A convergent Plug-and-Play Majorization-Minimization algorithm for Poisson inverse problems

Thibaut Modrzyk; Ane Etxebeste; Élie Bretin; Voichita Maxim

cs.CVMar 25, 2026

A convergent Plug-and-Play Majorization-Minimization algorithm for Poisson inverse problems

Authors: Thibaut Modrzyk, Ane Etxebeste, Élie Bretin, Voichita Maxim

Organizations: CREATIS · INSA-Lyon, CREATIS UMR 5220 U1294, Villeurbanne, F-69621, France · ICJ, MMCS · INSA-Lyon, CNRS UMR 5208, Institut Camille Jordan, Villeurbanne, F-69621, France.

Abstract

In this paper, we present a novel variational plug-and-play algorithm for Poisson inverse problems. Our approach minimizes an explicit functional which is the sum of a Kullback-Leibler data fidelity term and a regularization term based on a pre-trained neural network. By combining classical likelihood maximization methods with recent advances in gradient-based denoisers, we allow the use of pre-trained Gaussian denoisers without sacrificing convergence guarantees. The algorithm is formulated in the majorization-minimization framework, which guarantees convergence to a stationary point. Numerical experiments confirm state-of-the-art performance in deconvolution and tomography under moderate noise, and demonstrate clear superiority in high-noise conditions, making this method particularly valuable for nuclear medicine applications.

Figures & tables

Fig. 1: Left: Deconvolution on the Leaves image from the Set3c dataset with the Gaussian kernel and noise level $\zeta=60$ . Right: Same image and kernel, with noise level $\zeta=5$ . All images were produced from the same measurement data. Evolution of the PSNR and the gap between iterates have been plotted for theoretically convergent methods.

	Kernels	Gaussian			Levin1			Levin4			Uniform			Average
$\zeta$	Methods	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
60	RL-TV [ 11 ]	24.78	0.610	0.425	23.33	0.520	0.493	24.00	0.544	0.440	23.80	0.568	0.474	23.98	0.561	0.458
	DPS [ 36 ]	20.41	0.469	0.549	21.98	0.545	0.382	22.85	0.580	0.380	18.56	0.404	0.567	20.95	0.499	0.469
	Prox-DiffPIR [ 37 ]	26.04	0.688	0.337	25.31	0.632	0.307	26.31	0.676	0.255	24.97	0 .630	0.391	25.66	0.656	0.322
	PhD-Net [ 38 ]	26.00	0.685	0.420	24.56	0.608	0.424	25.81	0.665	0.424	24.70	0.618	0.503	25.26	0.644	0.455
	DPIR [ 17 ]	2 6.46	0 .701	0.403	2 6.00	0.663	0.420	2 7.03	0.711	0.350	2 4.78	0.618	0.515	2 6.07	0 .673	0.422

TABLE I: Comparison on the Kodak test dataset

Fig. 2: Left: Deconvolution on the Butterfly image from the Set3c dataset with the first Levin real world motion blur kernel and noise level $\zeta=60$ . Right: Same image and kernel, with noise level $\zeta=5$ .

Fig. 3: Top row: PET reconstructions of a representative mid-axial slice from the BrainWeb test dataset, including two hot lesions. Bottom row: MR slice and the corresponding relative error maps for each reconstruction algorithm with respect to the emission map.

Fig. 4: NRMSE values for each method and for different region of interest on the Brainweb test dataset

Fig. 5: Contrast-to-Noise values for each method on the Brainweb test dataset

Fig. 6: Reconstruction of a test slice with the different methods. Top row: 128 views with the lowest Gaussian noise level, which was used to train the fully-supervised methods PD-Net and FBP-SwinIR. Bottom row: 128 views with the highest Gaussian noise level, which was never seen during the training of the fully supervised methods.

		192 angles			128 angles			64 angles
$\sigma$	Methods	PSNR $\uparrow$	SSIM $\uparrow$	MAE (HU) $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	MAE (HU) $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	MAE (HU) $\downarrow$
0.005	FBP	17.39 $\pm$ 0.72	0.119 $\pm$ 0.023	290 $\pm$ 31.2	16.60 $\pm$ 0.71	0.103 $\pm$ 0.021	318 $\pm$ 34.0	15.07 $\pm$ 0.66	0.077 $\pm$ 0.017	378 $\pm$ 38.9
	MLEM [ 3 ]	28.72 $\pm$ 1.06	0.772 $\pm$ 0.028	62.7 $\pm$ 10.2	28.67 $\pm$ 1.06	0.765 $\pm$ 0.028	63.1 $\pm$ 10.3	28.27 $\pm$ 1.04	0.735 $\pm$ 0.032	66.7 $\pm$ 10.7
	PDNet [ 50 ]	3 7.28 $\pm$ 0.97	0 .897 $\pm$ 0.018	2 4.7 $\pm$ 3.2	3 7.05 $\pm$ 0.99	0 .893 $\pm$ 0.018	2 5.2 $\pm$ 3.2	3 4.58 $\pm$ 1.08	0 .853 $\pm$ 0.022	3 3.4 $\pm$ 4.3
	FBP-SwinIR [ 51 ]	37.66 $\pm$ 1.01	0.903 $\pm$ 0.018	23.50 $\pm$ 3.1	37.31 $\pm$ 1.01	0.898 $\pm$ 0.018	24.4 $\pm$ 3.2	31.82 $\pm$ 1.53	0.796 $\pm$ 0.040	45.3 $\pm$ 8.0
	PnP-MM	36.40 $\pm$ 1.09	0.892 $\pm$ 0.017	26.53 $\pm$ 3.3	36.44 $\pm$ 0.94	0.888 $\pm$ 0.016	26.9 $\pm$ 3.1	35.64 $\pm$ 1.19	0.873 $\pm$ 0.020	28.6 $\pm$ 4.0

TABLE II: Comparison on AAP Mayo test dataset

Fig. 7: Reconstruction of a test slice with the different methods. Top row: 128 views with the lowest Gaussian noise level, which was used to train the fully-supervised methods. Bottom row: 64 views with the lowest Gaussian noise level, which was never seen during the training of the fully supervised methods.

Panel of Table 10 — TABLE III: Comparison on the CBSD68 validation dataset

Noise level ( $\sigma$ )	1	5	10	20	30	40	50
NRMSE (%)	$2.18\pm 0.302$	$3.68\pm 0.414$	$6.08\pm 0.691$	$10.4\pm 1.52$	$13.8\pm 1.96$	$16.2\pm 2.27$	$18.6\pm 2.94$

TABLE IV: Quantitative denoising performances on the Brainweb test dataset

Table 12

		192 angles			128 angles			64 angles
Method	Parameter	$\sigma=0.5\%$	$\sigma=1\%$	$\sigma=2\%$	$\sigma=0.5\%$	$\sigma=1\%$	$\sigma=2\%$	$\sigma=0.5\%$	$\sigma=1\%$	$\sigma=2\%$
MLEM	Iterations	40	35	20	40	35	20	20	20	20
PnP-MM	$\lambda$	0.12	0.20	0.20	0.10	0.15	0.15	0.10	0.10	0.10
	$\sigma\times 255$	4	4	8	5	5	8	5	8	8
	$\tau$	1	1	1	1	1	1	1	1	1
	Iterations	600	600	600	600	600	600	600	600	600

TABLE VII: Hyperparameters for the CT reconstruction experiment.

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aDepartment of Engineering Sciences, Faculty of Advanced Technologies,Sep University of Mohaghegh Ardabili, Ardabil, Iran

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Images are important tools in various sciences. Despite the development of photo-taking tools, creating clear and image without noise remains challenging in practice. In particular, Poisson noise has an effect on medical and astronomical images, and reduces their quality. Additionally, blur is another factor that has an effect on image quality. The problem of image restoration becomes very complicated when we have no information about the Point Spread Function (PSF). These types of problems are known as blind case. However, in some images, such as some astronomical images, the type of PSF can be specified, and these types of problems are known as nonblind problems. Total Variation (TV) is a widely used method for solving such inverse problems, where the selection of the penalty function is the most critical factor that affects the method's performance. In this paper, to improve edge preservation, we employ a reweighted

\ell_1

-regularization of the fractional order derivative. Furthermore, we propose a nonblind and blind image deblurring approach under Poisson noise using the Minimax Concave Penalty (MCP), which is a continuous, sparsity promoting, and nearly unbiased regularizer. This formulation leads to a nonconvex optimization model. To solve the proposed model, we introduce an efficient numerical algorithm based on the Alternating Direction Method of Multipliers (ADMM) and provide an analysis of its convergence. Finally, the effectiveness of the proposed algorithm are demonstrated through extensive experiments on various images.

Abstract

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