A Simple but Universal Fully Linearized ADMM Algorithm for Optimization Based Image Reconstruction

Abstract

Abstract Background and Objective: Optimization based image reconstruction algorithm is an advanced algorithm in medical imaging. However, the corresponding solving algorithm is challenging because the optimization model is usually large-scale and non-smooth. This work aims to devise a simple but universal solver for optimization models. Methods: The alternating direction method of multipliers (ADMM) algorithm is a simple and effective solver of the optimization models. However, there always exists a sub-problem that has not closed-form solution. One may use gradient descent algorithm to solve this sub-problem, but the step-size selection via line search is time-consuming. Or, one may use fast Fourier transform (FFT) to get a closed-form solution if the system matrix and the sparse transform matrix are both of special structure. In this work, we propose a simple but universal fully linearized ADMM (FL-ADMM) algorithm that avoids line search to determine step-size and applies to system matrix and sparse transform of any structures. Results: We derive the FL-ADMM algorithm instances for three total variation (TV) models in 2D computed tomography (CT). Further, we validate and evaluate one FL-ADMM algorithm and explore how the two important factors impact convergence rate. Also, we compare this algorithm with the Chambolle-Pock algorithm via real CT phantom reconstructions. These studies show that the FL-ADMM algorithm may accurately solve optimization models in image reconstruction. Conclusion: The FL-ADMM algorithm is a simple, effective, convergent and universal solver of optimization models in image reconstruction. Compared to the existing ADMM algorithms, the new algorithm does not need time-consuming step-size line-search or special demand to system matrix and sparse transform. It is a rapid prototyping tool for optimization based image reconstruction.