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Learning Pseudo-Contractive Denoisers for Inverse Problems
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Proposes pseudo-contractive denoisers, Ishikawa-process plug-and-play solvers (PnPI-GD, PnPI-HQS, PnPI-FBS) with convergence guarantees, and a holomorphic-functional-calculus training loss to enforce the condition on DRUNet.
Problem
Plug-and-play methods need denoiser assumptions (non-expansiveness, averagedness, contractiveness) for convergence, but these constraints degrade denoising and recovery performance.
Approach
Replace strict Lipschitz-type denoiser constraints with the weaker pseudo-contractiveness (spectrum of the Jacobian's symmetric part in the half-plane real(z)<=1), pair it with Ishikawa iterations for strong convergence, and enforce it in training via the holomorphic map f(z)=z/(z-2) and a spectral-norm penalty.
Key results
- PC-DRUNet reaches 33.01 dB PSNR at sigma=15 on Set12, close to unconstrained DRUNet (33.08) and above NE-DRUNet (31.68) and Prox-DRUNet (31.71).
- PnPI-HQS achieves the best average deblurring PSNR of 27.69 dB (sigma=12.75) and 26.57 dB (sigma=17.85) over Levin's 8 kernels, outperforming DPIR's 27.64 and 26.48.
- PnPI-HQS gives the best super-resolution PSNR for s=2 at sigma=0 (31.87 dB) and the best Poisson denoising SSIM at peak=20 (0.8212).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use pseudo-contractive training instead of standard DRUNet training (ablation) | DRUNet · Set12 Gaussian denoising, σ=15 | PSNR: 33.08 -> 33.01 | Hurt |
| Use pseudo-contractive training instead of standard DRUNet training (ablation) | DRUNet · Set12 Gaussian denoising, σ=25 | PSNR: 30.80 -> 30.69 | Hurt |
| Use pseudo-contractive training instead of standard DRUNet training (ablation) | DRUNet · Set12 Gaussian denoising, σ=40 | PSNR: 28.76 -> 28.66 | Hurt |
| Use 1/2-strictly pseudo-contractive training instead of non-expansive training (ablation) | DRUNet · Set12 Gaussian denoising, σ=15 | PSNR: 31.68 -> 32.90 | Helped |
| Use pseudo-contractive training instead of 1/2-strictly pseudo-contractive training (ablation) | DRUNet · Set12 Gaussian denoising, σ=40 | PSNR: 28.44 -> 28.66 | Helped |
| Train PC-DRUNet with r=10^{-3} rather than r=10^{-4} (ablation) | PC-DRUNet · Set12, σ=40 | ||1/2 I + 1/2 J||_*: 1.4646 -> 0.9977 | Helped |
| Train SPC-DRUNet with r=10^{-3} rather than r=10^{-4} (ablation) | SPC-DRUNet · Set12, σ=40 | ||(S-2I)^{-1}S||_*: 1.0826 -> 0.9981 | Helped |
| Use PnPI-HQS instead of DPIR | Set12 deblurring with Levin kernels · σ=12.75 | Average PSNR: 27.64 -> 27.69 | Helped |
| Use PnPI-HQS instead of DPIR | Set12 deblurring with Levin kernels · σ=17.85 | Average PSNR: 26.48 -> 26.57 | Helped |
| Use PnPI-HQS instead of DPIR | Set12 super-resolution, s=2 · σ=0 | PSNR: 30.99 -> 31.87 | Helped |
| Use PnPI-HQS instead of DPIR | Set12 super-resolution, s=4 · σ=0 | PSNR: 26.56 -> 26.29 | Hurt |
| Use PnPI-FBS instead of DPIR | Set12 Poisson denoising · peak=20 | PSNR: 27.87 -> 28.16 | Helped |
Surprising findings
- DPIR, which has no convergence guarantee, is competitive with or better than the proposed convergent methods on several tasks (e.g., s=4 super-resolution and deblurring at sigma=17.85).
- PnPI-GD and PnPI-FBS are applied to Poisson denoising although the gradient is not cocoercive and convergence is not guaranteed; the authors report empirical convergence.
Limitations
- Convergence for Poisson denoising is not guaranteed because the gradient of the data term is not cocoercive.
- Pseudo-contractiveness is enforced only softly via a penalty with trade-off parameter r, so the condition holds approximately (norms slightly above 1 for smaller r).
- Experiments are limited to Set12 and DRUNet-based denoisers; the training uses a small dataset (800 DIV2K images).
Experimental setup
Datasets: DIV2K Set12 Levin camera shake kernels
Benchmarks: Set12
Metrics: PSNR SSIM spectral norm of Jacobian-based operators
Baselines: MMO-PnP-FBS NE-PnP-PGD Prox-PnP-DRS DPIR FFDNet DnCNN DRUNet NE-DRUNet Prox-DRUNet
Builds on: PnP-ADMM RED DRUNet Ishikawa iteration Ryu et al. contractive denoiser (RealSN)
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