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Author: Brian Patrick Preskitt Publisher: ISBN: Category : Languages : en Pages : 231
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.
Author: Brian Patrick Preskitt Publisher: ISBN: Category : Languages : en Pages : 231
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.
Author: Sami Eid Merhi Publisher: ISBN: 9781392157947 Category : Electronic dissertations Languages : en Pages : 123
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.
Author: Alexander H. Barnett Publisher: Cambridge University Press ISBN: 1316518876 Category : Mathematics Languages : en Pages : 321
Book Description
This book provides a theoretical foundation and conceptual framework for the problem of recovering the phase of the Fourier transform.
Author: Tim Salditt Publisher: Springer Nature ISBN: 3030344134 Category : Science Languages : en Pages : 634
Book Description
This open access book, edited and authored by a team of world-leading researchers, provides a broad overview of advanced photonic methods for nanoscale visualization, as well as describing a range of fascinating in-depth studies. Introductory chapters cover the most relevant physics and basic methods that young researchers need to master in order to work effectively in the field of nanoscale photonic imaging, from physical first principles, to instrumentation, to mathematical foundations of imaging and data analysis. Subsequent chapters demonstrate how these cutting edge methods are applied to a variety of systems, including complex fluids and biomolecular systems, for visualizing their structure and dynamics, in space and on timescales extending over many orders of magnitude down to the femtosecond range. Progress in nanoscale photonic imaging in Göttingen has been the sum total of more than a decade of work by a wide range of scientists and mathematicians across disciplines, working together in a vibrant collaboration of a kind rarely matched. This volume presents the highlights of their research achievements and serves as a record of the unique and remarkable constellation of contributors, as well as looking ahead at the future prospects in this field. It will serve not only as a useful reference for experienced researchers but also as a valuable point of entry for newcomers.
Author: Jamal Arif Publisher: ISBN: Category : Fiber optics Languages : en Pages : 274
Book Description
"Optical signals have two basic components that are power and phase, which can be demonstrated in both time and frequency domains. The common diagnostics which are available in the market only measure power component in time or frequency domain. However in certain areas phase measurements are also required. The phase retrieval techniques are used to calculate phase measurements from different methods. There are a number of applications where phase measurements are required like astronomy, wavefront sensing technique (James Webb space telescope), x-ray crystallography, fiber optic telecommunications etc. Various phase retrieval algorithms have been used in retrieving phase measurements in temporal and frequency domains. Gerchberg Saxton Algorithm technique is an iterative phase retrieval technique which has been used in phase retrieval methods. This iterative process involves iterative Fourier transformation back and forth between the object and Fourier domains with applications of the measured data or known constraints in each domain. We worked on developing an iterative phase retrieval technique keeping Gerchberg Saxton Algorithm as the basis of it and were able to successfully demonstrate phase retrieval in both temporal and spectral forms for a) Gaussian pulses having a wide range of initial educated guess phase; b) Chirped Gaussian pulses having various amounts of chirp; c) Chirped Super Gaussian pulses having various amounts of chirp. A metrics system was denied on which phase retrieval technique's success was based showing minimization of power, phase and instantaneous frequency metrics. During the study we found that chirped super Gaussian pulses of order 4 converge better than the chirped Gaussian pulses and also explored a way to choose a good educational phase without knowledge of the actual phase. Thus, this research provided a new foundation for further research on phase retrieval techniques of Gaussian and chirped Gaussian pulses."--Abstract.
Author: David Aaron Barmherzig Publisher: ISBN: Category : Languages : en Pages :
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.
Author: Christopher D. Manning Publisher: Cambridge University Press ISBN: 1139472100 Category : Computers Languages : en Pages :
Book Description
Class-tested and coherent, this textbook teaches classical and web information retrieval, including web search and the related areas of text classification and text clustering from basic concepts. It gives an up-to-date treatment of all aspects of the design and implementation of systems for gathering, indexing, and searching documents; methods for evaluating systems; and an introduction to the use of machine learning methods on text collections. All the important ideas are explained using examples and figures, making it perfect for introductory courses in information retrieval for advanced undergraduates and graduate students in computer science. Based on feedback from extensive classroom experience, the book has been carefully structured in order to make teaching more natural and effective. Slides and additional exercises (with solutions for lecturers) are also available through the book's supporting website to help course instructors prepare their lectures.
Author: Brian Patrick Preskitt
Author: Brian Patrick Preskitt
Author: Sami Eid Merhi
Author: Alexander H. Barnett
Author: Tim Salditt
Author: Oleh Melnyk
Author: Jamal Arif
Author: David Aaron Barmherzig
Author: 
Author: Christopher D. Manning
Author: Herbert Aaron Hauptman