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Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Inverse Problems

Deliang Wei, Peng Chen, Haobo Xu, Jiale Yao, Fang Li, Tieyong Zeng

TL;DR

Poisson inverse problems with Poisson noise pose convergence challenges for Plug-and-Play imaging using deep denoisers. The authors introduce cocoercive conservative (CoCo) denoisers learned via a Helmholtz-based training that enforces Hamiltonian regularization and spectral regularization, yielding a denoiser that is both $\gamma$-cocoercive and conservative. They prove that CoCo denoisers are proximal operators of a weakly convex implicit prior, and they establish global convergence of PnP methods to stationary points of the associated restoration model. Empirical results on photon-limited deconvolution, single-photon imaging, and low-dose CT demonstrate competitive visual quality and quantitative gains over closely related convergent methods, validating both the theory and practical performance.

Abstract

Plug-and-play (PnP) methods with deep denoisers have shown impressive results in imaging problems. They typically require strong convexity or smoothness of the fidelity term and a (residual) non-expansive denoiser for convergence. These assumptions, however, are violated in Poisson inverse problems, and non-expansiveness can hinder denoising performance. To address these challenges, we propose a cocoercive conservative (CoCo) denoiser, which may be (residual) expansive, leading to improved denoising. By leveraging the generalized Helmholtz decomposition, we introduce a novel training strategy that combines Hamiltonian regularization to promote conservativeness and spectral regularization to ensure cocoerciveness. We prove that CoCo denoiser is a proximal operator of a weakly convex function, enabling a restoration model with an implicit weakly convex prior. The global convergence of PnP methods to a stationary point of this restoration model is established. Extensive experimental results demonstrate that our approach outperforms closely related methods in both visual quality and quantitative metrics.

Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Inverse Problems

TL;DR

Poisson inverse problems with Poisson noise pose convergence challenges for Plug-and-Play imaging using deep denoisers. The authors introduce cocoercive conservative (CoCo) denoisers learned via a Helmholtz-based training that enforces Hamiltonian regularization and spectral regularization, yielding a denoiser that is both -cocoercive and conservative. They prove that CoCo denoisers are proximal operators of a weakly convex implicit prior, and they establish global convergence of PnP methods to stationary points of the associated restoration model. Empirical results on photon-limited deconvolution, single-photon imaging, and low-dose CT demonstrate competitive visual quality and quantitative gains over closely related convergent methods, validating both the theory and practical performance.

Abstract

Plug-and-play (PnP) methods with deep denoisers have shown impressive results in imaging problems. They typically require strong convexity or smoothness of the fidelity term and a (residual) non-expansive denoiser for convergence. These assumptions, however, are violated in Poisson inverse problems, and non-expansiveness can hinder denoising performance. To address these challenges, we propose a cocoercive conservative (CoCo) denoiser, which may be (residual) expansive, leading to improved denoising. By leveraging the generalized Helmholtz decomposition, we introduce a novel training strategy that combines Hamiltonian regularization to promote conservativeness and spectral regularization to ensure cocoerciveness. We prove that CoCo denoiser is a proximal operator of a weakly convex function, enabling a restoration model with an implicit weakly convex prior. The global convergence of PnP methods to a stationary point of this restoration model is established. Extensive experimental results demonstrate that our approach outperforms closely related methods in both visual quality and quantitative metrics.
Paper Structure (28 sections, 10 theorems, 98 equations, 10 figures, 10 tables, 1 algorithm)

This paper contains 28 sections, 10 theorems, 98 equations, 10 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. CoCo denoisers
  3. Cocoercive denoisers
  4. Conservative denoisers
  5. Characterization on CoCo denoisers
  6. Training strategy
  7. PnP methods with CoCo denoisers
  8. Experiments
  9. Conclusion and Limitation
  10. Technical Appendices and Supplementary Material
  11. Proof of Lemma \ref{['lemma 1']}
  12. Characterizations on the denoiser
  13. More general characterization on the CoCo denoiser
  14. Proof of Theorem \ref{['thm 1']}
  15. Calculation for $\|(2\gamma\operatorname{J}-\operatorname{I})\|_*$ and $\|\operatorname{J}-\operatorname{J}^\mathrm{T}\|_*$
  16. ...and 13 more sections

Key Result

Lemma 2.1

Let $\operatorname{D}:V\rightarrow V$, and $\operatorname{J}=\nabla \operatorname{D}$ be its Fréchet differential. Then $\operatorname{D}$ is $\gamma$-cocoercive ($\gamma\in[0,\infty)$), if and only if $\|2\gamma\operatorname{J}(x)-\operatorname{I}\|_*\le1$ for any $x\in V$.

Figures (10)

  • Figure 1: Left: Spectrum distributions of the Fréchet differential (Jacobian matrix) on the complex plane under different assumptions. (a) Firmly non-expansiveness, $\operatorname{Sp}(\operatorname{J})\subset\{z\in\mathbb{C}: |2z-1|\le 1\}$; (b) Non-expansiveness, $\operatorname{Sp}(\operatorname{J})\subset\{z\in\mathbb{C}: |z|\le 1\}$; (c) Residual non-expansiveness, $\operatorname{Sp}(\operatorname{J})\subset\{z\in\mathbb{C}:|z-1|\le1 \}$; (d) $\frac{1}{2}$-strictly pseudo-contractiveness, $\operatorname{Sp}(\operatorname{J})\subset\{z\in\mathbb{C}:|z+1|\le 2\}$; (e) $0.25$-cocoerciveness, $\operatorname{Sp}(\operatorname{J})\subset\{z\in\mathbb{C}: |0.5z-1|\le1\}$; (f) Conservativeness, $\operatorname{Sp}(\operatorname{J})\subset\mathbb{R}$. In general, a larger region means less restrictive assumption. The spectrum of $\gamma$-CoCo denoisers ($\gamma=0.25$) lies inside the interval $(\text{f})\cap(\text{e})=[0,4]$. Spectrum outside $\mathbb{R}$ corresponds to the Hamiltonian part of the denoiser, and does not contribute to the denoising performance. Right: A two-dimensional illustration of the Helmholtz decomposition of a denoiser $\operatorname{D}$. $x$ denotes the clean image point, $\xi$ denotes the Gaussian noise, and $x+\xi$ denotes the noisy image. The arrow "$\rightarrow$" represents the denoising direction. (i) Denoising field $\operatorname{D}$. (ii) Conservative field $\operatorname{D}_c$. (iii) Hamiltonian field $\operatorname{D}_h$.
  • Figure 2: Deconvolution results by different methods on the image 'Butterfly' from Set3c with kernel $2$ and $p=50$ Poisson noises. (a) Blur image. (b) DPIR, PSNR=$25.00$dB. (c) RMMO-DRS, PSNR=$21.86$dB. (d) Prox-DRS, PSNR=$22.57$dB. (e) DPS, PSNR=$21.47$dB. (f) DiffPIR, PSNR=$22.66$dB. (g) SNORE, PSNR=$25.14$dB. (h) PnPI-HQS, PSNR=$23.56$dB. (i) B-RED, PSNR=$23.34$dB. (j) CoCo-ADMM, PSNR=$25.25$dB. (k) CoCo-PEGD, PSNR=$25.17$dB. (l) Clean image. (m) PSNR curves. (n) Relative error curves. $x$-axis denotes the iteration number.
  • Figure 3: Single photon imaging results in a real-world low light setting by different methods.
  • Figure 4: Eight blur kernels from levin2009understanding.
  • Figure 5: CT test images.
  • ...and 5 more figures

Theorems & Definitions (20)

  • Lemma 2.1: Proof in Appendix \ref{['Appendix lemma1']}
  • Theorem 2.2: Proof in Appendix \ref{['Appendix thm1']}
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 4.1: Proof in Appendix \ref{['appendix main thm']}
  • Remark 4.2
  • Remark 4.3
  • Theorem 4.4: Proof in Appendix \ref{['Appendix thm3']}
  • Proposition A.1: Proposition 3.1 by bauschke2010baillon
  • ...and 10 more