基于原始对偶分裂方法求解一类约束可分离凸优化问题及其应用

Algorithm Based on Primal-dual Splitting Method for Solving a Class of Constrained Separable Convex Optimization Problem and its Application

本文研究一类具有代表性的约束可分离凸优化模型,其目标函数中的数据误差项满足可微性条件,许多图像恢复和图像重建等问题都可以归结为该模型的求解.为克服现有求解该模型方法的不足,文中首先借助指示函数,将原模型转化为无约束的凸优化模型;然后基于原始对偶分裂方法思想,提出一种新的迭代算法,该算法具有结构简单和参数选取容易的特点,同时证明所提算法的收敛性.最后,为验证算法的有效性,我们将其应用于CT图像重建问题,数值实验结果表明所提出的算法在重建时间和重建图像质量上优于现有的其他算法.

In this paper, we study a constrained separable convex optimization model, in which the data error term in the objective function is differentiable. Many problems arising in image restoration and image reconstruction are the special cases of this convex optimization problem.In order to overcome the shortcomings of existing methods for solving this model, we transform the original problem into an unconstrained convex optimization problem by using an indicator function. Then, we propose a new iterative algorithm, which is based on the idea of the primaldual splitting method. The proposed algorithm has a simple structure and is easy for parameter selection. At the same time, we prove the convergence of the new iterative algorithm. Finally,to verify its effectiveness, we apply the algorithm to CT image reconstruction. Numerical experiments show that the proposed algorithm outperforms existing iterative algorithms in terms of reconstruction time and reconstruction image quality.