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Phase Retrieval: From Computational Imaging to Machine Learning
Jonathan Dong, Lorenzo Valzania, Antoine Maillard, Thanh-an Pham, Sylvain Gigan, Michael Unser
TL;DR
Phase retrieval seeks a complex signal from intensity-only measurements, a problem that remains challenging across its many applications. This tutorial unifies application models and reviews algorithms and recent theory, including links to single-layer neural networks. It presents a classification of models and surveys current algorithmic and theoretical advances.
Problem
Phase retrieval seeks a complex-valued signal from intensity-only measurements, a longstanding problem that remains difficult to solve.
Source: §I · p. 1Method
The tutorial unifies phase-retrieval applications into matrix-dependent groups and reviews recent reconstruction algorithms and theoretical advances.
Source: §I · p. 2Results
The tutorial presents a four-group application framework and a concise overview of recent algorithms, while describing neural-network links as a source of theoretical progress and promise for future applications.
Source: §VI · p. 12Takeaways & Limitations
The classification is intended to help practitioners choose algorithms suited to applications and to encourage links between research fields.
Source: §I · p. 2, §VI · p. 12
Abstract
from arXiv · showhide
Phase retrieval consists in the recovery of a complex-valued signal from intensity-only measurements. As it pervades a broad variety of applications, many researchers have striven to develop phase-retrieval algorithms. Classical approaches involve techniques as varied as generic gradient-descent routines or specialized spectral methods, to name a few. Yet, the phase-recovery problem remains a challenge to this day. Recently, however, advances in machine learning have revitalized the study of phase retrieval in two ways: significant theoretical advances have emerged from the analogy between phase retrieval and single-layer neural networks; practical breakthroughs have been obtained thanks to deep-learning regularization. In this tutorial, we review phase retrieval under a unifying framework that encompasses classical and machine-learning methods. We focus on three key elements: applications, overview of recent reconstruction algorithms, and the latest theoretical results.
I. INTRODUCTION
Phase retrieval seeks a complex signal from intensity-only measurements, a problem that spans imaging applications and remains challenging in both reconstruction and theory. This tutorial connects its mathematical model to applications, algorithms, and machine-learning theory.
A. Mathematical model
- The model seeks x* ∈ C^d from known measurements y and matrix A, with element-wise modulus measurements making recovery harder than the linear equation y = Ax*.The measurement model can include additive Gaussian or Poissonian noise; the oversampling ratio α = n/d compares measurements with unknown parameters.Source: §I · p. 1, A. Mathematical model · p. 2
I. INTRODUCTION
- The missing phase makes recovery challenging, while real-world structured measurement models remain harder to understand theoretically than random models.For random matrices, results characterize solvability regions and algorithm choice; progress has also reached more structured matrices.Source: §I · p. 1
- Phase retrieval spans imaging from X-ray to optical and THz settings, as well as computer-generated holography and optical computing, under a shared phase-retrieval equation.The problem has a long history in crystallography and astronomy, and reconstruction algorithms range from alternating projections to convex relaxations and spectral methods.Source: §I · p. 1
- The tutorial groups applications into four matrix-dependent categories and reviews recent algorithms and theoretical advances without developing their technical details.It aims to foster interaction across fields and clarify the phase-retrieval equation.Source: §I · p. 2
A. Mathematical model
- The phase-retrieval solution set is non-convex in general, and global phase ambiguity can be removed by quotienting the solution space or estimating invariant quantities.The model can be extended with noise statistics used in maximum-likelihood and Bayesian algorithms.Source: A. Mathematical model · p. 2
B. Neural-network representation
Phase retrieval is equivalent to training a single-layer neural network with quadratic activation. This simple architecture still exhibits non-convex optimization difficulties.
B. Neural-network representation
- Each measurement presents input a_i to a network with weights x and target y, so training for regression amounts to recovering x* in phase retrieval.The network has a quadratic activation function.Source: B. Neural-network representation · p. 2
- The framework links the nonlinear phase-retrieval equation to single-layer neural-network optimization and places it among four A-dependent imaging-problem classes.The figure pairs the equation, imaging settings, neural-network optimization, and the classes determined by measurement matrix A.Source: Figure 1 · p. 3
- Gradient descent is not guaranteed to converge because it can become trapped at local minima or saddle points.The network has a non-convex loss functional.Source: B. Neural-network representation · p. 3
III. APPLICATIONS
The applications section classifies phase-retrieval models by measurement operator and describes Fourier and coded-illumination settings. These settings differ in their information, acquisition requirements, and current theoretical coverage.
A. Unifying Framework for Phase-Retrieval Applications
- The proposed framework groups phase retrieval into Fourier, coded-illumination, coded-detection, and random models, with the matrix A reflecting the imaging system's physical model.The classification is intended to support links across fields and help practitioners and theoreticians connect applications and models.Source: A. Unifying Framework for… · p. 3
B. Fourier Phase Retrieval
- Fourier phase retrieval uses far-field propagation modeled by A = F, but its unitary transform gives n = d and α = 1, leaving the problem ill-posed without additional information about x*.Additional information can include another intensity measurement, support and non-negativity constraints, or other regularization assumptions.Source: B. Fourier Phase Retrieval · p. 4
C. Coded-illumination Phase Retrieval
- Coded illumination concatenates measurements from different illuminations, providing complementary and partially redundant information that improves convergence at the cost of a more involved setup and longer measurement time.Here A_l = F D_l and the oversampling ratio α equals the number L of images.Source: C. Coded-illumination Phase Retrieval · p. 4
- Coded-illumination applications include coded diffraction imaging, ptychography, and Fourier ptychography, which use structured illumination or probes across different imaging settings.Ptychography's oversampling is linked to overlap between consecutive illuminated areas; greater overlap makes the problem easier to solve.Source: C. Coded-illumination Phase Retrieval · p. 4
- Specialized and gradient-based algorithms are used for coded-illumination applications, but sensing-matrix structure currently prevents thorough theoretical results.The Ptychographic Iterative Engine is one popular projection algorithm.Source: C. Coded-illumination Phase Retrieval · p. 4
D. Coded-detection Phase Retrieval
Coded-detection phase retrieval uses multiple measurements created by modulating detection, including defocused images; random models range from fully random to pseudo-random, with differing theoretical tractability.
D. Coded-detection Phase Retrieval
- When modulating illumination near the object is infeasible, such as in astronomical observations, coded-detection phase retrieval can use multiple images captured at different defocused positions, a phase-diversity technique also applied to lensless imaging.Source: D. Coded-detection Phase Retrieval · p. 4
- Angular-spectrum modeling represents defocus through operators formed by Fourier propagation and diagonal matrices, with curvature proportional to defocus distance.The same multi-plane constraint structure also appears in computer-generated holography, where a phase mask is optimized to produce a volumetric intensity pattern.Source: D. Coded-detection Phase Retrieval · pp. 4–5
- General coded-detection strategies can vary Fourier-space diagonal matrices using higher-order aberrations or an aperture, but their solvability and algorithm choice remain poorly understood.The theory would benefit particularly from further results on these questions.Source: D. Coded-detection Phase Retrieval · p. 5
E. Random Phase Retrieval
- Independent random sampling, including Gaussian matrices, supports asymptotic recovery guarantees and typical-algorithm analysis, and random phase retrieval is comparatively well understood.Random operators also arise in complex-media imaging through multiple light scattering or programmable random input patterns.Source: E. Random Phase Retrieval · p. 5
- Pseudo-random coded diffraction and ptychography retain row correlations, making precise guarantees harder, while motivating use of algorithms designed for fully random settings.The tutorial distinguishes i.i.d. random models with finite moments from right unitarily invariant matrices, whose distributions are unchanged by right multiplication by a unitary matrix.Source: E. Random Phase Retrieval · p. 5
IV. RECONSTRUCTION ALGORITHMS
The tutorial surveys reconstruction algorithms and their practical trade-offs, while projection methods illustrate both the use of measurement constraints and limitations in convergence.
IV. RECONSTRUCTION ALGORITHMS
- The overview presents algorithms with intuitive explanations and guidance on when to use them, addressing the diversity of phase-retrieval models.More technical descriptions are covered by dedicated surveys.Source: §IV · p. 5
A. Projection Algorithms
- Projection algorithms sequentially enforce measurement or prior constraints, with Gerchberg–Saxton using two measurement planes and error reduction combining Fourier-modulus and object-support constraints [18, 2].Ptychography, Fourier ptychography, and phase diversity project onto more than two sets to exploit additional information.Source: A. Projection Algorithms · p. 5
- For full-rank A, the map is injective for generic A when n ≥ 4d − 4; below that threshold it is non-injective for d ≤ 3, while the lower-threshold claim is disproved at d = 4.For large d, the threshold n/d ≃ 4 separates typical injectivity behavior, suggesting prior information can help recovery when n/d ≲ 4.Source: A. Projection Algorithms · p. 6
- Projection methods are easy to implement and flexible, but need not converge to a solution; relaxed projections and averaged strategies led to hybrid input-output, difference-map, and relaxed averaged alternating-reflections methods [2, 22].These alternatives aim to improve convergence or avoid loops.Source: A. Projection Algorithms · pp. 5–6
B. Gradient-based Optimization
Gradient-based optimization applies flexible iterative loss minimization to phase retrieval, but non-convexity and initialization affect its performance.
B. Gradient-based Optimization
- Gradient methods refine an estimate by stepping toward the negative gradient of a loss, with applications including ptychography, coded-diffraction imaging, and defocus-based implementations.The loss choice matters; when noise statistics are known, a suitable loss can be selected, with Poissonian and additive Gaussian models as examples.Source: B. Gradient-based Optimization · p. 6
- Non-convex optimization may get stuck at local minima or saddle points, motivating conjugate-gradient, Nesterov, second-order, and stochastic-gradient strategies.Stochastic gradient descent uses only a partial dataset per iteration; using one intensity image at a time introduces noise in updates.Source: B. Gradient-based Optimization · pp. 6–7
- The initial estimate greatly impacts gradient-based performance, and informed-initialization algorithms are described as one response to this sensitivity.The theoretical basis of gradient-based algorithms remains an active research topic.Source: B. Gradient-based Optimization · p. 7
C. Convex Relaxations
Convex relaxations exploit the quadratic phase-retrieval model by introducing auxiliary variables, but their semidefinite-programming implementations can be computationally prohibitive for real imaging.
C. Convex Relaxations
- Convex relaxations reformulate the problem with auxiliary variables to obtain a convex optimization problem with stronger convergence guarantees.PhaseLift lifts x* to X* = x*(x*)H, making the quadratic measurements linear in X* [10].Source: C. Convex Relaxations · p. 7
- PhaseLift replaces the non-convex rank-one constraint with minimal-trace optimization, then estimates the signal from the leading eigenvector of the solution matrix.The formulation is a semidefinite program solvable with an off-the-shelf solver; other convex relaxations are also available.Source: C. Convex Relaxations · p. 7
- For a 100 × 100 image, standard convex-relaxation methods require optimization in a 10^8-dimensional space, so semidefinite programming has been used only as proof of concept in real imaging.Sketching methods to keep computation tractable have been proposed.Source: C. Convex Relaxations · p. 7
D. Spectral Methods
Spectral methods provide low-cost initial estimates through weighted covariance matrices, while Bayesian algorithms use posterior information to estimate the signal. The section contrasts their computational properties and illustrates their performance in phase retrieval.
D. Spectral Methods
- Spectral methods estimate the signal with the leading eigenvector of a covariance matrix weighted using intensities and sampling vectors.The choice of preprocessing function T strongly affects performance; an optimal choice is derived for random models R.1 and R.2, including noisy settings.Source: D. Spectral Methods · p. 7
- Power iterations recover the leading eigenvector at low computational cost with guaranteed convergence, without the lifting used by convex relaxations.Spectral estimates are typically refined with an iterative method such as gradient descent.Source: D. Spectral Methods · p. 7
- The optimal spectral method yields almost perfect correlation with a real image even without gradient-descent refinement.The illustration uses noiseless phase retrieval with a complex Gaussian matrix.Source: D. Spectral Methods · p. 8
E. Bayesian Algorithms
- Bayesian algorithms estimate from the posterior distribution; the MMSE estimator is the posterior mean, but computing such estimates can be hard.Classical MCMC sampling from the posterior is often exponential-time.Source: E. Bayesian Algorithms · p. 8
- AMP can be analyzed in the high-dimensional limit, but is computationally heavier than spectral or gradient-based methods and depends on generative-model assumptions.Its theoretical predictions agree well with finite-size simulations; outside assumed models, performance and convergence are not guaranteed.Source: E. Bayesian Algorithms · p. 8
V. REGULARIZATION IN THE DEEP-LEARNING ERA
Regularization adds prior information to phase-retrieval reconstruction, from classical constraints and sparsity penalties to learned image priors. The section also reviews weak-recovery thresholds and the dependence of algorithmic guarantees on the measurement model.
A. Conventional Regularization
- Regularized reconstruction combines a data-fidelity term for intensity measurements with a penalty favoring suitable properties of the recovered image.The penalty can encode nonnegativity, support, sparsity, or transform-domain structure such as total variation; proximal algorithms handle potentially nondifferentiable terms.Source: A. Conventional Regularization · p. 8
- AMP achieves optimal polynomial-time performance for phase retrieval with random unitary matrices but performs very poorly with a vanilla Fourier matrix..Source: A. Conventional Regularization · p. 8
- Figure 2 compares spectral methods using different trimming thresholds with gradient descent initialized by the optimal spectral method or randomly.For image recovery, OS-initialized gradient descent shows zero error by α ≳ 4, whereas randomly initialized gradient descent has no positive correlation for α ≤ 5.Source: Figure 2 · p. 9
- Weak recovery begins at α_WR = 1 for noiseless complex phase retrieval with a Gaussian measurement matrix, and the optimal spectral method reaches that threshold.The threshold separates impossible from possible weak recovery under the stated random-design setting.Source: A. Conventional Regularization · p. 9
- The weak-recovery threshold can be specified through an implicit equation depending on the singular-value spectrum of the measurement matrix.For complex noiseless phase retrieval, the equation is given as a function of the eigenvalues of AA^H/d.Source: A. Conventional Regularization · p. 9
- CNN-based priors consistently outperform classical regularization on natural images, while providing a way to incorporate learned image structure into phase retrieval.The tutorial presents several ways to introduce this prior into phase-retrieval algorithms.Source: A. Conventional Regularization · p. 9
B. Plugging-in a Denoiser
Denoiser-based methods incorporate image priors into reconstruction algorithms, while the tutorial also relates their use to broader recovery results and theoretical guarantees. The cited results show strong performance in some settings alongside limits on convergence and on the scope of current theory.
B. Plugging-in a Denoiser
- AMP achieves perfect recovery for α ≳ 2.03 on synthetic signals and actual images, while falling short of information-theoretic performance for 2 < α ≲ 2.03.The figure compares state-evolution predictions with algorithm runs.Source: Figure 3 · p. 10
- Perfect recovery can occur at α ≃ 2 for Gaussian measurements and a uniformly sampled signal, well before the typical injectivity threshold α = 4.The discussion presents perfect-recovery studies as more relevant to practical settings where signals and measurement matrices are physically constrained.Source: B. Plugging-in a Denoiser · p. 10
- Plug-and-Play methods alternate data-fidelity updates with denoising CNN steps in place of a conventional proximal regularizer.The data-fidelity step can use a proximal operator or gradient descent, yielding ADMM or FISTA variants, respectively.Source: B. Plugging-in a Denoiser · p. 10
- PnP convergence is generally not assured and can be proven only in restricted settings, despite the method’s flexibility and reported success.A cited application uses PnP with a denoising network for computationally efficient large-scale phase retrieval.Source: B. Plugging-in a Denoiser · p. 10
- High-dimensional analyses explain several algorithms’ performance, but current asymptotic guarantees rely largely on random models and remain open for structured matrices.The reviewed tools include state evolution, random-matrix theory, and analyses of gradient-based algorithms.Source: B. Plugging-in a Denoiser · p. 11
- The RED formulation uses a denoiser to define a differentiable regularization term.The denoiser f maps from C^d to C^d.Source: B. Plugging-in a Denoiser · p. 11
C. Generative models
Generative approaches regularize phase retrieval by restricting estimates to network-generated images, either through a learned image manifold or a CNN prior. Their optimization choices differ, and Deep Image Prior requires early stopping to avoid fitting noise.
C. Generative models
- GANs train a generator to produce realistic images and a discriminator to distinguish generated images from dataset images.The generator maps a random latent vector z to an image x = G(z).Source: C. Generative models · p. 11
- GAN-based phase retrieval constrains the image to generator outputs and optimizes the latent variable, restricting the search to the learned manifold.Posterior sampling can also provide spatial variance estimates for the final image.Source: C. Generative models · p. 11
D. Deep Image Prior
- Deep Image Prior avoids a training dataset by representing the image as a CNN output and optimizing the network weights.The approach has been implemented in several phase-retrieval applications.Source: D. Deep Image Prior · p. 11
- Unlike GANs, DIP tunes CNN weights whose dimension is typically larger than the image dimension, so it can overfit even random noise.Its architecture provides regularization through an initial tendency toward smooth natural images, but early stopping is needed to avoid fitting noise.Source: D. Deep Image Prior · pp. 11–12
- The exact principle behind Deep Image Prior regularization remains an open question.Source: D. Deep Image Prior · p. 12
E. Discussion
The discussion contrasts application-specific supervised inversion with more flexible deep-learning strategies, while noting that neural-network instability and training-data needs remain concerns. The conclusion presents a unifying model framework and identifies theoretical gaps in realistic settings.
E. Discussion
- Supervised networks trained to map measurements directly to images are tied to a particular imaging model, noise level, or camera sampling, limiting applicability.The paper describes Sections V-B–V-D as a second generation with greater robustness and flexibility.Source: E. Discussion · p. 12
- Deep-learning methods can produce artifacts or remove features, and realistic-looking outputs may conceal failure; their instability is difficult to detect.Many methods also remain black boxes, and network-based approaches often require training data that may impede medical applications.Source: E. Discussion · p. 12
- DIP and other unsupervised schemes offer an alternative to training-data-dependent methods and may support more controlled learning-based regularization.The paper notes that the potential and limits of recent tools are not yet clearly established.Source: E. Discussion · p. 12
VI. CONCLUSION
- The tutorial classifies phase-retrieval models and surveys algorithms to help practitioners choose methods suited to their applications.Source: §VI · p. 12
- The neural-network analogy has enabled theoretical studies of phase retrieval, while neural networks’ image performance brings promise alongside instability challenges.Source: §VI · p. 12
- Randomness in illumination or detection could aid experimental phase recovery, but many realistic phase-retrieval models remain theoretically unexplored.The conclusion calls for further study of these models.Source: §VI · p. 12