Phase Retrieval from Short-Time Fourier Measurements and Applications to Ptychography
Author : Oleh Melnyk
Publisher :
Page : 0 pages
File Size : 47,82 MB
Release : 2023
Category :
ISBN :
Fourier Ptychographic Imaging
Author : Guoan Zheng
Publisher : Morgan & Claypool Publishers
Page : 94 pages
File Size : 27,96 MB
Release : 2016-06-30
Category : Science
ISBN : 168174273X
Book Description
This book demonstrates the concept of Fourier ptychography, a new imaging technique that bypasses the resolution limit of the employed optics. In particular, it transforms the general challenge of high-throughput, high-resolution imaging from one that is coupled to the physical limitations of the optics to one that is solvable through computation. Demonstrated in a tutorial form and providing many MATLAB® simulation examples for the reader, it also discusses the experimental implementation and recent developments of Fourier ptychography. This book will be of interest to researchers and engineers learning simulation techniques for Fourier optics and the Fourier ptychography concept.
Phase Retrieval from Locally Supported Measurements
Author : Brian Patrick Preskitt
Publisher :
Page : 231 pages
File Size : 17,74 MB
Release : 2018
Category :
ISBN :
Book Description
In this dissertation, we study a new approach to the problem of phase retrieval, which is the task of reconstructing a complex-valued signal from magnitude-only measurements. This problem occurs naturally in several specialized imaging applications such as electron microscopy and X-ray crystallography. Although solutions were first proposed for this problem as early as the 1970s, these algorithms have lacked theoretical guarantees of success, and phase retrieval has suffered from a considerable gap between practice and theory for almost the entire history of its study. A common technique in fields that use phase retrieval is that of ptychography, where measurements are collected by only illuminating small sections of the sample at any time. We refer to measurements designed in this way as local measurements, and in this dissertation, we develop and expand the theory for solving phase retrieval in measurement regimes of this kind. Our first contribution is a basic model for this setup in the case of a one-dimensional signal, along with an algorithm that robustly solves phase retrieval under this model. This work is unique in many ways that represent substantial improvements over previously existing solutions: perhaps most significantly, many of the recovery guarantees in recent work rely on the measurements being generated by a random process, while we devise a class of measurements for which the conditioning of the system is known and quickly checkable. These advantages constitute major progress towards producing theoretical results for phase retrieval that are directly usable in laboratory settings. Chapter 1 conducts a survey of the history of phase retrieval and its applications, as well as the recent literature on the subject. Chapter 2 presents co-authored results defining and establishing the setting and solution of the base model explored in this dissertation. Chapter 3 expands the theory on what measurement schemes are admissible in our model, including an analysis of conditioning and runtime. Chapter 4 introduces an alternate solution for angular synchronization that yields helpful theoretical results. Chapter 5 brings our model nearer to the actual practice of ptychography. Chapter 6 extends the base model to two dimensions.
Phase Retrieval from Continuous and Discrete Ptychographic Measurements
Author : Sami Eid Merhi
Publisher :
Page : 123 pages
File Size : 40,69 MB
Release : 2019
Category : Electronic dissertations
ISBN : 9781392157947
Book Description
In this dissertation, we present and study two novel approaches to phase retrieval -- an inverse problem in which one attempts to reconstruct a complex-valued function (or vector) from phaseless (or magnitude-only) measurements. Phase retrieval arises in several scientific areas including bio-chemistry, optics, astronomy, quantum mechanics, and speech signal processing. Early solutions to phase retrieval, although practical, lacked robustness guarantees. To this day, practitioners in scientific imaging are still seeking demonstrably stable and robust recovery algorithms. Ptychography is a form of coherent diffractive imaging where diffraction patterns are processed by algorithms to recover an image of a specimen. More specifically, small regions of a specimen are illuminated one-at-a-time, and a detector captures the intensities of the resulting diffraction patterns. As such, the measurements are local and phaseless. In this work, we present two algorithms to recover signals from ptychographic measurements. The first algorithm aims to recover a discrete one-dimensional signal from discrete spectrogram measurements via a modified Wigner distribution deconvolution (WDD) method. While the method is known to practitioners of scientific imaging, robustness and recovery guarantees are lacking, if not absent; our contribution is to supply such guarantees. The second algorithm aims to approximately recover a compactly supported function from continuous spectrogram measurements via lifting and angular synchronization. This setup can be interpreted as the infinite-dimensional equivalent of discrete ptychographic imaging. Our contribution is a model which assumes infinite-dimensional signals and measurements ab initio, as opposed to most recent algorithms in which discrete models are a necessity. Finally, we consider the worst-case noise robustness of any phase retrieval algorithm which aims to reconstruct all nonvanishing vectors from the magnitudes of an arbitrary collection of local correlation measurements. The robustness results provided therein apply to a wide range of ptychographic imaging scenarios. In particular, our contribution is to show that stable recovery of high-resolution images of extremely large samples is likely to require a vast number of measurements, independent of the recovery algorithm employed. The first chapter introduces the phase retrieval problem and presents historical context, as well as applications in which phase retrieval manifests. In addition, we introduce ptychography, discuss existing WDD formulations, and compare these to our contribution in the discrete setting. Chapter 2 provides recovery guarantees for using aliased WDD methods to solve the phase retrieval problem in a discrete setting with sub-sampled measurements. In Chapter 3 we provide lower Lipschitz bounds for generic phase retrieval algorithms from locally supported measurements. Finally, Chapter 4 presents a numerical method to recover compactly supported functions from local measurements via lifting and angular synchronization.
Springer Handbook of Microscopy
Author : Peter W. Hawkes
Publisher : Springer Nature
Page : 1561 pages
File Size : 42,91 MB
Release : 2019-11-02
Category : Technology & Engineering
ISBN : 3030000699
Book Description
This book features reviews by leading experts on the methods and applications of modern forms of microscopy. The recent awards of Nobel Prizes awarded for super-resolution optical microscopy and cryo-electron microscopy have demonstrated the rich scientific opportunities for research in novel microscopies. Earlier Nobel Prizes for electron microscopy (the instrument itself and applications to biology), scanning probe microscopy and holography are a reminder of the central role of microscopy in modern science, from the study of nanostructures in materials science, physics and chemistry to structural biology. Separate chapters are devoted to confocal, fluorescent and related novel optical microscopies, coherent diffractive imaging, scanning probe microscopy, transmission electron microscopy in all its modes from aberration corrected and analytical to in-situ and time-resolved, low energy electron microscopy, photoelectron microscopy, cryo-electron microscopy in biology, and also ion microscopy. In addition to serving as an essential reference for researchers and teachers in the fields such as materials science, condensed matter physics, solid-state chemistry, structural biology and the molecular sciences generally, the Springer Handbook of Microscopy is a unified, coherent and pedagogically attractive text for advanced students who need an authoritative yet accessible guide to the science and practice of microscopy.
The Phase Retrieval Problem
Author : David Aaron Barmherzig
Publisher :
Page : pages
File Size : 30,2 MB
Release : 2019
Category :
ISBN :
Book Description
The phase retrieval problem is an inverse problem which consists of recovering a signal from a set of squared magnitude measurements. One version of this problem, often known as Fourier phase retrieval, arises ubiquitously in scientific imaging fields (such as diffraction imaging, crystallography, and optics, etc.) where one seeks to recover an image or signal from squared magnitude measurements of its Fourier transform. Another version, known as Gaussian phase retrieval, is manifested as the study of solving random systems of quadratic equations, and constitutes an important problem in the field of nonconvex optimization. The first part of this thesis introduces a general mathematical framework for the holographic phase retrieval problem. In this problem, which arises in holographic coherent diffraction imaging, a "reference" portion of the signal to be recovered via (Fourier) phase retrieval is a priori known from experimental design. A general formula is also derived for the expected recovery error when the measurement data is corrupted by Poisson shot noise. This facilitates an optimization perspective towards reference design and analysis, which is then employed towards quantifying the performance of various known reference choices. Based on insights gained from these results, a new "dual-reference" design is proposed which consists of two reference portions - being "block" and "pinhole" shaped regions - adjacent to the imaging specimen. Expected error analysis on data following a Poisson shot noise model shows that the dual-reference scheme produces uniformly superior performance over the leading single-reference schemes. Numerical experiments on simulated data corroborate these theoretical results, and demonstrate the advantage of the dual-reference design. Based on this work, a prototype experiment for holographic coherent diffraction imaging using a dual-reference has been designed at the SLAC National Accelerator Laboratory. The second part studies the one-dimensional Fourier phase retrieval problem, as well as the closely related spectral factorization problem. In its first chapter, a comprehensive exposition of the problem theory is provided. This includes a full characterization of its general nonuniqueness, as well as the special cases for which unique solutions exists. In the second chapter, a semidefinite programming formulation is derived for the Fourier phase retrieval problem. It is shown that this approach provides guaranteed recovery whenever there exists a unique phase retrieval solution. A correspondence is also established between solutions of the phase retrieval SDP, and sum-of-squares decompositions of Laurent and trigonometric polynomials. In the third chapter, a least-squares formulation is presented for the one-dimensional Fourier phase retrieval and spectral factorization problems. This formulation allows for the successful implementation of numerous first- and second-order optimization methods. In the third part, a biconvex formulation of the Gaussian phase retrieval problem is introduced. This allows for alternating-projection algorithms, such as ADMM and block coordinate descent, to be successfully applied to Gaussian phase retrieval. Both theoretical guarantees and numerical simulations demonstrate the success of these methods.
Proximal Algorithms
Author : Neal Parikh
Publisher : Now Pub
Page : 130 pages
File Size : 50,29 MB
Release : 2013-11
Category : Mathematics
ISBN : 9781601987167
Book Description
Proximal Algorithms discusses proximal operators and proximal algorithms, and illustrates their applicability to standard and distributed convex optimization in general and many applications of recent interest in particular. Much like Newton's method is a standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous tool for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are especially well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at a higher level of abstraction than classical algorithms like Newton's method: the base operation is evaluating the proximal operator of a function, which itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting a point onto a convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods. Proximal Algorithms discusses different interpretations of proximal operators and algorithms, looks at their connections to many other topics in optimization and applied mathematics, surveys some popular algorithms, and provides a large number of examples of proximal operators that commonly arise in practice.
Compressed Sensing and its Applications
Author : Holger Boche
Publisher : Birkhäuser
Page : 402 pages
File Size : 25,40 MB
Release : 2018-01-17
Category : Mathematics
ISBN : 3319698028
Book Description
This contributed volume contains articles written by the plenary and invited speakers from the second international MATHEON Workshop 2015 that focus on applications of compressed sensing. Article authors address their techniques for solving the problems of compressed sensing, as well as connections to related areas like detecting community-like structures in graphs, curbatures on Grassmanians, and randomized tensor train singular value decompositions. Some of the novel applications covered include dimensionality reduction, information theory, random matrices, sparse approximation, and sparse recovery. This book is aimed at both graduate students and researchers in the areas of applied mathematics, computer science, and engineering, as well as other applied scientists exploring the potential applications for the novel methodology of compressed sensing. An introduction to the subject of compressed sensing is also provided for researchers interested in the field who are not as familiar with it.
Computational Optical Phase Imaging
Author : Cheng Liu
Publisher : Springer Nature
Page : 311 pages
File Size : 43,95 MB
Release : 2022-04-11
Category : Science
ISBN : 9811916411
Book Description
In this book, computational optical phase imaging techniques are presented along with Matlab codes that allow the reader to run their own simulations and gain a thorough understanding of the current state-of-the-art. The book focuses on modern applications of computational optical phase imaging in engineering measurements and biomedical imaging. Additionally, it discusses the future of computational optical phase imaging, especially in terms of system miniaturization and deep learning-based phase retrieval.
Phase Retrieval by Alternating Direction Method of Multipliers
Author : Mehdi Akhavan
Publisher :
Page : 0 pages
File Size : 49,91 MB
Release : 2021
Category : Fourier transformations
ISBN :
Book Description
This dissertation aims at reconstructing a signal from the magnitude of its Fourier transform, known as phase retrieval. The problem arises in variety of areas such as crystallography, astronomy, optics, voice recognition, and coherent diffraction imaging (CDI). In particular, we focus on two types of phaseless measurements: short-time Fourier transform (STFT) and frequency-resolved optical gating (FROG). STFT takes the Fourier transform when passing a short-time window over a signal. When the window function is given, the problem is referred to as non-blind STFT, while blind STFT means to simultaneously estimate both the signal and the window from the magnitude measurements. FROG is closely related to STFT in such a way that the window function in FROG is just the signal itself. We apply alternating direction method of multipliers (ADMM) to solve all the aforementioned problems: non-blind STFT, blind STFT, and FROG. Specifically for the blind STFT, we discuss three approaches to address the scaling ambiguity. We also consider a special type of signals that has only a few non-zero elements by minimizing the L1 norm to promote sparsity in the objective function. Numerical experiments are provided to demonstrate the proposed algorithms outperform the state-of-the-art in non-blind STFT and FROG. As the blind STFT is one of the first kind, we compare the performance of the three proposed approaches.
