Unified Framework to Construct Fast Row-Action-Type Iterative CT Reconstruction Methods with Total Variation Using Multi Proximal Splitting

Recently, to realize low-dose scan in CT imaging, a number of iterative reconstruction methods with Total Variation (TV) regularization have been investigated. In particular, reconstruction methods based on proximal splitting framework have been actively researched thanks to the following two reasons. First, they allow to use the TV regularization term, which is known to be a non-differentiable function that cannot be handled with classical optimization methods. Second, using the proximal splitting leads to a new class of iterative reconstruction methods which cannot be found in the literature. The major drawback of existing research in this direction is that most of them use the proximal splitting which divides the cost function into a sum of only two terms like famous Chambolle-Pock algorithm and FISTA. In this paper, we propose a unified framework to construct a class of row-action-type iterative methods which converge very fast using frameworks of multi proximal splitting which divides the cost function into a sum of arbitrary number of sub-cost functions (more than two). The use of multi proximal splitting naturally allows us to construct row-action-type iterative methods converging to an exact minimizer of the cost function very quickly. In mathematical literature, there exist only three different frameworks of multi proximal splitting, which are the Passty splitting, Dykstra-like splitting, and modified Dykstra-like splitting. We develop three new iterative methods, i.e. the Passty iterative method, Dykstra-like iterative method, and modified Dykstra-like iterative method, by using these frameworks, for the case where the cost function is a sum of the standard least-squares data fidelity and the TV regularization term. We have compared the proposed three iterative methods with an empirical standard method using ordered-subset technique called OS-SIRT-TV method. The results demonstrate that the performances of proposed methods significantly outperform OS-SIRT method in terms of image quality with a comparable convergence speed.