On the convergence of an adaptive denoiser driven iterative regularization with early stopping
This work addresses the lack of rigorous theoretical justification for denoiser-driven methods in regularization theory, which is crucial for researchers and practitioners in imaging and inverse problems, though it is incremental as it builds on existing RED frameworks.
The paper tackles the stability and convergence of denoiser-driven iterative regularization for inverse problems by introducing a novel scheme (DDIR) with an adaptive step-size and stopping rule, proving it is a stable and convergent regularization method, and demonstrating improved reconstruction accuracy and computational efficiency in image deblurring and phase retrieval CT experiments.
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.