Fixed-Point Convergence of Multi-Block PnP ADMM and Its Application to Hyperspectral Image Restoration

Coupling methods of integrating multiple priors have emerged as a pivotal research focus in hyperspectral image (HSI) restoration. Among these methods, the Plug-and-Play (PnP) framework stands out and pioneers a novel coupling approach, enabling flexible integration of diverse methods into model-based approaches. However, the current convergence analyses of the PnP framework are highly unexplored, as they are limited to 2-block composite optimization problems, failing to meet the need of coupling modeling for incorporating multiple priors. This paper focuses on the convergence analysis of PnP-based algorithms for multi-block composite optimization problems. In this work, under the PnP framework and utilizing the alternating direction method of multipliers (ADMM) of the continuation scheme, we propose a unified multi-block PnP ADMM algorithm framework for HSI restoration. Inspired by the fixed-point convergence theory of the 2-block PnP ADMM, we establish a similar fixed-point convergence guarantee for the multi-block PnP ADMM with extended condition and provide a feasible parameter tuning methodology. Based on this framework, we design an effective mixed noise removal algorithm incorporating global, nonlocal and deep priors. Extensive experiments validate the algorithm's superiority and competitiveness.