Model-based iterative reconstruction plays a key role in solving inverse problems. However, the associated minimization problems are generally large-scale, nonsmooth, and sometimes even nonconvex, which present challenges in designing efficient iterative solvers. Preconditioning methods can significantly accelerate the convergence of iterative methods. In some applications, computing preconditioners on-the-fly is beneficial. Moreover, forward models in image reconstruction are typically represented as operators, and the corresponding explicit matrices are often unavailable, which brings additional challenges in designing preconditioners. Therefore, for practical use, computing and applying preconditioners should be computationally inexpensive. This paper adapts the randomized Nystr\"om approximation to compute effective preconditioners that accelerate image reconstruction without requiring an explicit matrix for the forward model. We leverage modern GPU computational platforms to compute the preconditioner on-the-fly. Moreover, we propose efficient approaches for applying the preconditioners to problems with classical nonsmooth regularizers, i.e., wavelet, total variation, and Hessian Schatten-norm. Our numerical results on image deblurring, super-resolution with impulsive noise, and 2D computed tomography reconstruction illustrate the efficiency and effectiveness of the proposed preconditioner.

翻译：基于模型的迭代重建在求解反问题中起着关键作用。然而，相关的极小化问题通常具有大规模、非光滑甚至非凸的特性，这为设计高效的迭代求解器带来了挑战。预条件方法能够显著加速迭代方法的收敛速度。在某些应用中，实时计算预条件子是有益的。此外，图像重建中的前向模型通常表示为算子形式，对应的显式矩阵往往不可获取，这为预条件子的设计带来了额外困难。因此，在实际应用中，预条件子的计算与应用应具有较低的计算成本。本文采用随机化Nyström近似来计算有效的预条件子，以加速图像重建过程，且无需前向模型的显式矩阵。我们利用现代GPU计算平台实时计算预条件子。此外，我们针对具有经典非光滑正则化项（即小波、全变差及Hessian Schatten范数）的问题，提出了高效应用预条件子的方法。我们在图像去模糊、含脉冲噪声的超分辨率以及二维计算机断层扫描重建上的数值结果，验证了所提预条件子的高效性与有效性。