Prerequisites & Notation

Prerequisites for This Chapter

This chapter addresses the phase retrieval problem — recovering a complex-valued signal from magnitude-only measurements. Phase retrieval arises whenever the measurement device records intensity (power) rather than amplitude and phase, which is common in many RF imaging scenarios. The algorithms here build on the optimization foundations of Part I and the imaging framework of Part III.

Compressed Sensing Theory — Sparsity, RIP, recovery guarantees for underdetermined systems(Review ch11)
Self-check: Can you state the restricted isometry property and explain why it guarantees sparse recovery?
Sparse Recovery Algorithms — ISTA/FISTA, ADMM, proximal methods(Review ch13)
Self-check: Can you implement ISTA and explain the role of the proximal operator?
Diffraction Tomography — Fourier diffraction theorem, forward models for RF imaging(Review ch14)
Self-check: Can you write the linearized forward model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ for a multi-static RF imaging setup?
Semidefinite programming basics — positive semidefinite matrices, trace minimization
Self-check: Do you know what $\mathbf{X} \succeq 0$ means and why $\text{tr}(\mathbf{X})$ serves as a convex surrogate for rank?
Wirtinger derivatives for functions of complex variables
Self-check: Can you compute $\frac{\partial}{\partial \bar{z}} |z|^2$ ?

Notation Introduced in This Chapter

The following symbols are introduced in this chapter. Standard notation from earlier chapters (e.g., $\mathbf{A}$ for the sensing matrix, $\mathbf{c}$ for reflectivity) carries over unchanged.

Symbol	Meaning	Introduced
$\mathbf{x} \in \mathbb{C}^N$	Unknown complex signal (image vector)	s01
$\mathbf{a}_i \in \mathbb{C}^N$	$i$ -th measurement vector	s01
$y_i = \|\langle \mathbf{a}_i, \mathbf{x}\rangle\|^2$	$i$ -th intensity (phaseless) measurement	s01
$M$	Number of measurements	s01
$\mathbf{X} = \mathbf{x}\mathbf{x}^H$	Rank-1 PSD matrix (lifted variable for PhaseLift)	s02
$\nabla_{\bar{\mathbf{z}}} f$	Wirtinger gradient with respect to $\bar{\mathbf{z}}$	s03
$\mu_t$	Step size at iteration $t$	s03
$\mathbf{D}_\ell$	Diagonal phase mask ( $\ell$ -th coded measurement)	s04
$\mathcal{H}_s(\cdot)$	Hard-thresholding operator (keep $s$ largest entries)	s04
$\text{dist}(\hat{\mathbf{x}}, \mathbf{x}_0)$	Phase-aligned distance: $\min_\phi \\|\hat{\mathbf{x}} - e^{j\phi}\mathbf{x}_0\\|$	s01

← Ch 15 Why Phase Retrieval Matters