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Phase Retrieval: An Overview of Recent Developments
Kishore Jaganathan, Yonina C. Eldar, Babak Hassibi
TL;DR
Phase retrieval seeks to recover signals from Fourier magnitudes even though phase is unavailable and the problem is intrinsically ill-posed. The chapter surveys added signal structure and measurement redundancy, reviews identifiability conditions and algorithms, and reports robust and efficient recovery in several settings. Its conclusions also identify practical boundaries, including unresolved ambiguities and computational costs for some methods.
Problem
Fourier-magnitude measurements discard phase and generally leave an ill-posed reconstruction problem, requiring additional signal structure or measurement redundancy for meaningful identification.
Method
The chapter surveys sparse phase retrieval, masks, and STFT magnitudes, covering uniqueness guarantees and recovery algorithms including TSPR, GESPAR, STliFT, and optimization-based methods.
Results
The reviewed approaches demonstrate robust and efficient recovery from phaseless Fourier measurements across sparse, masked, and STFT settings.
Takeaways & Limitations
Phaseless measurements may support simpler and less costly optical, imaging, radar, and wireless systems when phase measurement is challenging.
Takeaways & Limitations
Some convex formulations are not sparse or rank one and can require expensive repeated high-dimensional convex programs, while certain sparse signals retain time-shift ambiguities.
Abstract
from arXiv · showhide
The problem of phase retrieval is a classic one in optics and arises when one is interested in recovering an unknown signal from the magnitude (intensity) of its Fourier transform. While there have existed quite a few approaches to phase retrieval, recent developments in compressed sensing and convex optimization-based signal recovery have inspired a host of new ones. This work presents an overview of these approaches. Since phase retrieval, by its very nature, is ill-posed, to make the problem meaningful one needs to either assume prior structure on the signal (e.g., sparsity) or obtain additional measurements (e.g., masks, structured illuminations). For both the cases, we review conditions for the identifiability of the signal, as well as practical algorithms for signal recovery. In particular, we demonstrate that it is possible to robustly and efficiently identify an unknown signal solely from phaseless Fourier measurements, a fact with potentially far-reaching implications.
1 Introduction
Phase retrieval reconstructs signals when measurements retain Fourier magnitudes but lose phase, making the problem ill-posed without additional structure or measurement redundancy. This overview surveys sparse signals, masks, and STFT magnitudes alongside uniqueness conditions and recovery algorithms.
- Motivation: Physical detectors often measure Fourier magnitude or power spectral density rather than phase, losing structural information needed for signal reconstruction.In far-field optics, the measured field is essentially the Fourier transform magnitude up to a known phase factor.
- Ill-posedness: Fourier phase can vary over an N-dimensional set for a fixed magnitude, so distinct signals generally form an N-dimensional feasible manifold.This makes phase retrieval intrinsically ill-posed and explains why directly ignoring phase gives unsatisfactory recovery.
- Classic approaches: In one dimension, oversampled Fourier magnitudes can still admit up to 2N non-equivalent solutions, whereas higher-dimensional recovery is well posed up to trivial ambiguities except for a measure-zero set.Time-shift, conjugate-flip, and global phase changes are trivial ambiguities because they preserve autocorrelation.
- Classic approaches: Unique one-dimensional recovery requires additional signal constraints such as sparsity or measurement redundancy, while alternating-projection methods can converge to local minima.The Gerchberg-Saxton algorithm imposes time-domain and Fourier-magnitude constraints, but its non-convex projection step limits recovery even without noise.
- Scope and recent methods: The chapter reviews uniqueness guarantees and practical algorithms for sparse phase retrieval, masked measurements, and STFT-magnitude recovery.It highlights TSPR, GESPAR, and STliFT, and discusses semidefinite-programming methods motivated by the quadratic constraints in phase retrieval.
2 Sparse Phase Retrieval
Sparse phase retrieval uses prior sparsity to reduce ambiguity in Fourier-magnitude reconstruction, while recovery algorithms address the remaining computational and robustness challenges. The section reviews uniqueness conditions and methods including SDP, TSPR, and GESPAR.
- Uniqueness: Most sparse signals with aperiodic support can be uniquely identified from their autocorrelation, whereas periodic support generally preserves the non-uniqueness of ordinary phase retrieval.Aperiodic support is not uniformly spaced; periodic sparse signals can be viewed as upsampled non-sparse signals, making uniqueness generally impossible.
- Uniqueness: Collision-free autocorrelation uniquely identifies 1D sparse signals when the sparsity k ≠ 6, while recovery remains possible only up to time-shift, conjugate-flip, and global phase ambiguities.These operations preserve both the measurements and sparsity, so equivalent solutions are treated as interchangeable.
- Algorithms: SDP-based sparse recovery often fails because trivial ambiguities make multiple shifted and conjugate-flipped rank-one solutions feasible, so convex combinations need not remain sparse or rank one.Although ℓ1 minimization promotes sparse solutions in related problems, the unresolved symmetry can cause the optimizer to lose the desired structure.
- Algorithms: TSPR first estimates the support through a Turnpike-style distance reconstruction, then estimates signal values using a reduced-dimensional convex program.Its support stage uses intersection and graph steps; with noise, generalized intersection steps provide stable recovery, and the lifted problem dimension decreases from N to k.
- Algorithms: TSPR provably recovers most O(N^1/2−ε)-sparse signals efficiently and most O(N^1/4−ε)-sparse signals robustly, while GESPAR uses iterative 2-opt support swaps for fast, accurate recovery.For TSPR, N = 12500 and k = 80, or N = 50000 and k = 160, gives a success probability of 0.5.
3 Phase Retrieval using Masks
Masked magnitude-only measurements add information that can resolve phase ambiguities, while recovery uses convex, gradient-based, or combinatorial algorithms with different guarantees and noise behavior.
- Uniqueness and algorithms: Two generic masks suffice to identify most signals up to global phase when an M ≥2N-point DFT is used.The corresponding recovery problem remains algorithmically nontrivial despite this identifiability result.
- Uniqueness and algorithms: R ≥c log^2 N random masks suffice for the convex program to recover the underlying signal up to global phase with high probability in the noiseless setting.This bound refines an earlier R ≥c log^4 N result.
- Uniqueness and algorithms: Two deterministic masks suffice for the convex program to recover non-vanishing signals up to global phase with an M ≥2N-point DFT.The masks are {I, D1}; non-vanishing means every signal entry is nonzero.
- Uniqueness and algorithms: Three masks can efficiently recover most signals up to global phase when their Fourier transform is non-vanishing and the mask shift is coprime with N.Relative Fourier phases are inferred from measurements, with one phase fixed as a reference.
- Uniqueness and algorithms: O(log N) polarization-based masks are sufficient to recover all signals.This guarantee contrasts with results that recover only most signals under generic or non-vanishing assumptions.
- Numerical simulations: Noisy simulations show stable recovery for random-mask and two-mask setups, although direct comparison is meaningless because the random-mask case uses four times as many measurements.The two-mask setup uses N = 32 and a 64-point DFT; the random-mask setup uses eight masks.
4 STFT Phase Retrieval
STFT phase retrieval adds overlapping, windowed Fourier-magnitude measurements to improve identifiability and robustness. The review develops uniqueness conditions and reports strong empirical recovery for both non-vanishing and sparse signals, including noisy settings.
- Overlapping STFT sections introduce measurement redundancy that can enable unique and robust recovery, outperforming oversampled Fourier magnitude measurements with the same measurement count.
- 4.1 Uniqueness: L < W is necessary for typical W < N because non-overlapping sections leave locations unmeasured or reduce recovery to separate 1D phase-retrieval problems.
- 4.1.1 Non-vanishing signals: For most non-vanishing signals, 1 ≤ L < W with a non-vanishing window and W ≤ N/2 gives identification up to global phase; time-shift and conjugate-flip ambiguities are resolvable.
- 4.1.2 Sparse signals: Sparse signals can remain ambiguous when separated supports or specially placed nonzeros prevent STFT measurements from relating signal components.
- 4.2 Algorithms: STliFT uniquely recovers non-vanishing signals under window and overlap conditions, and simulations show very high-probability recovery across 2L ≤ W ≤ N/2.
- 4.3 Numerical Simulations: STliFT exhibits stable noisy recovery, while STFT-GESPAR improves over oversampled DFT recovery whenever L < 16 and outperforms GL and PCGP in the reported sparse-signal experiments.
5 Conclusions
The chapter reviews recent phase-retrieval progress, covering sparse signals, masks, and STFT measurements. It reports robust recovery methods and practical relevance for optics and imaging.
- The chapter reviews sparse phase retrieval, masked measurements, and STFT phase retrieval, emphasizing identifiability and practical recovery algorithms.It presents TSPR and GESPAR for sparse signals, masks enabling robust recovery, and STliFT for STFT recovery.
- Most sparse signals can be uniquely identified from their autocorrelation, with TSPR and GESPAR providing efficient and robust recovery algorithms.
- Various masks allow unique and robust recovery of the unknown signal from phaseless measurements.
- Most signals can be uniquely identified from their STFT magnitude, and STliFT is suggested as an efficient and robust recovery algorithm.
- The reviewed results have practical applications in optics and imaging systems where measuring phase is challenging.