A Semi-Convergent Stage-Wise Framework with Provable Global Convergence for Adaptive Total Variation Regularization
Liang Luo, Lei Zhang
TL;DR
This work tackles image restoration under blur and noise by introducing a semi-convergent stage-wise framework that alternates between first-order TV ($TV_1(u)=\phi(Du)$) and higher-order TV ($TV_2(u)=\phi(DDu)$) regularizers within an ADMM scheme. A PSNR-based select-and-propagate strategy leverages stage-wise semi-convergence to choose the locally best iterate and initialize the next stage, effectively translating local semi-convergence into global convergence. The authors prove that the stage-wise iterates remain bounded and the objective decreases monotonically, yielding convergence to a neighborhood of the ground truth. Extensive denoising and deblurring experiments show improved quantitative and perceptual performance over classical TV-based methods and several learning-based approaches, while preserving interpretability and algorithmic simplicity. The framework offers a flexible, theoretically grounded path to balance edge preservation and artifact suppression and invites extension to additional regularizers and imaging models.
Abstract
Image restoration requires a careful balance between noise suppression and structure preservation. While first-order total variation (TV) regularization effectively preserves edges, it often introduces staircase artifacts, whereas higher-order TV removes such artifacts but oversmooths fine details. To reconcile these competing effects, we propose a semi-convergent stage-wise framework that sequentially integrates first- and higher-order TV regularizers within an iterative restoration process implemented via ADMM. Each stage exhibits semi-convergence behavior, i.e., the iterates initially approach the ground truth before being degraded by over-regularization. By monitoring this evolution, the algorithm adaptively selects the locally optimal iterate (e.g., with the highest PSNR) and propagates it as the initial point for the next stage. This select-and-propagate mechanism effectively transfers local semi-convergence into a globally convergent iterative process. We establish theoretical guarantees showing that the sequence of stage-wise iterates is bounded, the objective values decrease monotonically. Extensive numerical experiments on denoising and deblurring benchmarks confirm that the proposed method achieves superior quantitative and perceptual performance compared with conventional first-, higher-order, hybrid TV methods, and learning based methods, while maintaining theoretical interpretability and algorithmic simplicity.