object size in simulations using different spatial scale factors. Then, we propose joint estimation of activity and attenuation from scattered and nonscattered (true) emission data, combining MLEM-OSL or MLGA with scatter-MLEM as well as trues-MLEM and the maximum-likelihood transmission (MLTR) algorithm. RESULTS Shortcomings of MLEM-OSL inhibit convergence to the true solution with high attenuation; these shortcomings are related to the linearization of a nonlinear measurement equation and can be linked to a new numerical criterion allowing geometrical interpretations in terms of low and high attenuation. Comparisons using simulated data confirm that while MLGA converges largely independent of the attenuation scale, MLEM-OSL converges if low-attenuation data dominate, but not with high attenuation. Convergence of MLEM-OSL can be improved by isolating data satisfying the aforementioned low-attenuation criterion. In joint estimation of activity and attenuation, scattered data helps avoid local minima that nonscattered data alone cannot. Combining MLEM-OSL with trues-MLEM may be sufficient for low-attenuation objects, while MLGA, scatter-MLEM, and MLTR may additionally be needed with higher attenuation. CONCLUSIONS The performance of S2A algorithms depends on spatial scales. MLGA provides lower computational complexity and convergence in more diverse setups than MLEM-OSL. Finally, scattered data may provide additional information to joint estimation of activity and attenuation through S2A reconstruction.

中文翻译：

---

从正电子发射断层扫描散射估计联合活动衰减的算法。

背景技术在缺少测量的透射数据的情况下，正电子发射断层摄影术中的衰减校正仍然具有挑战性。散布的发射数据可能会导致丢失信息，但是定量散布到衰减（S2A）重建需要输入重建的活动图像。在这里，我们研究S2A重建作为联合估计活动和衰减的基础。方法我们研究了两种S2A重建算法，即最大似然期望最大化（MLEM）和单步延迟衰减（MLEM-OSL）和最大似然梯度上升（MLGA）。我们研究这些算法的理论特性，重点是收敛性和收敛速度，并在使用不同空间比例因子的仿真中比较收敛速度和对象大小的影响。然后，我们提出了结合散射和非散射（真实）发射数据的活度和衰减的联合估算，将MLEM-OSL或MLGA与散射-MLEM以及trues-MLEM和最大似然传输（MLTR）算法结合使用。结果MLEM-OSL的缺点抑制了收敛到真正解决方案的高衰减。这些缺点与非线性测量方程的线性化有关，并且可以链接到新的数值标准，从而可以根据低衰减和高衰减进行几何解释。使用模拟数据进行的比较证实，虽然MLGA的收敛在很大程度上与衰减范围无关，但如果低衰减数据占主导地位而MLEM-OSL收敛，而高衰减则不收敛。通过隔离满足上述低衰减标准的数据，可以提高MLEM-OSL的收敛性。在活动性和衰减的联合估计中，分散的数据有助于避免局部最小值，而单独的非分散数据则无法做到。将MLEM-OSL与trues-MLEM结合起来对于低衰减的对象可能就足够了，而可能另外需要具有较高衰减的MLGA，scatter-MLEM和MLTR。结论S2A算法的性能取决于空间尺度。与MLEM-OSL相比，MLGA在更多样化的设置中提供了更低的计算复杂度和收敛性。最后，分散的数据可能会提供其他信息，以通过S2A重建共同估计活动和衰减。并且可能需要更高衰减的MLTR。结论S2A算法的性能取决于空间尺度。与MLEM-OSL相比，MLGA在更多样化的设置中提供了更低的计算复杂度和收敛性。最后，分散的数据可能会提供其他信息，以通过S2A重建共同估计活动和衰减。并且可能需要更高衰减的MLTR。结论S2A算法的性能取决于空间尺度。与MLEM-OSL相比，MLGA在更多样化的设置中提供了更低的计算复杂度和收敛性。最后，分散的数据可能会提供其他信息，以通过S2A重建共同估计活动和衰减。