Inverse problems are fundamental in fields like medical imaging, geophysics, and computerized tomography, aiming to recover unknown quantities from observed data. However, these problems often lack stability due to noise and ill-conditioning, leading to inaccurate reconstructions. To mitigate these issues, regularization methods are employed, introducing constraints to stabilize the inversion process and achieve a meaningful solution. Recent research has shown that the application of regularizing filters to diagonal frame decompositions (DFD) yields regularization methods. These filters dampen some frame coefficients to prevent noise amplification. This paper introduces a non-linear filtered DFD method combined with a learning strategy for determining optimal non-linear filters from training data pairs. In our experiments, we applied this approach to the inversion of the Radon transform using 500 image-sinogram pairs from real CT scans. Although the learned filters were found to be strictly increasing, they did not satisfy the non-expansiveness condition required to link them with convex regularizers and prove stability and convergence in the sense of regularization methods in previous works. Inspired by this, the paper relaxes the non-expansiveness condition, resulting in weakly convex regularization. Despite this relaxation, we managed to derive stability, convergence, and convergence rates with respect to the absolute symmetric Bregman distance for the learned non-linear regularizing filters. Extensive numerical results demonstrate the effectiveness of the proposed method in achieving stable and accurate reconstructions.

翻译：逆问题在医学成像、地球物理和计算机断层扫描等领域具有基础性地位，其目标是从观测数据中恢复未知量。然而，由于噪声和病态性，这些问题通常缺乏稳定性，导致重建结果不准确。为缓解这些问题，正则化方法被引入，通过施加约束来稳定反演过程并获得有意义的解。最近的研究表明，将正则化滤波器应用于对角框架分解（DFD）可产生正则化方法。这些滤波器通过抑制部分框架系数来防止噪声放大。本文提出了一种非线性滤波DFD方法，并结合一种学习策略，用于从训练数据对中确定最优非线性滤波器。在我们的实验中，我们应用该方法对Radon变换进行反演，使用了来自真实CT扫描的500个图像-正弦图对。尽管学习到的滤波器被发现是严格递增的，但它们不满足先前工作中将其与凸正则化器关联并证明正则化方法意义下的稳定性和收敛性所需的非扩张性条件。受此启发，本文放宽了非扩张性条件，从而得到弱凸正则化。尽管进行了这一放宽，我们仍成功推导出了针对学习到的非线性正则化滤波器在绝对对称Bregman距离下的稳定性、收敛性以及收敛速率。大量数值结果证明了所提方法在实现稳定且准确重建方面的有效性。