Chapter Summary

Chapter 16 Summary: Phase Retrieval and Phaseless Imaging

Key Points

1.
Phase carries the dominant structural information. The Oppenheim--Lim experiment shows that edges, positions, and spatial layout are encoded in the phase spectrum, while amplitude controls only contrast. Losing the phase causes catastrophic reconstruction degradation ( $\sim$ 20 dB NMSE penalty), not gradual quality loss.
2.
Phase retrieval is a non-linear inverse problem. The measurements $y_i = |\langle \mathbf{a}_i, \mathbf{x}\rangle|^2$ are quadratic in the unknown signal, with inherent global phase ambiguity. For generic measurements, $M \geq 4N$ intensity samples suffice for unique recovery up to global phase.
3.
PhaseLift provides the first polynomial-time guarantee. Lifting converts the quadratic problem to a linear SDP over $N \times N$ PSD matrices, with guaranteed recovery via trace minimization. However, $O(N^2)$ memory and $O(N^6)$ computation make it impractical beyond $N \sim 256$ .
4.
Wirtinger flow is the practical workhorse. Non-convex gradient descent with spectral initialization converges linearly to the global optimum with $M = O(N\log N)$ measurements, at $O(MN)$ per iteration — orders of magnitude faster than PhaseLift. Truncated variants achieve optimal $M = O(N)$ sample complexity.
5.
Sparse phase retrieval requires $O(s^2\log N)$ measurements. The quadratic dependence on sparsity $s$ (vs. linear for compressed sensing) is a fundamental consequence of the non-linear measurement model.
6.
Coded measurements are essential for RF imaging. The structured (Fourier-like) sensing matrix in RF imaging breaks the generic-measurement assumption. Phase masks ( $L \geq 3$ ) inject sufficient randomness for recovery. With sparse priors, phaseless imaging achieves within 3--6 dB of the coherent baseline.

Looking Ahead

This chapter completes Part III of the book — the classical toolkit for RF image reconstruction:

Chapter 11: Compressed sensing theory (sparsity, RIP).
Chapter 12: Matched filter and beamforming baselines.
Chapter 13: Sparse recovery algorithms (ISTA, FISTA, ADMM).
Chapter 14: Diffraction tomography and Fourier-based methods.
Chapter 15: Regularization and total variation.
Chapter 16: Phase retrieval for phaseless imaging.

The reader now has a complete set of classical tools for RF imaging — from linear reconstruction (matched filter, FBP) through regularized inversion (sparse recovery) to non-linear problems (phase retrieval).

Part IV introduces message passing and Bayesian inference, where algorithms like belief propagation and approximate message passing (AMP) exploit the statistical structure of the scene and sensing operator to achieve near-optimal recovery. These probabilistic methods provide a bridge between the classical optimization of Part III and the deep learning methods of Part V.

Phase Retrieval in RF Imaging Exercises