Chapter Summary

Key Points

1.
PnP replaces the proximal step in ADMM or PGD with any off-the-shelf denoiser, exploiting the equivalence between proximal operators and MAP Gaussian denoising. This decouples the algorithm from the explicit prior and allows state-of-the-art denoisers to be plugged in without modification.
2.
PnP-ADMM and PnP-PGD are the two main PnP variants. PnP-ADMM is more efficient when the data-fidelity subproblem has a closed-form or FFT-based solver; PnP-PGD is simpler to implement.
3.
Deep denoisers (DnCNN, DRUNet, SwinIR) serve as implicit image priors that capture textures and edges beyond the reach of handcrafted priors. DRUNet with noise-level conditioning is the recommended default for PnP, enabling adaptive denoising via a decreasing noise schedule.
4.
Convergence requires non-expansiveness ( $L \leq 1$ ) for PnP-PGD, or strong monotonicity of $(\mathbf{I} - \mathcal{D}_\sigma)$ for PnP-ADMM. Gradient-step denoisers and ICNN denoisers provide stronger guarantees at the cost of reduced expressivity.
5.
RED defines the explicit regulariser $R_\text{RED}(\mathbf{x}) = \tfrac{1}{2}\mathbf{x}^T(\mathbf{x} - \mathcal{D}_\sigma(\mathbf{x}))$ with gradient $\nabla R = \mathbf{x} - \mathcal{D}_\sigma(\mathbf{x})$ under Jacobian symmetry. RED provides an explicit objective but the Jacobian symmetry assumption is rarely exact for deep denoisers.
6.
For RF imaging, PnP and RED offer modular reconstruction that reuses the same denoiser across sensing geometries. Zero-shot DRUNet improves over LASSO for structured scenes; fine-tuning on simulated RF data closes an additional 2–3 dB gap over OAMP.

Looking Ahead

Chapter 22 extends learned priors to score-based diffusion models, where the MMSE denoiser from this chapter becomes the engine of a reverse diffusion process. Diffusion posterior sampling replaces the fixed denoiser in PnP with a sequence of score function evaluations, enabling posterior sampling rather than just point estimation.

PnP and RED for RF Imaging Exercises