Provably Contractive and High-Quality Denoisers for Convergent Restoration

Shubhi Shukla; Pravin Nair

cs.CVMar 27, 2026

Provably Contractive and High-Quality Denoisers for Convergent Restoration

Authors: Shubhi Shukla, Pravin Nair

Organizations: Indian Institute of Technology Madras, India

Abstract

Image restoration, the recovery of clean images from degraded measurements, has applications in various domains like surveillance, defense, and medical imaging. While modern convolutional and attention-based networks achieve strong restoration quality, they typically lack explicit stability guarantees under small input perturbations. In this work, we develop provably contractive denoiser networks with global Lipschitz constant smaller than one. Our design composes proximal layers obtained from unfolding techniques, with Lipschitz-controlled convolutional refinements. By contractivity, the proposed denoiser guarantees that input perturbations of strength $\|δ\|\leq \varepsilon$ induce at most $\varepsilon$ change at the output, whereas strong unconstrained baselines such as DnCNN and Restormer can exhibit larger deviations under the same perturbations. On image denoising, our model substantially improves over prior provably 1-Lipschitz / contractive baselines and narrows the gap to strong unconstrained denoisers. In addition, when used as a prior in Plug-and-Play restoration, the proposed denoiser yields provable convergence and strong empirical performance. Overall, our results show that strict Lipschitz control need not preclude practically competitive restoration quality, and suggest that stable, verifiable restoration models can be achieved without a severe loss in performance. Codes and pretrained models are available at https://github.com/SHUBHI1553/Contractive-Denoisers

Figures & tables

Panel of Figure 1 — Figure 1: JPEG robustness at $90\%$ quality. Our contractive denoiser remains stable, while Restormer’s denoising quality drops under mild compression. PSNR values are shown in parentheses.

Panel of Figure 2 — Figure 2: Robustness to chroma subsampling. Our contractive denoiser retains stable reconstruction quality, while DnCNN suffers significant degradation under chroma perturbation. PSNR values are shown in parentheses.

Figure 3: Architecture of contractive layer $\mathcal{T}$ . The layer performs a gradient step and prox-wavelet operation, followed by a scaled convolution.

Panel of Figure 4 — Figure 4: Color Gaussian denoising ( $\sigma=15$ ). Our contractive denoiser preserves fine structures similar to unconstrained models, while closely matching Restormer.

Panel of Figure 5 — Figure 5: Color Gaussian denoising ( $\sigma=15$ ). Our contractive model preserves feather and eye-ring textures similar to the best-performing Restormer.

Method	CBSD68			Kodak24			Urban100
Method	$\sigma{=}15$	$\sigma{=}25$	$\sigma{=}50$	$\sigma{=}15$	$\sigma{=}25$	$\sigma{=}50$	$\sigma{=}15$	$\sigma{=}25$	$\sigma{=}50$
eSN-ReLU	$24.84/0.5919$	$22.54/0.5174$	$19.76/0.2189$	$24.76/0.5265$	$20.44/0.350$	$17.87/0.2713$	$24.95/0.6291$	$20.68/0.5802$	$17.17/0.2922$
eSN-GS	$24.98/0.6016$	$22.78/0.5765$	$20.43/0.2245$	$24.87/0.5364$	$20.86/0.350$	$17.95/0.2856$	$25.21/0.6267$	$20.85/0.5832$	$17.32/0.3214$
D-FBS	$30.12/0.8409$	$27.54/0.7577$	$23.87/0.6206$	$31.20/0.8315$	$28.44/0.7558$	$24.90/0.6380$	$29.53/0.8519$	$26.34/0.7729$	$22.02/0.6273$
D-DRS	$30.07/0.8526$	$27.51/0.7695$	$24.11/0.6266$	$31.05/0.8465$	$28.58/0.7710$	$25.15/0.6484$	$29.73/0.8687$	$26.70/0.7934$	$22.52/0.6438$
Ours	$\mathbf{32.74/0.9115}$	$\mathbf{29.78/0.8501}$	$\mathbf{25.61/0.7211}$	$\mathbf{33.42/0.9020}$	$\mathbf{30.67/0.8446}$	$\mathbf{26.49/0.7285}$	$\mathbf{31.96/0.9092}$	$\mathbf{28.94/0.8585}$	$\mathbf{24.40/0.7441}$

Table 1: Color denoising (PSNR(dB)/SSIM) . Our model dominates

1

-Lipschitz models (in top) and narrows the gap with unconstrained models (in bottom).

Method	BSD68			Set12			Urban100
Method	$\sigma{=}15$	$\sigma{=}25$	$\sigma{=}50$	$\sigma{=}15$	$\sigma{=}25$	$\sigma{=}50$	$\sigma{=}15$	$\sigma{=}25$	$\sigma{=}50$
eSN-ReLU	$24.89/0.6899$	$20.48/0.4159$	$14.91/0.2179$	$25.67/0.6618$	$21.34/0.3941$	$15.75/0.2625$	$24.89/0.6262$	$20.61/0.4774$	$14.78/0.2705$
eSN-GS	$24.93/0.6976$	$20.68/0.4313$	$15.21/0.2231$	$25.73/0.6743$	$21.43/0.3854$	$15.87/0.2634$	$24.91/0.6341$	$20.78/0.5236$	$14.86/0.2823$
D-FBS	$29.89/0.8347$	$27.31/0.7537$	$24.21/0.6290$	$30.41/0.8451$	$28.23/0.7924$	$24.53/0.6788$	$29.21/0.8478$	$26.48/0.7803$	$22.43/0.6461$
D-DRS	$29.79/0.8560$	$27.24/0.7600$	$24.18/0.6368$	$30.39/0.8535$	$28.15/0.8136$	$24.48/0.6815$	$29.18/0.8774$	$26.47/0.8058$	$22.39/0.6339$
Ours	$\mathbf{30.52/0.8690}$	$\mathbf{27.82/0.7883}$	$\mathbf{24.40/0.6460}$	$\mathbf{31.43/0.8816}$	$\mathbf{28.71/0.8219}$	$\mathbf{25.19/0.7160}$	$\mathbf{30.11/0.8851}$	$\mathbf{26.94/0.8118}$	$\mathbf{22.93/0.6690}$

Table 2: Grayscale denoising (PSNR(dB)/SSIM) . Our model is best among

1

-Lipschitz models (top) and competitive with unconstrained models (bottom).

Dataset	PGD	DRS-Diff	GS	FBS	DRS
CBSD68	$26.13/0.7062$	$26.03/0.663$	$26.23/0.713$	$26.18/0.707$	$24.26/0.623$
Kodak24	$27.01/0.7116$	$26.12/0.603$	$27.11/0.720$	$27.05/0.712$	$25.19/0.649$
Urban100	$23.41/0.7167$	$25.17/0.722$	$23.51/0.722$	$23.47/0.718$	$21.20/0.602$

Table 3: PnP deblurring with a

90\%

random sparse kernel (PSNR(dB)/SSIM). (a) Comparison of PnP algorithms using our denoiser. (b) Comparison of different denoisers within GS-PnP and DRS-PnP, including convergent baselines.

Panel of Figure 6 — Figure 6: PnP box deblurring with a $9\times 9$ kernel. IDF and Noise2VST oversmooth fine details, and Restormer introduces artifacts. Our method restores sharper structures.

Panel of Figure 7 — Figure 7: PnP deblurring with $90\%$ sparse kernel. Our model preserves edges and letters with fewer artifacts than DnCNN and Restormer, without D-DRS-like oversmoothing.

Panel of Figure 8 — Figure 8: PnP convergence. All $1$ -Lipschitz denoisers converge; unconstrained models diverge, while ours achieves the best PSNR.

Panel of Figure 9 — Figure 9: $2\times$ PnP-FBS superresolution ( $\mathbf{B}$ : $9\times 9$ Gaussian blur, $\sigma=1$ ). Our model restores sharper text and edges with fewer artifacts as compared to DnCNN and less oversmoothing.

Panel of Figure 10 — Figure 10: $2\times$ PnP-FBS superresolution ( $\mathbf{B}$ : Gaussian blur $25\times 25$ , $\sigma=1.6$ ). Our denoiser yields sharper details and fewer artifacts than unconstrained models.

Dataset	Diffusion-based PnP		PnP-FBS
Dataset	DiffPIR	SGPnP	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	OURS
CBSD68	$27.86/0.725$	$28.12/0.8001$	$26.62/0.7403$	$27.03/0.7438$	$24.26/0.6009$	$24.72/0.6130$	$27.11/0.7179$	$25.35/0.7013$	$\mathbf{28.40}/\mathbf{0.8063}$
Kodak24	$28.69/0.6967$	$29.08/0.8021$	$27.45/0.7456$	$27.91/0.7451$	$25.24/0.6430$	$25.82/0.6490$	$27.45/0.6772$	$25.24/0.6692$	$\mathbf{29.21}/\mathbf{0.8102}$
Urban100	$\mathbf{28.31}/0.8012$	$28.02/0.8059$	$24.59/0.7600$	$25.48/0.7741$	$22.56/0.6390$	$23.33/0.7369$	$25.86/0.7419$	$25.26/0.7397$	$26.33/\mathbf{0.8099}$

Table 4: PSNR/SSIM for deblurring with a

90\%

sparse random blur. Diffusion-based methods, which do not provide contractivity guarantees, are reported separately.

Dataset	Diffusion-based PnP		PnP-FBS
Dataset	DiffPIR	SGPnP	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	OURS
CBSD68	$27.44/0.7315$	$27.18/0.7773$	$26.14/0.7276$	$26.44/0.7300$	$24.47/0.6251$	$25.05/0.6479$	$26.41/0.6800$	$26.33/0.7161$	$\mathbf{27.50}/\mathbf{0.7797}$
Kodak24	$\mathbf{28.82}/0.7114$	$28.12/0.7761$	$27.03/0.7412$	$27.38/0.7410$	$25.30/0.6573$	$26.11/0.6779$	$26.83/0.6344$	$27.74/0.7339$	$28.53/\mathbf{0.7850}$
Urban100	$\mathbf{27.03}/0.8146$	$26.87/\mathbf{0.8472}$	$23.87/0.7414$	$24.39/0.7557$	$22.09/0.6263$	$23.66/0.7126$	$25.02/0.6908$	$25.50/0.7705$	$25.63/0.8009$

Table 5: PSNR/SSIM for deblurring with Levin kernel 1 [ 35 ] . Diffusion-based methods, which do not provide contractivity guarantees, are reported separately.

Dataset	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	OURS
CBSD68	$25.28/0.6835$	$25.36/0.6863$	$24.54/0.6267$	$24.94/0.6498$	$25.36/0.6670$	$24.51/0.5822$	$\mathbf{25.61/0.7046}$
Kodak24	$26.04/0.6994$	$26.12/0.7018$	$25.32/0.6575$	$25.73/0.6705$	$25.96/0.6580$	$25.05/0.5588$	$\mathbf{26.35/0.7184}$
Urban100	$22.56/0.6750$	$22.71/0.6830$	$21.79/0.6203$	$22.62/0.6700$	$22.83/0.6541$	$22.34/0.5822$	$\mathbf{22.89/0.6976}$

Table 6: PnP-FBS superresolution with Gaussian blur (

9\times 9

\sigma=2

) and noise level

5

. Results are reported as PSNR(dB)/SSIM.

Dataset	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	OURS
CBSD68	$25.17/0.6773$	$25.25/0.6800$	$24.42/0.6203$	$24.82/0.6427$	$25.30/0.6633$	$24.38/0.5731$	$\mathbf{25.51/0.6986}$
Kodak24	$25.95/0.6945$	$26.02/0.6965$	$25.20/0.6518$	$25.63/0.6652$	$25.92/0.6555$	$24.93/0.5498$	$\mathbf{26.26/0.7135}$
Urban100	$22.42/0.6666$	$22.57/0.6743$	$21.64/0.6114$	$22.48/0.6614$	$22.74/0.6484$	$22.21/0.5719$	$\mathbf{22.76/0.6896}$

Table 7: PnP-FBS superresolution with a

21\times 21

anisotropic blur (

\sigma_{x}=3

\sigma_{y}=1

\theta=0

). Results are reported as PSNR(dB)/SSIM.

Panel of Figure 11 — Figure 11: Ablation and complexity analysis on CBSD68. (a) Intermediate representations across layers; (b) architectural components; (c) learned step sizes $\alpha_{k}$ ; (d) network depth; (e) complexity and runtime for a $481\times 321$ RGB input; and (f) runtime across patch strides.

Appendix figures & tables30 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 12 — Figure 12: Perturbation analysis. Left: output variation versus perturbation norm. Right: spatial error heatmaps showing $\mathrm{D}(\mathbf{x}+\delta\mathbf{x})-\mathrm{D}(\mathbf{x})$ for different denoisers.

Panel of Figure 13 — Figure 13: JPEG robustness ( $90\%$ quality). Our contractive denoiser remains stable, while Restormer degrades under mild compression. Values indicate PSNR in dB.

Panel of Figure 14 — Figure 14: JPEG robustness ( $90\%$ quality). Our contractive denoiser remains stable, while DnCNN degrades. PSNR shown.

Panel of Figure 15 — Figure 15: JPEG robustness ( $90\%$ quality). Our method remains stable under compression, while Restormer is highly sensitive to the perturbation.

Panel of Figure 16 — Figure 16: Color Gaussian denoising on an image from the McMaster dataset ( $\sigma=15/255$ ). Our contractive model preserves fine fabric textures comparable to Restormer, while D-FBS and D-DRS exhibit over-smoothing.

Panel of Figure 17 — Figure 17: Color Gaussian denoising on kodim15 (Kodak24). Our method preserves fine textures better than $1$ -Lipschitz baselines while remaining competitive with DnCNN and Restormer. PSNR/SSIM shown.

Dataset, $\sigma$	$1$ -Lipschitz / Contractive Models			Unconstrained Models				Pretrained Models
Dataset, $\sigma$	D-FBS	D-DRS	Ours	IDF	DnCNN	Noise2VST	Restormer	CLIP	DIL
CBSD68, $\sigma{=}15$	$30.12/0.8409$	$30.07/0.8526$	$\mathbf{32.74/0.9115}$	$32.18/0.8915$	$33.63/0.9296$	$33.71/0.9288$	$\mathbf{33.85/0.9316}$	$33.72/0.921$	$32.50/0.906$
CBSD68, $\sigma{=}25$	$27.54/0.7577$	$27.51/0.7695$	$\mathbf{29.78/0.8501}$	$29.80/0.8435$	$30.72/0.8806$	$\mathbf{31.03/0.8877}$	$30.90/0.8837$	$31.02/0.878$	$29.98/0.843$
CBSD68, $\sigma{=}50$	$23.87/0.6206$	$24.11/0.6266$	$\mathbf{25.61/0.7211}$	$25.73/0.7188$	$26.52/0.7714$	$25.93/0.7450$	$\mathbf{26.65/0.7765}$	$26.69/0.7311$	$26.43/0.7170$
Kodak24, $\sigma{=}15$	$31.20/0.8315$	$31.05/0.8465$	$\mathbf{33.42/0.9020}$	$32.87/0.8839$	$34.37/0.9219$	$\mathbf{34.64/0.9221}$	$34.62/0.9237$	$34.54/0.914$	$33.37/0.912$
Kodak24, $\sigma{=}25$	$28.44/0.7558$	$28.58/0.7710$	$\mathbf{30.67/0.8446}$	$30.64/0.8410$	$31.70/0.8837$	$\mathbf{32.33/0.8898}$	$31.93/0.8814$	$31.86/0.871$	$30.99/0.856$

Appendix

Table 8: Color denoising (PSNR (dB)/SSIM). Our model outperforms the other

1

-Lipschitz/contractive models and narrows the gap with unconstrained models. CLIP and DIL results are taken directly from their respective papers due to the unavailability of the required pretrained models.

Dataset, $\sigma$	$1$ -Lipschitz / Contractive Models			Unconstrained Models		Pretrained Models
Dataset, $\sigma$	D-FBS	D-DRS	Ours	DnCNN	Restormer	MWCNN	CRR-NN ( $t$ -step)	CRR-NN (prox.)
BSD68, $\sigma{=}15$	$29.89/0.8347$	$29.79/0.8560$	$\mathbf{30.52/0.8690}$	$\mathbf{31.58/0.8934}$	$31.47/0.8930$	$31.86/0.8947$	$-$	$-$
BSD68, $\sigma{=}25$	$27.31/0.7537$	$27.24/0.7600$	$\mathbf{27.82/0.7883}$	$\mathbf{28.87/0.8254}$	$28.75/0.8233$	$29.41/0.8360$	$28.06/0.7838$	$28.05/0.7825$
BSD68, $\sigma{=}50$	$24.21/0.6290$	$24.18/0.6368$	$\mathbf{24.40/0.6460}$	$\mathbf{25.12/0.6846}$	$24.35/0.6288$	$26.53/0.7366$	$-$	$-$
Set12, $\sigma{=}15$	$30.41/0.8451$	$30.39/0.8535$	$\mathbf{31.43/0.8816}$	$\mathbf{32.82/0.9044}$	$32.69/0.9020$	$33.15/0.9088$	$-$	$-$
Set12, $\sigma{=}25$	$28.23/0.7924$	$28.15/0.8136$	$\mathbf{28.71/0.8219}$	$\mathbf{30.32/0.8609}$	$30.13/0.8574$	$30.79/0.8711$	$28.91/0.8157$	$28.88/0.8100$

Appendix

Table 9: Grayscale denoising (PSNR (dB)/SSIM). Our model delivers competitive denoising performance across all datasets and noise levels while retaining its theoretical contractivity guarantees. CRR-NN results are reported only at

\sigma=25

due to the availability of its publicly released pretrained grayscale model. MWCNN results are obtained using publicly available pretrained models.

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$25.28$	$25.36$	$24.54$	$24.94$	$25.36$	$24.51$	$24.66$	$24.67$	$24.36$	$23.39$	$\mathbf{25.61}$
CBSD68	SSIM	$0.6835$	$0.6863$	$0.6267$	$0.6498$	$0.6670$	$0.5822$	$0.6851$	$0.6857$	$0.6675$	$0.5861$	$\mathbf{0.7046}$
Kodak24	PSNR	$26.04$	$26.12$	$25.32$	$25.73$	$25.96$	$25.05$	$25.94$	$25.94$	$25.72$	$24.41$	$\mathbf{26.35}$
Kodak24	SSIM	$0.6994$	$0.7018$	$0.6575$	$0.6705$	$0.6580$	$0.5588$	$0.7023$	$0.7036$	$0.6946$	$0.6188$	$\mathbf{0.7184}$
Urban100	PSNR	$22.56$	$22.71$	$21.79$	$22.62$	$22.83$	$22.34$	$22.64$	$22.64$	$22.65$	$21.38$	$\mathbf{22.89}$
Urban100	SSIM	$0.6750$	$0.6830$	$0.6203$	$0.6700$	$0.6541$	$0.5822$	$0.6860$	$0.6864$	$0.6917$	$0.6007$	$\mathbf{0.6976}$

Appendix

Table 10: PnP-FBS super-resolution (

\times 2

) with a

9\times 9

Gaussian blur kernel (

\sigma=2

) and noise level

\sigma_{n}=5/255

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$25.17$	$25.25$	$24.42$	$24.82$	$25.30$	$24.38$	$24.92$	$24.93$	$24.68$	$23.61$	$\mathbf{25.51}$
CBSD68	SSIM	$0.6773$	$0.6800$	$0.6203$	$0.6427$	$0.6633$	$0.5731$	$0.6904$	$0.6911$	$0.6728$	$0.5892$	$\mathbf{0.6986}$
Kodak24	PSNR	$25.95$	$26.02$	$25.20$	$25.63$	$25.92$	$24.93$	$26.12$	$26.13$	$26.00$	$24.57$	26.26
Kodak24	SSIM	$0.6945$	$0.6965$	$0.6518$	$0.6652$	$0.6555$	$0.5498$	$0.7129$	$0.7133$	$0.7059$	$0.6243$	$\mathbf{0.7135}$
Urban100	PSNR	$22.42$	$22.57$	$21.64$	$22.48$	$22.74$	$22.21$	$22.55$	$22.54$	$22.69$	$21.26$	22.76
Urban100	SSIM	$0.6666$	$0.6743$	$0.6114$	$0.6614$	$0.6484$	$0.5719$	$0.6798$	$0.6803$	$0.6812$	$0.5957$	0.6896

Appendix

Table 11: PnP-FBS super-resolution (

\times 2

) with a

21\times 21

anisotropic Gaussian blur kernel (

\sigma_{x}=3

\sigma_{y}=1

) and noise level

\sigma_{n}=5/255

Panel of Figure 18 — Figure 18: PnP-FBS super-resolution under Gaussian blur ( $25\times 25$ , $\sigma=1.6$ ). From bicubic input, our contractive denoiser restores clearer structures and achieves the highest PSNR and competitive SSIM.

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$24.10$	$24.14$	$23.63$	$23.78$	$24.21$	$23.81$	$23.54$	$23.55$	$23.23$	$22.70$	24.24
CBSD68	SSIM	$0.6122$	$0.6123$	$0.5778$	$0.5874$	$0.6001$	$0.5468$	$0.6093$	$0.6100$	$0.5911$	$0.5371$	0.6219
Kodak24	PSNR	$24.93$	$24.97$	$24.43$	$24.66$	$24.92$	$24.52$	$24.93$	$24.94$	$24.69$	$23.88$	25.08
Kodak24	SSIM	$0.6404$	$0.6399$	$0.6149$	$0.6187$	$0.6050$	$0.5445$	$0.6478$	$0.6489$	$0.6346$	$0.5857$	0.6491
Urban100	PSNR	$21.86$	$21.92$	$21.48$	$21.66$	$22.15$	$21.93$	$21.35$	$21.36$	$21.19$	$20.28$	$\mathbf{22.20}$
Urban100	SSIM	$0.5618$	$0.5643$	$0.5279$	$0.5437$	$0.5698$	$0.5322$	$0.6047$	$0.6056$	$0.5980$	$0.5260$	0.6134

Appendix

Table 12: PnP-FBS super-resolution (

\times 2

) under defocus blur using a disk kernel with radius

5

, kernel size

11\times 11

, and noise level

\sigma_{n}=5/255

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$24.86$	$24.92$	$24.25$	$24.55$	$25.06$	$24.37$	$23.56$	$23.57$	$23.21$	$22.73$	$\mathbf{25.15}$
CBSD68	SSIM	$0.6581$	$0.6598$	$0.6110$	$0.6291$	$0.6474$	$0.5745$	$0.6098$	$0.6106$	$0.5880$	$0.5355$	$\mathbf{0.6757}$
Kodak24	PSNR	$25.68$	$25.74$	$25.04$	$25.41$	$25.78$	$25.04$	$24.99$	$25.00$	$24.70$	$23.88$	$\mathbf{25.97}$
Kodak24	SSIM	$0.6793$	$0.6805$	$0.6442$	$0.6538$	$0.6436$	$0.5594$	$0.6496$	$0.6508$	$0.6331$	$0.5847$	$\mathbf{0.6955}$
Urban100	PSNR	$22.72$	$22.85$	$22.06$	$22.67$	$23.15$	$22.72$	$21.46$	$21.46$	$21.28$	$20.30$	$\mathbf{23.19}$
Urban100	SSIM	$0.6563$	$0.6632$	$0.6103$	$0.6445$	$0.6377$	$0.5684$	$0.6102$	$0.6111$	$0.6013$	$0.5255$	$\mathbf{0.6696}$

Appendix

Table 13: PnP-FBS super-resolution (

\times 2

) under box blur using a

9\times 9

uniform kernel and noise level

\sigma_{n}=5/255

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRRNN( $t$ -step)	CRRNN(prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$24.25$	$24.31$	$24.10$	$24.50$	$23.95$	$23.41$	$24.13$	$23.95$	$24.08$	$23.50$	$\mathbf{24.52}$
CBSD68	SSIM	$0.6117$	$0.6248$	$0.6046$	$0.6305$	$0.5763$	$0.5213$	$0.6163$	$0.5940$	$0.6189$	$0.5794$	$\mathbf{0.6421}$
Kodak24	PSNR	$24.98$	$25.07$	$24.84$	$\mathbf{25.31}$	$24.52$	$23.95$	$25.00$	$24.77$	$25.11$	$24.38$	$25.25$
Kodak24	SSIM	$0.6244$	$0.6413$	$0.6363$	$0.6545$	$0.5660$	$0.5045$	$0.6349$	$0.6023$	$0.6519$	$0.6101$	$\mathbf{0.6565}$
Urban100	PSNR	$21.62$	$21.68$	$21.30$	$\mathbf{22.02}$	$21.51$	$21.19$	$21.65$	$21.62$	$21.82$	$21.10$	$21.82$
Urban100	SSIM	$0.5994$	$0.6141$	$0.5921$	$0.6299$	$0.5636$	$0.5166$	$0.6082$	$0.5892$	$\mathbf{0.6309}$	$0.5788$	$0.6273$

Appendix

Table 14: PnP-FBS super-resolution (

\times

4) under anisotropic Gaussian blur (

\sigma_{x}=2.5

\sigma_{y}=1.5

\theta=45^{\circ}

, kernel size

21\times 21

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$24.29$	$24.35$	$24.15$	$24.54$	$23.99$	$23.43$	$24.19$	$24.01$	$24.14$	$23.55$	$\mathbf{24.57}$
CBSD68	SSIM	$0.6133$	$0.6264$	$0.6070$	$0.6327$	$0.5768$	$0.5204$	$0.6180$	$0.5956$	$0.6213$	$0.5823$	$\mathbf{0.6439}$
Kodak24	PSNR	$25.00$	$25.09$	$24.90$	$\mathbf{25.33}$	$24.53$	$23.95$	$25.03$	$24.80$	$25.15$	$24.39$	$25.27$
Kodak24	SSIM	$0.6253$	$0.6426$	$0.6394$	$0.6560$	$0.5652$	$0.5025$	$0.6359$	$0.6030$	$0.6537$	$0.6129$	$\mathbf{0.6577}$
Urban100	PSNR	$21.67$	$21.73$	$21.36$	$\mathbf{22.07}$	$21.56$	$21.22$	$21.71$	$21.68$	$21.88$	$21.15$	$21.87$
Urban100	SSIM	$0.6024$	$0.6172$	$0.5954$	$\mathbf{0.6432}$	$0.5655$	$0.5172$	$0.6112$	$0.5919$	$0.6349$	$0.5826$	$0.6304$

Appendix

Table 15: PnP-FBS super-resolution (

\times

4) under Gaussian blur (

\sigma=2.0

, kernel size

11\times 11

Panel of Figure 19 — Figure 19: $2\times$ super-resolution with PnP-FBS under Gaussian blur ( $25\times 25$ , $\sigma=1.6$ ). Our contractive denoiser recovers sharper stripes and cleaner backgrounds, outperforming D-FBS/DRS and avoiding artifacts seen in Restormer.

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$23.22$	$23.29$	$23.08$	$23.22$	$22.91$	$22.68$	$22.58$	$22.40$	$22.51$	$22.11$	$\mathbf{23.38}$
CBSD68	SSIM	$0.5536$	$0.5613$	$0.5484$	$0.5592$	$0.5088$	$0.4808$	$0.5451$	$0.5285$	$0.5497$	$0.5241$	$\mathbf{0.5688}$
Kodak24	PSNR	$24.05$	$24.16$	$23.92$	$24.17$	$23.61$	$23.39$	$23.89$	$23.74$	$23.96$	$23.42$	$\mathbf{24.22}$
Kodak24	SSIM	$0.5829$	$0.5933$	$0.5903$	$0.5964$	$0.5127$	$0.4838$	$0.5864$	$0.5632$	$0.5965$	$0.5731$	$\mathbf{0.5988}$
Urban100	PSNR	$20.38$	$20.44$	$20.15$	$20.45$	$20.30$	$20.19$	$20.19$	$20.24$	$20.26$	$19.73$	$\mathbf{20.49}$
Urban100	SSIM	$0.5243$	$0.5337$	$0.5208$	$\mathbf{0.5409}$	$0.4834$	$0.4607$	$0.5239$	$0.5153$	$0.5385$	$0.4973$	$0.5384$

Appendix

Table 16: PnP-FBS super-resolution (

\times 4

) under defocus blur (disk kernel radius

=6

, kernel size

13\times 13

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$23.12$	$23.19$	$22.98$	$23.10$	$22.83$	$22.61$	$22.46$	$22.28$	$22.36$	$22.04$	$\mathbf{23.28}$
CBSD68	SSIM	$0.5485$	$0.5556$	$0.5431$	$0.5520$	$0.5046$	$0.4776$	$0.5372$	$0.5191$	$0.5401$	$0.5168$	$\mathbf{0.5633}$
Kodak24	PSNR	$23.95$	$24.05$	$23.82$	$24.05$	$23.52$	$23.31$	$23.79$	$23.65$	$23.84$	$23.35$	$\mathbf{24.11}$
Kodak24	SSIM	$0.5788$	$0.5885$	$0.5856$	$0.5907$	$0.5098$	$0.4825$	$0.5816$	$0.5583$	$0.5901$	$0.5675$	$\mathbf{0.5941}$
Urban100	PSNR	$20.29$	$20.35$	$20.04$	$20.33$	$20.23$	$20.13$	$20.10$	$20.15$	$20.14$	$19.64$	$\mathbf{20.41}$
Urban100	SSIM	$0.5191$	$0.5278$	$0.5145$	$0.5330$	$0.4803$	$0.4583$	$0.5173$	$0.5091$	$0.5293$	$0.4886$	$\mathbf{0.5333}$

Appendix

Table 17: PnP-FBS super-resolution (

\times 4

) under an

11\times 11

uniform blur.

Panel of Figure 20 — Figure 20: PnP-FBS deblurring with Levin09 kernel (idx3). Our method achieves the best reconstruction and avoids artifacts seen in DnCNN and Restormer.

Panel of Figure 21 — Figure 21: Sparse-kernel deblurring ( $15\times 15$ , $90\%$ zeros). Our method restores sharper facial details and cleaner reconstructions with fewer artifacts compared to other denoisers.

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$25.79$	$25.84$	$25.04$	$25.36$	$25.75$	$25.20$	$25.89$	$25.88$	$25.16$	$23.81$	$\mathbf{26.03}$
CBSD68	SSIM	$0.7170$	$0.7163$	$0.6576$	$0.6795$	$0.7167$	$0.6486$	$0.7178$	$0.7172$	$0.6714$	$0.5981$	$\mathbf{0.7329}$
Kodak24	PSNR	$26.48$	$26.58$	$25.76$	$26.13$	$26.55$	$25.86$	$26.60$	$26.59$	$26.06$	$24.29$	$\mathbf{26.76}$
Kodak24	SSIM	$0.7313$	$0.7300$	$0.6856$	$0.6976$	$0.7159$	$0.6330$	$0.7338$	$0.7333$	$0.6947$	$0.6230$	$\mathbf{0.7454}$
Urban100	PSNR	$23.28$	$23.41$	$22.47$	$23.20$	$23.67$	$23.26$	$23.25$	$23.17$	$23.13$	$21.25$	$\mathbf{23.69}$
Urban100	SSIM	$0.7244$	$0.7313$	$0.6665$	$0.7105$	$0.7259$	$0.6584$	$0.7154$	$0.7121$	$0.6980$	$0.6065$	$\mathbf{0.7420}$

Appendix

Table 18: PnP-FBS image deblurring under Gaussian blur (

\sigma=2.0

, kernel size

9\times 9

) and noise level

\sigma_{n}=5/255

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$24.59$	$24.66$	$23.98$	$24.17$	$24.88$	$23.88$	$24.69$	$24.69$	$24.07$	$23.08$	$\mathbf{24.91}$
CBSD68	SSIM	$0.6414$	$0.6404$	$0.5977$	$0.6111$	$0.6293$	$0.5330$	$0.6446$	$0.6443$	$0.6060$	$0.5535$	$\mathbf{0.6559}$
Kodak24	PSNR	$25.44$	$25.54$	$24.78$	$25.10$	$\mathbf{25.81}$	$24.67$	$25.60$	$25.59$	$25.08$	$23.96$	$\mathbf{25.81}$
Kodak24	SSIM	$0.6686$	$0.6672$	$0.6344$	$0.6416$	$0.6245$	$0.5152$	$0.6736$	$0.6734$	$0.6403$	$0.5913$	$\mathbf{0.6825}$
Urban100	PSNR	$21.80$	$21.98$	$21.15$	$21.70$	$\mathbf{22.60}$	$22.03$	$21.96$	$21.88$	$21.65$	$20.28$	$22.54$
Urban100	SSIM	$0.6278$	$0.6321$	$0.5838$	$0.6182$	$0.6196$	$0.5446$	$0.6362$	$0.6279$	$0.6136$	$0.5358$	$\mathbf{0.6515}$

Appendix

Table 19: PnP-FBS image deblurring under defocus blur using a disk kernel with radius

5

, kernel size

11\times 11

, and noise level

\sigma_{n}=5/255

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$26.62$	$27.03$	$24.26$	$24.72$	$27.11$	$25.35$	$25.35$	$25.27$	$23.71$	$22.35$	$\mathbf{28.40}$
CBSD68	SSIM	$0.7403$	$0.7438$	$0.6009$	$0.6130$	$0.7179$	$0.7013$	$0.6772$	$0.6724$	$0.5738$	$0.5070$	$\mathbf{0.8063}$
Kodak24	PSNR	$27.45$	$27.91$	$25.24$	$25.82$	$27.45$	$25.24$	$26.75$	$26.67$	$25.29$	$23.50$	$\mathbf{29.21}$
Kodak24	SSIM	$0.7456$	$0.7451$	$0.6430$	$0.6490$	$0.6772$	$0.6692$	$0.7121$	$0.7082$	$0.6334$	$0.5602$	$\mathbf{0.8102}$
Urban100	PSNR	$24.59$	$25.48$	$22.56$	$23.33$	$25.86$	$25.26$	$23.29$	$23.19$	$22.19$	$19.79$	$\mathbf{26.33}$
Urban100	SSIM	$0.7600$	$0.7741$	$0.6390$	$0.7369$	$0.7419$	$0.7397$	$0.7017$	$0.6971$	$0.6309$	$0.4953$	$\mathbf{0.8099}$

Appendix

Table 20: PnP-FBS image deblurring with a

90\%

sparse kernel and noise level

\sigma_{n}=5/255

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$26.14$	$26.44$	$24.47$	$25.05$	$26.41$	$26.33$	$26.70$	$26.63$	$25.16$	$23.78$	$\mathbf{27.50}$
CBSD68	SSIM	$0.7276$	$0.7300$	$0.6251$	$0.6479$	$0.6800$	$0.7161$	$0.7447$	$0.7415$	$0.6513$	$0.5820$	$\mathbf{0.7797}$
Kodak24	PSNR	$27.03$	$27.38$	$25.30$	$26.11$	$26.83$	$27.74$	$27.84$	$27.78$	$26.51$	$24.78$	$\mathbf{28.53}$
Kodak24	SSIM	$0.7412$	$0.7410$	$0.6573$	$0.6779$	$0.6344$	$0.7339$	$0.7619$	$0.7593$	$0.6905$	$0.6166$	$\mathbf{0.7850}$
Urban100	PSNR	$23.87$	$24.39$	$22.09$	$23.66$	$25.02$	$25.50$	$24.72$	$24.64$	$24.09$	$21.67$	$\mathbf{25.63}$
Urban100	SSIM	$0.7414$	$0.7557$	$0.6263$	$0.7126$	$0.6908$	$0.7705$	$0.7635$	$0.7606$	$0.7199$	$0.6035$	$\mathbf{0.8009}$

Appendix

Table 21: PnP-FBS image deblurring with kernel 1 from Levin et al. [ 35 ] and noise level

\sigma_{n}=5/255

Panel of Figure 22 — Figure 22: PnP-FBS deblurring under diagonal motion blur ( $15\times 15$ ). Our method restores clearer details while avoiding artifacts in DnCNN/Restormer and the over-smoothing of Noise2VST and IDF.

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$25.98$	$26.28$	$24.22$	$24.86$	$26.17$	$24.15$	$26.85$	$26.82$	$25.56$	$24.17$	$\mathbf{27.30}$
CBSD68	SSIM	$0.7146$	$0.7181$	$0.6079$	$0.6329$	$0.6691$	$0.5626$	$0.7511$	$0.7502$	$0.6830$	$0.6083$	$\mathbf{0.7683}$
Kodak24	PSNR	$26.90$	$27.27$	$25.08$	$25.94$	$26.59$	$24.61$	$27.69$	$27.67$	$26.55$	$24.81$	$\mathbf{28.33}$
Kodak24	SSIM	$0.7303$	$0.7317$	$0.6447$	$0.6643$	$0.6249$	$0.5188$	$0.7635$	$0.7629$	$0.7078$	$0.6298$	$\mathbf{0.7767}$
Urban100	PSNR	$23.86$	$24.38$	$21.51$	$23.50$	$24.77$	$23.45$	$24.81$	$24.77$	$24.19$	$21.93$	$\mathbf{25.08}$
Urban100	SSIM	$0.7159$	$0.7384$	$0.6097$	$0.6919$	$0.6747$	$0.5948$	$0.7677$	$0.7671$	$0.7314$	$0.6285$	$\mathbf{0.7743}$

Appendix

Table 22: PnP-FBS image deblurring with kernel 3 from Levin et al. [ 35 ] and noise level

\sigma_{n}=5/255

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$25.81$	$25.88$	$24.96$	$25.47$	$25.87$	$24.96$	$25.91$	$25.91$	$25.33$	$23.59$	$\mathbf{26.14}$
CBSD68	SSIM	$0.7246$	$0.7249$	$0.6608$	$0.6853$	$0.7180$	$0.6251$	$0.7264$	$0.7259$	$0.6787$	$0.6000$	$\mathbf{0.7441}$
Kodak24	PSNR	$26.70$	$26.78$	$25.81$	$26.37$	$26.78$	$25.72$	$26.84$	$26.83$	$26.37$	$24.02$	$\mathbf{27.03}$
Kodak24	SSIM	$0.7469$	$0.7467$	$0.6928$	$0.7106$	$0.7190$	$0.6112$	$0.7507$	$0.7503$	$0.7102$	$0.6289$	$\mathbf{0.7657}$
Urban100	PSNR	$23.01$	$23.17$	$22.10$	$23.16$	$23.32$	$22.80$	$23.17$	$23.13$	$23.16$	$20.93$	$\mathbf{23.36}$
Urban100	SSIM	$0.7080$	$0.7115$	$0.6496$	$0.7006$	$0.6945$	$0.6165$	$0.7144$	$0.7102$	$0.7008$	$0.6015$	$\mathbf{0.7280}$

Appendix

Table 23: PnP-FBS image deblurring under anisotropic Gaussian blur (kernel size

21\times 21

\sigma_{x}=3.0

\sigma_{y}=1.0

) and noise level

\sigma_{n}=5/255

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$24.47$	$24.63$	$23.46$	$23.72$	$\mathbf{25.08}$	$23.96$	$24.51$	$24.49$	$23.64$	$23.13$	$24.98$
CBSD68	SSIM	$0.6538$	$0.6570$	$0.5857$	$0.5945$	$\mathbf{0.6885}$	$0.6473$	$0.6551$	$0.6532$	$0.5925$	$0.5652$	$0.6851$
Kodak24	PSNR	$25.31$	$25.51$	$24.36$	$24.79$	$\mathbf{26.10}$	$24.96$	$25.45$	$25.43$	$24.84$	$23.99$	$25.89$
Kodak24	SSIM	$0.6748$	$0.6762$	$0.6250$	$0.6320$	$0.6962$	$0.6640$	$0.6791$	$0.6776$	$0.6342$	$0.5932$	$\mathbf{0.7012}$
Urban100	PSNR	$21.88$	$22.11$	$20.81$	$21.83$	$22.49$	$22.23$	$21.74$	$21.69$	$21.48$	$20.34$	$\mathbf{22.58}$
Urban100	SSIM	$0.6455$	$0.6562$	$0.5878$	$0.6250$	$0.6657$	$0.6325$	$0.6325$	$0.6312$	$0.6067$	$0.5440$	$\mathbf{0.6824}$

Appendix

Table 24: PnP-FBS image deblurring under diagonal motion blur using a

15\times 15

kernel with non-zero values along the main diagonal and noise level

\sigma_{n}=5/255

Dataset	Metric	FBS	DRS	IDF	Noise2VST	DnCNN	Restormer	CRR-NN ( $t$ -step)	CRR-NN (prox.)	MWCNN	CLIP	OURS
CBSD68	PSNR	$24.73$	$24.83$	$24.01$	$24.25$	$24.96$	$23.63$	$24.85$	$24.84$	$24.14$	$23.08$	$\mathbf{25.12}$
CBSD68	SSIM	$0.6478$	$0.6470$	$0.5980$	$0.6123$	$0.6282$	$0.5119$	$0.6512$	$0.6509$	$0.6070$	$0.5527$	$\mathbf{0.6663}$
Kodak24	PSNR	$25.63$	$25.77$	$24.84$	$25.22$	$25.96$	$24.42$	$25.83$	$25.83$	$25.20$	$23.96$	$\mathbf{26.11}$
Kodak24	SSIM	$0.6748$	$0.6736$	$0.6353$	$0.6434$	$0.6244$	$0.4908$	$0.6805$	$0.6802$	$0.6421$	$0.5910$	$\mathbf{0.6930}$
Urban100	PSNR	$22.03$	$22.24$	$21.21$	$21.92$	$22.66$	$22.06$	$22.18$	$22.16$	$21.85$	$20.28$	$\mathbf{22.69}$
Urban100	SSIM	$0.6535$	$0.6645$	$0.5950$	$0.6268$	$0.6240$	$0.5261$	$0.6476$	$0.6471$	$0.6220$	$0.5361$	$\mathbf{0.6672}$

Appendix

Table 25: PnP-FBS image deblurring under box blur using a

9\times 9

uniform kernel and noise level

\sigma_{n}=5/255

Panel of Figure 23 — Figure 23: PnP-FBS deblurring with the Levin09 kernel (idx3). Our contractive denoiser reconstructs sharper structures while avoiding ringing artifacts seen around the edges in DnCNN and Restormer and the over-smoothing of IDF and Noise2VST.

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Trainable Nonexpansive Denoisers for Contractive Image Reconstruction

Arghya Sinha, Aditya Banerjee, Trishit Mukherjee, Kunal N. Chaudhury

Indian Institute of Science, Bengaluru, India.

Image Reconstruction Lipschitz Neural Networks Image Inverse Problems Image Denoising

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Trainable denoisers with Lipschitz control have become central to convergent image reconstruction. However, training neural networks that simultaneously offer strong denoising performance and global Lipschitz guarantees is challenging. Existing approaches enforce Lipschitz control only empirically, providing no guarantees beyond the training data. In this work, we show that by exploiting the action of permutations on the image lattice, we can constrain a neural architecture that is globally nonexpansive (Lipschitz bound

\leqslant 1

). We integrate the proposed denoiser with forward imaging operators to develop a reconstruction mechanism that is provably contractive and therefore globally convergent. Experiments on standard inverse problems, such as superresolution and deblurring, demonstrate that our reconstruction performance is competitive with softly constrained baselines while providing Lipschitz guarantees.

Abstract

Image Restoration

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