We propose a preconditioned alternating projection algorithm (PAPA) for solving the maximum a posteriori (MAP) emission computed tomography (ECT) reconstruction problem. Specifically, we formulate the reconstruction problem as a constrained convex optimization problem with the total variation (TV) regularization. We then characterize the solution of the constrained convex optimization problem and show that it satisfies a system of fixed-point equations defined in terms of two proximity operators raised from the convex functions that define the TV-norm and the constrain involved in the problem. The characterization (of the solution) via the proximity operators that define two projection operators naturally leads to an alternating projection algorithm for finding the solution. For efficient numerical computation, we introduce to the alternating projection algorithm a preconditioning matrix (the EM-preconditioner) for the dense system matrix involved in the optimization problem. We prove theoretically convergence of the preconditioned alternating projection algorithm. In numerical experiments, performance of our algorithms, with an appropriately selected preconditioning matrix, is compared with performance of the conventional MAP expectation-maximization (MAP-EM) algorithm with TV regularizer (EM-TV) and that of the recently developed nested EM-TV algorithm for ECT reconstruction. Based on the numerical experiments performed in this work, we observe that the alternating projection algorithm with the EM-preconditioner outperforms significantly the EM-TV in all aspects including the convergence speed, the noise in the reconstructed images and the image quality. It also outperforms the nested EM-TV in the convergence speed while providing comparable image quality.

中文翻译：

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用于最大后验重建的预处理交替投影算法

我们提出了一种预处理交替投影算法 (PAPA)，用于解决最大后验 (MAP) 发射计算机断层扫描 (ECT) 重建问题。具体来说，我们将重建问题表述为具有总变差 (TV) 正则化的约束凸优化问题。然后，我们描述了约束凸优化问题的解决方案，并表明它满足一个定点方程组，该方程组由定义 TV 范数和问题中涉及的约束的凸函数提出的两个邻近算子定义。通过定义两个投影算子的邻近算子来表征（解决方案）自然会导致用于寻找解决方案的交替投影算法。为了有效的数值计算，我们向交替投影算法引入了一个预处理矩阵（EM 预处理器），用于优化问题中涉及的密集系统矩阵。我们证明了预处理交替投影算法的理论收敛性。In numerical experiments, performance of our algorithms, with an appropriately selected preconditioning matrix, is compared with performance of the conventional MAP expectation-maximization (MAP-EM) algorithm with TV regularizer (EM-TV) and that of the recently developed nested EM-用于 ECT 重建的 TV 算法。基于在这项工作中进行的数值实验，我们观察到带有 EM 预处理器的交替投影算法在包括收敛速度在内的所有方面都明显优于 EM-TV，重建图像中的噪声和图像质量。它还在收敛速度方面优于嵌套 EM-TV，同时提供可比的图像质量。