总变差约束的数据分离最小图像重建模型及其Chambolle-Pock求解算法

基于优化的迭代法，可以结合压缩感知和低秩矩阵等稀疏优化技术高精度地重建图像.其中，总变差最小（total variation minimization，TV）模型是一种简单有效的优化模型.传统的约束TV模型，使用数据保真项为约束项，TV正则项为目标函数.本文研究TV约束的、数据分离最小（TV constrained，data divergence minimization，TVcDM）新型TV模型及其求解算法.详细推导了TVcDM模型的Chambolle-Pock（CP）算法，验证了模型及算法的正确性；分析了算法的收敛行为；评估了模型的稀疏重建能力；分析了模型参数的选择对重建的影响及算法参数对收敛速率的影响.研究表明，TVcDM模型有高精度稀疏重建能力；TVcDM-CP算法确保收敛，但迭代过程中有振荡现象；TV限对重建有重要影响，参数值过大会引入噪声而过小会模糊图像细节；算法参数的不同选取会导致不同的收敛速率.

The total variation constrained data divergence minimization model for image reconstruction and its Chambolle-Pock solving algorithm

Image reconstruction is an important inverse problem to reconstruct images from its transform. The two main reconstruction methods are the analytic method and the iterative method. The analytic method, for example, the filtered backprojection algorithm, needs complete projection data, so it is not competent to accurately reconstruct an image from sparse data. Thus the iterative method combined with optimization techniques has received more and more attention. The optimization-based iterative image reconstruction algorithm may accurately reconstruct images by the use of compressed sensing, low rank matrix and other sparse optimization techniques. Among them, the total variation (TV) minimization model is a simple but effective optimization model. The traditional, constrained TV model employs the data fidelity term as the constraint term and the TV regularization term as the objective function. In the present work, we study a novel, TV constrained, data divergence minimization (TVcDM) model and its solver. We derive in detail the Chambolle-Pock (CP) algorithm for solving the TVcDM model, verify the correctness of the model and its solver, analyze the convergence behavior of the algorithm, evaluate the sparse reconstruction ability of the TVcDM-CP algorithm and finally analyze the influence of the model parameters on reconstruction and the effect of algorithm parameters on convergence rate. The studies show that the TVcDM model may accurately reconstruct images from sparse-view projections. The TVcDM-CP algorithm may ensure convergence but the vibration phenomena may be observed in the convergence process. The model parameter, TV tolerance, has important influence on reconstruction quality, i. e. too big a value introduces noise whereas too small a value may smoothen the image details. Also, the studies reveal that different algorithm-parameter selections may lead to different convergence rates. The TVcDM-CP algorithm may be tailored and applied to other computed tomography scanning configurations and other imaging modalities. The necessary key work is just to design the corresponding system matrix and select the optimal model parameters and algorithm parameters according to the insights gained in the work.