Plug-and-Play Priors

In a splitting method such as ADMM, the regulariser only enters through its Proximal Operator:

$$z^{k+1}={\mathrm{prox}}_{\beta \mathcal{R}/\rho }(x^{k+1}+u^k)=\arg \underset{z}{\min }\beta \mathcal{R}(z)+\frac{\rho }{2}\Vert z-(x^{k+1}+u^k){\Vert }^2.$$

This is a MAP denoising problem: remove Gaussian noise of variance $\beta /\rho$ under the prior $e^{-\mathcal{R}}$. Plug-and-Play (PnP) replaces this step with an arbitrary denoiser $D$ with noise level $\sqrt{\beta /\rho }$, such as BM3D or a trained CNN like DRUNet:

$$z^{k+1}=D_{\sqrt{\beta /\rho }}(x^{k+1}+u^k).$$

The prior is never written down; it is defined implicitly by the denoiser.

Properties.

• Modular: the physics ($x$-step, with an EIT data prox) and the prior ($z$-step) are completely decoupled. One trained denoiser works for any forward operator (operator-agnostic; see Learned Regularization).
• Convergence: a general denoiser is not the prox of any function, so convergence needs assumptions. Examples are non-expansive or averaged denoisers, or MMSE denoisers, for which nonconvex PnP-ADMM convergence has been shown (Park et al. 2023). Without such conditions iterates can oscillate. Decreasing the noise level of the denoiser over the iterations helps in practice.
• Variants exist for half-quadratic splitting (DPIR; Zhang et al. 2021) and proximal gradient methods.

The diffusion-model analogue is DiffPIR. A related explicit construction is Regularization by Denoising.

In ModularEIT.jl: ProximalMap, ADMM.

References

1. S. V. Venkatakrishnan, C. A. Bouman, B. Wohlberg (2013). Plug-and-Play priors for model based reconstruction. IEEE GlobalSIP 2013, 945–948. doi:10.1109/GlobalSIP.2013.6737048
2. K. Zhang, Y. Li, W. Zuo, L. Zhang, L. Van Gool, R. Timofte (2021). Plug-and-Play Image Restoration with Deep Denoiser Prior. IEEE TPAMI 44(10). arXiv:2008.13751
3. C. Park, S. Shoushtari, W. Gan, U. S. Kamilov (2023). Convergence of Nonconvex PnP-ADMM with MMSE Denoisers. IEEE CAMSAP 2023, 511–515. doi:10.1109/CAMSAP58249.2023.10403463
4. E. K. Ryu, J. Liu, S. Wang, X. Chen, Z. Wang, W. Yin (2019). Plug-and-Play Methods Provably Converge with Properly Trained Denoisers. ICML 2019. arXiv:1905.05406