Fixed-Point and Objective Convergence of Plug-and-Play Algorithms

TL;DR

This paper proves convergence guarantees for a class of linear denoisers in plug-and-play algorithms, ensuring both iterate and objective convergence in image reconstruction tasks.

Contribution

It provides the first simultaneous convergence analysis for PnP algorithms with non-symmetric linear denoisers like nonlocal means.

Findings

Convergence of PnP algorithms is established for non-symmetric linear denoisers.

The analysis covers convex data-fidelity terms and specific inner product spaces.

Experimental validation confirms the theoretical convergence results.

Abstract

A standard model for image reconstruction involves the minimization of a data-fidelity term along with a regularizer, where the optimization is performed using proximal algorithms such as ISTA and ADMM. In plug-and-play (PnP) regularization, the proximal operator (associated with the regularizer) in ISTA and ADMM is replaced by a powerful image denoiser. Although PnP regularization works surprisingly well in practice, its theoretical convergence -- whether convergence of the PnP iterates is guaranteed and if they minimize some objective function -- is not completely understood even for simple linear denoisers such as nonlocal means. In particular, while there are works where either iterate or objective convergence is established separately, a simultaneous guarantee on iterate and objective convergence is not available for any denoiser to our knowledge. In this paper, we establish both forms of convergence for a special class of linear denoisers. Notably, unlike existing works where the focus is on symmetric denoisers, our analysis covers non-symmetric denoisers such as nonlocal means and almost any convex data-fidelity. The novelty in this regard is that we make use of the convergence theory of averaged operators and we work with a special inner product (and norm) derived from the linear denoiser; the latter requires us to appropriately define the gradient and proximal operators associated with the data-fidelity term. We validate our convergence results using image reconstruction experiments.