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

Abstract

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.