Abstract. 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 (DFDs) yields regularization methods. These filters dampen certain frame coefficients to prevent noise amplification. This paper introduces a nonlinear filtered DFD method, inspired by a learning strategy. In experiments, we learned optimal nonlinear filters from training data pairs without imposing constraints. Although the learned filters were found to be strictly increasing, they did not satisfy the nonexpansiveness condition required to link them with convex regularizers and prove stability and convergence in the sense of regularization methods in previous works. Motivated by this, the paper relaxes the nonexpansiveness condition, resulting in weakly convex regularization. Despite the relaxed assumptions, we managed to derive stability, convergence, and convergence rates with respect to the absolute symmetric Bregman distance for the learned nonlinear regularizing filters. We illustrate our findings by applying the method to the inversion of the Radon transform using 500 image-sinogram training pairs from real CT scans. Our results show that the learned optimal nonlinear filters in the filtered DFD not only yield accurate reconstructions but also closely align with our theoretical assumptions.

Non-uniqueness and instability are characteristic features of image reconstruction methods. As a result, it is necessary to develop regularization methods that can be used to compute reliable approximate solutions. A regularization method provides a family of stable reconstructions that converge to a specific solution of the noise-free problem as the noise level tends to zero. The standard regularization technique is defined by a variational image reconstruction that minimizes a data discrepancy augmented by a regularizer. The actual numerical implementation makes use of iterative methods, often involving proximal mappings of the regularizer. In recent years, Plug-and-Play (PnP) image reconstruction has been developed as a new powerful generalization of variational methods based on replacing proximal mappings by more general image denoisers. While PnP iterations yield excellent results, neither stability nor convergence in the sense of regularization have been studied so far. In this work, we extend the idea of PnP by considering families of PnP iterations, each accompanied by its own denoiser. As our main theoretical result, we show that such PnP reconstructions lead to stable and convergent regularization methods. This shows for the first time that PnP is as mathematically justified for robust image reconstruction as variational methods.