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Phase Retrieval with Application to Optical Imaging

Yoav Shechtman, Yonina C. Eldar, Oren Cohen, Henry N. Chapman, Jianwei Miao, Mordechai Segev

arXiv:1402.7350v1cs.ITmath.IT

TL;DR

Optical phase retrieval addresses the fact that the human eye cannot measure a light wave’s phase. This review links optical physics with signal-processing methods and algorithms, surveys the state of the art, and identifies challenges and areas for future work.

  • Problem

    The human eye cannot measure the phase of a light wave, motivating phase retrieval in optical imaging.

    Source: §I · p. 1

  • Method

    The review connects relevant optical physics with signal-processing methods and algorithms.

    Source: §I · p. 1

  • Results

    The review aims to describe the field’s current state of the art and identify its challenges.

    Source: §I · p. 1

  • Takeaways & Limitations

    It suggests vision and areas for further work in phase retrieval.

    Source: §I · p. 1

Abstract

from arXiv · show

This review article provides a contemporary overview of phase retrieval in optical imaging, linking the relevant optical physics to the information processing methods and algorithms. Its purpose is to describe the current state of the art in this area, identify challenges, and suggest vision and areas where signal processing methods can have a large impact on optical imaging and on the world of imaging at large, with applications in a variety of fields ranging from biology and chemistry to physics and engineering.

I. INTRODUCTION

This introduction frames phase retrieval as a means of recovering phase from intensity or far-field diffraction measurements, connecting optical physics with signal-processing methods across optical imaging and coherent X-ray imaging. It surveys the field’s development and contributions to nonlinear sparse recovery while identifying challenges in uniqueness and algorithmic recovery.

I. INTRODUCTION

  • Optical detectors cannot directly record light-wave phase and instead measure photon flux proportional to the field magnitude squared.Phase retrieval addresses this by recovering phase from the measured magnitude of the optical far-field.Source: §I · p. 1
  • The review links optical physics to signal-processing methods and algorithms, describes the state of the art, identifies challenges, and proposes directions for future impact in optical imaging and beyond.It covers applications spanning biology, chemistry, physics, and engineering.Source: §I · p. 1

II. HISTORICAL BACKGROUND

  • Hard X-rays offer wavelengths comparable to atomic resolution, but lenses in this spectral region suffer large aberrations and are difficult to make; algorithmic retrieval avoids dependence on lens quality but requires very low-noise detectors.Source: §II · p. 2
  • Required X-ray exposure rises as (1/d)^4, while absorbed energy damages samples and can make measured intensity reflect damage rather than true molecular structure [5].Source: §II · p. 2
  • Although ultrashort pulses were proposed to outrun atomic-scale damage [7], direct X-ray imaging requires optical components unavailable at molecular resolution.Hard-X-ray lenses would require picometer-scale fabrication, while existing X-ray mirrors resolve only many-nanometer features.Source: §II · p. 3
  • The sample-support constraint enables image recovery at nanometric resolution, with phase information retrieved numerically and experimentally in various examples,,,,,.Source: §II · p. 3
  • Coherent diffractive imaging (CDI) reconstructs an object from its coherent far-field diffraction intensity, while determining 3D structures of large proteins remains a major challenge,.Source: §II · p. 3
  • Phase retrieval may lack a unique solution, and even when one exists, there is no guarantee an algorithm can find it.Source: §II · p. 3
  • Signal-processing research applies optimization and sparsity to phase retrieval, with sparsity-based methods reported to resolve features smaller than 1/5 of the wavelength,,.Source: §II · p. 3
  • Research on nonlinear sparse recovery draws worldwide contributions to its theory, algorithms, and practice as coherent X-ray sources rapidly develop.The passages mention both the rapid development of coherent X-ray sources and more researchers contributing to nonlinear sparse recovery.Source: §II · p. 3

A. Problem Formulation

Fourier phase retrieval seeks an object or signal from Fourier-magnitude measurements, as in coherent diffractive imaging. The missing phase makes recovery difficult, though practical reconstructions can be excellent even without theoretical uniqueness guarantees.

  • In forward-scattering CDI, a coherent wave illuminates the object and far-field intensity corresponds to Fourier magnitude under the stated small-object, far-measurement conditions.Recovering the object is equivalent to recovering the wave at the object plane from the measured intensity.Source: A. Problem Formulation · p. 4
  • Fourier phase retrieval seeks x from |X[k]| alone, so the Fourier phase is unmeasured.Source: A. Problem Formulation · p. 4
  • Swapping the Fourier phases of Cameraman and Lenna produces inverse transforms that demonstrate phase's significant information about the images.Source: A. Problem Formulation · p. 5
  • Phase retrieval can lack a unique solution, but that lack of guarantees does not prevent excellent reconstructions in many settings.Source: A. Problem Formulation · p. 5

B. Uniqueness

Uniqueness depends on measurement structure and signal conditions: Fourier magnitudes are nonunique in 1D but generically unique in higher dimensions, while general measurements admit separate uniqueness and stability conditions.

1) Fourier measurements:

  • 1D Fourier magnitude is generally nonunique, even with known support, because distinct signals can share an autocorrelation and thus the same magnitude.Source: 1) Fourier measurements · p. 5
  • For d ≥ 2, prior results,, establish uniqueness from continuous Fourier magnitude for all but a measure-zero set, up to trivial ambiguities.Source: 1) Fourier measurements · p. 5
  • Exact guarantees linking sample counts to image types remain an open question in practice.Source: 1) Fourier measurements · p. 5
  • One bit of Fourier-phase sign information yields uniqueness only for real signals whose Z transform has no unit-circle zeros.Source: 1) Fourier measurements · pp. 5–6
  • For a 1D k-sparse real signal, full autocorrelation suffices for uniqueness under the stated k ≠ 6 and collision-free conditions [72].Source: 1) Fourier measurements · p. 6

2) General measurements:

  • With noisy general measurements, order N log(N) measurements, or k log(N) in the k-sparse case, suffice for stable uniqueness; the stated least-squares minimization yields the correct solution under these conditions.Source: 2) General measurements · p. 6
  • For real general measurements, injectivity holds if and only if the measurement vectors satisfy the complement property.The section states that this imposes a lower limit M > 2N − 1 measurements.Source: 2) General measurements · p. 6

IV. ALGORITHMS

This section surveys phase-retrieval algorithms, since uniqueness guarantees alone do not provide a general way to recover the signal, covering nonconvex methods and alternating projections along with their benefits and limitations. It also discusses initialization sensitivity in Fienup methods, the recommendation to try multiple starting points, and a method for selecting an initial point that yields iterations arbitrarily close to the true vector under random Gaussian measurements.

A. General algorithms

  • The general phase-retrieval objective is nonconvex, so finding a global minimum is unclear even when one exists.The review surveys approaches and prior information intended to increase the probability of convergence to the true solution.Source: A. General algorithms · p. 7

1) Alternating projections:

  • Gerchberg–Saxton alternates between enforcing real-space and Fourier magnitudes; its recovery error decreases monotonically, but it may converge to a local minimum.The method was pioneered for recovering a complex image from magnitude measurements at two planes.Source: 1) Alternating projections · p. 7
  • Fienup methods replace the real-space magnitude constraint with other constraints or corrections, and HIO applies a correction where the estimate violates real-space constraints.Possible constraints include non-negativity and known support; HIO is a commonly used variant in optical phase retrieval applications such as CDI.Source: 1) Alternating projections · pp. 7–8
  • HIO is empirically reported to reach a global minimum for noise-free oversampled diffraction, but high noise can cause stagnation, and the method requires a predefined support.Its convergence is unproven and it is sensitive to prior accuracy; support must be tightly known, especially for complex signals.Source: 1) Alternating projections · pp. 8–9
  • OSS is reported to outperform HIO and its variants in many settings by applying a Gaussian smoothing filter to the off-support region, with filter variance decreasing across iterations.OSS is based on Fienup iterations and replaces HIO’s fourth step.Source: 1) Alternating projections · p. 9
  • Fienup method performance depends on initialization, making it possible and recommended to try several starting points.Source: 1) Alternating projections · p. 9
  • For random Gaussian measurements, the initialization method in [61] yields iterations that converge arbitrarily close to the true vector.Source: 1) Alternating projections · p. 9

2) Semi-Definite Programming (SDP) based algorithms:

SDP methods lift quadratic phase-retrieval measurements into linear constraints on a higher-dimensional matrix, then relax the rank-one requirement; PhaseLift has a stated high-probability recovery guarantee under random Gaussian measurements.

B. Sparsity based algorithms

This section surveys sparsity-based phase retrieval, where priors restrict objects to few nonzero coefficients in a known representation. It covers alternating-projection, SDP, sparse Fourier, and greedy approaches, including GESPAR for recovering sparse vectors from quadratic measurements.

Alternating projections with sparsity prior

  • Fienup sparsity iterations retain the first three steps, then project into Ψ, keep the k largest coefficients, and project back; the error is nonincreasing toward a local minimum.For non-invertible Ψ, the passage proposes replacing the projection and thresholding steps with a sparse-solution method [57].Source: B. Sparsity based algorithms · p. 10

SDP based methods with sparsity prior

  • QCS adds sparsity constraints to rank minimization by encouraging row sparsity in X, then uses iterative log-det with thresholding to approximate the NP-hard problem and extracts a rank-1 estimate.When x is sparse, X = xx* has a small number of nonzero rows.Source: B. Sparsity based algorithms · p. 10
  • A related SDP formulation [50] relaxes rank minimization to trace minimization with an l1 penalty, and its solution is unique in the noiseless case.The objective is min Tr(X) + λ||X||1 subject to measurement constraints within noise tolerance ϵ.Source: B. Sparsity based algorithms · p. 10
  • A two-step SDP method for sparse 1D Fourier phase retrieval determines the support from autocorrelation, then recovers the signal experimentally from O(k^2) measurements for k-sparse signals.The signal is estimated after its support has been identified.Source: B. Sparsity based algorithms · p. 11

Greedy methods with sparsity prior

  • Because lifting algorithms increase the problem dimension, the section introduces greedy methods with sparsity priors as a response to that drawback.Source: B. Sparsity based algorithms · p. 11

B. Sparsity based algorithms

  • Applications include CDI of 1D objects, temporally varying objects, waveguide-array phase retrieval [56], and joint phase retrieval and sub-wavelength imaging.Source: B. Sparsity based algorithms · p. 12
  • Over 90% successful recovery through k=15, GESPAR [54] outperformed SDP and Sparse-Fienup, whose corresponding limits were k=8 and k=7, for N=64, M=128.The comparison used 100 simulations at each sparsity level.Source: Figure 6 · p. 13, B. Sparsity based algorithms · p. 13
  • GESPAR recovered a sparse 195 × 195 image from 38,025 2D-Fourier magnitude measurements in 80 seconds; sparse Fienup failed, and SDP was impractical at that matrix size.The image comprised 15 circles selected from a 225-element dictionary.Source: Figure 6 · p. 13, Figure 7 · p. 13

V. APPLICATIONS IN LENSLESS IMAGING

Lensless imaging seeks to recover images from experimental measurements, where phase retrieval faces challenges beyond those in simulations or digital-data processing. The section outlines these experimental complications while noting that successful retrieval has been widely achieved.

V. APPLICATIONS IN LENSLESS IMAGING

  • The section presents several coherent diffraction imaging applications connected to phase retrieval algorithms discussed earlier.Source: §V · p. 14
  • Phase retrieval from experimentally obtained patterns presents additional challenges compared with numerical simulations or digital-data processing.The section frames lensless imaging applications around recovering images from experimental measurements.Source: §V · p. 14
  • Experimental diffraction measurements introduce noise, parasitic scattering, incomplete intensity data, nonuniform pixel sampling, and sometimes partial incoherence.The missing center can prevent direct recording of very low spatial frequencies; pixels integrate photons over their subtended solid angle rather than uniformly sampling the diffraction signal.Source: §V · p. 14
  • Experimental phase retrieval in optical imaging has nevertheless been widely achieved,,,,,,,,.Source: §V · p. 14

A. Quantitative comparison of alternating-projection algo-

This section compares four algorithms using experimental X-ray CDI reconstructions, reporting OSS as strongest among those compared and describing how combining independent reconstructions can suppress artifacts. It also notes that X-ray free-electron laser CDI enables lensless imaging of moving objects and reliable phase recovery from high-contrast diffraction patterns.

A. Quantitative comparison of alternating-projection algo-

  • For simulated noisy vesicle data, OSS had the lowest E (0.07), versus NR-HIO (0.16), ER-HIO (0.24), and HIO (0.28).Visual inspection also identified OSS as the most faithful reconstruction among the four algorithms.Source: Figure 8 · p. 14, A. Quantitative comparison of… · p. 14
  • For experimental yeast-spore diffraction data, OSS produced the most consistent reconstructions among the four algorithms.Each algorithm was tested in five independent trials of 100 runs with different random initial phases; the lowest-error reconstruction was selected in each trial.Source: A. Quantitative comparison of… · p. 15

B. X-ray free electron laser CDI

  • CDI works without optical components, avoiding damage from intense X-ray pulses and enabling reliable imaging of moving objects.The lensless setup suits destructive pulses that permit only a single measurement before the object disintegrates.Source: B. X-ray free electron laser CDI · p. 15
  • For reproducible particles such as viruses, randomly oriented diffraction patterns can yield a 3D image when orientations are determined; Loh et al demonstrated proof of concept.Source: B. X-ray free electron laser CDI · p. 15
  • High-contrast X-ray diffraction patterns with near-zero intensity minima improve phase-recovery reliability, and PRTF is often near unity except in low-signal-to-noise regions.PRTF assesses consistency of recovered phases across patterns reconstructed from different starting guesses.Source: B. X-ray free electron laser CDI · p. 15
  • The experimental diffraction pattern comparison reports five-trial average reconstruction and variance for HIO, ER-HIO, NR-HOP, and OSS, alongside average RF and consistency.The pattern was measured from a S. pombe yeast spore cell.Source: Figure 9 · p. 16
  • Combining ten independent reconstructions yields complex-valued sums whose Fourier spectrum is modulated by the PRTF, making noise- or truncation-related artifacts unlikely to appear in the recovered image.The passage describes truncating data to a lower resolution as one possible source of artifacts.Source: B. X-ray free electron laser CDI · p. 16

C. Tabletop short wavelength CDI

Tabletop coherent extreme-UV sources extend CDI beyond large synchrotron and X-ray free-electron laser facilities. Sparsity priors are also used to recover sub-wavelength features and reduce acquisition time in simulations of changing objects.

C. Tabletop short wavelength CDI

  • A tabletop coherent extreme-UV CDI experiment used 29 nm high-harmonic radiation and reconstructed a masked carbon film at 214 nm resolution.The setup selected a single harmonic and focused it onto the sample with multilayer mirrors.Source: C. Tabletop short wavelength CDI · p. 16, Figure 11 · p. 17

D. Sub-wavelength CDI using sparsity

  • The greedy recovery iteratively updates object support, finds a local minimum, and removes the weakest circle until convergence.The sparsity prior can regularize the transform and address loss of high spatial frequencies for features smaller than approximately λ/2.Source: D. Sub-wavelength CDI using sparsity · pp. 16–17
  • Long acquisition times needed for sufficient signal-to-noise ratio limit temporal resolution and the dynamical phenomena accessible to X-ray CDI.The passage describes non-destructive X-ray CDI acquisition times on the order of tens of seconds.Source: D. Sub-wavelength CDI using sparsity · p. 17
  • 2 orders of magnitude improvement in acquisition time is reported in simulations for an object with 5 randomly varying pixels between frames.The approach uses sparsity in the change between sequential CDI frames to denoise Fourier-magnitude measurements.Source: D. Sub-wavelength CDI using sparsity · p. 17
  • A greedy algorithm recovered an arrangement of 100 nm nano-holes from low-pass-filtered Fourier magnitude by exploiting a sparse-circle prior.The prior specifies a small number of 100 nm circles on a grid, while their number, locations, and amplitudes are unknown.Source: D. Sub-wavelength CDI using sparsity · p. 16, Figure 12 · p. 18

VI. OTHER PHYSICAL SETTINGS, BOTTLENECKS, AND VISION

The review extends phase retrieval beyond standard CDI and Fourier measurements to scanning and other non-Fourier optical systems, including propagation through spatially varying media and specialized imaging settings. It also discusses challenges and bottlenecks and offers a future vision for the field.

A. Non-Fourier Measurements

  • Measurements between the image plane and far field produce patterns unlike the Fourier magnitude, requiring revised algorithms and raising uniqueness and convergence questions.The review describes propagation using transfer functions beyond the Fourier transform, including the Fresnel regime.Source: A. Non-Fourier Measurements · pp. 18–19
  • Measurements at multiple non-Fourier planes add signal information, unlike additional far-field measurements, which repeat Fourier-magnitude data.This broader measurement setting may relax requirements for oversampling or prior knowledge of the support.Source: A. Non-Fourier Measurements · p. 19
  • Non-Fourier measurements can provide multiple observations that may relax requirements on oversampling or advance knowledge of the support.Measurements at different planes each add information about the signal.Source: A. Non-Fourier Measurements · p. 19
  • Multi-frequency measurements require well-separated, narrow-band frequencies to meet the high coherence needed for CDI.Source: A. Non-Fourier Measurements · p. 19
  • The multi-frequency idea also applies to continuous broad-bandwidth pulses centered on a single frequency, provided the pulse power spectrum is known in advance,, [117].Source: A. Non-Fourier Measurements · p. 19
  • Scanning coherent diffraction imaging (CDI) scans overlapping beam regions across a sample to image extended objects, a method known as ptychography,,,.Source: A. Non-Fourier Measurements · p. 19
  • In spatially varying media such as photonic crystals, phase retrieval uses a transfer function different from free-space propagation; when that function is known, modified algorithms can solve the problem.Photonic crystal fibers are identified as a potential imaging medium for endoscopy, where measurements are the squared field magnitude at the measurement plane rather than the sought image’s Fourier magnitude.Source: A. Non-Fourier Measurements · pp. 19–20
  • Other CDI extensions include Bragg CDI for reconstructing nano-crystal structure and strain,,,, and sparsity-based phase retrieval and super-resolution in optical waveguide arrays [56].Source: A. Non-Fourier Measurements · p. 20

B. Phase Retrieval Combining Holographic Methods

Holography records interference between an image wave and a known reference wave, embedding image information in the measured pattern. In phase retrieval, a known Fourier-domain wave can add prior information that may improve recovery resolution or relax support constraints.

B. Phase Retrieval Combining Holographic Methods

  • Holography interferes an image-carrying electromagnetic field with a same-frequency field of known structure, the reference wave.The reference is typically simple, such as an approximately constant-amplitude, constant-phase plane wave.Source: B. Phase Retrieval Combining… · p. 20
  • A hologram records the squared magnitude of the sum of image and reference waves, embedding image information in its transmission function.Illuminating the hologram with the known reference wave reconstructs the image multiplied by a constant.Source: B. Phase Retrieval Combining… · p. 20
  • Fourier holography adds information to phase-retrieval measurements by including a known wave in the far-field intensity.A tiny hole at a predetermined position creates a wave with a tilted phase, which interferes with the sought image's Fourier spectrum.Source: B. Phase Retrieval Combining… · p. 20
  • The added prior information can improve algorithmic recovery resolution or relax prior support constraints.Holographic methods have been used with X-rays and electrons,,.Source: B. Phase Retrieval Combining… · p. 20

C. Challenges, Bottlenecks and Vision

The review identifies unmet goals in resolution, object complexity, and real-time imaging, alongside unresolved physical challenges at short wavelengths and ultrashort timescales. It points to new signal-processing methods for recovering information from measurements as a route toward imaging complex biological systems at atomic scale and in real time.

C. Challenges, Bottlenecks and Vision

  • Current imaging has not reached atomic, sub-nanometer resolution, and nanometric imaging cannot handle objects whose support is extremely large relative to resolution.The stated vision includes deciphering atomic constituents and their interactions with other molecules.Source: C. Challenges, Bottlenecks and Vision · p. 20
  • Ultrafast methods have probed electron dynamics and tunneling, but have not yet imaged even a simple molecule.The review identifies complex biological structures as further beyond current ultrafast imaging.Source: C. Challenges, Bottlenecks and Vision · p. 20
  • Generating coherent radiation in the hard X-ray regime remains a major obstacle and often requires very large facilities.The review also notes that fundamental limits on coherent X-ray flux with current methods are not known.Source: C. Challenges, Bottlenecks and Vision · p. 21
  • Attosecond CDI faces a broad-bandwidth diffraction pattern that superposes Fourier contents and requires new algorithms.Light-matter interactions during scattering at short wavelengths and ultrashort timescales are not yet fully understood.Source: C. Challenges, Bottlenecks and Vision · p. 21
  • The review envisions imaging complex biological systems at atomic scale in real time, a task complicated by simultaneous spatial and temporal scales and process redundancy.It identifies new signal-processing methods for recovering information from experimental measurements as a potential contribution toward that vision.Source: C. Challenges, Bottlenecks and Vision · p. 21

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