  Plug-and-play (PnP) method is a recent paradigm for image regularization,
where the proximal operator (associated with some given regularizer) in an
iterative algorithm is replaced with a powerful denoiser. Algorithmically, this
involves repeated inversion (of the forward model) and denoising until
convergence. Remarkably, PnP regularization produces promising results for
several restoration applications. However, a fundamental question in this
regard is the theoretical convergence of the PnP iterations, since the
algorithm is not strictly derived from an optimization framework. This question
has been investigated in recent works, but there are still many unresolved
problems. For example, it is not known if convergence can be guaranteed if we
use generic kernel denoisers (e.g. nonlocal means) within the ISTA framework
(PnP-ISTA). We prove that, under reasonable assumptions, fixed-point
convergence of PnP-ISTA is indeed guaranteed for linear inverse problems such
as deblurring, inpainting and superresolution (the assumptions are verifiable
for inpainting). We compare our theoretical findings with existing results,
validate them numerically, and explain their practical relevance.
