Study on convergence of plug-and-play ISTA with adaptive-kernel denoisers

Plug-and-play (PnP) is a powerful framework that applies off-the-shelf denoisers to regularize imaging inverse problems in an iterative style. Remarkably, in several restoration applications, this kind of regularization exhibits promising behaviors. Among the algorithms derived from the framework, the ISTA-based (PnP-ISTA) has attracted much attentions due to the effectiveness and simple update rule in iterations. And its convergence analysis has become a fundamental topic. Most recently, the theoretical convergence of PnP-ISTA with kernel denoisers has been established, where the kernel is generic (relaxing the reliance on special properties) and thus beneficial for wider applications. In there, the convergence proof focuses on fixed kernels. Note that, the denoisers with adaptive kernels usually achieve more powerful performance than those with fixed ones. Inspired by the preliminary observation, we extend the fixed kernels to adaptive ones for the denoisers in PnP-ISTA. Under a mild assumption, we prove the convergence theoretically. Meanwhile, for inpainting, an important scenario, we broaden the interval of the step size from 01to02, which still guarantees the convergence. Experimental results agree with our theoretical conclusions.