Phase Retrieval Problem Research Articles

AutoPhaseNN: unsupervised physics-aware deep learning of 3D nanoscale Bragg coherent diffraction imaging

The problem of phase retrieval underlies various imaging methods from astronomy to nanoscale imaging. Traditional phase retrieval methods are iterative and are therefore computationally expensive. Deep learning (DL) models have been developed to either provide learned priors or completely replace phase retrieval. However, such models require vast amounts of labeled data, which can only be obtained through simulation or performing computationally prohibitive phase retrieval on experimental datasets. Using 3D X-ray Bragg coherent diffraction imaging (BCDI) as a representative technique, we demonstrate AutoPhaseNN, a DL-based approach which learns to solve the phase problem without labeled data. By incorporating the imaging physics into the DL model during training, AutoPhaseNN learns to invert 3D BCDI data in a single shot without ever being shown real space images. Once trained, AutoPhaseNN can be effectively used in the 3D BCDI data inversion about 100× faster than iterative phase retrieval methods while providing comparable image quality.

On the effect of zero-flipping on the stability of the phase retrieval problem in the Paley-Wiener class

In the classical phase retrieval problem in the Paley-Wiener class \(PW_L\) for \(L>0\), i.e. to recover \(f\in PW_L\) from |f|, Akutowicz, Walther, and Hofstetter independently showed that all such solutions can be obtained by flipping an arbitrary set of complex zeros across the real line. This operation is called zero-flipping and we denote by \({\mathfrak {F}}_a f\) the resulting function. The operator \({\mathfrak {F}}_a\) is defined even if a is not a genuine zero of f, that is if we make an error on the location of the zero. Our main goal is to investigate the effect of \({\mathfrak {F}}_a\). We show that \({\mathfrak {F}}_af\) is no longer bandlimited but is still wide-banded. We then investigate the effect of \({\mathfrak {F}}_a\) on the stability of phase retrieval by estimating the quantity \(\inf _{|c|=1}\Vert cf-{\mathfrak {F}}_af\Vert _2\). We then investigate the stability of a double zero-flipping \(\inf _{|c|=1}\Vert cf-{\mathfrak {F}}_{{\bar{a}}}{\mathfrak {F}}_bf\Vert _2\) which turns out to be also the stability between two zero-flipping \(\inf _{|c|=1}\Vert c{\mathfrak {F}}_bf-{\mathfrak {F}}_af\Vert _2\). We show that this quantity is dominated by the distance between a and b.

Solving phase retrieval with random initial guess is nearly as good as by spectral initialization

The problem of recovering a signal x∈Rn from a set of magnitude measurements yi=|〈ai,x〉|,i=1,…,m is referred as phase retrieval, which has many applications in fields of physical sciences and engineering. In this paper we show that the smoothed amplitude flow based model for phase retrieval has benign geometric structure under the optimal sampling complexity. In particular, we show that when the measurements ai∈Rn are Gaussian random vectors and the number of measurements m≥Cn, our smoothed amplitude flow based model has no spurious local minimizers with high probability, i.e., the target solution x is the unique global minimizer (up to a global phase) and the loss function has a negative directional curvature around each saddle point. Due to this benign geometric landscape, the phase retrieval problem can be solved by the gradient descent algorithms without spectral initialization. Numerical experiments show that the gradient descent algorithm with random initialization performs well even comparing with state-of-the-art algorithms with spectral initialization in empirical success rate and convergence speed.

Fourier Ptychographic Microscopy via Alternating Direction Method of Multipliers

Fourier ptychographic microscopy (FPM) has risen as a promising computational imaging technique that breaks the trade-off between high resolution and large field of view (FOV). Its reconstruction is normally formulated as a blind phase retrieval problem, where both the object and probe have to be recovered from phaseless measured data. However, the stability and reconstruction quality may dramatically deteriorate in the presence of noise interference. Herein, we utilized the concept of alternating direction method of multipliers (ADMM) to solve this problem (termed ADMM-FPM) by breaking it into multiple subproblems, each of which may be easier to deal with. We compared its performance against existing algorithms in both simulated and practical FPM platform. It is found that ADMM-FPM method belongs to a global optimization algorithm with a high degree of parallelism and thus results in a more stable and robust phase recovery under noisy conditions. We anticipate that ADMM will rekindle interest in FPM as more modifications and innovations are implemented in the future.

Phase retrieval from multiple FRFT measurements based on nonconvex low-rank minimization

Phase retrieval (PR) problems aim to recover the phase from intensity measurements and are relevant in many applications such as X-ray crystallography and diffraction imaging. Most existing phase retrieval algorithms apply mask-based Fourier transform (FT) measurements to provide redundant information for signal reconstruction. This gives rise to a series of problems with masks such as the choice of suitable masks. In this work, we prove the rationality of multiple fractional Fourier transform (MFRFT) measurements, which can be applied to replace mask-based FT measurements. In addition, based on the low rank property (rank-1) of the desired matrix, we present a new low-rank model for PR problems using truncated nuclear norm regularization (TNNR), called TNNR-MFRFT. Extensive experiments validate the superiority of our approach over the state-of-the-art algorithms.

Reweighted L1-norm regularized phase retrieval for x-ray differential phase contrast radiograph.

Talbot-Lau x-ray grating interferometry greatly decreases the requirements on x-ray sources to realize differential phase contrast imaging and has found many applications in industrial and medical imaging. Phase retrieval from the noisy differential signal is crucial for quantitative analysis, comparison, and fusion with other imaging modalities. In this paper, we introduce a reweighted L1-norm based nonlinear regularization method for the phase retrieval problem. Both simulation and experimental results demonstrated that, comparing with the widely used L1-norm based regularization method and Wiener filter method, the proposed method is more effective both in eliminating the strip noises and in preserving the image detail.

On Foundational Discretization Barriers in STFT Phase Retrieval

We prove that there exists no window function g in {L^2(mathbb {R})} and no lattice {mathcal {L}} subset mathbb {R}^2 such that every f in {L^2(mathbb {R})} is determined up to a global phase by spectrogram samples |V_gf({mathcal {L}})| where V_gf denotes the short-time Fourier transform of f with respect to g. Consequently, the forward operator f↦|Vgf(L)|\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} f \\mapsto |V_gf({\\mathcal {L}})| \\end{aligned}$$\\end{document}mapping a square-integrable function to its spectrogram samples on a lattice is never injective on the quotient space with f sim h identifying two functions which agree up to a multiplicative constant of modulus one. We will further elaborate this result and point out that under mild conditions on the lattice {mathcal {L}}, functions which produce identical spectrogram samples but do not agree up to a unimodular constant can be chosen to be real-valued. The derived results highlight that in the discretization of the STFT phase retrieval problem from lattice measurements, a prior restriction of the underlying signal space to a proper subspace of {L^2(mathbb {R})} is inevitable.

Circumcentering Reflection Methods for Nonconvex Feasibility Problems

Recently, circumcentering reflection method (CRM) has been introduced for solving the feasibility problem of finding a point in the intersection of closed constraint sets. It is closely related with Douglas–Rachford method (DR). We prove local convergence of CRM in the same prototypical settings of most theoretical analysis of regular nonconvex DR, whose consideration is made natural by the geometry of the phase retrieval problem. For the purpose, we show that CRM is related to the method of subgradient projections. For many cases when DR is known to converge to a feasible point, we establish that CRM locally provides a better convergence rate. As a root finder, we show that CRM has local convergence whenever Newton–Raphson method does, has quadratic rate whenever Newton–Raphson method does, and exhibits superlinear convergence in many cases when Newton–Raphson method fails to converge at all. We also obtain explicit regions of convergence. As an interesting aside, we demonstrate local convergence of CRM to feasible points in cases when DR converges to fixed points that are not feasible. We demonstrate an extension in higher dimensions, and use it to obtain convergence rate guarantees for sphere and subspace feasibility problems. Armed with these guarantees, we experimentally discover that CRM is highly sensitive to compounding numerical error that may cause it to achieve worse rates than those guaranteed by theory. We then introduce a numerical modification that enables CRM to achieve the theoretically guaranteed rates. Any future works that study CRM for product space formulations of feasibility problems should take note of this sensitivity and account for it in numerical implementations.

Phase retrieval from incomplete data via weighted nuclear norm minimization

Recovering an unknown object from the magnitude of its Fourier transform is a phase retrieval problem. Here, we consider a much difficult case, where those observed intensity values are incomplete and contaminated by both salt-and-pepper and random-valued impulse noise. To take advantage of the low-rank property within the image of the object, we use a regularization term which penalizes high weighted nuclear norm values of image patch groups. For outliers (impulse noise) in the observation, the ℓ1−2 metric is adopted as the data fidelity term. Then we break down the resulting optimization problem into smaller ones, for example, weighted nuclear norm proximal mapping and ℓ1−2 minimization, because the nonconvex and nonsmooth subproblems have available closed-form solutions. The convergence results are also presented, and numerical experiments are provided to demonstrate the superior reconstruction quality of the proposed method.

One-Bit Phase Retrieval: More Samples Means Less Complexity?

The classical problem of phase retrieval has found a wide array of applications in optics, imaging and signal processing. In this paper, we consider the phase retrieval problem in a one-bit setting, where the signals are sampled using one-bit analog-to-digital converters (ADCs). A significant advantage of deploying one-bit ADCs in signal processing systems is their superior sampling rates as compared to their high-resolution counterparts. This leads to an enormous amount of one-bit samples gathered at the output of the ADC in a short period of time. We demonstrate that this advantage pays extraordinary dividends when it comes to convex phase retrieval formulations, namely that the often encountered matrix semi-definiteness constraints as well as rank constraints (that are computationally prohibitive to enforce), become redundant for phase retrieval in the face of a growing sample size. Several numerical results are presented to illustrate the effectiveness of the proposed methodologies.

Total Least Squares Phase Retrieval

We address the phase retrieval problem with errors in the sensing vectors. A number of recent methods for phase retrieval are based on least squares (LS) formulations which assume errors in the quadratic measurements. We extend this approach to handle errors in the sensing vectors by adopting the total least squares (TLS) framework that is used in linear inverse problems with operator errors. We show how gradient descent and the specific geometry of the phase retrieval problem can be used to obtain a simple and efficient TLS solution. Additionally, we derive the gradients of the TLS and LS solutions with respect to the sensing vectors and measurements which enables us to calculate the solution errors. By analyzing these error expressions we determine conditions under which each method should outperform the other. We run simulations to demonstrate that our method can lead to more accurate solutions. We further demonstrate the effectiveness of our approach by performing phase retrieval experiments on real optical hardware which naturally contains both sensing vector and measurement errors.

Learning the Proximity Operator in Unfolded ADMM for Phase Retrieval

This paper considers the phase retrieval (PR) problem, which aims to reconstruct a signal from phaseless measurements such as magnitude or power spectrograms. PR is generally handled as a minimization problem involving a quadratic loss. Recent works have considered alternative discrepancy measures, such as the Bregman divergences, but it is still challenging to tailor the optimal loss for a given setting. In this paper we propose a novel strategy to automatically learn the optimal metric for PR. We unfold a recently introduced ADMM algorithm into a neural network, and we emphasize that the information about the loss used to formulate the PR problem is conveyed by the proximity operator involved in the ADMM updates. Therefore, we replace this proximity operator with trainable activation functions: learning these in a supervised setting is then equivalent to learning an optimal metric for PR. Experiments conducted with speech signals show that our approach outperforms the baseline ADMM, using a light and interpretable neural architecture.

Sparse signal recovery from phaseless measurements via hard thresholding pursuit

In this paper, we consider the sparse phase retrieval problem, recovering an $s$-sparse signal $\bm{x}^{\natural}\in\mathbb{R}^n$ from $m$ phaseless samples $y_i=|\langle\bm{x}^{\natural},\bm{a}_i\rangle|$ for $i=1,\ldots,m$. Existing sparse phase retrieval algorithms are usually first-order and hence converge at most linearly. Inspired by the hard thresholding pursuit (HTP) algorithm in compressed sensing, we propose an efficient second-order algorithm for sparse phase retrieval. Our proposed algorithm is theoretically guaranteed to give an exact sparse signal recovery in finite (in particular, at most $O(\log m + \log(\|\bm{x}^{\natural}\|_2/|x_{\min}^{\natural}|))$) steps, when $\{\bm{a}_i\}_{i=1}^{m}$ are i.i.d. standard Gaussian random vector with $m\sim O(s\log(n/s))$ and the initialization is in a neighborhood of the underlying sparse signal. Together with a spectral initialization, our algorithm is guaranteed to have an exact recovery from $O(s^2\log n)$ samples. Since the computational cost per iteration of our proposed algorithm is the same order as popular first-order algorithms, our algorithm is extremely efficient. Experimental results show that our algorithm can be several times faster than existing sparse phase retrieval algorithms.

Sample-Efficient Sparse Phase Retrieval via Stochastic Alternating Minimization

In this work we propose a nonconvex two-stage \underline{s}tochastic \underline{a}lternating \underline{m}inimizing (SAM) method for sparse phase retrieval. The proposed algorithm is guaranteed to have an exact recovery from $O(s\log n)$ samples if provided the initial guess is in a local neighbour of the ground truth. Thus, the proposed algorithm is two-stage, first we estimate a desired initial guess (e.g. via a spectral method), and then we introduce a randomized alternating minimization strategy for local refinement. Also, the hard-thresholding pursuit algorithm is employed to solve the sparse constraint least square subproblems. We give the theoretical justifications that SAM find the underlying signal exactly in a finite number of iterations (no more than $O(\log m)$ steps) with high probability. Further, numerical experiments illustrates that SAM requires less measurements than state-of-the-art algorithms for sparse phase retrieval problem.

PDMM: A Novel Primal-Dual Majorization-Minimization Algorithm for Poisson Phase-Retrieval Problem

In this paper, we introduce a novel iterative algorithm for the problem of phase-retrieval where the measurements consist of only the magnitude of linear function of the unknown signal, and the noise in the measurements follow Poisson distribution. The proposed algorithm is based on the principle of majorization-minimization (MM); however, the application of MM here is very novel and distinct from the way MM has been usually used to solve optimization problems in the literature. More precisely, we reformulate the original minimization problem into a saddle point problem by invoking Fenchel dual representation of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\log (\cdot)$</tex-math></inline-formula> term in the Poisson likelihood function. We then propose tighter surrogate functions over both primal and dual variables resulting in a double-loop MM algorithm, which we have named as Primal-Dual Majorization-Minimization (PDMM) algorithm. The iterative steps of the resulting algorithm are simple to implement and involve only computing matrix vector products. We also extend our algorithm to handle various <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1}$</tex-math></inline-formula> regularized Poisson phase-retrieval problems (which exploit sparsity). The proposed algorithm is compared with previously proposed algorithms such as wirtinger flow (WF), MM (conventional), and alternating direction methods of multipliers (ADMM) for the Poisson data model. The simulation results under different experimental settings show that PDMM is faster than the competing methods, and its performance in recovering the original signal is at par with the state-of-the-art algorithms.

PGPAL: A Monotonic Iterative Algorithm for Phase-Retrieval Under the Presence of Poisson-Gaussian Noise

In this paper, we propose an iterative algorithm named PGPAL for the phase-retrieval problem under the scenario when the measurements follow Poisson plus Gaussian (PG) distribution, which is a more realistic model in applications like astronomy, microscopy, medical imaging, and remote sensing. The proposed algorithm is based on majorization-minimization (MM) framework, in which a simple surrogate function that tightly upperbounds the challenging underlying maximum-likelihood (ML) objective is constructed and minimized iteratively. The proposed algorithm monotonically decreases the ML objective and the algorithm is guaranteed to converge to a stationary point of the ML objective. Numerical simulations are performed for the one-dimensional phase-retrieval case, and the performance of the proposed method is compared with the recently proposed Poisson phase-retrieval algorithm, which assumes the measurement model to be only Poisson distributed. The performance of the proposed MM-based algorithm in terms of the normalized root mean square error (NRMSE) is found to be much better in comparison to the method for Poisson phase-retrieval.

Global convergence of model function based Bregman proximal minimization algorithms

Lipschitz continuity of the gradient mapping of a continuously differentiable function plays a crucial role in designing various optimization algorithms. However, many functions arising in practical applications such as low rank matrix factorization or deep neural network problems do not have a Lipschitz continuous gradient. This led to the development of a generalized notion known as the L-smad property, which is based on generalized proximity measures called Bregman distances. However, the L-smad property cannot handle nonsmooth functions, for example, simple nonsmooth functions like vert x^4-1 vert and also many practical composite problems are out of scope. We fix this issue by proposing the MAP property, which generalizes the L-smad property and is also valid for a large class of structured nonconvex nonsmooth composite problems. Based on the proposed MAP property, we propose a globally convergent algorithm called Model BPG, that unifies several existing algorithms. The convergence analysis is based on a new Lyapunov function. We also numerically illustrate the superior performance of Model BPG on standard phase retrieval problems and Poisson linear inverse problems, when compared to a state of the art optimization method that is valid for generic nonconvex nonsmooth optimization problems.

Carrier fringe pattern analysis: Links between methods

Introducing a carrier into a fringe pattern makes the phase retrieval problem well-posed, and has become one of the pillar techniques in optical metrology. The carrier fringe pattern analysis for high-accuracy phase retrieval is hence important, for which, many well-known methods have been developed, such as the Fourier transform method, the windowed Fourier transform based methods (the windowed Fourier filtering and the windowed Fourier ridges), the spatial carrier phase-shifting method, and the sampling moiré method. These methods were developed independently and seemingly different. The purpose of this paper is to reveal the close relationships between them, and to connect these methods into a tight-knit methodology family.

On the sampling requirements for ptychography

We investigate how the distance between probe positions and the resolution of measured diffraction patterns affects iterative ptychographic reconstruction. The ptychographic phase retrieval problem where both the object and probe are reconstructed is analyzed theoretically, by introducing a 4D object to describe the 4D ptychopgraphical data set. This is similar to introducing the 3D Ewald sphere to describe the field in a 3D focal volume. Conclusions drawn on the basis of the theoretical analysis are verified using simulations with a simplified 2D ptychographic problem where the probe and object are 1D. Though the 2D problem is a simplification compared to the typical 4D ptychography problem of lensless imaging, it still has practical applications for pulse reconstruction.

Asymptotic Properties of Stationary Solutions of Coupled Nonconvex Nonsmooth Empirical Risk Minimization

This paper has two main goals: (a) establish several statistical properties—consistency, asymptotic distributions, and convergence rates—of stationary solutions and values of a class of coupled nonconvex and nonsmooth empirical risk-minimization problems and (b) validate these properties by a noisy amplitude-based phase-retrieval problem, the latter being of much topical interest. Derived from available data via sampling, these empirical risk-minimization problems are the computational workhorse of a population risk model that involves the minimization of an expected value of a random functional. When these minimization problems are nonconvex, the computation of their globally optimal solutions is elusive. Together with the fact that the expectation operator cannot be evaluated for general probability distributions, it becomes necessary to justify whether the stationary solutions of the empirical problems are practical approximations of the stationary solution of the population problem. When these two features, general distribution and nonconvexity, are coupled with nondifferentiability that often renders the problems “non-Clarke regular,” the task of the justification becomes challenging. Our work aims to address such a challenge within an algorithm-free setting. The resulting analysis is, therefore, different from much of the analysis in the recent literature that is based on local search algorithms. Furthermore, supplementing the classical global minimizer-centric analysis, our results offer a promising step to close the gap between computational optimization and asymptotic analysis of coupled, nonconvex, nonsmooth statistical estimation problems, expanding the former with statistical properties of the practically obtained solution and providing the latter with a more practical focus pertaining to computational tractability.