AdaptivePnP: Fast Auto-Tuned Plug-and-Play for Deconvolution

Haley Duba-Sullivan; Emma J. Reid; Charles A. Bouman; Gregery T. Buzzard

eess.IVNov 6, 2026

AdaptivePnP: Fast Auto-Tuned Plug-and-Play for Deconvolution

Authors: Haley Duba-Sullivan, Emma J. Reid, Charles A. Bouman, Gregery T. Buzzard

Organizations: Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA · National Security Sciences Directorate, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA · School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA

Abstract

Plug-and-play (PnP) methods reconstruct images by alternating a data-fitting step with an image denoiser. Their performance depends on two parameters: the step size $γ$ and the measurement noise level $σ$ . In practice, $γ$ is often tuned by grid search over validation images and must be retuned whenever the problem setting changes, and $σ$ is assumed known or estimated separately. We propose AdaptivePnP, a PnP framework that jointly estimates $σ$ and $γ$ at every iteration from the data-fitting residual. To do so, we apply the PnP virtual model to the data-fitting proximal map and show that the residual is zero-mean Gaussian with covariance parameterized by $γ$ and $σ$ . However, when one parameter is much larger than the other, the residual carries little information about the smaller one and using the maximum likelihood (ML) estimator can degrade the PnP iterations. We therefore introduce the identifiable maximum likelihood (IML) estimator, which restricts the ML estimator to an identifiable window in which the normalized Cramér--Rao bound of every parameter is bounded by a user-specified tolerance. For the virtual model, this window is an interval of ratios $ρ= γ/σ$ , so computing the IML estimate reduces to a one-dimensional search over $ρ$ . We apply AdaptivePnP to two standard PnP methods, yielding algorithms that require neither a tuned $γ$ nor a known $σ$ . On image deblurring and super-resolution, AdaptivePnP matches or outperforms PnP baselines tuned with the true $σ$ . Code is available at https://github.com/hdubasullivan/AdaptivePnP.

Figures & tables

Fig. 1: Overview of the proposed AdaptivePnP framework. Standard PnP methods alternate a data-fitting proximal map with an image denoiser (gray boxes); AdaptivePnP adds a parameter estimation step in each iteration (blue boxes). Namely, AdaptivePnP applies the identifiable maximum likelihood (IML) estimator to the data-fitting residual, resulting in a joint estimate of the noise level $\sigma$ and the step size $\gamma$ . The estimates are fed back to both the denoiser and the data-fitting proximal map, so neither a tuned step size nor a known noise level is required.

Figure 2

Fig. 2: The twelve $25\times 25$ blur kernels used in experiments.

		Isotropic Gaussian		Anisotropic Gaussian		Motion
		Tuned PnP	AdaptivePnP	Tuned PnP	AdaptivePnP	Tuned PnP	AdaptivePnP
Deblurring	PGM	27.83 / 0.768	28.39 / 0.807	25.49 / 0.701	25.83 / 0.719	26.64 / 0.719	28.01 / 0.777
Deblurring	ADMM	28.27 / 0.793	28.32 / 0.808	25.77 / 0.722	25.79 / 0.719	28.02 / 0.800	27.96 / 0.782
2 $\times$ Super-Resolution	PGM	24.07 / 0.603	26.69 / 0.737	24.83 / 0.634	25.23 / 0.675	22.24 / 0.470	25.47 / 0.650
2 $\times$ Super-Resolution	ADMM	23.88 / 0.623	26.55 / 0.737	25.13 / 0.671	25.28 / 0.685	23.03 / 0.578	25.79 / 0.701

TABLE I: Mean PSNR / SSIM of AdaptivePnP and tuned PnP reconstructions, averaged over the CBSD68 dataset with four kernels in each family (G1–G4, A1–A4, M1–M4) and three noise levels (1%, 3%, and 5%). Tuned PnP uses the true

\sigma

and a step size selected by grid search, whereas AdaptivePnP uses neither.The best result within each algorithm and kernel family is bolded.

Panel of Figure 5 — Fig. 3: Mean reconstruction PSNR and ratio $\rho_{k}=\gamma_{k}/\sigma_{k}$ of AdaptivePnP and tuned PnP over 10 iterations for (a) deblurring and (b) $2\times$ SR. Results are averaged over the CBSD68 dataset with all 12 kernels and three noise levels. AdaptivePnP rapidly adapts $\rho_{k}$ and reaches its final reconstruction quality within the first few iterations without requiring a validation-tuned parameter choice.

Fig. 4: Deblurring reconstructions using AdaptivePnP and tuned PnP; rows show an isotropic Gaussian kernel (G4) with $3\%$ measurement noise, an anisotropic Gaussian kernel (A1) with $1\%$ noise, and a motion blur kernel (M1) with $5\%$ noise. The ground truth image and blurred noisy measurement are shown for reference, and the highlighted regions show magnified image details. AdaptivePnP produces visual quality comparable to or better than the tuned PnP baselines despite estimating both $\sigma$ and $\gamma$ directly from the reconstruction.

Fig. 5: $2\times$ SR reconstructions using AdaptivePnP and tuned PnP; rows show an isotropic Gaussian kernel (G3) with $1\%$ measurement noise, an anisotropic Gaussian kernel (A3) with $3\%$ noise, and a motion blur kernel (M3) with $5\%$ noise. The ground truth image and low-resolution noisy measurement are shown for reference, and the highlighted regions show magnified image details. AdaptivePnP recovers visibly sharper image detail than the tuned PnP baselines, consistent with the PSNR and SSIM improvements in Table I .

Panel of Figure 8 — Fig. 6: Mean PSNR of AdaptivePnP after 10 iterations using varying tolerances $\tau$ for (a) deblurring and (b) $2\times$ SR. Results are averaged over the first 20 images from CBSD68 with all 12 kernels and three noise levels. Reconstruction quality is insensitive to the precise choice of the tolerance once it is sufficiently large; we use $\tau=0.05$ (dashed line) for all experiments.

Panel of Figure 9 — Fig. 7: Identifiable windows $\mathcal{W}_{\tau}$ at $\tau=0.05$ for all kernels under (a) deblurring and (b) $2\times$ SR. Horizontal intervals show $[\rho_{\inf},\rho_{\sup}]$ , with arrows indicating $\rho_{\sup}>10^{6}$ . All twelve windows are nonempty in both settings.

Panel of Figure 10 — Fig. 8: Mean PSNR of AdaptivePnP over 10 iterations with the proposed IML estimator and the unconstrained ML estimator for (a) deblurring and (b) $2\times$ SR. Results are averaged over the first 20 images from CBSD68 with all 12 kernels and three noise levels. At each iteration, the mean PSNR excludes any runs that have diverged to a non-finite PSNR, which occurs only for the ML estimator. With IML, both algorithms converge within a few iterations, while using ML degrades quality over iterations.

Panel of Figure 11 — Fig. 9: Mean PSNR of AdaptivePnP over 10 iterations with estimated $\sigma_{k}$ and known $\sigma$ for (a) deblurring and (b) $2\times$ SR. Results are averaged over the first 20 images of CBSD68 with all 12 kernels and three noise levels; $\gamma$ is estimated by the IML estimator in both settings. AdaptivePnP removes the need for a known $\sigma$ at no cost in reconstruction quality.

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