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Nonlinear Approaches To The Inverse Problem Of Phase Retrieval From Single Measurement X Ray Intensity Data
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Book Synopsis Nonlinear Approaches to the Inverse Problem of Phase Retrieval from Single-measurement X-ray Intensity Data by : Julian Moosmann
Download or read book Nonlinear Approaches to the Inverse Problem of Phase Retrieval from Single-measurement X-ray Intensity Data written by Julian Moosmann and published by . This book was released on 2014 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Nonlinear Approaches for Phase Retrieval in the Fresnel Region for Hard X-ray Imaging by : Valentina Ion
Download or read book Nonlinear Approaches for Phase Retrieval in the Fresnel Region for Hard X-ray Imaging written by Valentina Ion and published by . This book was released on 2013 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The development of highly coherent X-ray sources offers new possibilities to image biological structures at different scales exploiting the refraction of X-rays. The coherence properties of the third-generation synchrotron radiation sources enables efficient implementations of phase contrast techniques. One of the first measurements of the intensity variations due to phase contrast has been reported in 1995 at the European Synchrotron Radiation Facility (ESRF). Phase imaging coupled to tomography acquisition allows threedimensional imaging with an increased sensitivity compared to absorption CT. This technique is particularly attractive to image samples with low absorption constituents. Phase contrast has many applications, ranging from material science, paleontology, bone research to medicine and biology. Several methods to achieve X-ray phase contrast have been proposed during the last years. In propagation based phase contrast, the measurements are made at different sample-to-detector distances. While the intensity data can be acquired and recorded, the phase information of the signal has to be "retrieved" from the modulus data only. Phase retrieval is thus an illposed nonlinear problem and regularization techniques including a priori knowledge are necessary to obtain stable solutions. Several phase recovery methods have been developed in recent years. These approaches generally formulate the phase retrieval problem as a linear one. Nonlinear treatments have not been much investigated. The main purpose of this work was to propose and evaluate new algorithms, in particularly taking into account the nonlinearity of the direct problem. In the first part of this work, we present a Landweber type nonlinear iterative scheme to solve the propagation based phase retrieval problem. This approach uses the analytic expression of the Fréchet derivative of the phase-intensity relationship and of its adjoint, which are presented in detail. We also study the effect of projection operators on the convergence properties of the method. In the second part of this thesis, we investigate the resolution of the linear inverse problem with an iterative thresholding algorithm in wavelet coordinates. In the following, the two former algorithms are combined and compared with another nonlinear approach based on sparsity regularization and a fixed point algorithm. The performance of theses algorithms are evaluated on simulated data for different noise levels. Finally the algorithms were adapted to process real data sets obtained in phase CT at the ESRF at Grenoble.
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 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 from X-ray Intensity Measurements by : David Montiel
Download or read book Phase Retrieval from X-ray Intensity Measurements written by David Montiel and published by . This book was released on 2009 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Deep Learning Based Phase Retrieval for X-ray Phase Contrast Imaging by : Kannara Mom
Download or read book Deep Learning Based Phase Retrieval for X-ray Phase Contrast Imaging written by Kannara Mom and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The development of highly coherent X-ray sources, such as third-generation synchrotron radiation facilities, has significantly contributed to the advancement of phase contrast imaging. The high degree of coherence of these sources enables efficient implementation of phase contrast techniques, and can increase sensitivity by several orders of magnitude. This novel imaging technique has found applications in a wide range of fields, including material science, paleontology, bone research, medicine, and biology. It enables the imaging of samples with low absorption constituents, where traditional absorption-based methods may fail to provide sufficient contrast. Several phase-sensitive imaging techniques have been developed, among them, propagation-based imaging requires no equipment other than the source, object and detector. Although the intensity can be measured at one or several propagation distances, the phase information is lost and must be estimated from those diffraction patterns, a process called phase retrieval. Phase retrieval in this context is a nonlinear ill-posed inverse problem. Various classical methods have been proposed to retrieve the phase, either by linearizing the problem to obtain an analytical solution, or by iterative algorithms. The main purpose of this thesis was to study what new deep learning approaches could bring to this phase retrieval problem. Various deep learning algorithms have been proposed and evaluated to address this problem. In the first part of this work, we show how neural networks can be used to reconstruct directly from measurements data, without model information. The architecture of the Mixed Scale Dense Network (MS-D Net) is introduced, combining dilated convolution and dense connection. In the second part of this thesis, we propose a nonlinear primal-dual algorithm for the retrieval of phase shift and absorption from a single X-ray in-line phase contrast. We showed that choosing different regularizers for absorption and phase can improve the reconstructions. In the third part, we propose to integrate neural networks into an existing optimization scheme using so-called unrolling approaches, in order to give the convolutional neural networks a specific role in the reconstruction. The performance of theses algorithms are evaluated using simulated noisy data as well as images acquired at NanoMAX (MAX IV, Lund, Sweden).
Book Synopsis PHASE RETRIEVAL FROM TWO DEFOCUSED IMAGES BY THE TRANSPORT OF INTENSITY EQUATION FORMALISM WITH FAST FOURIER TRANSFORM. by :
Download or read book PHASE RETRIEVAL FROM TWO DEFOCUSED IMAGES BY THE TRANSPORT OF INTENSITY EQUATION FORMALISM WITH FAST FOURIER TRANSFORM. written by and published by . This book was released on 2001 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The problem of phase retrieval from intensity measurements plays an important role in many fields of physical research, e.g. optics, electron and x-ray microscopy, crystallography, diffraction tomography and others. In practice the recorded images contain information only on the intensity distribution I(x, y)=[Psi]*[Psi]=[vert-bar]A[vert-bar][sup 2] of the imaging wave function[Psi]= A*exp( -i[var-phi]) and the phase information[var-phi](x, y) is usually lost. In general, the phase problem can be solved either by special holographic/interferometric methods, or by non-interferometric approaches based on intensity measurements in far Fraunhofer zone or in the Fresnel zone at two adjacent planes orthogonal to the optical axis. The latter approach uses the transport-of-intensity equation (TIE) formalism, introduced originally by Teague[1] and developed later in[2]. Applications of TIE to nonmagnetic materials and magnetic inductance mapping were successfully made in[3,4]. However, this approach still needs further improvement both in mathematics and in practical solutions, since the result is very sensitive to many experimental parameters.
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.
Download or read book Optics Index written by and published by . This book was released on 1990 with total page 700 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Phase Retrieval from Continuous and Discrete Ptychographic Measurements by : Sami Eid Merhi
Download or read book Phase Retrieval from Continuous and Discrete Ptychographic Measurements written by Sami Eid Merhi and published by . This book was released on 2019 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Computational Imaging written by and published by . This book was released on 2006 with total page 464 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.
Book Synopsis Journal of the Optical Society of America by :
Download or read book Journal of the Optical Society of America written by and published by . This book was released on 1991 with total page 694 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Phase Retrieval from Locally Supported Measurements by : Brian Patrick Preskitt
Download or read book Phase Retrieval from Locally Supported Measurements written by Brian Patrick Preskitt and published by . This book was released on 2018 with total page 231 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Feedback Control Theory by : John C. Doyle
Download or read book Feedback Control Theory written by John C. Doyle and published by Courier Corporation. This book was released on 2013-04-09 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: An excellent introduction to feedback control system design, this book offers a theoretical approach that captures the essential issues and can be applied to a wide range of practical problems. Its explorations of recent developments in the field emphasize the relationship of new procedures to classical control theory, with a focus on single input and output systems that keeps concepts accessible to students with limited backgrounds. The text is geared toward a single-semester senior course or a graduate-level class for students of electrical engineering. The opening chapters constitute a basic treatment of feedback design. Topics include a detailed formulation of the control design program, the fundamental issue of performance/stability robustness tradeoff, and the graphical design technique of loopshaping. Subsequent chapters extend the discussion of the loopshaping technique and connect it with notions of optimality. Concluding chapters examine controller design via optimization, offering a mathematical approach that is useful for multivariable systems.
Book Synopsis Compressive Phase Retrieval by : Lei Tian (Ph. D.)
Download or read book Compressive Phase Retrieval written by Lei Tian (Ph. D.) and published by . This book was released on 2013 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recovering a full description of a wave from limited intensity measurements remains a central problem in optics. Optical waves oscillate too fast for detectors to measure anything but time{averaged intensities. This is unfortunate since the phase can reveal important information about the object. When the light is partially coherent, a complete description of the phase requires knowledge about the statistical correlations for each pair of points in space. Recovery of the correlation function is a much more challenging problem since the number of pairs grows much more rapidly than the number of points. In this thesis, quantitative phase imaging techniques that works for partially coherent illuminations are investigated. In order to recover the phase information with few measurements, the sparsity in each underly problem and ecient inversion methods are explored under the framework of compressed sensing. In each phase retrieval technique under study, diffraction during spatial propagation is exploited as an effective and convenient mechanism to uniformly distribute the information about the unknown signal into the measurement space. Holography is useful to record the scattered field from a sparse distribution of particles; the ability of localizing each particles using compressive reconstruction method is studied. When a thin sample is illuminated with partially coherent waves, the transport of intensity phase retrieval method is shown to be eective to recover the optical path length of the sample and remove the eect of the illumination. This technique is particularly suitable for X-ray phase imaging since it does not require a coherent source or any optical components. Compressive tomographic reconstruction, which makes full use of the priors that the sample consists of piecewise constant refractive indices, are demonstrated to make up missing data. The third technique, known as the phase space tomography (PST), addresses the correlation function recovery problem. Implementing the PST involves measuring many intensity images under spatial propagation. Experimental demonstration of a compressive reconstruction method, which finds the sparse solution by decomposing the correlation function into a few mutually uncorrelated coherent modes, is presented to produce accurate reconstruction even when the measurement suers from the 'missing cone' problem in the Fourier domain.
Book Synopsis Coherent X-Ray Optics by : David Paganin
Download or read book Coherent X-Ray Optics written by David Paganin and published by Oxford University Press on Demand. This book was released on 2006-01-12 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: X-ray optics is undergoing a renaissance, which may be paralleled to that experienced by visible-light optics following the invention of the laser. The associated surge of activity in "coherent" x-ray optics has been documented in this monograph, the first of its type in the field.
