References & Further Reading

References

1. G. Caire, RF Imaging: From Signals to Images, Ferkans Interactive Textbook, TU Berlin, 2026

   The primary reference for this book. Chapter 16 material on phaseless RF imaging, xPRA-LM, and hybrid approaches draws directly from this source.

2. E. J. Candes, T. Strohmer, and V. Voroninski, PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming, 2013

   The foundational PhaseLift paper — introduces the lifting trick and proves exact recovery via trace minimization SDP. Section s02 follows this development closely.

3. E. J. Candes, X. Li, and M. Soltanolkotabi, Phase Retrieval via Wirtinger Flow: Theory and Algorithms, 2015

   Introduces Wirtinger flow with spectral initialization and proves linear convergence for Gaussian measurements. Section s03 presents this algorithm and its convergence theory.

4. I. Waldspurger, A. d'Aspremont, and S. Mallat, Phase Recovery, MaxCut and Complex Semidefinite Programming, 2015

   Introduces PhaseCut — a max-cut-based SDP relaxation that operates directly on the phases rather than lifting to an $N \times N$ matrix. More scalable than PhaseLift.

5. Y. Chen and E. J. Candes, Solving Random Quadratic Systems of Equations is Nearly as Easy as Solving Linear Systems, 2017

   Introduces truncated Wirtinger flow with optimal $M = O(N)$ sample complexity. The truncation idea — discarding outlier gradient contributions — is key to practical robustness.

6. K. Jaganathan, Y. C. Eldar, and B. Hassibi, STFT Phase Retrieval: Uniqueness Guarantees and Recovery Algorithms, 2016

   Phase retrieval from short-time Fourier transform magnitudes. The STFT measurement model is relevant to time-frequency RF imaging and provides natural redundancy for phase recovery.

7. P. Netrapalli, P. Jain, and S. Sanghavi, Phase Retrieval Using Alternating Minimization, 2015

   Alternating minimization approach to phase retrieval with improved spectral initialization using truncation. Provides an alternative convergence analysis that complements the Wirtinger flow framework.

8. G. Wang, L. Zhang, G. B. Giannakis, M. Akcakaya, and J. Chen, Sparse Phase Retrieval via Truncated Amplitude Flow, 2017

   The SPARTA algorithm for sparse phase retrieval — combines truncation, amplitude loss, and hard thresholding. Achieves $M = O(s^2\log N)$ sample complexity for $s$-sparse signals.

9. M. Cheney and B. Borden, Fundamentals of Radar Imaging, SIAM, CBMS-NSF Regional Conference Series, 2009

   Comprehensive treatment of radar imaging with chapters on non-coherent processing and phaseless measurement models. Provides the physical motivation for RF phase retrieval.

10. A. Dremeau, A. Liutkus, D. Martina, O. Katz, C. Schulke, F. Krzakala, S. Gigan, and L. Daudet, Reference-less Measurement of the Transmission Matrix of a Highly Scattering Material Using a DMD and Phase Retrieval Techniques, 2015

    Demonstrates phase retrieval in a scattering medium using coded illumination patterns — the optical analogue of coded measurements in RF imaging. Validates the practical feasibility of phaseless imaging.

11. R. Balan, P. Casazza, and D. Eddins, On Signal Reconstruction Without Phase, 2006

    Establishes the fundamental uniqueness result for phase retrieval: $4N - 4$ generic measurements suffice for unique recovery (up to global phase) of a complex signal in $\mathbb{C}^N$.

12. A. V. Oppenheim and J. S. Lim, The Importance of Phase in Signals, 1981

    The classic paper demonstrating that phase carries more perceptual information than amplitude in images. Motivates the entire study of phase retrieval.

Further Reading

• Phase retrieval meets deep learning

  C. A. Metzler, P. Schniter, A. Veeraraghavan, and R. G. Baraniuk, *prDeep: Robust Phase Retrieval with a Flexible Deep Network*, ICML, 2018

  Bridges this chapter (classical phase retrieval) and Part V (deep learning for imaging) by using a learned denoiser within the Wirtinger flow iterations — a plug-and-play approach for phaseless imaging that often outperforms purely classical methods.

• Comprehensive survey of phase retrieval

  Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, *Phase Retrieval with Application to Optical Imaging*, IEEE Signal Processing Magazine, 2015

  Provides a broad overview of phase retrieval algorithms across optics, X-ray crystallography, and signal processing. Covers both convex and non-convex methods with emphasis on practical applications — useful for understanding the broader context beyond RF imaging.

• Fourier phase retrieval uniqueness

  T. Bendory, R. Beinert, and Y. C. Eldar, *Fourier Phase Retrieval: Uniqueness and Algorithms*, in Compressed Sensing and its Applications, Springer, 2017

  Detailed treatment of when Fourier phase retrieval has a unique solution and when it does not. Essential for understanding the additional ambiguities (conjugate inversion, shifts) specific to Fourier measurements that do not arise with generic measurement vectors.

• Non-convex optimization landscape

  J. Sun, Q. Qu, and J. Wright, *A Geometric Analysis of Phase Retrieval*, Foundations of Computational Mathematics, 2018

  Provides the deepest analysis of why non-convex gradient descent succeeds for phase retrieval: the loss landscape has no spurious local minima, and all saddle points have negative curvature directions. This explains the empirical success of Wirtinger flow beyond what the original convergence proof guarantees.

• Ptychographic imaging

  A. M. Maiden and J. M. Rodenburg, *An Improved Ptychographical Phase Retrieval Algorithm for Diffractive Imaging*, Ultramicroscopy, 2009

  Ptychography uses overlapping illumination patterns as coded measurements for phase retrieval — directly analogous to the phase mask approach in Section 16.4. Understanding the optical implementation provides physical intuition for the RF setting.