In this paper, we are concerned with efficiently solving the sequences of regularized linear least squares problems associated with employing Tikhonov-type regularization with regularization operators designed to enforce edge recovery. An optimal regularization parameter, which balances the fidelity to the data with the edge-enforcing constraint term, is typically not known a priori. This adds to the total number of regularized linear least squares problems that must be solved before the final image can be recovered. Therefore, in this paper, we determine effective multigrid preconditioners for these sequences of systems. We focus our approach on the sequences that arise as a result of the edge-preserving method introduced in [6], where we can exploit an interpretation of the regularization term as a diffusion operator; however, our methods are also applicable in other edge-preserving settings, such as iteratively reweighted least squares problems. Particular attention is paid to the selection of components of the multigrid preconditioner in order to achieve robustness for different ranges of the regularization parameter value. In addition, we present a parameter culling approach that, when used with the L-curve heuristic, reduces the total number of solves required. We demonstrate our preconditioning and parameter culling routines on examples in computed tomography and image deblurring.

翻译：本文关注高效求解与采用旨在强制边缘恢复的正则化算子进行Tikhonov型正则化相关的正则化线性最小二乘问题序列。一个平衡数据保真度与边缘强制约束项的最优正则化参数通常先验未知，这增加了最终图像恢复前必须求解的正则化线性最小二乘问题总数。因此，本文为这些系统序列确定了有效的多网格预条件子。我们将方法聚焦于文献[6]中引入的边缘保持方法所产生的序列，该研究可将正则化项解释为扩散算子从而加以利用；然而，我们的方法同样适用于其他边缘保持场景，例如迭代重加权最小二乘问题。特别关注多网格预条件子组件的选取，以实现对不同范围正则化参数的鲁棒性。此外，我们提出一种参数剔除方法，当与L曲线启发式算法结合使用时，可减少所需求解的总次数。我们通过计算机断层扫描和图像去模糊的示例验证了所提出的预条件与参数剔除策略。