Search Menu
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information
Search Scholarly Publications

Title: Efficient ptychographic phase retrieval via a matrix-free Levenberg-Marquardt algorithm

Abstract

The phase retrieval problem, where one aims to recover a complex-valued image from far-field intensity measurements, is a classic problem encountered in a range of imaging applications. Modern phase retrieval approaches usually rely on gradient descent methods in a nonlinear minimization framework. Calculating closed-form gradients for use in these methods is tedious work, and formulating second order derivatives is even more laborious. Additionally, second order techniques often require the storage and inversion of large matrices of partial derivatives, with memory requirements that can be prohibitive for data-rich imaging modalities. We use a reverse-mode automatic differentiation (AD) framework to implement an efficient matrix-free version of the Levenberg-Marquardt (LM) algorithm, a longstanding method that finds popular use in nonlinear least-square minimization problems but which has seen little use in phase retrieval. Furthermore, we extend the basic LM algorithm so that it can be applied for more general constrained optimization problems (including phase retrieval problems) beyond just the least-square applications. Since we use AD, we only need to specify the physics-based forward model for a specific imaging application; the first and second-order derivative terms are calculated automatically through matrix-vector products, without explicitly forming the large Jacobian or Gauss-Newton matrices typically required for themore » LM method. We demonstrate that this algorithm can be used to solve both the unconstrained ptychographic object retrieval problem and the constrained “blind” ptychographic object and probe retrieval problems, under the popular Gaussian noise model as well as the Poisson noise model. We compare this algorithm to state-of-the-art first order ptychographic reconstruction methods to demonstrate empirically that this method outperforms best-in-class first-order methods: it provides excellent convergence guarantees with (in many cases) a superlinear rate of convergence, all with a computational cost comparable to, or lower than, the tested first-order algorithms.« less

Authors:
ORCiD logo; ORCiD logo; ORCiD logo; ORCiD logo; ORCiD logo;
Publication Date:
Research Org.:
Argonne National Laboratory (ANL), Argonne, IL (United States); SLAC National Accelerator Laboratory (SLAC), Menlo Park, CA (United States)
Sponsoring Org.:
USDOE; National Institutes of Health (NIH); USDOE Office of Science (SC), Basic Energy Sciences (BES). Materials Sciences & Engineering Division
OSTI Identifier:
1806256
Alternate Identifier(s):
OSTI ID: 1813106; OSTI ID: 1814924
Grant/Contract Number:  
AC02-06CH11357; R01 GM104530; R01 MH115265; AC02-76SF00515
Resource Type:
Published Article
Journal Name:
Optics Express
Additional Journal Information:
Journal Name: Optics Express Journal Volume: 29 Journal Issue: 15; Journal ID: ISSN 1094-4087
Publisher:
Optical Society of America
Country of Publication:
United States
Language:
English
Subject:
36 MATERIALS SCIENCE; fourier optics; image and signal processing

Citation Formats

Kandel, Saugat, Maddali, S., Nashed, Youssef S. G., Hruszkewycz, Stephan O., Jacobsen, Chris, and Allain, Marc. Efficient ptychographic phase retrieval via a matrix-free Levenberg-Marquardt algorithm. United States: N. p., 2021. Web. doi:10.1364/OE.422768.
Copy to clipboard
Kandel, Saugat, Maddali, S., Nashed, Youssef S. G., Hruszkewycz, Stephan O., Jacobsen, Chris, & Allain, Marc. Efficient ptychographic phase retrieval via a matrix-free Levenberg-Marquardt algorithm. United States. https://doi.org/10.1364/OE.422768
Copy to clipboard
Kandel, Saugat, Maddali, S., Nashed, Youssef S. G., Hruszkewycz, Stephan O., Jacobsen, Chris, and Allain, Marc. Wed . "Efficient ptychographic phase retrieval via a matrix-free Levenberg-Marquardt algorithm". United States. https://doi.org/10.1364/OE.422768.
Copy to clipboard
@article{osti_1806256,
title = {Efficient ptychographic phase retrieval via a matrix-free Levenberg-Marquardt algorithm},
author = {Kandel, Saugat and Maddali, S. and Nashed, Youssef S. G. and Hruszkewycz, Stephan O. and Jacobsen, Chris and Allain, Marc},
abstractNote = {The phase retrieval problem, where one aims to recover a complex-valued image from far-field intensity measurements, is a classic problem encountered in a range of imaging applications. Modern phase retrieval approaches usually rely on gradient descent methods in a nonlinear minimization framework. Calculating closed-form gradients for use in these methods is tedious work, and formulating second order derivatives is even more laborious. Additionally, second order techniques often require the storage and inversion of large matrices of partial derivatives, with memory requirements that can be prohibitive for data-rich imaging modalities. We use a reverse-mode automatic differentiation (AD) framework to implement an efficient matrix-free version of the Levenberg-Marquardt (LM) algorithm, a longstanding method that finds popular use in nonlinear least-square minimization problems but which has seen little use in phase retrieval. Furthermore, we extend the basic LM algorithm so that it can be applied for more general constrained optimization problems (including phase retrieval problems) beyond just the least-square applications. Since we use AD, we only need to specify the physics-based forward model for a specific imaging application; the first and second-order derivative terms are calculated automatically through matrix-vector products, without explicitly forming the large Jacobian or Gauss-Newton matrices typically required for the LM method. We demonstrate that this algorithm can be used to solve both the unconstrained ptychographic object retrieval problem and the constrained “blind” ptychographic object and probe retrieval problems, under the popular Gaussian noise model as well as the Poisson noise model. We compare this algorithm to state-of-the-art first order ptychographic reconstruction methods to demonstrate empirically that this method outperforms best-in-class first-order methods: it provides excellent convergence guarantees with (in many cases) a superlinear rate of convergence, all with a computational cost comparable to, or lower than, the tested first-order algorithms.},
doi = {10.1364/OE.422768},
journal = {Optics Express},
number = 15,
volume = 29,
place = {United States},
year = {Wed Jul 07 00:00:00 EDT 2021},
month = {Wed Jul 07 00:00:00 EDT 2021}
}
Copy to clipboard
Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1364/OE.422768
Copyright Statement
Other availability
Search WorldCat Search WorldCat to find libraries that may hold this journal
Save / Share:
Export Metadata
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.

Works referenced in this record:

Phase retrieval with transverse translation diversity: a nonlinear optimization approach
journal, January 2008

  • Guizar-Sicairos, Manuel; Fienup, James R.
  • Optics Express, Vol. 16, Issue 10
  • DOI: 10.1364/OE.16.007264

Maximum-likelihood refinement for coherent diffractive imaging
journal, June 2012

Unconstrained Optimization of Real Functions in Complex Variables
journal, January 2012

  • Sorber, Laurent; Barel, Marc Van; Lathauwer, Lieven De
  • SIAM Journal on Optimization, Vol. 22, Issue 3
  • DOI: 10.1137/110832124

Experimental robustness of Fourier ptychography phase retrieval algorithms
journal, January 2015

  • Yeh, Li-Hao; Dong, Jonathan; Zhong, Jingshan
  • Optics Express, Vol. 23, Issue 26
  • DOI: 10.1364/OE.23.033214

Ptychography and Related Diffractive Imaging Methods
book, May 2008

Use of coherent X-ray diffraction to map strain fields in nanocrystals
journal, October 2001

Fourier ptychographic microscopy with sparse representation
journal, August 2017

Distributed Automatic Differentiation for Ptychography
journal, January 2017

Using automatic differentiation as a general framework for ptychographic reconstruction
journal, January 2019

  • Kandel, Saugat; Maddali, S.; Allain, Marc
  • Optics Express, Vol. 27, Issue 13
  • DOI: 10.1364/OE.27.018653

Conjugate gradient method for phase retrieval based on the Wirtinger derivative
journal, January 2017

  • Wei, Zhun; Chen, Wen; Qiu, Cheng-Wei
  • Journal of the Optical Society of America A, Vol. 34, Issue 5
  • DOI: 10.1364/JOSAA.34.000708

On Fienup Methods for Sparse Phase Retrieval
journal, February 2018

  • Pauwels, Edouard Jean Robert; Beck, Amir; Eldar, Yonina C.
  • IEEE Transactions on Signal Processing, Vol. 66, Issue 4
  • DOI: 10.1109/TSP.2017.2780044

Blind Ptychographic Phase Retrieval via Convergent Alternating Direction Method of Multipliers
journal, January 2019

  • Chang, Huibin; Enfedaque, Pablo; Marchesini, Stefano
  • SIAM Journal on Imaging Sciences, Vol. 12, Issue 1
  • DOI: 10.1137/18M1188446

Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging
journal, January 2012

  • Humphry, M. J.; Kraus, B.; Hurst, A. C.
  • Nature Communications, Vol. 3, Issue 1
  • DOI: 10.1038/ncomms1733

Movable Aperture Lensless Transmission Microscopy: A Novel Phase Retrieval Algorithm
journal, July 2004

Convergence Properties of the Inexact Levenberg-Marquardt Method under Local Error Bound Conditions
journal, January 2002

  • Dan, Hiroshige; Yamashita, Nobuo; Fukushima, Masao
  • Optimization Methods and Software, Vol. 17, Issue 4
  • DOI: 10.1080/1055678021000049345

Dynamische Theorie der Kristallstrukturanalyse durch Elektronenbeugung im inhomogenen Primärstrahlwellenfeld
journal, November 1970

  • Hegerl, R.; Hoppe, W.
  • Berichte der Bunsengesellschaft für physikalische Chemie, Vol. 74, Issue 11
  • DOI: 10.1002/bbpc.19700741112

Direct three-dimensional tomographic reconstruction and phase retrieval of far-field coherent diffraction patterns
journal, February 2019

  • Ramos, Tiago; Grønager, Bastian Ellegård; Andersen, Martin Skovgaard
  • Physical Review A, Vol. 99, Issue 2
  • DOI: 10.1103/PhysRevA.99.023801

On the Levenberg-Marquardt methods for convex constrained nonlinear equations
journal, January 2013

Globally Convergent Levenberg-Marquardt Method for Phase Retrieval
journal, April 2019

  • Ma, Chao; Liu, Xin; Wen, Zaiwen
  • IEEE Transactions on Information Theory, Vol. 65, Issue 4
  • DOI: 10.1109/TIT.2018.2881187

Noise models for low counting rate coherent diffraction imaging
journal, January 2012

  • Godard, Pierre; Allain, Marc; Chamard, Virginie
  • Optics Express, Vol. 20, Issue 23
  • DOI: 10.1364/OE.20.025914

An inexact Levenberg-Marquardt method for large sparse nonlinear least squres
journal, April 1985

  • Wright, S. J.; Holt, J. N.
  • The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, Vol. 26, Issue 4
  • DOI: 10.1017/S0334270000004604

Relaxed averaged alternating reflections for diffraction imaging
journal, November 2004

Blind ptychography: uniqueness and ambiguities
journal, February 2020

Levenberg–Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints
journal, December 2004

  • Kanzow, Christian; Yamashita, Nobuo; Fukushima, Masao
  • Journal of Computational and Applied Mathematics, Vol. 172, Issue 2
  • DOI: 10.1016/j.cam.2004.02.013

Extended Levenberg-Marquardt Method for Composite Function Minimization
journal, June 2017

Phase retrieval by iterated projections
journal, January 2003

A complex gradient operator and its application in adaptive array theory
journal, January 1983

  • Brandwood, D. H.
  • IEE Proceedings F Communications, Radar and Signal Processing, Vol. 130, Issue 1
  • DOI: 10.1049/ip-f-1.1983.0003

Proximal Heterogeneous Block Implicit-Explicit Method and Application to Blind Ptychographic Diffraction Imaging
journal, January 2015

  • Hesse, Robert; Luke, D. Russell; Sabach, Shoham
  • SIAM Journal on Imaging Sciences, Vol. 8, Issue 1
  • DOI: 10.1137/14098168X

Invited Article: A unified evaluation of iterative projection algorithms for phase retrieval
journal, January 2007

  • Marchesini, S.
  • Review of Scientific Instruments, Vol. 78, Issue 1
  • DOI: 10.1063/1.2403783

An estimator for the diagonal of a matrix
journal, November 2007

Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields
journal, January 2004

  • Guizar-Sicairos, Manuel; Gutiérrez-Vega, Julio C.
  • Journal of the Optical Society of America A, Vol. 21, Issue 1
  • DOI: 10.1364/JOSAA.21.000053

Implicit alternating direction methods
journal, January 1959

Probe retrieval in ptychographic coherent diffractive imaging
journal, March 2009

An Algorithm for Least-Squares Estimation of Nonlinear Parameters
journal, June 1963

  • Marquardt, Donald W.
  • Journal of the Society for Industrial and Applied Mathematics, Vol. 11, Issue 2
  • DOI: 10.1137/0111030

A phase retrieval algorithm for shifting illumination
journal, November 2004

  • Rodenburg, J. M.; Faulkner, H. M. L.
  • Applied Physics Letters, Vol. 85, Issue 20
  • DOI: 10.1063/1.1823034

Phase retrieval via Sparse Wirtinger Flow
journal, August 2019

  • Yuan, Ziyang; Wang, Hongxia; Wang, Qi
  • Journal of Computational and Applied Mathematics, Vol. 355
  • DOI: 10.1016/j.cam.2019.01.009

An improved ptychographical phase retrieval algorithm for diffractive imaging
journal, September 2009

Iterative least-squares solver for generalized maximum-likelihood ptychography
journal, January 2018

  • Odstrčil, Michal; Menzel, Andreas; Guizar-Sicairos, Manuel
  • Optics Express, Vol. 26, Issue 3
  • DOI: 10.1364/OE.26.003108

A method for the solution of certain non-linear problems in least squares
journal, January 1944

  • Levenberg, Kenneth
  • Quarterly of Applied Mathematics, Vol. 2, Issue 2
  • DOI: 10.1090/qam/10666

Proximal Algorithms
journal, January 2014

  • Parikh, Neal
  • Foundations and Trends® in Optimization, Vol. 1, Issue 3
  • DOI: 10.1561/2400000003

Quantitative Nanoscale Imaging of Lattice Distortions in Epitaxial Semiconductor Heterostructures Using Nanofocused X-ray Bragg Projection Ptychography
journal, September 2012

  • Hruszkewycz, S. O.; Holt, M. V.; Murray, C. E.
  • Nano Letters, Vol. 12, Issue 10
  • DOI: 10.1021/nl303201w

Fast Curvature Matrix-Vector Products for Second-Order Gradient Descent
journal, July 2002

Phase retrieval algorithms: a comparison
journal, January 1982

Three dimensions, two microscopes, one code: Automatic differentiation for x-ray nanotomography beyond the depth of focus limit
journal, March 2020

  • Du, Ming; Nashed, Youssef S. G.; Kandel, Saugat
  • Science Advances, Vol. 6, Issue 13
  • DOI: 10.1126/sciadv.aay3700

Ptychographic X-ray computed tomography at the nanoscale
journal, September 2010

  • Dierolf, Martin; Menzel, Andreas; Thibault, Pierre
  • Nature, Vol. 467, Issue 7314
  • DOI: 10.1038/nature09419

Sparse Phase Retrieval via Truncated Amplitude Flow
journal, January 2018

  • Wang, Gang; Zhang, Liang; Giannakis, Georgios B.
  • IEEE Transactions on Signal Processing, Vol. 66, Issue 2
  • DOI: 10.1109/TSP.2017.2771733

Adorym: a multi-platform generic X-ray image reconstruction framework based on automatic differentiation
journal, January 2021

  • Du, Ming; Kandel, Saugat; Deng, Junjing
  • Optics Express, Vol. 29, Issue 7
  • DOI: 10.1364/OE.418296

Optical ptychography: a practical implementation with useful resolution
journal, January 2010

  • Maiden, Andrew M.; Rodenburg, John M.; Humphry, Martin J.
  • Optics Letters, Vol. 35, Issue 15
  • DOI: 10.1364/OL.35.002585

Further improvements to the ptychographical iterative engine
journal, January 2017

Reconstruction of an object from the modulus of its Fourier transform
journal, January 1978

Hard-X-Ray Lensless Imaging of Extended Objects
journal, January 2007

Applications of algorithmic differentiation to phase retrieval algorithms
journal, January 2014

  • Jurling, Alden S.; Fienup, James R.
  • Journal of the Optical Society of America A, Vol. 31, Issue 7
  • DOI: 10.1364/JOSAA.31.001348

Fast Exact Multiplication by the Hessian
journal, January 1994

Global Complexity Bound of the Inexact Levenberg–Marquardt Method
journal, December 2017

  • Huang, Jian-Chao; Fan, Jin-Yan
  • Journal of the Operations Research Society of China, Vol. 6, Issue 3
  • DOI: 10.1007/s40305-017-0184-0

Phase Retrieval via Wirtinger Flow: Theory and Algorithms
journal, April 2015

  • Candes, Emmanuel J.; Li, Xiaodong; Soltanolkotabi, Mahdi
  • IEEE Transactions on Information Theory, Vol. 61, Issue 4
  • DOI: 10.1109/TIT.2015.2399924

Wide-field, high-resolution Fourier ptychographic microscopy
journal, July 2013

Efficient maximum likelihood estimator fitting of histograms
journal, May 2010

    Sort by:
    [ × clear filter / sort ]

    Similar Records in DOE PAGES and OSTI.GOV collections: