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Paper · 2104.10348 · 2021

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

arXiv · PDF · Open in the Atlas

Code that ran

We lifted 4 functions out of this paper's own repositories and ran 2 of them in a sandbox. "Ran" means the function executed on a synthesized input and returned a value. It is not a reproduction of the paper's results.

RepositoryRoleRan
pravin1390/ScaledPnP canonical 2 of 4
FunctionStatusWhere it lives
proj Ran pravin1390/ScaledPnP/pnp_fbs_superres.py
pointer only (licence: NONE) · get_code("f0ecb9f6504fb30e")
psnr Ran pravin1390/ScaledPnP/pnp_fbs_superres.py
pointer only (licence: NONE) · get_code("3c1cd4ef5aa47de2")
dsg_nlm Not yet run pravin1390/ScaledPnP/denoisers/kernel_filters.py
pointer only (licence: NONE) · get_code("ed7a7667aad4e6c5")
nlm Not yet run pravin1390/ScaledPnP/denoisers/kernel_filters.py
pointer only (licence: NONE) · get_code("87665ea84595714d")

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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.

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