Signal Reconstruction from Modulo Observations

We consider the problem of reconstructing a signal from under-determined modulo observations (or measurements). This observation model is inspired by a (relatively) less well-known imaging mechanism called modulo imaging, which can be used to extend the dynamic range of imaging systems; variations of this model have also been studied under the category of phase unwrapping. Signal reconstruction in the under-determined regime with modulo observations is a challenging ill-posed problem, and existing reconstruction methods cannot be used directly. In this paper, we propose a novel approach to solving the inverse problem limited to two modulo periods, inspired by recent advances in algorithms for phase retrieval under sparsity constraints. We show that given a sufficient number of measurements, our algorithm perfectly recovers the underlying signal and provides improved performance over other existing algorithms. We also provide experiments validating our approach on both synthetic and real data to depict its superior performance

Alternating Phase Projected Gradient Descent with Generative Priors for Solving Compressive Phase Retrieval

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.Comment: Published in ICASSP 201

Phase Retrieval by Alternating Minimization with Random Initialization

We consider a phase retrieval problem, where the goal is to reconstruct a $n$-dimensional complex vector from its phaseless scalar products with $m$ sensing vectors, independently sampled from complex normal distributions. We show that, with a random initialization, the classical algorithm of alternating minimization succeeds with high probability as $n,m\rightarrow\infty$ when ${m}/{\log^3m}\geq Mn^{3/2}\log^{1/2}n$ for some $M>0$. This is a step toward proving the conjecture in \cite{Waldspurger2016}, which conjectures that the algorithm succeeds when $m=O(n)$. The analysis depends on an approach that enables the decoupling of the dependency between the algorithmic iterates and the sensing vectors

Compressive Phase Retrieval via Reweighted Amplitude Flow

The problem of reconstructing a sparse signal vector from magnitude-only measurements (a.k.a., compressive phase retrieval), emerges naturally in diverse applications, but it is NP-hard in general. Building on recent advances in nonconvex optimization, this paper puts forth a new algorithm that is termed compressive reweighted amplitude flow and abbreviated as CRAF, for compressive phase retrieval. Specifically, CRAF operates in two stages. The first stage seeks a sparse initial guess via a new spectral procedure. In the second stage, CRAF implements a few hard thresholding based iterations using reweighted gradients. When there are sufficient measurements, CRAF provably recovers the underlying signal vector exactly with high probability under suitable conditions. Moreover, its sample complexity coincides with that of the state-of-the-art procedures. Finally, substantial simulated tests showcase remarkable performance of the new spectral initialization, as well as improved exact recovery relative to competing alternatives

Accelerated Wirtinger Flow: A fast algorithm for ptychography

This paper presents a new algorithm, Accelerated Wirtinger Flow (AWF), for ptychographic image reconstruction from phaseless diffraction pattern measurements. AWF is based on combining Nesterov's acceleration approach with Wirtinger gradient descent. Theoretical results enable prespecification of all AWF algorithm parameters, with no need for computationally-expensive line searches and no need for manual parameter tuning. AWF is evaluated in the context of simulated X-ray ptychography, where we demonstrate fast convergence and low per-iteration computational complexity. We also show examples where AWF reaches higher image quality with less computation than classical algorithms. AWF is also shown to have robustness to noise and probe misalignment

Learning Illumination Patterns for Coded Diffraction Phase Retrieval

Signal recovery from nonlinear measurements involves solving an iterative optimization problem. In this paper, we present a framework to optimize the sensing parameters to improve the quality of the signal recovered by the given iterative method. In particular, we learn illumination patterns to recover signals from coded diffraction patterns using a fixed-cost alternating minimization-based phase retrieval method. Coded diffraction phase retrieval is a physically realistic system in which the signal is first modulated by a sequence of codes before the sensor records its Fourier amplitude. We represent the phase retrieval method as an unrolled network with a fixed number of layers and minimize the recovery error by optimizing over the measurement parameters. Since the number of iterations/layers are fixed, the recovery incurs a fixed cost. We present extensive simulation results on a variety of datasets under different conditions and a comparison with existing methods. Our results demonstrate that the proposed method provides near-perfect reconstruction using patterns learned with a small number of training images. Our proposed method provides significant improvements over existing methods both in terms of accuracy and speed

Phase retrieval of complex-valued objects via a randomized Kaczmarz method

This paper investigates the convergence of the randomized Kaczmarz algorithm for the problem of phase retrieval of complex-valued objects. While this algorithm has been studied for the real-valued case}, its generalization to the complex-valued case is nontrivial and has been left as a conjecture. This paper establishes the connection between the convergence of the algorithm and the convexity of an objective function. Based on the connection, it demonstrates that when the sensing vectors are sampled uniformly from a unit sphere and the number of sensing vectors $m$ satisfies $m>O(n\log n)$ as $n, m\rightarrow\infty$, then this algorithm with a good initialization achieves linear convergence to the solution with high probability

Solving Almost all Systems of Random Quadratic Equations

This paper deals with finding an $n$-dimensional solution $x$ to a system of quadratic equations of the form $y_i=|\langle{a}_i,x\rangle|^2$ for $1\le i \le m$, which is also known as phase retrieval and is NP-hard in general. We put forth a novel procedure for minimizing the amplitude-based least-squares empirical loss, that starts with a weighted maximal correlation initialization obtainable with a few power or Lanczos iterations, followed by successive refinements based upon a sequence of iteratively reweighted (generalized) gradient iterations. The two (both the initialization and gradient flow) stages distinguish themselves from prior contributions by the inclusion of a fresh (re)weighting regularization technique. The overall algorithm is conceptually simple, numerically scalable, and easy-to-implement. For certain random measurement models, the novel procedure is shown capable of finding the true solution $x$ in time proportional to reading the data $\{(a_i;y_i)\}_{1\le i \le m}$. This holds with high probability and without extra assumption on the signal $x$ to be recovered, provided that the number $m$ of equations is some constant $c>0$ times the number $n$ of unknowns in the signal vector, namely, $m>cn$. Empirically, the upshots of this contribution are: i) (almost) $100\%$ perfect signal recovery in the high-dimensional (say e.g., $n\ge 2,000$) regime given only an information-theoretic limit number of noiseless equations, namely, $m=2n-1$ in the real-valued Gaussian case; and, ii) (nearly) optimal statistical accuracy in the presence of additive noise of bounded support. Finally, substantial numerical tests using both synthetic data and real images corroborate markedly improved signal recovery performance and computational efficiency of our novel procedure relative to state-of-the-art approaches.Comment: 27 pages, 8 figure

Sample-Efficient Algorithms for Recovering Structured Signals from Magnitude-Only Measurements

We consider the problem of recovering a signal $\mathbf{x}^* \in \mathbf{R}^n$, from magnitude-only measurements $y_i = |\left\langle\mathbf{a}_i,\mathbf{x}^*\right\rangle|$ for $i=[m]$. Also called the phase retrieval, this is a fundamental challenge in bio-,astronomical imaging and speech processing. The problem above is ill-posed; additional assumptions on the signal and/or the measurements are necessary. In this paper we first study the case where the signal $\mathbf{x}^*$ is $s$-sparse. We develop a novel algorithm that we call Compressive Phase Retrieval with Alternating Minimization, or CoPRAM. Our algorithm is simple; it combines the classical alternating minimization approach for phase retrieval with the CoSaMP algorithm for sparse recovery. Despite its simplicity, we prove that CoPRAM achieves a sample complexity of $O(s^2\log n)$ with Gaussian measurements $\mathbf{a}_i$, matching the best known existing results; moreover, it demonstrates linear convergence in theory and practice. Additionally, it requires no extra tuning parameters other than signal sparsity $s$ and is robust to noise. When the sorted coefficients of the sparse signal exhibit a power law decay, we show that CoPRAM achieves a sample complexity of $O(s\log n)$, which is close to the information-theoretic limit. We also consider the case where the signal $\mathbf{x}^*$ arises from structured sparsity models. We specifically examine the case of block-sparse signals with uniform block size of $b$ and block sparsity $k=s/b$. For this problem, we design a recovery algorithm Block CoPRAM that further reduces the sample complexity to $O(ks\log n)$. For sufficiently large block lengths of $b=\Theta(s)$, this bound equates to $O(s\log n)$. To our knowledge, this constitutes the first end-to-end algorithm for phase retrieval where the Gaussian sample complexity has a sub-quadratic dependence on the signal sparsity level

Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow

This paper presents a new algorithm, termed \emph{truncated amplitude flow} (TAF), to recover an unknown vector $\bm{x}$ from a system of quadratic equations of the form $y_i=|\langle\bm{a}_i,\bm{x}\rangle|^2$, where $\bm{a}_i$'s are given random measurement vectors. This problem is known to be \emph{NP-hard} in general. We prove that as soon as the number of equations is on the order of the number of unknowns, TAF recovers the solution exactly (up to a global unimodular constant) with high probability and complexity growing linearly with both the number of unknowns and the number of equations. Our TAF approach adopts the \emph{amplitude-based} empirical loss function, and proceeds in two stages. In the first stage, we introduce an \emph{orthogonality-promoting} initialization that can be obtained with a few power iterations. Stage two refines the initial estimate by successive updates of scalable \emph{truncated generalized gradient iterations}, which are able to handle the rather challenging nonconvex and nonsmooth amplitude-based objective function. In particular, when vectors $\bm{x}$ and $\bm{a}_i$'s are real-valued, our gradient truncation rule provably eliminates erroneously estimated signs with high probability to markedly improve upon its untruncated version. Numerical tests using synthetic data and real images demonstrate that our initialization returns more accurate and robust estimates relative to spectral initializations. Furthermore, even under the same initialization, the proposed amplitude-based refinement outperforms existing Wirtinger flow variants, corroborating the superior performance of TAF over state-of-the-art algorithms.Comment: 37 Pages, 16 figure