  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.
