Prerequisites & Notation

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

Neural scene representations: NeRF, signed distance functions, volume rendering, and the rendering equation (Chapter 24) (Review ch24)
Self-check: Can you write the NeRF volume rendering integral and explain what the MLP learns?
Differentiable rendering and the analysis-through-synthesis pipeline for inverse problems (Chapter 25) (Review ch25)
Self-check: Can you explain how gradients flow through a differentiable renderer to update scene parameters?
The unified forward model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ and Kronecker structure of $\mathbf{A}$ (Chapter 7) (Review ch07)
Self-check: Can you describe the measurement model for a MIMO-OFDM sensing system and identify the Kronecker factors?
Scene representation and scattering models: voxel grids, reflectivity fields, and the Born approximation (Chapter 8) (Review ch08)
Self-check: Can you explain the difference between specular and diffuse scattering in the context of RF imaging?

Symbols introduced in this chapter. See also the NGlobal Notation Table master table in the front matter.

Symbol	Meaning	Introduced
$\mathcal{G} = \{(\boldsymbol{\mu}_k, \boldsymbol{\Sigma}_k, \alpha_k, \mathbf{f}_k)\}_{k=1}^N$	Set of $N$ 3D Gaussians with centres, covariances, opacities, and feature vectors	s01
$\boldsymbol{\mu}_k \in \mathbb{R}^3$	Centre (mean) of the $k$ -th Gaussian	s01
$\boldsymbol{\Sigma}_k \in \mathbb{R}^{3 \times 3}$	Covariance matrix of the $k$ -th Gaussian	s01
$\alpha_k \in [0,1]$	Opacity of the $k$ -th Gaussian	s01
$\mathbf{f}_k$	Feature vector (colour or RF attribute) of the $k$ -th Gaussian	s01
$G_k(\mathbf{u})$	2D projected Gaussian evaluated at pixel $\mathbf{u}$	s01
$P_{\text{dB}}(\mathbf{u})$	Rendered received power in dB at pixel/location $\mathbf{u}$	s02
$\mathcal{T}(\boldsymbol{\mu}_k, \boldsymbol{\Sigma}_{k})$	Tensorial RF field decomposition of a Gaussian	s03
$Y_\ell^m(\hat{\mathbf{d}})$	Spherical harmonic of degree $\ell$ and order $m$ in direction $\hat{\mathbf{d}}$	s03
$\hat{P}_k(\hat{\mathbf{d}})$	Directional power pattern of the $k$ -th Gaussian scatterer	s03
$\sigma_{\text{RCS},k}$	Radar cross-section attribute of the $k$ -th Gaussian in RadarSplat	s04