Closed-Form Nonlocal Shrinkage for Multiplicative Image Denoising and SAR Despeckling

Xuran Hu; Mingzhe Zhu; Djordje Stanković; Yujie Zhu; Zhenpeng Feng; Yifang Ban; Ljubiša Stanković

cs.CVAug 15, 2026

Closed-Form Nonlocal Shrinkage for Multiplicative Image Denoising and SAR Despeckling

Authors: Xuran Hu, Mingzhe Zhu, Djordje Stanković, Yujie Zhu, Zhenpeng Feng, Yifang Ban, Ljubiša Stanković

Organizations: School of Electronic Engineering, Xidian University, Xi’an, China · Kunshan Innovation Institute of Xidian University, Kunshan, China · Division of Geoinformatics, KTH Royal Institute of Technology, Stockholm, Sweden · Department of Electrical Engineering, University of Montenegro, Podgorica, Montenegro ·

Abstract

Multiplicative noise poses a challenge in coherent and signal-dependent imaging owing to its intensity-dependent variance and frequently non-Gaussian distribution. We propose a deterministic nonlocal estimator that combines a logarithmic Yeo--Johnson transformation, patch grouping, an adaptive singular basis, and sparse shrinkage. The orthonormal group dictionary makes the weighted Lasso separable and yields an exact coefficient-wise soft-threshold solution. This solution replaces the iterative inner solver and expresses patch reliability and atom importance through a single threshold field. Since the dictionary is estimated from the noisy group, we introduce a random-matrix correction governed by the group aspect ratio $γ=p^2/K$ . The correction links patch size, group size, and shrinkage strength. Experiments cover gamma-corrupted images from three standard benchmarks and real synthetic aperture radar (SAR) imagery from five sensors. The method gives the best result in 18 of 24 PSNR/SSIM comparisons with twelve published methods and the lowest mean ratio-image deviation across six real SAR configurations. These results support geometry-calibrated nonlocal modeling for structure-preserving image restoration, with SAR despeckling serving as a demanding application. Code is available here.

Figures & tables

Figure 1: Overview of the proposed image-restoration pipeline. The logarithmic Yeo–Johnson transformation reduces the asymmetry of multiplicative noise. Similar patches are grouped and represented in their orthonormal singular basis $D=U$ , which yields an analytic coefficient-wise threshold.

Figure 2: One-look Set12 results. Panels are ordered as labeled. Insets show the marked regions, with PSNR/SSIM below. DIP, CL-SAR, and S3DIP used different half-resolution scenes and are omitted.

Figure 3: Two-look McMaster results. Panels are ordered as labeled. Insets show the marked regions, with PSNR/SSIM below.

Figure 4: Four-look Kodak24 comparison. Panels are ordered as labeled, and insets show the marked regions. PSNR in decibels and SSIM multiplied by 100 are shown below each panel.

Figure 5: Sentinel-1 results on two scenes. Panels are ordered as labeled, and insets show the marked regions. Values below the panels are the ratio-image deviations from Table 3 . Lower values are better.

Method	1-Look		2-Look		4-Look		8-Look
	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
Set12
ANLM	14.11	27.86	17.97	40.04	23.04	60.08	22.94	66.98
DIP	17.08	48.69	18.18	56.81	19.04	61.86	20.83	68.97
DnCNN	18.36	39.91	20.72	46.02	22.43	51.99	23.44	58.01
SARCAM	16.25	27.15	22.56	57.74	24.29	68.34	23.41	70.61

Table 2: Synthetic image-denoising results under gamma-distributed multiplicative noise (PSNR/SSIM).

Figure 6: Gaofen-3 results. Panels are ordered as labeled, and insets show the marked regions. Values below the panels are the ratio-image deviations from Table 3 .

Figure 7: miniSAR results. Insets show the marked regions. Values below the panels are the ratio-image deviations from Table 3 .

Figure 8: Standardized gamma-noise distributions at one and eight looks in the log domain and after the logarithmic Yeo–Johnson transformation, group-mean removal, and projection. The curve is the standard normal density.

Figure 9: Threshold-factor ablation at four looks. The panels show the reference, observation, both factors fixed, $w_{2}$ fixed, $w_{1}$ fixed, and the full model. Insets show texture preservation, and PSNR/SSIM values refer to this scene.

	$w_{1},w_{2}$	LYJ	Synthetic (4 looks)		Real
			PSNR	SSIM	ENL	EPI	EPD
(a)	$\times$	$\times$	22.62	65.94	2.486	0.571	9.85
(b)	$\times$	$\checkmark$	23.21	67.21	2.537	0.599	10.32
(c)	$\checkmark$	$\times$	25.60	74.79	2.861	0.658	13.78
(d)	$\checkmark$	$\checkmark$	25.98	77.13	2.784	0.733	14.33

Table 4: Module ablation. LYJ denotes the logarithmic Yeo–Johnson transform. The symbols

\checkmark

and

\times

indicate enabled and disabled components.

Figure 10: Four-look Kodak24 ratio-image distributions pooled over 24 images. The reference ratio includes quantization effects, and the dotted curve represents ideal gamma noise. Narrower and broader distributions indicate residual noise and structure leakage, respectively.

	1-Look		2-Look		4-Look		8-Look
	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
Full $\tau$	20.30	57.87	23.45	67.68	26.01	75.21	28.12	81.11
$w_{1}$ frozen	20.28	58.32	23.36	67.50	25.88	74.89	27.95	80.67
$w_{2}$ frozen	20.10	56.92	22.66	63.91	24.62	70.39	26.49	76.61
Frozen	20.06	56.74	22.51	63.18	24.29	69.16	26.04	75.25

Table 5: Ablation of

\tau_{ik}=w_{2,i}/w_{1,k}^{2}

. A fixed factor is set to its group mean. Differences between rows isolate the sparse-coding stage.

Method	$\mathrm{E}[G]$	$\theta$	Method	$\mathrm{E}[G]$	$\theta$
ANLM Xing et al. [2020]	1.0178	0.1085	DIP Ulyanov et al. [2018]	0.9673	0.2973
DnCNN Zhang et al. [2017]	0.9591	0.0716	SARCAM Ko and Lee [2022]	0.9700	0.1790
SAR2SAR Dalsasso et al. [2021]	0.9817	0.2724	SARtrans Perera et al. [2022]	0.9050	0.2675
AGSDNet Thakur and Maji [2022a]	0.9446	0.1243	SIFSDNet Thakur and Maji [2022b]	0.9629	0.1886
MONet Vitale et al. [2023]	0.8932	0.2047	RDDPM Hu et al. [2024]	1.3241	0.3990
CL-SAR Fang et al. [2024]	0.8025	0.1894	S3DIP Albisani et al. [2025]	0.7473	0.2162

Table 7: Gamma-fit consistency on four-look Kodak24. The ideal values are

\mathrm{E}[G]=1

and

\theta=0.25

Image Denoising

Image denoising removes unwanted noise while preserving the edges, textures, and colors needed for faithful images.

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This letter introduces a physics-informed self-supervised framework for sonar image despeckling that reformulates despeckling as residual consistency in the homomorphic log domain. By constraining the log-ratio residual to obey multiplicative speckle statistics, the proposed method eliminates the need for clean supervision while preventing degenerate identity solutions. A variance-targeted statistical loss combined with edge-aware structural regularization and median-guided curriculum stabilization enables effective speckle suppression with preserved structural fidelity. This formulation along with a lightweight neural network achieves state-of-the-art performance across multiple real sonar datasets and demonstrates excellent cross-dataset robustness, while remaining suitable for real-time deployment.

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