In this paper, we develop a framework of generalized phase retrieval in which one aims to reconstruct a vector ${\mathbf x}$ in ${\mathbb R}^d$ or ${\mathbb C}^d$ through quadratic samples ${\mathbf x}^*A_1{\mathbf x}, \dots, {\mathbf x}^*A_N{\mathbf x}$. The generalized phase retrieval includes as special cases the standard phase retrieval as well as the phase retrieval by orthogonal projections. We first explore the connections among generalized phase retrieval, low-rank matrix recovery and nonsingular bilinear form. Motivated by the connections, we present results on the minimal measurement number needed for recovering a matrix that lies in a set $W\in {\mathbb C}^{d\times d}$. Applying the results to phase retrieval, we show that generic $d \times d$ matrices $A_1,\ldots, A_N$ have the phase retrieval property if $N\geq 2d-1$ in the real case and $N \geq 4d-4$ in the complex case for very general classes of $A_1,\ldots,A_N$, e.g. matrices with prescribed ranks or orthogonal projections. Our method also leads to a novel proof for the classical Stiefel-Hopf condition on nonsingular bilinear form. We also give lower bounds on the minimal measurement number required for generalized phase retrieval. For several classes of dimensions $d$ we obtain the precise values of the minimal measurement number. Our work unifies and enhances results from the standard phase retrieval, phase retrieval by projections and low-rank matrix recovery.
In this work we analyze the problem of phase retrieval from Fourier measurements with random diffraction patterns. To this end, we consider the recently introduced PhaseLift algorithm, which expresses the problem in the language of convex optimization. We provide recovery guarantees which require O(log^2 d) different diffraction patterns, thus improving on recent results by Candes et al. [arXiv:1310.3240], which require O(log^4 d) different patterns.
We consider the classical 1D phase retrieval problem. In order to overcome the difficulties associated with phase retrieval from measurements of the Fourier magnitude, we treat recovery from the magnitude of the short-time Fourier transform (STFT). We first show that the redundancy offered by the STFT enables unique recovery for arbitrary nonvanishing inputs, under mild conditions. An efficient algorithm for recovery of a sparse input from the STFT magnitude is then suggested, based on an adaptation of the recently proposed GESPAR algorithm. We demonstrate through simulations that using the ST…
This contribution proposes a two stage strategy to allow for phase retrieval in state of the art sub-Nyquist sampling schemes for sparse multiband signals. The proposed strategy is based on data acquisition via modulated wideband converters known from sub-Nyquist sampling. This paper describes how the modulators have to be modified such that signal recovery from sub-Nyquist amplitude samples becomes possible and a corresponding recovery algorithm is given which is computational efficient. In addition, the proposed strategy is fairly general, allowing for several constructions and recovery algo…
The aim of this paper is to study the stability of the $\ell_1$ minimization for the compressive phase retrieval and to extend the instance-optimality in compressed sensing to the real phase retrieval setting. We first show that the $m={\mathcal O}(k\log(N/k))$ measurements is enough to guarantee the $\ell_1$ minimization to recover $k$-sparse signals stably provided the measurement matrix $A$ satisfies the strong RIP property. We second investigate the phaseless instance-optimality with presenting a null space property of the measurement matrix $A$ under which there exists a decoder $Δ$ so th…
We present the optimal design of a spectral method widely used to initialize nonconvex optimization algorithms for solving phase retrieval and other signal recovery problems. Our work leverages recent results that provide an exact characterization of the performance of the spectral method in the high-dimensional limit. This characterization allows us to map the task of optimal design to a constrained optimization problem in a weighted $L^2$ function space. The latter has a closed-form solution. Interestingly, under a mild technical condition, our results show that there exists a fixed design t…
The support recovery problem consists of determining a sparse subset of variables that is relevant in generating a set of observations. In this paper, we study the support recovery problem in the phase retrieval model consisting of noisy phaseless measurements, which arises in a diverse range of settings such as optical detection, X-ray crystallography, electron microscopy, and coherent diffractive imaging. Our focus is on information-theoretic fundamental limits under an approximate recovery criterion, considering both discrete and Gaussian models for the sparse non-zero entries, along with G…
The problem of recovering a signal from its Fourier magnitude is of paramount importance in various fields of engineering and applied physics. Due to the absence of Fourier phase information, some form of additional information is required in order to be able to uniquely, efficiently and robustly identify the underlying signal. Inspired by practical methods in optical imaging, we consider the problem of signal reconstruction from the Short-Time Fourier Transform (STFT) magnitude. We first develop conditions under which the STFT magnitude is an almost surely unique signal representation. We the…
This paper considers the recovery of continuous time signals from the magnitude of its samples. It uses a combination of structured modulation and oversampling and provides sufficient conditions on the signal and the sampling system such that signal recovery is possible. In particular, it is shown that an average sampling rate of four times the Nyquist rate is sufficient to reconstruct a signal from its magnitude measurements.
We consider the recovery of a (real- or complex-valued) signal from magnitude-only measurements, known as phase retrieval. We formulate phase retrieval as a convex optimization problem, which we call PhaseMax. Unlike other convex methods that use semidefinite relaxation and lift the phase retrieval problem to a higher dimension, PhaseMax is a "non-lifting" relaxation that operates in the original signal dimension. We show that the dual problem to PhaseMax is Basis Pursuit, which implies that phase retrieval can be performed using algorithms initially designed for sparse signal recovery. We dev…
The problem of signal recovery from its Fourier transform magnitude is of paramount importance in various fields of engineering and has been around for over 100 years. Due to the absence of phase information, some form of additional information is required in order to be able to uniquely identify the signal of interest. In this work, we focus our attention on discrete-time sparse signals (of length $n$). We first show that, if the DFT dimension is greater than or equal to $2n$, almost all signals with {\em aperiodic} support can be uniquely identified by their Fourier transform magnitude (up t…
The paper considers the phase retrieval problem in N-dimensional complex vector spaces. It provides two sets of deterministic measurement vectors which guarantee signal recovery for all signals, excluding only a specific subspace and a union of subspaces, respectively. A stable analytic reconstruction procedure of low complexity is given. Additionally it is proven that signal recovery from these measurements can be solved exactly via a semidefinite program. A practical implementation with 4 deterministic diffraction patterns is provided and some numerical experiments with noisy measurements co…
In this paper, we study the problem of robust phase recovery. We investigate a novel approach based on extremely quantized (one-bit) phase-less measurements and a corresponding recovery scheme. The proposed approach has surprising robustness and stability properties and, unlike currently available methods, allows to efficiently perform phase recovery from measurements affected by severe (possibly unknown) non-linear perturbations, such as distortions (e.g. clipping). Beyond robustness, we show how our approach can be used within greedy approaches based on alternating minimization. In particula…
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 addit…
Phase retrieval aims at recovering a complex-valued signal from magnitude-only measurements, which attracts much attention since it has numerous applications in many disciplines. However, phase recovery involves solving a system of quadratic equations, indicating that it is a challenging nonconvex optimization problem. To tackle phase retrieval in an effective and efficient manner, we apply coordinate descent (CD) such that a single unknown is solved at each iteration while all other variables are kept fixed. As a result, only minimization of a univariate quartic polynomial is needed which is …
In phase retrieval we want to recover an unknown signal $\boldsymbol x\in\mathbb C^d$ from $n$ quadratic measurements of the form $y_i = |\langle{\boldsymbol a}_i,{\boldsymbol x}\rangle|^2+w_i$ where $\boldsymbol a_i\in \mathbb C^d$ are known sensing vectors and $w_i$ is measurement noise. We ask the following weak recovery question: what is the minimum number of measurements $n$ needed to produce an estimator $\hat{\boldsymbol x}(\boldsymbol y)$ that is positively correlated with the signal $\boldsymbol x$? We consider the case of Gaussian vectors $\boldsymbol a_i$. We prove that - in the hig…
We prove new results about the robustness of well-known convex noise-blind optimization formulations for the reconstruction of low-rank matrices from underdetermined linear measurements. Our results are applicable for symmetric rank-one measurements as used in a formulation of the phase retrieval problem. We obtain these results by establishing that with high probability rank-one measurement operators defined by i.i.d. Gaussian vectors exhibit the so-called Schatten-1 quotient property, which corresponds to a lower bound for the inradius of their image of the nuclear norm (Schatten-1) unit bal…
Phase retrieval consists in the recovery of an unknown signal from phaseless measurements of its usually complex-valued Fourier transform. Without further assumptions, this problem is notorious to be severe ill posed such that the recovery of the true signal is nearly impossible. In certain applications like crystallography, speckle imaging in astronomy, or blind channel estimation in communications, the unknown signal has a specific, sparse structure. In this paper, we exploit these sparse structure to recover the unknown signal uniquely up to inevitable ambiguities as global phase shifts, tr…
We consider the task of recovering two real or complex $m$-vectors from phaseless Fourier measurements of their circular convolution. Our method is a novel convex relaxation that is based on a lifted matrix recovery formulation that allows a nontrivial convex relaxation of the bilinear measurements from convolution. We prove that if the two signals belong to known random subspaces of dimensions $k$ and $n$, then they can be recovered up to the inherent scaling ambiguity with $m >> (k+n) \log^2 m$ phaseless measurements. Our method provides the first theoretical recovery guarantee for this prob…
A new sparse signal recovery algorithm for multiple-measurement vectors (MMV) problem is proposed in this paper. The sparse representation is iteratively drawn based on the idea of zero-point attracting projection (ZAP). In each iteration, the solution is first updated along the negative gradient direction of an approximate $\ell_{2,0}$ norm to encourage sparsity, and then projected to the solution space to satisfy the under-determined equation. A variable step size scheme is adopted further to accelerate the convergence as well as to improve the recovery accuracy. Numerical simulations demons…
Phase retrieval in dynamical sampling is a novel research direction, where an unknown signal has to be recovered from the phaseless measurements with respect to a dynamical frame, i.e. a sequence of sampling vectors constructed by the repeated action of an operator. The loss of the phase here turns the well-posed dynamical sampling into a severe ill-posed inverse problem. In the existing literature, the involved operator is usually completely known. In this paper, we combine phase retrieval in dynamical sampling with the identification of the system. For instance, if the dynamical frame is bas…
National Sun Yat-sen University（国立中山大学）；National Chiao Tung University（国立交通大学）；Southeast University（东南大学）
Phase retrieval (PR) is an important component in modern computational imaging systems. Many algorithms have been developed over the past half-century. Recent advances in deep learning have introduced new possibilities for a robust and fast PR. An emerging technique called deep unfolding provides a systematic connection between conventional model-based iterative algorithms and modern data-based deep learning. Unfolded algorithms, which are powered by data learning, have shown remarkable performance and convergence speed improvement over original algorithms. Despite their potential, most existi…
We consider the task of recovering two real or complex $m$-vectors from phaseless Fourier measurements of their circular convolution. Our method is a novel convex relaxation that is based on a lifted matrix recovery formulation that allows a nontrivial convex relaxation of the bilinear measurements from convolution. We prove that if the two signals belong to known random subspaces of dimensions $k$ and $n$, then they can be recovered up to the inherent scaling ambiguity with $m \gg (k+n) \log^2 m$ phaseless measurements. Our method provides the first theoretical recovery guarantee for this pro…
In this paper, we will introduce the notion of {\it conjugate phase retrieval}, which is a relaxed definition of phase retrieval allowing recovery of signals up to conjugacy as well as a global phase factor. It is known that frames of real vectors are never phase retrievable on ${\mathbb C}^M$ in the ordinary sense, but we show that they can be conjugate phase retrievable in complex vector spaces. We continue to develop the theory on conjugate phase retrievable real frames. In particular, a complete characterization of conjugate phase retrievable real frames on ${\mathbb C}^2$ and ${\mathbb C}…
In this note we show that stable recovery of complex-valued signals $x\in\mathbb{C}^n$ up to global sign can be achieved from the magnitudes of $4n-1$ Fourier measurements when a certain "symmetrization and zero-padding" is performed before measurement ($4n-3$ is possible in certain cases). For real signals, symmetrization itself is linear and therefore our result is in this case a statement on uniform phase retrieval. Since complex conjugation is involved, such measurement procedure is not complex-linear but recovery is still possible from magnitudes of linear measurements on, for example, $(…
In this paper we study a realistic setup for phase retrieval, where the signal of interest is modulated or masked and then for each modulation or mask a diffraction pattern is collected, producing a coded diffraction pattern (CDP) [CLM13]. We are interested in the setup where the resolution of the collected CDP is limited by the Fraunhofer diffraction limit of the imaging system. We investigate a novel approach based on a geometric quantization scheme of phase-less linear measurements into (one-bit) coded diffraction patterns, and a corresponding recovery scheme. The key novelty in this approa…
In this paper, we propose a generalized expectation consistent signal recovery algorithm to estimate the signal $\mathbf{x}$ from the nonlinear measurements of a linear transform output $\mathbf{z}=\mathbf{A}\mathbf{x}$. This estimation problem has been encountered in many applications, such as communications with front-end impairments, compressed sensing, and phase retrieval. The proposed algorithm extends the prior art called generalized turbo signal recovery from a partial discrete Fourier transform matrix $\mathbf{A}$ to a class of general matrices. Numerical results show the excellent agr…
In this paper, we analyze the non-convex framework of Wirtinger Flow (WF) for phase retrieval and identify a novel sufficient condition for universal exact recovery through the lens of low rank matrix recovery theory. Via a perspective in the lifted domain, we show that the convergence of the WF iterates to a true solution is attained geometrically under a single condition on the lifted forward model. As a result, a deterministic relationship between the accuracy of spectral initialization and the validity of {the regularity condition} is derived. In particular, we determine that a certain con…
We explicitly give a frame of cardinality $5n-6$ such that every signal in $\mathbb{C}^n$ can be recovered up to a phase from its associated intensity measurements via the PhaseLift approach. Furthermore, we give explicit linear measurements with $4r(n-r)+n-2r$ outcomes that enable the recovery of every positive semidefinite $n\times n$ matrix of rank at most $r$.
The recovery of a signal from the intensity measurements with some entries being known in advance is termed as {\em affine phase retrieval}. In this paper, we prove that a natural least squares formulation for the affine phase retrieval is strongly convex on the entire space under some mild conditions, provided the measurements are complex Gaussian random vecotrs and the measurement number $m \gtrsim d \log d$ where $d$ is the dimension of signals. Based on the result, we prove that the simple gradient descent method for the affine phase retrieval converges linearly to the target solution with…
Understanding the limits of list-decoding and list-recovery of Reed-Solomon (RS) codes is of prime interest in coding theory and has attracted a lot of attention in recent decades. However, the best possible parameters for these problems are still unknown, and in this paper, we take a step in this direction. We show the existence of RS codes that are list-decodable or list-recoverable beyond the Johnson radius for \emph{any} rate, with a polynomial field size in the block length. In particular, we show that for any $ε\in (0,1)$ there exist RS codes that are list-decodable from radius $1-ε$ and…
Conventional sparse phase retrieval schemes can recover sparse signals from the magnitude of linear measurements only up to a global phase ambiguity. This work proposes a novel approach that instead utilizes the magnitude of affine measurements to achieve ambiguity-free signal reconstruction. The proposed method relies on two-stage approach that consists of support identification followed by the exact recovery of nonzero signal entries. In the noise-free case, perfect support identification using a simple counting rule is guaranteed subject to a mild condition on the signal sparsity, and subse…
This paper considers the problem of recovering a $k$-sparse, $N$-dimensional complex signal from Fourier magnitude measurements. It proposes a Fourier optics setup such that signal recovery up to a global phase factor is possible with very high probability whenever $M \gtrsim 4k\log_2(N/k)$ random Fourier intensity measurements are available. The proposed algorithm is comprised of two stages: An algebraic phase retrieval stage and a compressive sensing step subsequent to it. Simulation results are provided to demonstrate the applicability of the algorithm for noiseless and noisy scenarios.
Sparse signal recovery has been dominated by the basis pursuit denoise (BPDN) problem formulation for over a decade. In this paper, we propose an algorithm that outperforms BPDN in finding sparse solutions to underdetermined linear systems of equations at no additional computational cost. Our algorithm, called WSPGL1, is a modification of the spectral projected gradient for $\ell_1$ minimization (SPGL1) algorithm in which the sequence of LASSO subproblems are replaced by a sequence of weighted LASSO subproblems with constant weights applied to a support estimate. The support estimate is derive…
In this paper, we investigate a new method for phase recovery when prior information on the missing phases is available. In particular, we propose to take into account this information in a generic fashion by means of a multivariate Von Mises dis- tribution. Building on a Bayesian formulation (a Maximum A Posteriori estimation), we show that the problem can be expressed using a Mahalanobis distance and be solved by a lifting optimization procedure.
This paper considers the phase retrieval problem in which measurements consist of only the magnitude of several linear measurements of the unknown, e.g., spectral components of a time sequence. We develop low-complexity algorithms with superior performance based on the majorization-minimization (MM) framework. The proposed algorithms are referred to as PRIME: Phase Retrieval vIa the Majorization-minimization techniquE. They are preferred to existing benchmark methods since at each iteration a simple surrogate problem is solved with a closed-form solution that monotonically decreases the origin…
Donoho and Stark have shown that a precise deterministic recovery of missing information contained in a time interval shorter than the time-frequency uncertainty limit is possible. We analyze this signal recovery mechanism from a physics point of view and show that the well-known Shannon-Nyquist sampling theorem, which is fundamental in signal processing, also uses essentially the same mechanism. The uncertainty relation in the context of information theory, which is based on Fourier analysis, provides a criterion to distinguish Shannon-Nyquist sampling from compressed sensing. A new signal re…
We consider the problem of high-dimensional misspecified phase retrieval. This is where we have an $s$-sparse signal vector $\mathbf{x}_*$ in $\mathbb{R}^n$, which we wish to recover using sampling vectors $\textbf{a}_1,\ldots,\textbf{a}_m$, and measurements $y_1,\ldots,y_m$, which are related by the equation $f(\left<\textbf{a}_i,\textbf{x}_*\right>) = y_i$. Here, $f$ is an unknown link function satisfying a positive correlation with the quadratic function. This problem was analyzed in a recent paper by Neykov, Wang and Liu, who provided recovery guarantees for a two-stage algorithm with samp…
"Phase retrieval" refers to the recovery of signals from the magnitudes (and not the phases) of linear measurements. While there has been a recent explosion in development of phase retrieval methods, the lack of a common interface has made it difficult to compare new methods against the current state-of-the-art. PhasePack is a software library that creates a common interface for a wide range of phase retrieval schemes. PhasePack also provides a test bed for phase retrieval methods using both synthetic data and publicly available empirical datasets.
In deep tissue photoacoustic imaging the spatial resolution is inherently limited by the acoustic wavelength. Recently, it was demonstrated that it is possible to surpass the acoustic diffraction limit by analyzing fluctuations in a set of photoacoustic images obtained under unknown speckle illumination patterns. Here, we purpose an approach to boost reconstruction fidelity and resolution, while reducing the number of acquired images by utilizing a compressed sensing computational reconstruction framework. The approach takes into account prior knowledge of the system response and sparsity of t…
The task of reconstructing a low rank matrix from incomplete linear measurements arises in areas such as machine learning, quantum state tomography and in the phase retrieval problem. In this note, we study the particular setup that the measurements are taken with respect to rank one matrices constructed from the elements of a random tight frame. We consider a convex optimization approach and show both robustness of the reconstruction with respect to noise on the measurements as well as stability with respect to passing to approximately low rank matrices. This is achieved by establishing a ver…
LORIA（洛林计算机研究与应用实验室）；Université de Lorraine（洛林大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；University of California, Berkeley（加州大学伯克利分校）；Linköping University（林雪平大学）
This paper deals with sparse phase retrieval, i.e., the problem of estimating a vector from quadratic measurements under the assumption that few components are nonzero. In particular, we consider the problem of finding the sparsest vector consistent with the measurements and reformulate it as a group-sparse optimization problem with linear constraints. Then, we analyze the convex relaxation of the latter based on the minimization of a block l1-norm and show various exact recovery and stability results in the real and complex cases. Invariance to circular shifts and reflections are also discuss…
We consider an $\ell_2$-regularized non-convex optimization problem for recovering signals from their noisy phaseless observations. We design and study the performance of a message passing algorithm that aims to solve this optimization problem. We consider the asymptotic setting $m,n \rightarrow \infty$, $m/n \rightarrow δ$ and obtain sharp performance bounds, where $m$ is the number of measurements and $n$ is the signal dimension. We show that for complex signals the algorithm can perform accurate recovery with only $m= \left(\frac{64}{π^2}-4\right)n \approx 2.5n$ measurements. Also, we provi…
