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Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Imaging Inverse Problems
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Trains a cocoercive conservative (CoCo) Gaussian denoiser via spectral and Hamiltonian regularization, proves it is a proximal operator of a weakly convex prior, and uses it in PnP-ADMM/PEGD for Poisson inverse problems with global convergence guarantees.
Problem
Plug-and-play restoration for Poisson inverse problems, where the fidelity is not smooth and existing convergent PnP denoisers are either restricted by non-expansiveness (hurting denoising) or are not proximal operators.
Approach
Relaxes non-expansiveness to gamma-cocoerciveness, enforces conservativeness via a Hamiltonian (anti-symmetric Jacobian) penalty derived from the generalized Helmholtz decomposition, and shows the resulting denoiser (with averaging parameter t) is a proximal operator of a weakly convex function, giving convergent PnP-ADMM/PEGD for convex Poisson fidelity.
Key results
- On photon-limited deconvolution (CBSD68, Levin kernels, p=50), CoCo-ADMM reaches 26.00 dB PSNR vs 25.38 dB for DPIR and 25.10 dB for RMMO-DRS.
- On low-dose CT (Mayo, p=100), CoCo-ADMM achieves 36.68 dB vs 35.98 dB for WNet.
- The 0.25-CoCo-DRUNet denoiser gives 34.00 dB at sigma=15 on CBSD68, close to DRUNet's 34.14 dB, while being cocoercive.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use 0.25-CoCo-DRUNet instead of DRUNet | Gaussian denoising · CBSD68, σ=15 | PSNR: 34.14 -> 34.00 | Hurt |
| Use 0.25-CoCo-DRUNet instead of DRUNet | Gaussian denoising · CBSD68, σ=25 | PSNR: 31.54 -> 31.38 | Hurt |
| Use 0.25-CoCo-DRUNet instead of DRUNet | Gaussian denoising · CBSD68, σ=40 | PSNR: 29.33 -> 29.16 | Hurt |
| Use γ=0.25 instead of γ=0.50 | CoCo-DRUNet Gaussian denoiser · CBSD68, σ=15 | PSNR: 33.38 -> 34.00 | Helped |
| Use γ=0.25 instead of γ=0.50 | CoCo-DRUNet Gaussian denoiser · CBSD68, σ=25 | PSNR: 30.65 -> 31.38 | Helped |
| Use γ=0.25 instead of γ=0.50 | CoCo-DRUNet Gaussian denoiser · CBSD68, σ=40 | PSNR: 28.25 -> 29.16 | Helped |
| Use CoCo-ADMM instead of DPIR | Poisson deconvolution · CBSD68, p=100 | PSNR: 26.51 -> 26.89 | Helped |
| Use CoCo-ADMM instead of DPIR | Poisson deconvolution · CBSD68, p=100 | SSIM: 0.7419 -> 0.7358 | Hurt |
| Use CoCo-ADMM instead of PnPI-HQS | Poisson deconvolution · CBSD68, p=50 | PSNR: 25.61 -> 26.00 | Helped |
| Use CoCo-ADMM instead of PnPI-HQS | Poisson deconvolution · CBSD68, p=50 | SSIM: 0.6956 -> 0.7026 | Helped |
| Use CoCo-ADMM instead of WNet | Low-dose CT reconstruction · Mayo, p=500 | PSNR: 37.12 -> 37.63 | Helped |
| Use CoCo-ADMM instead of WNet | Low-dose CT reconstruction · Mayo, p=500 | SSIM: 0.9266 -> 0.9403 | Helped |
Surprising findings
- The CoCo denoiser is slightly weaker than unregularized DRUNet in Gaussian denoising, so the theoretical guarantees cost a little denoising quality.
Limitations
- Training regularizers only encourage cocoerciveness and conservativeness; the trained denoiser can still violate them on some images.
- Slightly worse Gaussian denoising PSNR than the non-regularized DRUNet.
- Global convergence requires averaging parameter t below a threshold (e.g. t<0.3333 for gamma=0.25).
Experimental setup
Datasets: DIV2K CBSD68 Set3c Mayo
Benchmarks: CBSD68 Set3c Mayo
Metrics: PSNR SSIM spectral norm of symmetry error cocoercivity spectral norm
Baselines: DPIR RMMO-DRS Prox-DRS DPS DiffPIR SNORE PnPI-HQS B-RED FBP PWLS-TGV PWLS-CSCGR UNet
Builds on: DRUNet gradient step denoiser Prox-DRUNet spectral regularization for Lipschitz denoisers PnP-ADMM
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