Convergent Plug-and-Play Image Restoration with Annealed Noise Levels

Samuel Hurault

math.OCSep 26, 2026

Convergent Plug-and-Play Image Restoration with Annealed Noise Levels

Authors: Samuel Hurault

Organizations: LIGM, CNRS, Univ. Gustave Eiffel

Abstract

Plug-and-Play (PnP) methods solve imaging inverse problems by incorporating deep denoisers into iterative optimization algorithms. Although practical implementations often decrease the denoiser noise level $σ$ along iterations, most existing convergence analyses assume a fixed denoiser. In this work, we establish convergence guarantees for a broad family of Plug-and-Play algorithms with annealed noise level, spanning deterministic methods (RED--GD and PnP--PGD) and stochastic methods (SNORE, equivariant RED, and a variant of PnP--Flow). For each method, we identify an explicit, nonconvex objective associated with the terminal denoising level and prove asymptotic stationarity of the iterates with respect to this objective. Our analysis does not prescribe any decay rate for the noise schedule, and our assumptions cover both learned gradient-step denoisers and exact MMSE denoisers. Overall, our theoretical results bridge the gap between existing PnP convergence theory and the decreasing-denoising practices used by state-of-the-art image restoration methods. We empirically demonstrate the benefits of such schedules and illustrate the predicted convergence behavior on several imaging inverse problems, including inpainting, super-resolution, demosaicing and tomography.

Figures & tables

Figure 1: Inpainting of a $64\times 64$ hole in a $128\times 128$ face image using RED gradient descent with the same denoiser (GS–DiffUNet; see Section 4 ). With a constant noise level, a small $\sigma$ leaves the initial flat patch, while larger values fill the hole at the cost of missing fine details. Decreasing geometrically $\sigma_{k}$ along iterates, from a large initial noise level to a small terminal one, produces a sharp face.

Method	Objective	Stepsize conditions	Guarantee
RED–GD (Section 3.1 )	$f+\lambda g_{\epsilon}$	$\gamma(L_{f}+\lambda L_{g})<2$	Stationarity and point convergence
SNORE (Section 3.2 )	$f+\lambda\mathbb{E}_{Z}[g_{\epsilon}(\cdot+\epsilon Z)]$	$\gamma_{k}(L_{f}+\lambda L_{g})<2$ ; $\sum_{k}\gamma_{k}=\infty$ , $\sum_{k}\gamma_{k}^{2}<\infty$	Stationarity a.s.
ERED (Appendix B.3 )	$f+\lambda\mathbb{E}_{Q}[g_{\epsilon}(Q\cdot)]$	$\gamma_{k}(L_{f}+\lambda L_{g})<2$ ; $\sum_{k}\gamma_{k}=\infty$ , $\sum_{k}\gamma_{k}^{2}<\infty$	Stationarity a.s.
PnP–PGD (Section 3.3 )	$f+\gamma^{-1}\phi_{\epsilon}$	$\gamma L_{f}<1$	Stationarity and point convergence
SNOPnP / PnP–Flow (Appendix B.5 )	$f+\gamma^{-1}\phi_{\epsilon}$	$\gamma L_{f}<1$ ; $\sum_{k}\tau_{k}^{2}<\infty$	Stationarity a.s.

Table 1: Convergence guarantees for the algorithms introduced in Section 2 with

\sigma_{k}\downarrow\epsilon

	Inpainting		Demosaicing		SR $\times 4$
Method	Constant	Annealed	Constant	Annealed	Constant	Annealed
RED–GD	24.22	25.36	33.54	37.89	27.17	27.51
PnP–PGD	20.03	25.09	35.35	37.76	26.49	27.25
SNORE	24.34	24.98	33.32	38.03	27.14	27.34
SNOPnP	21.93	24.42	36.22	37.78	27.18	27.29
ERED	24.08	25.44	33.55	38.00	27.24	27.56

Table 2: PSNR (dB) on

100

FFHQ face images after

500

iterations with constant or annealed ( 15 ) noise level. RED methods and DPIR use the GS–DiffUNet denoiser; PnP methods use Prox–DiffUNet. Best value per problem in bold. The corresponding CBSD68 results are in Table 4 of Appendix C .

Panel of Figure 2 — Figure 2: Inpainting of a $64\times 64$ hole on FFHQ $128\times 128$ . Top: the constant- $\sigma$ versions (and DPIR) leave a smooth patch, while the annealed versions fill the hole. Bottom: PSNR, target objective $F_{\epsilon}(x_{k})$ and residual $\|x_{k+1}-x_{k}\|/\|x_{0}\|$ of the annealed runs, along the iterations, for the image shown above. Note that every algorithm has its own target objective $F_{\epsilon}$ , so only the decrease of each curve is meaningful, not the levels.

Panel of Figure 3 — Figure 3: Demosaicing of a Bayer color filter array on FFHQ $128\times 128$ . The constant- $\sigma$ versions (and DPIR) keep the color fringes of the initial interpolation along the edges, the annealed versions remove them. Bottom: PSNR, terminal objective and residual along the iterations.

Panel of Figure 4 — Figure 4: $\times 4$ super-resolution on FFHQ $128\times 128$ . PSNR differences are smaller than in the previous experiments, but the annealed methods yield visibly sharper and more detailed reconstructions. Bottom: PSNR, terminal objective and residual along the iterations.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

PSNR (dB)
$\sigma$	0.01	0.05	0.2	0.5	1	5
Pretrained DiffUNet [ 3 ]	44.61	35.54	28.60	24.45	21.49	15.48
GS–DiffUNet (fine-tuned to $\sigma\leq 5$ )	44.54	35.69	28.87	24.87	22.08	15.75
Prox–DiffUNet	44.42	35.33	28.28	24.36	21.72	16.12
$\max_{x}\\|\nabla^{2}g_{\sigma}(x)\\|_{2}$
GS–DiffUNet	1.090	2.117	4.155	3.062	3.096	1.207

Appendix

Table 3: Denoising PSNR and spectral norm

\|\nabla^{2}g_{\sigma}\|

16

noisy validation images. GS and Prox denote the gradient-step and proximal denoisers. Bold norms are below

1

CBSD68 (68 images), denoiser GS-DRUNet (RED algorithms) / Prox-DRUNet (PnP algorithms)
	Inpainting		Tomography (60 views)		SR $\times 4$
	constant $\sigma$	annealed	constant $\sigma$	annealed	constant $\sigma$	annealed
RED–GD	29.71	30.99	27.93	29.02	24.64	24.70
PnP–PGD	30.30	30.70	23.58	27.42	24.68	24.70
SNORE	29.93	30.93	27.89	28.98	24.64	24.73
SNOPnP	30.26	30.64	22.40	27.33	24.65	24.69

Appendix

Table 4: PSNR (dB) on the

68

CBSD68 test images (

128\times 128

) for inpainting, tomography with

60

views, and super-resolution, same layout as Table 2 : constant noise level

\sigma_{c}

versus annealed schedule

\sigma_{k}\to\epsilon

, both tuned, and DPIR. RED algorithms use GS–DRUNet, PnP algorithms Prox–DRUNet, both finetuned to

\sigma\leq 2

. Best annealed value per problem in bold.

Panel of Figure 5 — Figure 5: Tomography with $60$ views on CBSD68, same layout as Figure 2 ; the observation is shown through its filtered back-projection.

Panel of Figure 6 — Figure 6: Inpainting of $50\%$ random missing pixels on CBSD68, same layout as Figure 2 .

dataset	problem	algorithm	$\varepsilon$ ( $\sigma_{c}$ )	$\lambda$	step	$q$
FFHQ	Inpainting	RED–GD	0.014 (2)	0.3	$\gamma_{\rm rel}=1$	–
FFHQ	Inpainting	PnP–PGD	0.01 (0.04)	–	$c=0.3$	–
FFHQ	Inpainting	SNORE	0.01 (2)	0.3	$\gamma_{\rm rel}=1$	0.55
FFHQ	Inpainting	SNOPnP	0.01 (0.16)	–	$c=0.95$	1
FFHQ	Inpainting	ERED	0.01 (2)	0.5	$\gamma_{\rm rel}=1$	1
FFHQ	Demosaicing	RED–GD	0.01 (0.08)	0.1	$\gamma_{\rm rel}=1$	–

Appendix

Table 5: Selected hyper-parameters of the different algorithms for each problem and dataset (validation on

16

images)

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FlowADMM: Plug-and-play ADMM with Flow-based Renoise-Denoise Priors

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