Abstract：
Most penalized maximum likelihood methods for tomographic image reconstruction based on Bayes\' law include a freely adjustable hyperparameter to balance the data fidelity term and the prior/penalty term for a specific noise-resolution tradeoff. The hyperparameter is determined empirically via a trial-and-error fashion in many applications, which then selects the optimal result from multiple iterative reconstructions. These penalized methods are not only time-consuming by their iterative nature, but also require manual adjustment. This study aims to investigate a theory-based strategy for Bayesian image reconstruction without a freely adjustable hyperparameter, to substantially save time and computational resources. The Bayesian image reconstruction problem is formulated by two probability density functions (PDFs), one for the data fidelity term and the other for the prior term. When formulating these PDFs, we introduce two parameters. While these two parameters ensure the PDFs completely describe the data and prior terms, they cannot be determined by the acquired data; thus, they are called complete but unobservable parameters. Estimating these two parameters becomes possible under the conditional expectation and maximization for the image reconstruction, given the acquired data and the PDFs. This leads to an iterative algorithm, which jointly estimates the two parameters and computes the to-be reconstructed image by maximizing a posteriori probability, denoted as joint-parameter-Bayes. In addition to the theoretical formulation, comprehensive simulation experiments are performed to analyze the stopping criterion of the iterative joint-parameter-Bayes method. Finally, given the data, an optimal reconstruction is obtained without any freely adjustable hyperparameter by satisfying the PDF condition for both the data likelihood and the prior probability, and by satisfying the stopping criterion. Moreover, the stability of joint-parameter-Bayes is investigated through factors such as initialization, the PDF specification, and renormalization in an iterative manner. Both phantom simulation and clinical patient data results show that joint-parameter-Bayes can provide comparable reconstructed image quality compared to the conventional methods, but with much less reconstruction time. To see the response of the algorithm to different types of noise, three common noise models are introduced to the simulation data, including white Gaussian noise to post-log sinogram data, Poisson-like signal-dependent noise to post-log sinogram data and Poisson noise to the pre-log transmission data. The experimental outcomes of the white Gaussian noise reveal that the two parameters estimated by the joint-parameter-Bayes method agree well with simulations. It is observed that the parameter introduced to satisfy the prior\'s PDF is more sensitive to stopping the iteration process for all three noise models. A stability investigation showed that the initial image by filtered back projection is very robust. Clinical patient data demonstrated the effectiveness of the proposed joint-parameter-Bayes and stopping criterion.
摘要：
基于贝叶斯定律的层析成像图像重建的大多数惩罚最大似然方法包括可自由调整的超参数，以平衡数据保真度项和特定噪声分辨率权衡的先验/惩罚项。在许多应用中，超参数是通过反复试验的方式根据经验确定的，然后从多次迭代重建中选择最优结果。这些惩罚方法不仅因其迭代性质而耗时，还需要手动调整。本研究旨在研究一种基于理论的贝叶斯图像重建策略，无需可自由调整的超参数，大大节省时间和计算资源。贝叶斯图像重建问题由两个概率密度函数(PDF)表示，一个用于数据保真度项，另一个用于先前项。在制定这些PDF时，我们引入两个参数。虽然这两个参数确保PDF完全描述了数据和先前的术语，它们不能由获取的数据确定；因此，它们被称为完全但不可观察的参数。估计这两个参数成为可能的条件期望和最大化的图像重建，给定获得的数据和PDF。这导致了一个迭代算法，联合估计两个参数，并通过最大化后验概率来计算待重建图像，表示为联合参数贝叶斯。除了理论上的表述，进行了综合仿真实验，分析了迭代联合参数贝叶斯方法的停止准则。最后,鉴于数据，通过满足数据似然性和先验概率的PDF条件，在没有任何可自由调整的超参数的情况下获得最佳重建，并满足停止标准。此外，通过初始化等因素研究了联合参数贝叶斯的稳定性，PDF规范，并以迭代方式重新归一化。体模模拟和临床患者数据结果表明，与传统方法相比，联合参数贝叶斯可以提供相当的重建图像质量。但重建时间要少得多。要查看算法对不同类型噪声的响应，在仿真数据中引入了三种常见的噪声模型，包括高斯白噪声到对数正弦图数据，泊松信号相关噪声对对数后正弦图数据和泊松噪声对对数前传输数据。高斯白噪声的实验结果表明，联合参数贝叶斯方法估计的两个参数与仿真吻合良好。观察到，为满足先前的PDF而引入的参数对停止所有三个噪声模型的迭代过程更敏感。稳定性研究表明，通过滤波反投影获得的初始图像非常鲁棒。临床患者数据证明了所提出的联合参数贝叶斯和停止标准的有效性。
