Prerequisites & Notation

This chapter assumes familiarity with the following topics.

Neural network architectures (CNNs, U-Nets) and training via backpropagation (Chapter 20) (Review ch20)
Self-check: Can you describe the U-Net architecture and explain the role of skip connections?
Denoising as a building block for inverse problems; Tweedie's formula (Chapter 21) (Review ch21)
Self-check: Can you write the Gaussian denoising MMSE estimator as a conditional expectation?
Generative models for imaging: score-based diffusion and DDPM (Chapter 22) (Review ch22)
Self-check: Can you explain why diffusion models require paired or at least distribution-level training data?
Linear inverse problem formulation $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ (Chapters 12--14) (Review ch12)
Self-check: Can you describe the forward model for RF imaging and the role of the sensing matrix $\mathbf{A}$ ?

Symbols introduced or heavily used in this chapter.

Symbol	Meaning	Introduced
$f_\theta(\mathbf{z})$	Generator network with parameters $\theta$ and fixed random input $\mathbf{z}$ (DIP)	s01
$\\mathbf{z}$	Fixed random input to the DIP network (not optimised)	s01
$\\text{SURE}(f_\\theta)$	Stein's Unbiased Risk Estimate for denoiser $f_\theta$	s03
$\\operatorname{div}(f_\\theta)$	Divergence of the denoiser: $\sum_i \partial [f_\theta(\mathbf{y})]_i / \partial y_i$	s03
$T_g$	Group transformation operator (rotation, shift, flip) for $g \in \mathcal{G}$	s04
$\\mathcal{G}$	Group of transformations for equivariant imaging	s04
$\\mathcal{L}_{\\text{EI}}$	Equivariance loss	s04
$\\mathcal{L}_{\\text{DC}}$	Data-consistency loss	s04