Inverse problems are core issues in several scientific areas, including signal processing and medical imaging. As 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 has proven to be effective and computationally efficient. However, the existing convergence analysis applies only to linear filters and a few non-linear filters such as soft thresholding. In this paper, we analyze the filtered diagonal frame decomposition with general non-linear filters. In particular, our results generalize SVD-based spectral filtering from linear to non-linear filters as a special case. We present three strategies to demonstrate convergence. The first two strategies relate non-linear diagonal frame filtering to variational regularization and plug-and-play regularization, respectively. The third strategy allows us to relax the assumptions involved and still obtain a full convergence analysis.

翻译：反问题是包括信号处理和医学成像在内的多个科学领域的核心问题。由于反问题通常存在对数据扰动的不稳定性，研究者已提出多种正则化技术。其中，使用滤波对角框架分解已被证明有效且计算高效。然而，现有的收敛性分析仅适用于线性滤波器和少数非线性滤波器（如软阈值）。本文分析了采用一般非线性滤波器的滤波对角框架分解。特别地，我们的结果将基于奇异值分解的谱滤波从线性情形推广到非线性情形作为特例。我们提出了三种收敛性证明策略：前两种策略分别将非线性对角框架滤波与变分正则化及即插即用正则化相关联；第三种策略使我们能够放宽所涉及的假设，同时仍能获得完整的收敛性分析。