Prerequisites & Notation

Before You Begin

This chapter builds on 3D representations from Chapter 24 and the matched-filter imaging framework from Chapter 13. If any item feels unfamiliar, revisit the linked material first.

Signed distance functions, occupancy networks, and neural implicit representations (Chapter 24, Sections 24.2--24.3) (Review ch24)
Self-check: Can you write the analytical SDF for a sphere and verify the Eikonal equation?
Matched-filter imaging and backpropagation: the image $\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^H \mathbf{D}^{-1} \mathbf{y}$ (Chapter 13) (Review ch13)
Self-check: Can you explain how the matched-filter image relates to the sensing matrix?
The unified forward model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ and its Kronecker structure (Chapter 7) (Review ch07)
Self-check: Can you describe the sensing matrix in terms of steering vectors and beamforming gains?
Neural network training with backpropagation: loss functions, gradient computation, and SGD (Chapter 20) (Review ch20)
Self-check: Can you train an MLP on a regression task and compute gradients via autodiff?

Notation for This Chapter

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

Symbol	Meaning	Introduced
$f(\mathbf{p})$	Signed distance function (SDF) at point $\mathbf{p}$	s01
$f_\\theta(\\mathbf{p})$	Neural SDF parameterised by $\theta$	s01
$o_\\theta(\\mathbf{p})$	Occupancy network output at point $\mathbf{p}$	s03
$\\mathcal{L}_{\\text{eik}}$	Eikonal regularization loss	s04
$\\Gamma(\\mathbf{p})$	Surface reflectivity function	s02
$P_{\\text{MF}}(\\mathbf{p})$	Matched-filter power image at point $\mathbf{p}$	s02
$\\hat{P}_{\\text{MF}}(\\mathbf{p}; \\theta)$	Rendered MF power from neural SDF model	s02
$\\epsilon_r(\\mathbf{p})$	Relative permittivity at point $\mathbf{p}$	s04

← Ch 24 Signed Distance Functions for RF