Program computational optimization research at lehigh. In this paper part i, it is demonstrated that the spp gs and gshio algorithms are both much better at avoiding stagnation during phase retrieval, allowing. Randomized firstorder methods for saddle point optimization. Ts optimization from point close to ts if you are close to a saddlepoin already, then starting a simple optts job will do the trick if you are lucky uses an approximate hessian.
We develop a convex optimization based framework to retrieve the signal support from the. A unified evaluation of iterative projection algorithms for phase retrieval. Maxima, minima, and saddle points article khan academy. We develop and test a nonlinear optimization algorithm for solving the problem of phase retrieval with transverse translation diversity, where the diverse farfield intensity measurements are taken after translating the object relative to a known illumination pattern. Phase retrieval is one of the toughest challenges in optimization, requiring the solution of a largescale nonlinear, nonconvex and nonsmooth constrained problem. The relaxation for optimal complicance design is independent of whether the underlying elastic problem is formulated in terms of displacements or strains.
An alternating optimization approach for phase retrieval huaiping ming1, dongyan huang2, lei xie1, haizhou li2 and minghui dong2 1school of computer science, northwestern polytechnical university, xian, china 2institute for infocomm research, a. For the first time, this paper investigates the phase retrieval problem with the assumption that the phase of the complex signal is sparse in contrast to the sparsity assumption on the signal itself as considered. In mathematics, the method of steepest descent or stationary phase method or saddle point method is an extension of laplaces method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point saddle point, in roughly the direction of steepest descent or stationary phase. Recent studies in compressive sensing have successfully shown that under certain conditions, a linear program, namely. Scientific computing research program under contract number deac0206ch157.
Marchesini s 2007 phase retrieval and saddlepoint optimization j. For the purposes of numerical experimentation and computation, it may be advantageous to formulate optimal design problems in terms of displacements as is done in ref. Alternating direction methods for classical and ptychographic phase. Sheridan1, 1school of electrical, electronic and communication engineering, communications and. We help companies unlock the hidden value in their supply chain with our operations research, ai and ml based supply chain planning and optimization suite. We study convex relaxation algorithms for phase retrieval on imaging problems.
Compressive phase retrieval via generalized approximate. Phase retrieval using hio matlab answers matlab central. We thus need to show that any feasible x with rankx 1, obeysf x f x0, a fact which follows from the strong concavity of f of the logarithm. In this note, we focus on smooth nonconvex optimization problems that obey. C m only needs to recover a vector of length m o s log. Optimization of phase retrieval in in line holography by. The method is provably stable for a class of saddlepoint problems, and improves performance of numerous gans. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Softwaretesting process will be quantified and impact of quantified steps on the overall quality of the software will be decided. It can also produce comparisons between algorithms, and.
We show that exploiting structural assumptions on the signal and the observations, such as sparsity, smoothness or positivity. Scientific computing research program under contract number deac02 06ch157. Phasepack is a software library that implements a wide range of different phase retrieval algorithms and initialization methods. Time and time again developers will go off to make something faster without systematically proving the cause of slowness and come back two weeks later with 700 lines of hand. Saddle point systems in optimal control martin stoll. Software testing efforts will be optimized in this phase using suitable optimization techniques. In this paper, we take a step toward bridging this gap. Lagrangian methods for semidefinite programming math. Software test automation, new strategy testing, complexity, and quality parameters. Browse other questions tagged optimization linearprogramming or ask your own question.
Gradient descent with random initialization princeton university. Phase retrieval for sparse signals using rank minimization. Phase retrieval, nonconvex optimization, function landscape. Shankar sastry department of electrical engineering and computer sciences, university of california at berkeley, ca, usa, email. Iterative algorithms with feedback are among the most powerful and versatile optimization methods for phase retrieval.
The paper deals with the problem of finding sparse solutions to systems of polynomial equations possibly perturbed by noise. The relaxed problem delivered by the displacementbased formulation is. Marchesini submitted on 24 nov 2006, last revised 7 jun 2007 this version, v5 abstract. Given a linear system in a real or complex domain, linear regression aims to recover the model parameters from a set of observations. Phase retrieval is the process of algorithmically finding solutions to the phase problem. Aug 14, 2017 this generalized phase retrieval gpr problem is a fundamental task in various disciplines and has been the subject of much recent investigation. Algorithms and theory for clustering and nonconvex quadratic program. Natural nonconvex heuristics often work remarkably well for gpr in practice, but lack clear theoretic explanations. Signal recovery from the amplitudes of the fourier transform, or equivalently from the autocorrelation function is a classical problem. If youre behind a web filter, please make sure that the domains. Phase retrieval consists of finding the phase that satisfies a set of constraints for a measured amplitude. Phase retrieval and saddlepoint optimization nasaads. We propose a new algorithm for recovering both complex field phase and amplitude and source distribution illumination spatial coherence from. A saddlepoint theorem with application to structural.
Phase retrieval and support estimation in xray diffraction. Phase retrieval with transverse translation diversity. Added to a standard smlm system, our setup features a dm placed in the fourier plane of the microscope, which allows one to spatially modulate the phase of the. The relaxed problem delivered by the displacementbased. Among these, the hybrid inputoutput algorithm has demonstrated practical solutions to gigaelement nonlinear phase retrieval problems, escaping local minima and producing images at resolutions beyond the capabilities of lensbased optical methods. Among these, the hybrid inputoutput algorithm has demonstrated. Solving composite optimization problems, with applications to phase retrieval john duchi based on joint work with feng ruan. The saddlepoint approximation is used with integrals in the. Analysis of asymptotic escape of strict saddle sets in. Visualizing and improving the robustness of phase retrieval algorithms. We also show the potential application of our analysis for a broader range of manifold optimization problems. Identifying and attacking the saddle point problem in high. Among these, the hybrid inputoutput algorithm has demonstrated practical solutions to gigaelement nonlinear phase retrieval problems, escaping local minima and producing images at. Fixed point equations and generalized interior feedback.
Efficient algorithms are being used in astronomical imaging, electron microscopy, lensless xray imaging and xray crystallography, substituting lenses and other optical elements in the imageforming process. International workshop on phase retrieval and coherent scattering desyesrfsls 15 june, 2005 igesa, island of porquerolles, france. The saddle point approximation is used with integrals in the. It has been long recognized that gradient methods or more generally subgradient methods for the nondi. Added to a standard smlm system, our setup features a dm placed in the fourier plane of the microscope, which allows one to. In general, students want to chose a software platform that will be both stateoftheart and accessible longterm. These include, for example, matrix completion 24, phase retrieval 42, and. In particular, we formulate the ptychographic phase retrieval problem as an unconstrained nonlinear minimization problem in section 2, and compare the con. This allows phase retrieval approaches based on, e. Optimization theory and algorithms uc davis tetrapods. We propose a new algorithm for recovering both complex field phase and amplitude and source distribution illumination spatial coherence from a stack of intensity images captured through focus. Gradient descent, smooth optimization, saddle points, local minimum, dynamical systems. If youre seeing this message, it means were having trouble loading external resources on our website. Randomized firstorder methods for saddle point optimization cong d.
A geometric analysis of phase retrieval springerlink. How to escape saddle points efficiently proceedings of machine. Phase retrieval algorithms proceed by iteratively applying a projectionbased operator to a random initial guess u, which ultimately drives the current iterate toward a. There are many software tools available to solve optimization problems ranging from free and opensource to proprietary commercial packages. Phase transitions for optimality gaps in optimal power flows pascal van hentenryck abstract. Solving composite optimization problems, with applications. Increasingly, the focus is shifting to nonconvex optimization, in which the functions to be minimized may have a plethora of local solutions and saddle points. Nonconvex problems appear in deep learning, as well as in important applications in signal and image processing. Here the phase retrieval problem is applied to the estimation of the phase of an electromagnetic wave field based on knowledge of its magnitude.
Phase retrieval and support estimation in xray diffraction james r. If x0 is the unique solution to phase retrieval up to global phase, then x0 x0x0. Solving optimization problems requires some familiarity with a computer programming language. Lei tian, laura waller, 3d intensity and phase imaging from light field measurements in an led array microscope, optica 2, 104111 2015. An optimization approach to eo phase retrieval chris kreucher and james a. Hopkins professor of optics university of rochester institute of optics presented to coherence 2005. Saddle point problems arise in a number of areas such as constrained optimization duality, zerosum games, and general equilibrium theory. A generalization for optimized phase retrieval algorithms daniel e. Iterative algorithms for ptychographic phase retrieval. Escaping saddle points in constrained optimization neurips. We further formally define strict saddle points for the constrained optimization. Phase retrieval and saddlepoint optimization article pdf available in journal of the optical society of america a 2410.
Dang y guanghui lan z november, 2015 abstract in this paper, we present novel randomized algorithms for solving saddle point problems whose dual feasible region is given by the direct product of many convex sets. Robust sparse phase retrieval made easy sciencedirect. We apply this saddle analysis to the phase retrieval problem on the lowrank matrix manifold, prove that there are only a finite number of saddles, and they are strict saddles with high probability. Georges iii integrity applications incorporated 900 victors way, suite 220, ann arbor, mi 48108 abstract this paper describes an optimization approach to optical phase retrieval. The second is a combined gs hybrid inputoutput algorithm gshioa. Phase retrieval and saddle point optimization marchesini, stefano. Nonconvex optimization, phase retrieval, levenbergmarquardt method. In particular, we show how these solutions can be recovered from groupsparse solutions of a derived system of linear equations. A geometric analysis of phase retrieval columbia university. Nonlinear optimization algorithm for partially coherent. Examples include image classification, data clustering, and phase retrieval. It can also produce comparisons between algorithms, and comes with empirical datasets for testing on realworld problems. Phase retrieval with sparse phase constraint hieu thao nguyen, d. May 23, 2016 nonlinear optimization algorithm for partially coherent phase retrieval and source recovery abstract.
Oct 18, 2019 increasingly, the focus is shifting to nonconvex optimization, in which the functions to be minimized may have a plethora of local solutions and saddle points. Quantitative dpc phase imaging on led array microscope. Matlab code implements a tikhonov deconvolution based phase reconstruction algorithm from single or multiaxis dpc data. Specifically, if an eigenvalue is negative, the newton step moves along the eigenvector in a direction opposite to the gradient descent step, and thus moves in the direction of. Iterative algorithms with feedback are amongst the most powerful and versatile optimization methods for phase retrieval. In mathematics, the method of steepest descent or stationaryphase method or saddlepoint method is an extension of laplaces method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point saddle point, in roughly the direction of steepest descent or stationary phase. The first we refer to as the spatial phase perturbation gs algorithm spp gsa. Cn from the magnitudes y z of the linear often fourierbased transformation z ax0. On monotone nonexpansive mapping and their approximation fixed point results buthinah bin dehaish abstract. Russell luke, oleg soloviev, michael verhaegen siam j. Two modified gerchbergsaxton gs iterative phase retrieval algorithms are proposed. Phase retrieval, error reduction algorithm, and fienup variants.
Properties of the saddle point system k a bt b 0 kis nonsingular if b has full rank and a is positive on kerb kis symmetric and inde nite. July 11, 2012 compressive phase retrieval from squared output measurements via semidefinite programming henrik ohlsson, allen y. A generalization for optimized phase retrieval algorithms. Due to the absence of phase information, signal recovery requires some form of additional prior information. This generalized phase retrieval gpr problem is a fundamental task in various disciplines and has been the subject of much recent investigation. Multigrid optimization for largescale ptychographic phase. A threelevel optimization model for fuelsupply strategies of natural gasfired units bining zhao abstract. Concrete applications such as dictionary learning, generalized phase retrieval, and orthogonal tensor decomposition are known to induce such structures. Zola3d allows flexible 3d localization microscopy over an. Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery abstract.
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