Inverse problems are key issues in several scientific areas, including signal processing and medical imaging. Since inverse problems typically suffer from instability with respect to data perturbations, a variety of regularization techniques have been proposed. In particular, the use of filtered diagonal frame decompositions (DFDs) has proven to be effective and computationally efficient. However, existing convergence analysis applies only to linear filters and a few non-linear filters such as soft thresholding. In this paper, we analyze filtered DFDs with general non-linear filters. In particular, our results generalize singular value decomposition-based spectral filtering from linear to non-linear filters as a special case. As a first approach, we establish a connection between non-linear diagonal frame filtering and variational regularization, allowing us to use results from variational regularization to derive the convergence of non-linear spectral filtering. In the second approach, as our main theoretical results, we relax the assumptions involved in the variational case while still deriving convergence. Furthermore, we discuss connections between non-linear filtering and plug-and-play regularization and explore potential benefits of this relationship.

中文翻译：

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正则逆问题非线性对角框滤波的收敛性

反问题是信号处理和医学成像等多个科学领域的关键问题。由于逆问题通常遭受数据扰动的不稳定性，因此已经提出了各种正则化技术。特别是，过滤对角帧分解（DFD）的使用已被证明是有效的且计算效率高。然而，现有的收敛分析仅适用于线性滤波器和一些非线性滤波器，例如软阈值处理。在本文中，我们使用一般非线性滤波器来分析滤波后的 DFD。特别是，我们的结果将基于奇异值分解的谱滤波从线性滤波器推广到非线性滤波器作为一种特殊情况。作为第一种方法，我们在非线性对角帧滤波和变分正则化之间建立联系，使我们能够使用变分正则化的结果来推导非线性谱滤波的收敛性。在第二种方法中，作为我们的主要理论结果，我们放松了变分情况中涉及的假设，同时仍然导出收敛性。此外，我们讨论了非线性滤波和即插即用正则化之间的联系，并探讨了这种关系的潜在好处。