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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: 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: 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: 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: Carolin Homann Publisher: Göttingen University Press ISBN: 3863952103 Category : Languages : en Pages : 126
Book Description
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.
Author: Henry Stark Publisher: Elsevier ISBN: 0323145973 Category : Technology & Engineering Languages : en Pages : 565
Book Description
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.
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: N.E. Hurt Publisher: Springer Science & Business Media ISBN: 9781402003370 Category : Mathematics Languages : en Pages : 328
Book Description
'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.
Author: Alexander H. Barnett
Author: Alexander H. Barnett
Author: David Aaron Barmherzig
Author: Tim Salditt
Author: Keith Allen Rinaldi
Author: Yaocheng Tian
Author: Peter Ebenfelt
Author: Carolin Homann
Author: Henry Stark
Author: Christopher D. Manning
Author: N.E. Hurt