Prerequisites & Notation

Prerequisites for This Chapter

This chapter connects the optical computer-vision pipeline — multi-view geometry, differentiable rendering, and physics-informed networks — to the RF imaging framework developed in earlier chapters. We show how the same analysis-through-synthesis philosophy that powers NeRF and 3DGS can be adapted to electromagnetic wave propagation, and how multi-modal fusion combines RF with camera and LiDAR sensing.

Electromagnetic scattering and the Born approximation(Review ch06)
Self-check: Can you write the volume integral equation for a scattered field under the Born approximation?
MIMO radar and virtual aperture(Review ch11)
Self-check: Can you explain how $N_t + N_r$ antennas create $N_t N_r$ virtual array elements?
3D scene representations (NeRF, SDF, 3DGS)(Review ch24)
Self-check: Can you describe how NeRF represents a scene as a neural radiance field and renders it via volume rendering?
Differentiable rendering and inverse rendering (basics)(Review ch25)
Self-check: Can you state the rendering equation and explain what makes a renderer differentiable?

Notation for This Chapter

Symbols introduced or heavily used in this chapter. See also the global notation table in the front matter.

Symbol	Meaning	Introduced
$\mathbf{F}$	Fundamental matrix (epipolar geometry)	s01
$\mathbf{E}$	Essential matrix ( $\mathbf{E} = [\mathbf{t}]_\times \mathbf{R}$ )	s01
$\mathbf{K}$	Camera intrinsic matrix	s01
$L_o, L_i, L_e$	Outgoing, incoming, emitted radiance	s02
$f_r$	Bidirectional reflectance distribution function (BRDF)	s02
$\chi(\mathbf{r})$	Contrast function ( $\epsilon_r(\mathbf{r}) - 1$ )	s03
$G(\mathbf{r}, \mathbf{r}')$	Free-space Green's function	s03
$\mathcal{F}[u]$	PDE residual operator (PINN loss)	s05
$\mathcal{K}_\theta$	Fourier integral kernel operator (FNO)	s05

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