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：本文针对采用Tikhonov型正则化方法且正则化算子设计为强制边缘恢复时所关联的正则化线性最小二乘问题序列的高效求解展开研究。平衡数据保真度与边缘约束项的最优正则化参数通常无法先验已知，这增加了在最终图像重建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有效的多重网格预处理器。我们重点关注文献[6]中提出的边缘保持方法所衍生的序列——在该方法中，正则化项可被解释为扩散算子；然而，我们的方法同样适用于其他边缘保持场景，如迭代重加权最小二乘问题。为在不同正则化参数值范围内实现鲁棒性，我们特别关注多重网格预处理器组件的选择策略。此外，我们提出一种参数剔除方法，当与L曲线准则联合使用时，可减少总求解次数。最后通过计算机断层扫描与图像去模糊实例验证了所提预处理与参数剔除流程的有效性。