Phasebook: a survey of selected open problems in phase retrieval

The phase retrieval problem, where one aims to recover a complex-valued image from far-field intensity measurements, is a classic problem encountered in a range of imaging applications. Modern phase retrieval approaches usually rely on gradient descent methods in a nonlinear minimization framework. Calculating closed-form gradients for use in these methods is tedious work, and formulating second order derivatives is even more laborious. Additionally, second order techniques often require the storage and inversion of large matrices of partial derivatives, with memory requirements that can be prohibitive for data-rich imaging modalities. We use a reverse-mode automatic differentiation (AD) framework to implement an efficient matrix-free version of the Levenberg-Marquardt (LM) algorithm, a longstanding method that finds popular use in nonlinear least-square minimization problems but which has seen little use in phase retrieval. Furthermore, we extend the basic LM algorithm so that it can be applied for more general constrained optimization problems (including phase retrieval problems) beyond just the least-square applications. Since we use AD, we only need to specify the physics-based forward model for a specific imaging application; the first and second-order derivative terms are calculated automatically through matrix-vector products, without explicitly forming the large Jacobian or Gauss-Newton matrices typically required for the LM method. We demonstrate that this algorithm can be used to solve both the unconstrained ptychographic object retrieval problem and the constrained "blind" ptychographic object and probe retrieval problems, under the popular Gaussian noise model as well as the Poisson noise model. We compare this algorithm to state-of-the-art first order ptychographic reconstruction methods to demonstrate empirically that this method outperforms best-in-class first-order methods: it provides excellent convergence guarantees with (in many cases) a superlinear rate of convergence, all with a computational cost comparable to, or lower than, the tested first-order algorithms.

Effective phase retrieval of sparse signals with convergence guarantee

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. The signal recovery will be accomplished via an efficient One-bit Phase Retrieval Algorithm, referred to as OPeRA. Several numerical results are presented to illustrate the effectiveness of the proposed methodologies.

Phase retrieval from single interference fringe is important and effective method in obtaining the real phase distribution. The original phase can be retrieved by the line integral of its gradient expressed as sine and cosine components, which were gained by the Hilbert transform twice from a single interference fringe pattern. However, this method fails when the phase transformation of the interference fringe is too fast. In this paper, a novel method to recover the continuous phase of the whole field is proposed to solve the above problems. The shear interference technique is introduced into the phase retrieval method to build an exponential 2-D complex light field of natural base for the phase slope obtained by the Hilbert transform. Then, the expressions of phase slopes in x- and y-directions are constructed as a discrete Poisson equation. Therefore, the calculation of phase retrieval is equivalent to solve the discrete Poisson equation mathematically. Finally, the real phase is gotten by the weighted discrete cosine transform (WDCT) of the discrete Poisson equation. The simulation results verify the validity of this method and show that the proposed method can achieve the phase retrieval of the phase discontinuity in x- and y-directions, which leads to the under-sampled problem. It can restore the whole field phase distribution rapidly and accurately. Moreover, this method is applied to phase retrieval of interferometric synthetic aperture radar (InSAR) with the under-sampled problem in this paper. The experimental results show that this method can recover the phase of InSAR with the under-sampled problems caused by terrain abrupt change and so on. Compared with other commonly used methods, it achieved satisfactory results. This method provides a new idea for solving the under-sampled problem in the phase retrieval from a single-frame interference fringe.

The twin-image problem in phase retrieval is characterized by the simultaneous occurrence of features from the original object and its inversion about the origin (twin image). This problem can occur in reconstructions for which the object support is centrosymmetric or loose, and in severe cases it can greatly hinder image quality. In this paper we examine this problem and find that it arises when the retrieved Fourier-domain phase is divided into sets of regions, some of which reconstruct the object while others the twin. We examine sample reconstructions that present the twin-image problem to different extents and find that, even when the twin-image problem is not visually evident, it can exist in small regions of the retrieved Fourier phase. The reduced-support constraint approach is shown to be effective in escaping stagnation caused by the twin-image problem.

Stable phase retrieval from locally stable and conditionally connected measurements

The three-dimensional imaging properties of a light microscope are traditionally described through an intensity point spread function (PSF) or its Fourier transform, the optical transfer function (OTF). However, the imaging properties can be more compactly described by a generalized two-dimensional pupil function. Use of the pupil function allows easy modification of an observed PSF to introduce known aberrations, a much more difficult task when using a PSF or OTF. Unfortunately, it is not straightforward to determine the complex-valued pupil function from the measured intensity PSF, because of the lack of phase information. This is the problem of phase retrieval. Several phase retrieval algorithms have been developed for two-dimensional imaging in astronomy. We have modified one such algorithm to be appropriate for the high-aperture, non-paraxial case of high resolution light microscopy. PSFs reconstructed from phase-retrieved pupil functions, modified by calculated aberrations closely reproduce the features of measured, aberrated PSFs.

The problem of phase retrieval is to determine a signal fin mathcal {H}, with mathcal {H} a Hilbert space, from intensity measurements |F(omega )|, where F(omega ):=langle f, varphi _omega rangle are measurements of f with respect to a measurement system (varphi _omega )_{omega in Omega }subset mathcal {H}. Although phase retrieval is always stable in the finite-dimensional setting whenever it is possible (i.e. injectivity implies stability for the inverse problem), the situation is drastically different if mathcal {H} is infinite-dimensional: in that case phase retrieval is never uniformly stable (Alaifari and Grohs in SIAM J Math Anal 49(3):1895–1911, 2017; Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016); moreover, the stability deteriorates severely in the dimension of the problem (Cahill et al. 2016). On the other hand, all empirically observed instabilities are of a certain type: they occur whenever the function |F| of intensity measurements is concentrated on disjoint sets D_jsubset Omega , i.e. when F= sum _{j=1}^k F_j where each F_j is concentrated on D_j (and k ge 2). Motivated by these considerations, we propose a new paradigm for stable phase retrieval by considering the problem of reconstructing F up to a phase factor that is not global, but that can be different for each of the subsets D_j, i.e. recovering F up to the equivalence F∼∑j=1keiαjFj.\\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}$$\\begin{aligned} F \\sim \\sum _{j=1}^k e^{\\mathrm {i}\\alpha _j} F_j. \\end{aligned}$$\\end{document}We present concrete applications (for example in audio processing) where this new notion of stability is natural and meaningful and show that in this setting stable phase retrieval can actually be achieved, for instance, if the measurement system is a Gabor frame or a frame of Cauchy wavelets.

Linear algorithms for phase retrieval in the Fresnel region

We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main result states that also in this framework, the problem of phase retrieval is never uniformly stable in infinite dimensions. On the other hand, we show weak stability of the problem. This complements recent work by Cahill, Casazza, and Daubechies [Trans. Amer. Math. Soc. Ser. B, 3 (2016), pp. 63--76], where it has been shown that phase retrieval is always unstable for the setting of discrete frames in Hilbert spaces. In particular, our result implies that the stability properties cannot be improved by oversampling the underlying discrete frame. We generalize the notion of complement property (CP) to the setting of continuous frames for Banach spaces (over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$) and verify that it is a necessary condition for uniqueness of the p...

In this paper we show that the problem of phase retrieval can be efficiently and provably solved via an alternating minimization algorithm suitably initialized. Our initialization is based on One Bit Phase Retrieval that we introduced in [1], where we showed that O(n log(n)) Gaussian phase-less measurements ensure robust recovery of the phase. In this paper we improve the sample complexity bound to O(n) measurements for sufficiently large n, using a variant of Matrix Bernstein concentration inequality that exploits the intrinsic dimension, together with properties of one bit phase retrieval.

There are several important remote sensing applications where the development of Ground Penetrating Synthetic Aperture Radar (GPENSAR) is the logical approach, e.g., searching for buried military facilities, minefield mapping, survey of underground pipelines. Penetration of sufficient soil depth for useful results require a SAR to operate at VHF/UHF frequencies, e.g., 200 - 300 MHz. At these frequencies a satellite SAR will encounter substantial distortion in the double passage of the SAR signal through the ionosphere. One of the ionospheric distortions is equivalent the phase aberrations caused in imaging through the turbulent atmosphere, and the problem of phase retrieval for the GPENSAR becomes a necessity. For GPENSR there are imaging concepts that exploit dual polarization radiation of the SAR pulse. The phase retrieval problem then becomes one of compensation for the phase aberrations induced in each of the polarization components returned to the satellite receiver. We discuss the use of the two polarizations to cancel the ionospheric phase aberrations. Unfortunately, the resulting signal has only relative phase of the two polarizations. We discuss an algorithm for the retrieval of the absolute phase. The algorithm is based on an optimization approach. Although phase retrieval by optimization is difficult because of local minima, the retrieval of absolute phase in the dual polarization case is substantially less difficult, because the two polarizations constrain the solution sufficiently to eliminate many local minima.

The classical problem of phase retrieval arises in various signal acquisition systems. Due to the ill-posed nature of the problem, the solution requires assumptions on the structure of the signal. In the last several years, sparsity and support-based priors have been leveraged successfully to solve this problem. In this work, we propose replacing the sparsity/support priors with generative priors and propose two algorithms to solve the phase retrieval problem. Our proposed algorithms combine the ideas from AltMin approach for non-convex sparse phase retrieval and projected gradient descent approach for solving linear inverse problems using generative priors. We empirically show that the performance of our method with projected gradient descent is superior to the existing approach for solving phase retrieval under generative priors. We support our method with an analysis of sample complexity with Gaussian measurements.

This paper considers the problem of sparse phase retrieval for Frequency Domain Optical Coherence Tomography (FDOCT). Existing phase retrieval algorithms typically require larger number of measurements and the reconstruction is accurate only up to some ambiguities that are inherent in Fourier phase retrieval. In this paper, we overcome these drawbacks by developing a compressive differential FDOCT (dFDOCT) technique that uses li minimization for reconstructing the signal from a pair of differential phaseless measurements. Theoretical guarantees are developed which establish that the proposed method requires minimal number of measurements for sparse phase retrieval. Numerical results demonstrate the superior performance of this method for signals of large length and sparsity without any ambiguities.

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.