Prerequisites & Notation

This chapter assumes familiarity with the following topics.

Scattering physics and the Born approximation forward model (Chapters 5-6) (Review ch06)
Self-check: Can you write the Born-approximation forward model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ and explain each term?
Neural network fundamentals: MLPs, backpropagation, gradient-based optimisation (Chapter 20) (Review ch20)
Self-check: Can you explain how automatic differentiation computes gradients through a multi-layer network?
Differentiable rendering and physics-informed neural networks (Chapter 23) (Review ch23)
Self-check: Can you describe how a differentiable forward model enables end-to-end training from measurements?

Symbols introduced or emphasised in this chapter.

Symbol	Meaning	Introduced
$\sigma(\mathbf{x})$	Volume density at position $\mathbf{x} \in \mathbb{R}^3$ (opacity per unit length)	s01
$\mathbf{c}(\mathbf{x}, \mathbf{d})$	View-dependent colour (RGB) at position $\mathbf{x}$ , direction $\mathbf{d}$	s01
$T(t)$	Transmittance: accumulated survival probability along a ray up to distance $t$	s01
$\gamma(\mathbf{x})$	Positional encoding mapping $\mathbb{R}^3 \to \mathbb{R}^{6L}$	s01
$\rho(\mathbf{x}, \mathbf{d}, f)$	Complex-valued RF reflectivity (replaces colour in RF rendering)	s02
$\alpha(\mathbf{x}, f)$	Frequency-dependent attenuation coefficient (Np/m)	s02
$F_\theta$	Neural radiance field parameterised by weights $\theta$	s01
$\hat{S}(\mathbf{r}, f)$	Rendered complex received signal along ray $\mathbf{r}$ at frequency $f$	s05