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Geometry Of The Phase Retrieval Problem
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Book Synopsis Geometry of the Phase Retrieval Problem by : Alexander H. Barnett
Download or read book Geometry of the Phase Retrieval Problem written by Alexander H. Barnett and published by Cambridge University Press. This book was released on 2022-05-05 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a theoretical foundation and conceptual framework for the problem of recovering the phase of the Fourier transform.
Book Synopsis The Phase Retrieval Problem by : David Aaron Barmherzig
Download or read book The Phase Retrieval Problem written by David Aaron Barmherzig and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Phase Retrieval and Zero Crossings by : N.E. Hurt
Download or read book Phase Retrieval and Zero Crossings written by N.E. Hurt and published by Springer Science & Business Media. This book was released on 2001-11-30 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: 'Et moi, ... , si j'avait su comment en :revenir, One scrvice mathematics has rendered the je n'y scrais point alle.' human race. lt has put common sense back Jules Veme where it bdongs, on the topmost shelf next to the dusty canister labclled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Erle T. Bc1l 0. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'.All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.
Book Synopsis The Phase Retrieval Problem by : Keith Allen Rinaldi
Download or read book The Phase Retrieval Problem written by Keith Allen Rinaldi and published by . This book was released on 1986 with total page 80 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Phase retrieval problems in x-ray physics by : Carolin Homann
Download or read book Phase retrieval problems in x-ray physics written by Carolin Homann and published by Göttingen University Press. This book was released on 2015 with total page 126 pages. Available in PDF, EPUB and Kindle. Book excerpt: In phase retrieval problems that occur in imaging by coherent x-ray diffraction, one tries to reconstruct information about a sample of interest from possibly noisy intensity measurements of the wave fi eld traversing the sample. The mathematical formulation of these problems bases on some assumptions. Usually one of them is that the x-ray wave fi eld is generated by a point source. In order to address this very idealized assumption, it is common to perform a data preprocessing step, the so-called empty beam correction. Within this work, we study the validity of this approach by presenting a quantitative error estimate. Moreover, in order to solve these phase retrieval problems, we want to incorporate a priori knowledge about the structure of the noise and the solution into the reconstruction process. For this reason, the application of a problem adapted iteratively regularized Newton-type method becomes particularly attractive. This method includes the solution of a convex minimization problem in each iteration step. We present a method for solving general optimization problems of this form. Our method is a generalization of a commonly used algorithm which makes it efficiently applicable to a wide class of problems. We also proof convergence results and show the performance of our method by numerical examples.
Book Synopsis Discrete Variational Problems with Interfaces by : Roberto Alicandro
Download or read book Discrete Variational Problems with Interfaces written by Roberto Alicandro and published by Cambridge University Press. This book was released on 2023-12-31 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many materials can be modeled either as discrete systems or as continua, depending on the scale. At intermediate scales it is necessary to understand the transition from discrete to continuous models and variational methods have proved successful in this task, especially for systems, both stochastic and deterministic, that depend on lattice energies. This is the first systematic and unified presentation of research in the area over the last 20 years. The authors begin with a very general and flexible compactness and representation result, complemented by a thorough exploration of problems for ferromagnetic energies with applications ranging from optimal design to quasicrystals and percolation. This leads to a treatment of frustrated systems, and infinite-dimensional systems with diffuse interfaces. Each topic is presented with examples, proofs and applications. Written by leading experts, it is suitable as a graduate course text as well as being an invaluable reference for researchers.
Book Synopsis Nanoscale Photonic Imaging by : Tim Salditt
Download or read book Nanoscale Photonic Imaging written by Tim Salditt and published by Springer Nature. This book was released on 2020-06-09 with total page 634 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Music Through Fourier Space by : Emmanuel Amiot
Download or read book Music Through Fourier Space written by Emmanuel Amiot and published by Springer. This book was released on 2016-10-26 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explains the state of the art in the use of the discrete Fourier transform (DFT) of musical structures such as rhythms or scales. In particular the author explains the DFT of pitch-class distributions, homometry and the phase retrieval problem, nil Fourier coefficients and tilings, saliency, extrapolation to the continuous Fourier transform and continuous spaces, and the meaning of the phases of Fourier coefficients. This is the first textbook dedicated to this subject, and with supporting examples and exercises this is suitable for researchers and advanced undergraduate and graduate students of music, computer science and engineering. The author has made online supplementary material available, and the book is also suitable for practitioners who want to learn about techniques for understanding musical notions and who want to gain musical insights into mathematical problems.
Book Synopsis High-Dimensional Data Analysis with Low-Dimensional Models by : John Wright
Download or read book High-Dimensional Data Analysis with Low-Dimensional Models written by John Wright and published by Cambridge University Press. This book was released on 2022-01-13 with total page 718 pages. Available in PDF, EPUB and Kindle. Book excerpt: Connecting theory with practice, this systematic and rigorous introduction covers the fundamental principles, algorithms and applications of key mathematical models for high-dimensional data analysis. Comprehensive in its approach, it provides unified coverage of many different low-dimensional models and analytical techniques, including sparse and low-rank models, and both convex and non-convex formulations. Readers will learn how to develop efficient and scalable algorithms for solving real-world problems, supported by numerous examples and exercises throughout, and how to use the computational tools learnt in several application contexts. Applications presented include scientific imaging, communication, face recognition, 3D vision, and deep networks for classification. With code available online, this is an ideal textbook for senior and graduate students in computer science, data science, and electrical engineering, as well as for those taking courses on sparsity, low-dimensional structures, and high-dimensional data. Foreword by Emmanuel Candès.
Book Synopsis Computational Geometry by : Franco P. Preparata
Download or read book Computational Geometry written by Franco P. Preparata and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 413 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the reviews: "This book offers a coherent treatment, at the graduate textbook level, of the field that has come to be known in the last decade or so as computational geometry. ... ... The book is well organized and lucidly written; a timely contribution by two founders of the field. It clearly demonstrates that computational geometry in the plane is now a fairly well-understood branch of computer science and mathematics. It also points the way to the solution of the more challenging problems in dimensions higher than two." #Mathematical Reviews#1 "... This remarkable book is a comprehensive and systematic study on research results obtained especially in the last ten years. The very clear presentation concentrates on basic ideas, fundamental combinatorial structures, and crucial algorithmic techniques. The plenty of results is clever organized following these guidelines and within the framework of some detailed case studies. A large number of figures and examples also aid the understanding of the material. Therefore, it can be highly recommended as an early graduate text but it should prove also to be essential to researchers and professionals in applied fields of computer-aided design, computer graphics, and robotics." #Biometrical Journal#2
Book Synopsis Multiple View Geometry in Computer Vision by : Richard Hartley
Download or read book Multiple View Geometry in Computer Vision written by Richard Hartley and published by Cambridge University Press. This book was released on 2004-03-25 with total page 676 pages. Available in PDF, EPUB and Kindle. Book excerpt: A basic problem in computer vision is to understand the structure of a real world scene given several images of it. Techniques for solving this problem are taken from projective geometry and photogrammetry. Here, the authors cover the geometric principles and their algebraic representation in terms of camera projection matrices, the fundamental matrix and the trifocal tensor. The theory and methods of computation of these entities are discussed with real examples, as is their use in the reconstruction of scenes from multiple images. The new edition features an extended introduction covering the key ideas in the book (which itself has been updated with additional examples and appendices) and significant new results which have appeared since the first edition. Comprehensive background material is provided, so readers familiar with linear algebra and basic numerical methods can understand the projective geometry and estimation algorithms presented, and implement the algorithms directly from the book.
Book Synopsis Sparse Solutions of Underdetermined Linear Systems and Their Applications by : Ming-Jun Lai
Download or read book Sparse Solutions of Underdetermined Linear Systems and Their Applications written by Ming-Jun Lai and published by SIAM. This book was released on 2021-06-25 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook presents a special solution to underdetermined linear systems where the number of nonzero entries in the solution is very small compared to the total number of entries. This is called a sparse solution. Since underdetermined linear systems can be very different, the authors explain how to compute a sparse solution using many approaches. Sparse Solutions of Underdetermined Linear Systems and Their Applications contains 64 algorithms for finding sparse solutions of underdetermined linear systems and their applications for matrix completion, graph clustering, and phase retrieval and provides a detailed explanation of these algorithms including derivations and convergence analysis. Exercises for each chapter help readers understand the material. This textbook is appropriate for graduate students in math and applied math, computer science, statistics, data science, and engineering. Advisors and postdoctoral scholars will also find the book interesting and useful.
Book Synopsis Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses by : Rick Trebino
Download or read book Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses written by Rick Trebino and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 428 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Frequency-Resolved Optical-Gating (FROG) technique has revolutionized our ability to measure and understand ultrashort laser pulses. This book contains everything you need to know to measure even the shortest, weakest, or most complex ultrashort laser pulses. Whether you're an undergrad or an advanced researcher, you'll find easy-to-understand descriptions of all the key ideas behind all the FROG techniques, all the practical details of pulse measurement, and many new directions of research. This book is not like any other scientific book. It is a lively discussion of the basic concepts. It is an advanced treatment of research-level issues.
Book Synopsis Applications of Complex Analysis to the Phase Retrieval Problem by : Rolando III. Perez
Download or read book Applications of Complex Analysis to the Phase Retrieval Problem written by Rolando III. Perez and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of phase retrieval involves the recovery of a function f in some functionspace from given data about the magnitude of |f| (phaseless information) and other assumptions on f, where these other assumptions can be in terms of some transform of f. Phase retrieval problems are widely studied because of their physical applications in fields of science and engineering.In this thesis, our central objective is to apply complex analytic tools to determine the solutions and investigate the stability of certain phase retrieval problems. Firstly, we solve the phase retrieval problem for wide-band signals, which are functions with mildly decreasing Fourier transforms. To do so, we first translate the problem to functions in the Hardy spaces on the disc via a conformal bijection, and take advantage of the inner-outer factorization. We also consider the same problem coupled with additional magnitude constraints, and determine if these constraints force uniqueness of the solution. Secondly, we extend some uniqueness results on the phase retrieval problem on the Hardy space on the disc to more general situations. More precisely, we show that certain holomorphic functions are uniquely determined by their moduli on two intersecting segments or on two concentric circles. Finally, we investigate the effect of zero-flipping on the stability of the phase retrieval problem for functions in the Paley-Wiener class, where zero-flipping refers to the replacement of zeros by their complex conjugates. We represent zero-flipping as an operator, and use its Fourier analytic properties to show our stability results.
Book Synopsis Introduction to Information Retrieval by : Christopher D. Manning
Download or read book Introduction to Information Retrieval written by Christopher D. Manning and published by Cambridge University Press. This book was released on 2008-07-07 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis 1-D and 2-D Phase Retrieval by Solving Linear Systems of Equations and by Using the Wavelet Transform by : Amy Elise Bell
Download or read book 1-D and 2-D Phase Retrieval by Solving Linear Systems of Equations and by Using the Wavelet Transform written by Amy Elise Bell and published by . This book was released on 1997 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Image Recovery: Theory and Application by : Henry Stark
Download or read book Image Recovery: Theory and Application written by Henry Stark and published by Elsevier. This book was released on 2013-04-25 with total page 565 pages. Available in PDF, EPUB and Kindle. Book excerpt: Image Recovery: Theory and Application focuses on signal recovery and synthesis problems. This book discusses the concepts of image recovery, including regularization, the projection theorem, and the pseudoinverse operator. Comprised of 13 chapters, this volume begins with a review of the basic properties of linear vector spaces and associated operators, followed by a discussion on the Gerchberg-Papoulis algorithm. It then explores image restoration and the basic mathematical theory in image restoration problems. The reader is also introduced to the problem of obtaining artifact-free computed tomographic reconstruction. Other chapters consider the importance of Bayesian approach in the context of medical imaging. In addition, the book discusses the linear programming method, which is particularly important for images with large number of pixels with zero value. Such images are usually found in medical imaging, microscopy, electron microscopy, and astronomy. This book can be a valuable resource to materials scientists, engineers, computed tomography technologists, and astronomers.
