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Truncated amplitude flow with coded diffraction patterns

INVERSE PROBLEMS 2025年第1期41卷

作者： Li, Huiping Li, Jiayi Hangzhou Normal Univ Sch Math Hangzhou Peoples R China Hong Kong Univ Sci & Technol Dept Math Hong Kong Peoples R China

The phase retrieval problem, which involves Fourier transforms with random masks, has a wide range of applications in signal and image processing, such as diffraction imaging. In recent years, various first-order methods have been explored, including truncated amplitude flow (TAF). These methods are considered to be simple and effective for solving the phase retrieval problem. However, existing convergence results for TAF mainly focus on random Gaussian measurements, which is very far from practical scenarios. In this paper, we analyze the convergence of a revised TAF (RTAF) for the phase retrieval problem. We focus on coded diffraction patterns composed of Fourier transforms and random masks. Then we justify the linear convergence behavior of such algorithm when near the true solution. By utilizing the local initializer found by a non-truncated spectral method, we establish the exact convergence rates in both the noiseless case and the bounded noise case. Further, we have shown that a general signal x is an element of C-n can be estimated on an optimal sampling order (specifically, O(n log n)). Moreover, our demonstrations indicate that sharp stability can be maintained even when the observations are contaminated by bounded noise. Our experiments have revealed that the RTAF algorithm can reliably estimate the original signal using fewer measurements, aligning with our theoretical analysis. Additionally, numerical experiments have highlighted that the RTAF algorithm outperforms the truncated Wirtinger flow and truncated Riemannian gradient descent algorithms in terms of efficiency and stability, showcasing its superior performance over state-of-the-art algorithms.

关键词： phase retrieval coded diffraction pattern truncated amplitude flow non-convex optimization sample complexity

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Uniqueness theorems for tomographic phase retrieval with few coded diffraction patterns

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INVERSE PROBLEMS 2022年第8期38卷 085008-085008页

作者： Fannjiang, Albert Univ Calif Davis Dept Math Davis CA 95616 USA

3D tomographic phase retrieval under the Born approximation for discrete objects supported on a n x n x n grid is analyzed. It is proved that n projections are sufficient and necessary for unique determination by computed tomography with full projected field measurements and that n + 1 coded projected diffraction patterns are sufficient for unique determination, up to a global phase factor, in tomographic phase retrieval. Hence n + 1 is nearly, if not exactly, the minimum number of diffractions patterns needed for 3D tomographic phase retrieval under the Born approximation.

关键词： discrete tomography phase retrieval coded diffraction pattern uniqueness

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Super-Resolution Phase Retrieval From Designed coded diffraction patterns

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IEEE TRANSACTIONS ON IMAGE PROCESSING 2020年 29卷 2598-2609页

作者： Bacca, Jorge Pinilla, Samuel Arguello, Henry Univ Ind Santander Dept Comp Sci Bucaramanga 680002 Colombia Univ Ind Santander Dept Elect Engn Bucaramanga 680002 Colombia

Super-resolution phase retrieval is an inverse problem that appears in diffractive optical imaging (DOI) and consists in estimating a high-resolution image from low-resolution phaseless measurements. DOI has three diffraction zones where the data can be acquired, known as near, middle, and far fields. Recent works have studied super-resolution phase retrieval under a setup that records coded diffraction patterns at the near and far fields. However, the attainable resolution of the image is mainly governed by the sensor characteristics, whose cost increases in proportion to the resolution. Also, these methodologies lack theoretical analysis. Hence, this work derives super-resolution models from low-resolution coded phaseless measurements at any diffraction zone that in contrast to prior contributions, the attainable resolution of the image is determined by the resolution of the coded aperture. For the proposed models, the existence of a unique solution (up to a global unimodular constant) is guaranteed with high probability, which can be increased by designing the coded aperture. Therefore, a strategy that designs the spatial distribution of the coded aperture is developed. Additionally, a super-resolution phase retrieval algorithm that minimizes a smoothed nonconvex least-squares objective function is proposed. The method first approximates the image by a spectral algorithm, which is then refined based upon a sequence of alternate steps. Simulation results show that the proposed algorithm overcomes state-of-the-art methods in reconstructing the high-resolution image. In addition, the reconstruction quality using designed coded apertures is higher than that of the non-designed ensembles.

关键词： diffraction Apertures Optical diffraction Optical imaging Super-resolution phase retrieval smoothed function coded aperture design coded diffraction pattern

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Spectral Method for Phase Retrieval: An Expectation Propagation Perspective

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IEEE TRANSACTIONS ON INFORMATION THEORY 2021年第2期67卷 1332-1355页

作者： Ma, Junjie Dudeja, Rishabh Xu, Ji Maleki, Arian Wang, Xiaodong Columbia Univ Dept Stat New York NY 10027 USA Columbia Univ Dept Elect Engn New York NY 10027 USA Chinese Acad Sci Inst Computat Math & Sci Engn Comp Acad Math & Syst Sci Beijing 100190 Peoples R China Columbia Univ Dept Comp Sci New York NY 10027 USA

Phase retrieval refers to the problem of recovering a signal x(star) is an element of C-n from its phaseless measurements y(i) = vertical bar a(i)(H) x(star)vertical bar, where {alpha(i)}(i=1)(m) are the measurement vectors. Spectral method is widely used for initialization in many phase retrieval algorithms. The quality of spectral initialization can have a major impact on the overall algorithm. In this paper, we focus on the model where A = [alpha(1), ... , alpha(m)](H) has orthonormal columns, and study the spectral initialization under the asymptotic setting m, n -> infinity with m/n infinity -> delta is an element of (1, infinity). We use the expectation propagation framework to characterize the performance of spectral initialization for Haar distributed matrices. Our numerical results confirm that the predictions of the EP method are accurate for not-only Haar distributed matrices, but also for realistic Fourier based models (e.g. the coded diffraction model). The main findings of this paper are the following: 1) There exists a threshold on delta(denoted as delta(weak)) below which the spectral method cannot produce a meaningful estimate. We show that delta(weak) = 2 for the column-orthonormal model. In contrast, previous results by Mondelli and Montanari show that delta(weak) = 1 for the i.i.d. Gaussian model. 2) The optimal design for the spectral method coincides with that for the i.i.d. Gaussian model, where the latter was recently introduced by Luo, Alghamdi and Lu.

关键词： Phase measurement Message passing Tools Signal processing algorithms Prediction algorithms Numerical models Approximation algorithms Phase retrieval spectral method coded diffraction pattern expectation propagation (EP) approximate message passing (AMP) state evolution orthogonal AMP vector AMP

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High-Dimensional Asymptotics for Phase Retrieval with Structured Sensing Matrices

High-Dimensional Asymptotics for Phase Retrieval with Struct...

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作者： Dudeja, Rishabh Columbia University

学位级别：Ph.D., Doctor of Philosophy

Phase Retrieval is an inference problem where one seeks to recover an unknown complex-valued n-dimensional signal vector from the magnitudes of m linear measurements. The linear measurements are specified using a m × n sensing matrix. This problem is a mathematical model for imaging systems arising in X-ray crystallography and other applications where it is infeasible to acquire the phase of the measurements. This dissertation presents some results regarding the analysis of this problem in the high-dimensional asymptotic regime where the number of measurements and the signal dimension diverge proportionally so that their ratio remains fixed. A limitation of existing high-dimensional analyses of this problem is that they model the sensing matrix as a random matrix with independent and identically (i.i.d.) distributed Gaussian entries. In practice, this matrix is highly structured with limited randomness. This work studies a correction to the i.i.d. Gaussian sensing model, known as the sub-sampled Haar sensing model which faithfully captures a crucial orthogonality property of realistic sensing matrices. The first result of this thesis provides a precise asymptotic characterization of the performance of commonly used spectral estimators for phase retrieval in the sub-sampled Haar sensing model. This result can be leveraged to tune certain parameters involved in the spectral estimator optimally. The second part of this dissertation studies the information-theoretic limits for better-than-random (or weak) recovery in the sub-sampled Haar sensing model. The main result in this part shows that appropriately tuned spectral methods achieve weak recovery with the information-theoretically optimal number of measurements. Simulations indicate that the performance curves derived for the sub-sampled Haar sensing model accurately describe the empirical performance curves for realistic sensing matrices such as randomly sub-sampled Fourier sensing matrices and coded diffraction Pat

关键词： High-dimensional statistics Phase retrieval Random matrices coded diffraction pattern

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Information Theoretic Limits for Phase Retrieval With Subsampled Haar Sensing Matrices

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IEEE TRANSACTIONS ON INFORMATION THEORY 2020年第12期66卷 8002-8045页

作者： Dudeja, Rishabh Ma, Junjie Maleki, Arian Columbia Univ Dept Stat New York NY 10027 USA

We study information theoretic limits of recovering an unknown n dimensional, complex signal vector x(star) with unit norm from m magnitude-only measurements of the form y(i) = vertical bar(Ax(star))(i)vertical bar(2), i = 1, 2 ..., m, where A is the sensing matrix. This is known as the Phase Retrieval problem and models practical imaging systems where measuring the phase of the observations is difficult. Since in a number of applications, the sensing matrix has orthogonal columns, we model the sensing matrix as a subsampled Haar matrix formed by picking n columns of a uniformly random m x m unitary matrix. We study this problem in the high dimensional asymptotic regime, where m, n -> infinity, while m/n -> delta with delta being a fixed number, and show that if m < (2 - o(n)(1)) center dot n, then any estimator is asymptotically orthogonal to the true signal vector x(star). This lower bound is sharp since when m > (2 + o(n)(1)) center dot n, estimators that achieve a non trivial asymptotic correlation with the signal vector are known from previous works.

关键词： Phase retrieval random orthogonal matrices subsampled Haar matrices coded diffraction pattern large deviation theory random matrix theory phase transition weak recovery threshold

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Analysis of Spectral Methods for Phase Retrieval With Random Orthogonal Matrices

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IEEE TRANSACTIONS ON INFORMATION THEORY 2020年第8期66卷 5182-5203页

作者： Dudeja, Rishabh Bakhshizadeh, Milad Ma, Junjie Maleki, Arian Columbia Univ Dept Stat New York NY 10027 USA

Phase retrieval refers to algorithmic methods for recovering a signal from its phaseless measurements. There has been recent interest in understanding the performance of local search algorithms that work directly on the non-convex formulation of the problem. Due to the non-convexity of the problem, the success of these local search algorithms depends heavily on their starting points. The most widely used initialization scheme is the spectral method, in which the leading eigenvector of a data-dependent matrix is used as a starting point. Recently, the performance of the spectral initialization was characterized accurately for measurement matrices with independent and identically distributed entries. This paper aims to obtain the same level of knowledge for isotropically random column-orthogonal matrices, which are substantially better models for practical phase retrieval systems. Towards this goal, we consider the asymptotic setting in which the number of measurements m, and the dimension of the signal, n, diverge to infinity with m/n = delta is an element of(1, infinity), and obtain a simple expression for the overlap between the spectral estimator and the true signal vector.

关键词： Phase retrieval spectral initialization random orthogonal matrices coded diffraction pattern phase transition random matrix theory

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FASPR: A fast sparse phase retrieval algorithm via the epigraph concept

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DIGITAL SIGNAL PROCESSING 2018年 80卷 12-26页

作者： Shi, Baoshun Chen, Shuzhen Tian, Ye Fan, Xiaoyu Lian, Qiusheng Yanshan Univ Sch Informat Sci & Engn 438 Hebei Ave Qinhuang Dao 066004 Hebei Peoples R China Hebei Key Lab Informat Transmiss & Signal Proc Qinhuang Dao 066004 Hebei Peoples R China

In this paper, we propose a fast sparsity-based approach to phase retrieval (PR) that refers to recovering an image of interest from the recorded magnitudes or intensities of its linear transform. The corresponding non-linear sampling operator makes the recovery task a challenging one. Recent efforts to solve this problem exploit sparse priors via fusing a data fidelity term with an elaborate sparsity-induced regularization term. However, two drawbacks exist in these sparsity-based PR algorithms. One is high computation complexity, and the other is the issue of many finely-tuned parameters, i.e., regular parameters and thresholding values. To address these issues, we propose a fast sparse PR algorithm (FASPR) based on the epigraph concept of the total variation (TV) function. We exploit the constraint based on the TV model, namely that the underlying image should be sparse in the gradient domain, to formulate a PR problem, and then solve the formulated problem in two steps: a phase-recovering step and an image-updating step. The fast iterative shrinkage-thresholding algorithm (FISTA) is utilized for solving the corresponding sub-problem in the image-updating step. The step size in FISTA is selected in an adaptive manner by solving a step size optimization problem. In the projection step of FISTA, the epigraph set of the TV function is defined, and the projection onto this set is calculated. For both real-valued images and complex-valued images, diffraction imaging from under-sampled and noisy coded diffraction patterns is simulated. Experimental results validate the effectiveness and the efficiency of the proposed FASPR algorithm. Most importantly, it does not need many finely-tuned parameters, and only the iteration times need to be specified. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Phase retrieval coded diffraction pattern Total variation Epigraph set FISTA

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Fourier phase retrieval with a single mask by Douglas-Rachford algorithms

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APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2018年第3期44卷 665-699页

作者： Chen, Pengwen Fannjiang, Albert Natl Chung Hsing Univ Appl Math Taichung 402 Taiwan Univ Calif Davis Dept Math Davis CA 95616 USA

The Fourier-domain Douglas-Rachford (FDR) algorithm is analyzed for phase retrieval with a single random mask. Since the uniqueness of phase retrieval solution requires more than a single oversampled coded diffraction pattern, the extra information is imposed in either of the following forms: 1) the sector condition on the object;2) another oversampled diffraction pattern, coded or uncoded. For both settings, the uniqueness of projected fixed point is proved and for setting 2) the local, geometric convergence is derived with a rate given by a spectral gap condition. Numerical experiments demonstrate global, power-law convergence of FDR from arbitrary initialization for both settings as well as for 3 or more coded diffraction patterns without oversampling. In practice, the geometric convergence can be recovered from the power-law regime by a simple projection trick, resulting in highly accurate reconstruction from generic initialization. (c) 2016 Elsevier Inc. All rights reserved.

关键词： Phase retrieval coded diffraction pattern Douglas-Rachford algorithm Geometric convergence Spectral gap

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