In positron emission tomography (PET), it is indispensable to perform attenuation correction in order to obtain the quantitatively accurate activity map (tracer distribution) in the body. Generally, this is carried out based on the estimated attenuation map obtained from computed tomography or magnetic resonance imaging. However, except for errors in the attenuation correction factors obtained, the additional scan not only brings in new radiation doses and/or increases the scanning time but also leads to severe misalignment induced by various motions during and between the two sequential scans. To address these issues, based on maximum likelihood estimation, we propose a new mathematical model for simultaneously reconstructing the activity and attenuation sinogram from the time-of-flight (TOF)-PET emission data only. Particularly, we make full use of the exclusively exponential form for the attenuation correction factors, and consider the constraint of a total amount of the activity in some mask region in the proposed model. Furthermore, we prove its well-posedness, including the existence, uniqueness and stability of the solution. We propose an alternating update algorithm to solve the model, and also analyze its convergence. Finally, numerical experiments with various TOF-PET emission data demonstrate that the proposed method is of numerical convergence and robust to noise, and outperforms some state-of-the-art methods in terms of accuracy and efficiency, and has the capability of autonomous attenuation correction.

翻译：在正电子发射断层扫描（PET）中，为获得体内定量准确的活动图（示踪剂分布），执行衰减校正是不可或缺的。通常，这一校正是基于计算机断层扫描或磁共振成像所获得的估计衰减图进行的。然而，除了衰减校正因子本身可能存在的误差外，额外的扫描不仅会带来新的辐射剂量和/或增加扫描时间，还会因两次连续扫描期间及扫描之间各种运动导致严重的错位问题。为解决这些问题，基于最大似然估计，我们提出了一种新的数学模型，仅利用飞行时间（TOF）-PET发射数据即可同时重建活动与衰减正弦图。特别地，我们充分利用了衰减校正因子特有的指数形式，并在所提模型中考虑了特定掩膜区域内活动总量约束。此外，我们证明了该模型的适定性，包括解的存在性、唯一性和稳定性。我们提出了一种交替更新算法来求解该模型，并分析了其收敛性。最后，基于多种TOF-PET发射数据的数值实验表明，所提方法具有数值收敛性且对噪声具有鲁棒性，在精度与效率方面优于一些现有先进方法，并具备自主衰减校正能力。