On the convergence of an adaptive denoiser driven iterative regularization with early stopping

TL;DR

This paper introduces a new denoiser-driven iterative regularization method called DDIR, with proven stability and convergence, and demonstrates its effectiveness in image deblurring and CT reconstruction.

Contribution

It extends RED by incorporating an adaptive step-size and stopping rule, providing the first rigorous convergence proof for DDIR within regularization theory.

Findings

The method achieves stable, convergent reconstructions.

Numerical experiments show improved accuracy and efficiency.

Effective with multiple denoisers like median, TNRD, and TV proximal.

Abstract

Solving inverse problems requires appropriate regularization techniques to ensure well-posedness and stability. In recent years, denoiser-driven methods have emerged as effective regularization strategies, achieving state-of-the-art performance in various imaging applications. However, their stability and convergence within iterative regularization frameworks remain largely unexplored. In this work, we extend the framework of Regularization by Denoising (RED) by introducing a novel denoiser-driven iterative regularization scheme, referred to as \texttt{DDIR}, that incorporates a new regularization functional based on averaged denoisers. The proposed approach employs an adaptive step-size strategy together with an \emph{a posteriori} stopping rule to ensure stability while alleviating oscillatory behavior and semi-convergence effects induced by noise. As our main theoretical contribution, we prove that the resulting reconstruction method constitutes a stable and convergent regularization scheme in the classical sense. To the best of our knowledge, this provides the first rigorous justification of \texttt{DDIR} within the framework of regularization theory. Finally, we demonstrate the performance of the proposed method through numerical experiments on image deblurring and phase retrieval Computed Tomography (CT) using three denoisers, namely median, TNRD, and TV proximal. The results highlight the effectiveness of the method in terms of reconstruction accuracy and computational efficiency.