Convergence of non-linear diagonal frame filtering for regularizing inverse problems
Andrea Ebner, Markus Haltmeier
TL;DR
This work extends diagonal frame filtering for ill-posed inverse problems from linear to general non-linear filters. By formulating nonlinear regularizing filters φ_α and showing their proximity-operator structure, the authors connect non-linear filtered DFDs to variational regularization and to plug-and-play regularization, establishing stability and weak convergence under broad conditions. They present two analytical routes: a stationary reduction to variational regularization and a direct analysis that relaxes the stationarity assumptions, both yielding convergence results to the true solution x^+ as the noise level δ and parameter α vanish with δ^2/α → 0. The paper also discusses a PnP perspective, showing how nonlinear DFD can be interpreted as a PnP scheme with diagonal denoisers, and provides numerical demonstrations comparing nonlinear filters to soft-thresholding. Collectively, the results broaden the theoretical foundation for nonlinear, frame-based regularization techniques in inverse problems and highlight practical connections to PnP methods.
Abstract
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 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 diagonal frame decompositions with general non-linear filters. In particular, our results generalize SVD-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.