Image restoration is typically addressed through non-convex inverse problems, which are often solved using first-order block-wise splitting methods. In this paper, we consider a general type of non-convex optimisation model that captures many inverse image problems and present an inertial block proximal linearised minimisation (iBPLM) algorithm. Our new method unifies the Jacobi-type parallel and the Gauss-Seidel-type alternating update rules, and extends beyond these approaches. The inertial technique is also incorporated into each block-wise subproblem update, which can accelerate numerical convergence. Furthermore, we extend this framework with a plug-and-play variant (PnP-iBPLM) that integrates deep gradient denoisers, offering a flexible and robust solution for complex imaging tasks. We provide comprehensive theoretical analysis, demonstrating both subsequential and global convergence of the proposed algorithms. To validate our methods, we apply them to multi-block dictionary learning problems in image denoising and deblurring. Experimental results show that both iBPLM and PnP-iBPLM significantly enhance numerical performance and robustness in these applications.

翻译：图像复原通常通过非凸逆问题来解决，这些问题常采用一阶分块分裂方法求解。本文考虑一种通用类型的非凸优化模型，该模型涵盖了许多图像逆问题，并提出了一种惯性块近端线性化最小化（iBPLM）算法。我们的新方法统一了雅可比型并行更新规则和高斯-赛德尔型交替更新规则，并超越了这些传统方法。惯性技术也被纳入每个块子问题的更新过程中，这能够加速数值收敛。此外，我们通过即插即用变体（PnP-iBPLM）扩展了该框架，该变体集成了深度梯度去噪器，为复杂成像任务提供了灵活而鲁棒的解决方案。我们提供了全面的理论分析，证明了所提算法的子序列收敛性和全局收敛性。为验证方法有效性，我们将其应用于图像去噪和去模糊中的多块字典学习问题。实验结果表明，iBPLM和PnP-iBPLM在这些应用中均显著提升了数值性能和鲁棒性。