Prerequisites & Notation

Prerequisites for This Chapter

This chapter develops diffraction tomography -- the family of imaging methods that reconstruct a scene by inverting the Fourier relationship between scattered fields and the object spectrum. We begin with computed tomography (CT) as the pedagogical template, then generalize to RF diffraction tomography under the Born approximation, extend to multi-frequency configurations, and finally treat near-field corrections for extremely large apertures (XL-MIMO).

Wave Propagation and Scattering(Review ch06)
Self-check: Can you write the Born approximation integral for the scattered field?
The Sensing Operator(Review ch08)
Self-check: Can you describe how Tx/Rx positions map measurements to k-space?
Sparse Recovery Algorithms(Review ch13)
Self-check: Can you explain FISTA and ADMM for regularized inversion?

Notation and Conventions

Key symbols used throughout this chapter. All formulations assume 2D imaging geometry (extension to 3D is straightforward) and time-harmonic fields with the $e^{-j\omega t}$ convention.

Symbol	Meaning	Introduced
$\chi(\mathbf{r})$	Object contrast function (proportional to dielectric contrast)
$\tilde{\chi}(\mathbf{K})$	Spatial Fourier transform of the contrast function
$k_0 = 2\pi / \lambda$	Free-space wavenumber at the operating frequency
$\hat{\mathbf{k}}_i, \hat{\mathbf{k}}_s$	Unit vectors for incident and scattered wave directions
$\mathbf{K} = k_0(\hat{\mathbf{k}}_s - \hat{\mathbf{k}}_i)$	Ewald vector (spatial frequency probed by the measurement)
$E_s$	Scattered electric field
$G(\mathbf{r}, \mathbf{r}')$	Free-space scalar Green's function
$N_v$	Number of view angles (Tx illumination directions)
$N_f$	Number of frequencies
$W$	Signal bandwidth
$f_0$	Carrier frequency
$d_{\mathrm{ff}} = 2D^2/\lambda$	Fraunhofer far-field distance for aperture diameter $D$

← Ch 14 Computed Tomography as the Canonical Inverse Problem