Prerequisites & Notation

This chapter assumes familiarity with the following topics. If any item feels unfamiliar, revisit the linked material first.

Proximal operators and ADMM (Chapter 4) (Review ch04)
Self-check: Can you state $\operatorname{prox}_f(\mathbf{v}) = \arg\min_\mathbf{x} \tfrac{1}{2}\|\mathbf{x} - \mathbf{v}\|^2 + f(\mathbf{x})$ and compute it for $f = \lambda\|\cdot\|_1$ ?
OAMP/VAMP and message-passing algorithms (Chapter 17) (Review ch17)
Self-check: Can you explain the denoiser role in OAMP and how it connects to the MMSE denoiser under a separable prior?
Convolutional neural networks for imaging (Chapter 20) (Review ch20)
Self-check: Can you describe DnCNN's residual architecture and why the noise residual is easier to learn than the clean image?

Symbols introduced or specialised in this chapter. See the global notation table for the full library conventions.

Symbol	Meaning	Introduced
$\mathcal{D}_\sigma$	Denoiser operating at noise standard deviation $\sigma$	s01
$\operatorname{prox}_f$	Proximal operator of function $f$	s01
$L(\mathcal{D})$	Lipschitz constant of denoiser $\mathcal{D}$	s03
$R_{\text{RED}}(\mathbf{x})$	RED regulariser: $\tfrac{1}{2}\mathbf{x}^T(\mathbf{x} - \mathcal{D}_\sigma(\mathbf{x}))$	s04
$\mathbf{J}_\mathcal{D}(\mathbf{x})$	Jacobian of the denoiser evaluated at $\mathbf{x}$	s04
$\lambda$	Regularisation parameter	s01
$\mathbf{A}$	Sensing / measurement matrix	s01
$\mathbf{c}$	Discretised reflectivity vector	s01
$\mathbf{w}$	Additive noise vector	s01