Experimental robustness of Fourier ptychography phase retrieval algorithms

Li-Hao Yeh; Jonathan Dong; Jingshan Zhong; Lei Tian; Michael Chen; Gongguo Tang; Mahdi Soltanolkotabi; Laura Waller

doi:10.1364/OE.23.033214

Experimental robustness of Fourier ptychography phase retrieval algorithms

Li-Hao Yeh, Jonathan Dong, Jingshan Zhong, Lei Tian, Michael Chen, Gongguo Tang, Mahdi Soltanolkotabi, and Laura Waller

Optics Express
Vol. 23,
Issue 26,
pp. 33214-33240
(2015)
•https://doi.org/10.1364/OE.23.033214

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Li-Hao Yeh, Jonathan Dong, Jingshan Zhong, Lei Tian, Michael Chen, Gongguo Tang, Mahdi Soltanolkotabi, and Laura Waller, "Experimental robustness of Fourier ptychography phase retrieval algorithms," Opt. Express 23, 33214-33240 (2015)

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Related Topics
Table of Contents Category
- Image Processing
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Phase retrieval (100.5070)
- Computational imaging (110.1758)

About this Article
History
- Original Manuscript: November 3, 2015
- Revised Manuscript: December 6, 2015
- Manuscript Accepted: December 6, 2015
- Published: December 16, 2015
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 11, Iss. 2

Accessible

Open Access

Abstract

Fourier ptychography is a new computational microscopy technique that provides gigapixel-scale intensity and phase images with both wide field-of-view and high resolution. By capturing a stack of low-resolution images under different illumination angles, an inverse algorithm can be used to computationally reconstruct the high-resolution complex field. Here, we compare and classify multiple proposed inverse algorithms in terms of experimental robustness. We find that the main sources of error are noise, aberrations and mis-calibration (i.e. model mis-match). Using simulations and experiments, we demonstrate that the choice of cost function plays a critical role, with amplitude-based cost functions performing better than intensity-based ones. The reason for this is that Fourier ptychography datasets consist of images from both brightfield and darkfield illumination, representing a large range of measured intensities. Both noise (e.g. Poisson noise) and model mis-match errors are shown to scale with intensity. Hence, algorithms that use an appropriate cost function will be more tolerant to both noise and model mis-match. Given these insights, we propose a global Newton’s method algorithm which is robust and accurate. Finally, we discuss the impact of procedures for algorithmic correction of aberrations and mis-calibration.

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References

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Year
|
Author
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Publication

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photonics 7, 739–745 (2013).
[Crossref]
J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Applied Physics Letters 85, 4795–4797 (2004).
[Crossref]
G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Optics express 17, 624–639 (2009).
[Crossref] [PubMed]
A. Williams, J. Chung, X. Ou, G. Zheng, S. Rawal, Z. Ao, R. Datar, C. Yang, and R. Cote, “Fourier ptychographic microscopy for filtration-based circulating tumor cell enumeration and analysis,” Journal of biomedical optics 19, 066007(2014).
[Crossref] [PubMed]
R. Horstmeyer, X. Ou, G. Zheng, P. Willems, and C. Yang, “Digital pathology with fourier ptychography,” Computerized Medical Imaging and Graphics 42, 38–43 (2015).
[Crossref]
J. Chung, X. Ou, R. P. Kulkarni, and C. Yang, “Counting White Blood Cells from a Blood Smear Using Fourier Ptychographic Microscopy,” PloS one10 (2015).
L. Tian, Z. Liu, L.-H. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
[Crossref]
L. Tian and L. Waller, “3D intensity and phase imaging from light field measurements in an LED array microscope,” Optica 2, 104–111 (2015).
[Crossref]
R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).
J. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Optics Letters 327–29 (1978).
[Crossref] [PubMed]
J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref] [PubMed]
V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[Crossref]
S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[Crossref]
M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Optics Express 16, 7264–7278 (2008).
[Crossref] [PubMed]
A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]
P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]
C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative Algorithms for Ptychographic Phase Retrieval,” arXiv:1105.5628v1 (2011).
X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
[Crossref] [PubMed]
L. Tian, X. Li, K. Ramchandran, and L. Waller, “Multiplexed coded illumination for Fourier ptychography with an LED array microscope,” Biomed. Opt. Express 5, 2376–2389 (2014).
[Crossref] [PubMed]
E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: Theory and algorithms,” arXiv:1407.1065 (2014).
L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Wirtinger flow optimization,” Optics Express 23, 4856–4866 (2015).
[Crossref] [PubMed]
J. Nocedal and S. Wright, “Numerical Optimization,” Springer Science & Business Media (2006).
B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization,” SIAM Review 52, 471–501 (2010).
[Crossref]
E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase Retrieval via Matrix Completion,” SIAM Journal on Imaging Sciences 6, 199–225 (2013).
[Crossref]
E. J. Candès, T. Strohmer, and V. Voroninski, “PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming,” Communications on Pure and Applied Math 66, 1241–1274 (2013).
[Crossref]
S. Burer and R. Monteiro, “A Nonlinear Programming Algorithm for Solving Semidefinite Programs via Low-Rank Factorization,” Mathematical Programming 95, 329–357 (2003).
[Crossref]
R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]
E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” arXiv:1310.3240 (2013).
P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New Journal of Physics 14, 063004 (2012).
[Crossref]
K. Kreutz-Delgado, “The Complex Gradient Operator and the CR-Calculus,” arXiv:0906.4835v1 (2009).
J. Nocedal and S. J. Wright, “Conjugate gradient methods,” Springer (2006).
A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[Crossref] [PubMed]
F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21, 13592–13606 (2013).
[Crossref] [PubMed]
A. Tripathi, I. McNulty, and O. G. Shpyrko, “Ptychographic overlap constraint errors and the limits of their numerical recovery using conjugate gradient descent methods,” Opt. Express 22, 1452–1466 (2014).
[Crossref] [PubMed]
R. Horstmeyer, X. Ou, J. Chung, G. Zheng, and C. Yang, “Overlapped Fourier coding for optical aberration removal,” Optics Express 22, 24062–24080 (2014).
[Crossref] [PubMed]
http://www.laurawaller.com/opensource/

2015 (5)

R. Horstmeyer, X. Ou, G. Zheng, P. Willems, and C. Yang, “Digital pathology with fourier ptychography,” Computerized Medical Imaging and Graphics 42, 38–43 (2015).
[Crossref]

L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Wirtinger flow optimization,” Optics Express 23, 4856–4866 (2015).
[Crossref] [PubMed]

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]

L. Tian and L. Waller, “3D intensity and phase imaging from light field measurements in an LED array microscope,” Optica 2, 104–111 (2015).
[Crossref]

L. Tian, Z. Liu, L.-H. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
[Crossref]

2014 (5)

R. Horstmeyer, X. Ou, J. Chung, G. Zheng, and C. Yang, “Overlapped Fourier coding for optical aberration removal,” Optics Express 22, 24062–24080 (2014).
[Crossref] [PubMed]

A. Tripathi, I. McNulty, and O. G. Shpyrko, “Ptychographic overlap constraint errors and the limits of their numerical recovery using conjugate gradient descent methods,” Opt. Express 22, 1452–1466 (2014).
[Crossref] [PubMed]

X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
[Crossref] [PubMed]

L. Tian, X. Li, K. Ramchandran, and L. Waller, “Multiplexed coded illumination for Fourier ptychography with an LED array microscope,” Biomed. Opt. Express 5, 2376–2389 (2014).
[Crossref] [PubMed]

A. Williams, J. Chung, X. Ou, G. Zheng, S. Rawal, Z. Ao, R. Datar, C. Yang, and R. Cote, “Fourier ptychographic microscopy for filtration-based circulating tumor cell enumeration and analysis,” Journal of biomedical optics 19, 066007(2014).
[Crossref] [PubMed]

2013 (4)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photonics 7, 739–745 (2013).
[Crossref]

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase Retrieval via Matrix Completion,” SIAM Journal on Imaging Sciences 6, 199–225 (2013).
[Crossref]

E. J. Candès, T. Strohmer, and V. Voroninski, “PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming,” Communications on Pure and Applied Math 66, 1241–1274 (2013).
[Crossref]

F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21, 13592–13606 (2013).
[Crossref] [PubMed]

2012 (2)

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New Journal of Physics 14, 063004 (2012).
[Crossref]

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[Crossref] [PubMed]

2010 (1)

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization,” SIAM Review 52, 471–501 (2010).
[Crossref]

2009 (3)

G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Optics express 17, 624–639 (2009).
[Crossref] [PubMed]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

2008 (1)

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Optics Express 16, 7264–7278 (2008).
[Crossref] [PubMed]

2007 (1)

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[Crossref]

2004 (1)

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Applied Physics Letters 85, 4795–4797 (2004).
[Crossref]

2003 (2)

S. Burer and R. Monteiro, “A Nonlinear Programming Algorithm for Solving Semidefinite Programs via Low-Rank Factorization,” Mathematical Programming 95, 329–357 (2003).
[Crossref]

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[Crossref]

1982 (1)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref] [PubMed]

1978 (1)

J. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Optics Letters 327–29 (1978).
[Crossref] [PubMed]

1971 (1)

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Ames, B.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]

Ao, Z.

Bean, R.

Berenguer, F.

Bian, L.

L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Wirtinger flow optimization,” Optics Express 23, 4856–4866 (2015).
[Crossref] [PubMed]

Brady, G. R.

Bunk, O.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

Burer, S.

S. Burer and R. Monteiro, “A Nonlinear Programming Algorithm for Solving Semidefinite Programs via Low-Rank Factorization,” Mathematical Programming 95, 329–357 (2003).
[Crossref]

Candès, E. J.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase Retrieval via Matrix Completion,” SIAM Journal on Imaging Sciences 6, 199–225 (2013).
[Crossref]

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” arXiv:1310.3240 (2013).

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: Theory and algorithms,” arXiv:1407.1065 (2014).

Chen, B.

Chen, F.

L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Wirtinger flow optimization,” Optics Express 23, 4856–4866 (2015).
[Crossref] [PubMed]

Chen, M.

L. Tian, Z. Liu, L.-H. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
[Crossref]

Chen, R. Y.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]

Chung, J.

R. Horstmeyer, X. Ou, J. Chung, G. Zheng, and C. Yang, “Overlapped Fourier coding for optical aberration removal,” Optics Express 22, 24062–24080 (2014).
[Crossref] [PubMed]

J. Chung, X. Ou, R. P. Kulkarni, and C. Yang, “Counting White Blood Cells from a Blood Smear Using Fourier Ptychographic Microscopy,” PloS one10 (2015).

Cote, R.

Dai, Q.

L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Wirtinger flow optimization,” Optics Express 23, 4856–4866 (2015).
[Crossref] [PubMed]

Datar, R.

Diaz, A.

Dierolf, M.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

Eldar, Y. C.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase Retrieval via Matrix Completion,” SIAM Journal on Imaging Sciences 6, 199–225 (2013).
[Crossref]

Elser, V.

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[Crossref]

Faulkner, H. M.

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Applied Physics Letters 85, 4795–4797 (2004).
[Crossref]

Fazel, M.

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization,” SIAM Review 52, 471–501 (2010).
[Crossref]

Fienup, J.

J. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Optics Letters 327–29 (1978).
[Crossref] [PubMed]

Fienup, J. R.

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Optics Express 16, 7264–7278 (2008).
[Crossref] [PubMed]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref] [PubMed]

Gerchberg, R.

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Guizar-Sicairos, M.

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New Journal of Physics 14, 063004 (2012).
[Crossref]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Optics Express 16, 7264–7278 (2008).
[Crossref] [PubMed]

Guo, K.

L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Wirtinger flow optimization,” Optics Express 23, 4856–4866 (2015).
[Crossref] [PubMed]

Horstmeyer, R.

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]

R. Horstmeyer, X. Ou, G. Zheng, P. Willems, and C. Yang, “Digital pathology with fourier ptychography,” Computerized Medical Imaging and Graphics 42, 38–43 (2015).
[Crossref]

R. Horstmeyer, X. Ou, J. Chung, G. Zheng, and C. Yang, “Overlapped Fourier coding for optical aberration removal,” Optics Express 22, 24062–24080 (2014).
[Crossref] [PubMed]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photonics 7, 739–745 (2013).
[Crossref]

Humphry, M. J.

Kraus, B.

Kreutz-Delgado, K.

K. Kreutz-Delgado, “The Complex Gradient Operator and the CR-Calculus,” arXiv:0906.4835v1 (2009).

Kulkarni, R. P.

J. Chung, X. Ou, R. P. Kulkarni, and C. Yang, “Counting White Blood Cells from a Blood Smear Using Fourier Ptychographic Microscopy,” PloS one10 (2015).

Li, X.

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: Theory and algorithms,” arXiv:1407.1065 (2014).

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” arXiv:1310.3240 (2013).

Liu, Z.

L. Tian, Z. Liu, L.-H. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
[Crossref]

Maia, F.

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative Algorithms for Ptychographic Phase Retrieval,” arXiv:1105.5628v1 (2011).

Maiden, A. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

Marchesini, S.

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[Crossref]

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative Algorithms for Ptychographic Phase Retrieval,” arXiv:1105.5628v1 (2011).

McNulty, I.

Menzel, A.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

Monteiro, R.

S. Burer and R. Monteiro, “A Nonlinear Programming Algorithm for Solving Semidefinite Programs via Low-Rank Factorization,” Mathematical Programming 95, 329–357 (2003).
[Crossref]

Nocedal, J.

J. Nocedal and S. J. Wright, “Conjugate gradient methods,” Springer (2006).

J. Nocedal and S. Wright, “Numerical Optimization,” Springer Science & Business Media (2006).

Ou, X.

R. Horstmeyer, X. Ou, G. Zheng, P. Willems, and C. Yang, “Digital pathology with fourier ptychography,” Computerized Medical Imaging and Graphics 42, 38–43 (2015).
[Crossref]

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]

R. Horstmeyer, X. Ou, J. Chung, G. Zheng, and C. Yang, “Overlapped Fourier coding for optical aberration removal,” Optics Express 22, 24062–24080 (2014).
[Crossref] [PubMed]

X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
[Crossref] [PubMed]

J. Chung, X. Ou, R. P. Kulkarni, and C. Yang, “Counting White Blood Cells from a Blood Smear Using Fourier Ptychographic Microscopy,” PloS one10 (2015).

Parrilo, P. A.

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization,” SIAM Review 52, 471–501 (2010).
[Crossref]

Peterson, I.

Pfeiffer, F.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

Qian, J.

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative Algorithms for Ptychographic Phase Retrieval,” arXiv:1105.5628v1 (2011).

Ramchandran, K.

Rawal, S.

Recht, B.

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization,” SIAM Review 52, 471–501 (2010).
[Crossref]

Robinson, I. K.

Rodenburg, J. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Applied Physics Letters 85, 4795–4797 (2004).
[Crossref]

Sarahan, M. C.

Saxton, W.

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Schirotzek, A.

C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative Algorithms for Ptychographic Phase Retrieval,” arXiv:1105.5628v1 (2011).

Shpyrko, O. G.

Soltanolkotabi, M.

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” arXiv:1310.3240 (2013).

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: Theory and algorithms,” arXiv:1407.1065 (2014).

Strohmer, T.

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L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Wirtinger flow optimization,” Optics Express 23, 4856–4866 (2015).
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P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New Journal of Physics 14, 063004 (2012).
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L. Tian and L. Waller, “3D intensity and phase imaging from light field measurements in an LED array microscope,” Optica 2, 104–111 (2015).
[Crossref]

L. Tian, Z. Liu, L.-H. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
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Tropp, J. A.

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E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase Retrieval via Matrix Completion,” SIAM Journal on Imaging Sciences 6, 199–225 (2013).
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R. Horstmeyer, X. Ou, J. Chung, G. Zheng, and C. Yang, “Overlapped Fourier coding for optical aberration removal,” Optics Express 22, 24062–24080 (2014).
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L. Tian, Z. Liu, L.-H. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
[Crossref]

Zhang, F.

Zheng, G.

L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Wirtinger flow optimization,” Optics Express 23, 4856–4866 (2015).
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X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
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G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photonics 7, 739–745 (2013).
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L. Tian, Z. Liu, L.-H. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
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Nature Photonics (1)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photonics 7, 739–745 (2013).
[Crossref]

New J. Phys. (1)

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17, 053044 (2015).
[Crossref] [PubMed]

New Journal of Physics (1)

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New Journal of Physics 14, 063004 (2012).
[Crossref]

Opt. Express (3)

X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
[Crossref] [PubMed]

Optica (2)

L. Tian and L. Waller, “3D intensity and phase imaging from light field measurements in an LED array microscope,” Optica 2, 104–111 (2015).
[Crossref]

L. Tian, Z. Liu, L.-H. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
[Crossref]

Optics Express (3)

R. Horstmeyer, X. Ou, J. Chung, G. Zheng, and C. Yang, “Overlapped Fourier coding for optical aberration removal,” Optics Express 22, 24062–24080 (2014).
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E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase Retrieval via Matrix Completion,” SIAM Journal on Imaging Sciences 6, 199–225 (2013).
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C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative Algorithms for Ptychographic Phase Retrieval,” arXiv:1105.5628v1 (2011).

K. Kreutz-Delgado, “The Complex Gradient Operator and the CR-Calculus,” arXiv:0906.4835v1 (2009).

J. Nocedal and S. J. Wright, “Conjugate gradient methods,” Springer (2006).

http://www.laurawaller.com/opensource/

J. Nocedal and S. Wright, “Numerical Optimization,” Springer Science & Business Media (2006).

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” arXiv:1310.3240 (2013).

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: Theory and algorithms,” arXiv:1407.1065 (2014).

J. Chung, X. Ou, R. P. Kulkarni, and C. Yang, “Counting White Blood Cells from a Blood Smear Using Fourier Ptychographic Microscopy,” PloS one10 (2015).

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Fig. 1 Fourier ptychographic reconstruction (amplitude only) of a test object with the algorithms discussed here, all using the same experimental dataset. Algorithms derived from the same cost function (amplitude-based, intensity-based, and Poisson-likelihood) give similar performance, and first-order methods (Gerchberg-Saxton) suffer artifacts.

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Fig. 2 (a) Experimental setup for Fourier ptychography with an LED array microscope. (b) The sample’s Fourier space is synthetically enlarged by capturing multiple images from different illumination angles. Each circle represents the spatial frequency coverage of the image captured by single-LED illumination. Brightfield images have orders of magnitude higher intensity than darkfield (see histograms), resulting in different noise levels.

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Fig. 3 To explain the artifacts in our experimental results, as well as evaluate the robustness of various algorithms under common types of errors, we simulate several FPM datasets with different types of known error: (1) Ideal data, (2) Poisson noise data, (3) aberrated data, (4) LED misaligned data (×: original position, ○: perturbed position).

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Fig. 4 Reconstructed amplitude from simulated datasets with three types of errors, using different algorithms. The intensity-based algorithms suffer from high frequency artifacts under both noise and model mis-match errors. The percentage on the top left corner of each image is the relative error of each reconstruction.

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Fig. 5 Reconstructed phase from simulated datasets with three types of errors, using different algorithms. The intensity-based algorithms suffer from phase wrapping artifacts under both noise and model mis-match errors. The percentage on the top left corner of each image is the relative error of each reconstruction.

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Fig. 6 Phase relative error as a function of iteration number for different algorithms with the (a) ideal data, (b) Poisson noise data, (c) aberrated data and (d) LED misaligned data. When the data is not perfect, some of the algorithms may not converge to a correct solution.

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Fig. 7 Both Poisson noise and model mis-match (aberrations, LED misalignment) cause errors that scale with mean intensity. Here, histograms show the intensity deviations under Poisson noise, aberration, and misalignment for a brightfield and darkfield image.

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Fig. 8 The intensity-based cost function gives higher weighting to images in the low spatial frequency region of the Fourier domain, resulting in high-frequency artifacts. Here, we show the gradient of the amplitude-based, Poisson-likelihood-based and intensity-based cost functions at the tenth iteration, using experimental data.

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Fig. 9 Object and pupil reconstruction results using different algorithms, with and without pupil estimation. The second-order method (sequential Gauss-Newton) with pupil estimation gives the best result, as expected. In this case, we find that the second-order method without pupil estimation is already better than first-order method (sequential gradient descent) with pupil estimation.

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Fig. 10 (a) Adding LED misalignment correction improves the reconstruction results (sequential Gauss-Newton method). (b) The original, perturbed, and corrected LED positions in angular coordinates. LED correction accurately retrieves the actual LED positions.

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Fig. 11 Experimental reconstructions with and without LED misalignment correction (sequential Gauss-Newton method). (a) The reconstructed object and pupil. (b) The original and corrected LED positions, in angular coordinates.

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Tables (2)

Table 1 Tuning Parameters

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Table 2 Convergence Speed

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Equations (45)

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I_{ℓ} (r) = {| ℱ^{- 1} {P (u) O (u - u_{ℓ})} |}^{2} .

min_{O (u)} f_{A} (O (u)) = min_{O (u)} \sum_{ℓ} \sum_{r} {| \sqrt{I_{ℓ} (r)} - | ℱ^{- 1} {P (u) O (u - u_{ℓ})} | |}^{2} .

p [I_{ℓ} (r) | O (u)] = \frac{1}{\sqrt{2 π σ_{w}^{2}}} exp [- \frac{{(I_{ℓ} (r) - {\hat{I}}_{ℓ} (r))}^{2}}{2 σ_{w}^{2}}],

\begin{array}{l} ℒ_{Gaussian} (O (u)) = - log \prod_{ℓ} \prod_{r} p [I_{ℓ} (r) | O (u)] \\ = \sum_{ℓ} \sum_{r} [\frac{1}{2} log (2 π σ_{w}^{2}) + \frac{{(I_{ℓ} (r) - {\hat{I}}_{ℓ} (r))}^{2}}{2 σ_{w}^{2}}] . \end{array}

min_{O (u)} f_{I} (O (u)) = min_{O (u)} \sum_{ℓ} \sum_{r} {| I_{ℓ} (r) - {| ℱ^{- 1} {P (u) O (u - u_{ℓ})} |}^{2} |}^{2} .

p [I_{ℓ} (r) | O (u)] = \frac{{[{\hat{I}}_{ℓ} (r)]}^{I_{ℓ} (r)} exp [- {\hat{I}}_{ℓ} (r)]}{I_{ℓ} (r)!} \approx \frac{1}{\sqrt{2 π σ_{ℓ, r}^{2}}} exp [- \frac{{(I_{ℓ} (r) - {\hat{I}}_{ℓ} (r))}^{2}}{2 σ_{ℓ, r}^{2}}] .

\begin{array}{l} min_{O (u)} ℒ_{Poisson} (O (u)) = min_{O (u)} \sum_{ℓ} \sum_{r} (- I_{ℓ} (r) log [{\hat{I}}_{ℓ} (r)] + {\hat{I}}_{ℓ} (r) + log [I_{ℓ} (r)!]) \\ \approx min_{O (u)} \sum_{ℓ} \sum_{r} \frac{{(I_{ℓ} (r) - {\hat{I}}_{ℓ} (r))}^{2}}{2 σ_{ℓ, r}^{2}} . \end{array}

{\hat{I}}_{ℓ} = {| g_{ℓ} |}^{2} = {| F^{- 1} diag (P) Q_{ℓ} O |}^{2} .

min_{O} f_{A} (O) = min_{O} \sum_{ℓ} {(\sqrt{I_{ℓ}} - | g_{ℓ} |)}^{†} (\sqrt{I_{ℓ}} - | g_{ℓ} |),

min_{O} f_{I} (O) = min_{O} \sum_{ℓ} {(\sqrt{I_{ℓ}} - {| g_{ℓ} |}^{2})}^{†} (I_{ℓ} - {| g_{ℓ} |}^{2}) .

{| g_{ℓ} |}^{2} = diag ({\bar{g}}_{ℓ}) F^{- 1} diag (P) Q_{ℓ} O = A_{ℓ} O = [\begin{matrix} a_{ℓ, 1}^{†} \\ ⋮ \\ a_{ℓ, m^{2}}^{†} \end{matrix}] O,

min_{O} ℒ_{Poisson} (O) = \sum_{ℓ} \sum_{j} [- I_{ℓ, j} log (a_{ℓ, j}^{†} O) a_{ℓ, j}^{†} O + log (I_{ℓ, j})!] .

f_{A, ℓ} (O) = {(\sqrt{I_{ℓ}} - | g_{ℓ} |)}^{†} (\sqrt{I_{ℓ}} - | g_{ℓ} |),

\nabla_{O} f_{A, ℓ} (O) = - Q_{ℓ}^{†} diag (\bar{P}) [F diag (\frac{\sqrt{I_{ℓ}}}{| g_{ℓ} |}) g_{ℓ} - diag (P) Q_{ℓ} O] .

O^{(i, ℓ + 1)} = O^{(i, ℓ)} - \frac{1}{{| P |}_{max}^{2}} \nabla_{O} f_{A, ℓ + 1} (O^{(i, ℓ)}),

O^{(i + 1)} = O^{(i)} - α^{(i)} \nabla_{O} f_{I} (O^{(i)}),

α^{(i)} = \frac{min (1 - e^{- i / i_{0}}, θ_{max})}{{(O^{(0)})}^{†} O^{(0)}},

\begin{array}{l} H_{cc, ℓ}^{A} & \approx {(\frac{\partial f_{A ℓ}}{\partial c})}^{†} (\frac{\partial f_{A ℓ}}{\partial c}) \\ = [\begin{matrix} \frac{1}{2} Q_{ℓ}^{†} diag ({| P |}^{2}) Q_{ℓ} & Q_{ℓ}^{†} diag (\bar{P}) F diag (\frac{g_{ℓ}^{2}}{{| g_{ℓ} |}^{2}}) {\bar{F}}^{- 1} diag (\bar{P}) {\bar{Q}}_{ℓ} \\ Q_{ℓ}^{T} diag (P) \bar{F} diag (\frac{{\bar{g}}_{ℓ}^{2}}{{| g_{ℓ} |}^{2}}) F^{- 1} diag (P) Q_{ℓ} & \frac{1}{2} Q_{ℓ}^{T} diag ({| P |}^{2}) {\bar{Q}}_{ℓ} \end{matrix}] \end{array},

{(H_{cc, ℓ}^{A})}^{- 1} \approx [\begin{matrix} 2 Q_{ℓ}^{†} diag (\frac{1}{{| P |}^{2} + Δ}) Q_{ℓ} & 0 \\ 0 & 2 Q_{ℓ}^{T} diag (\frac{1}{{| P |}^{2} + Δ}) {\bar{Q}}_{ℓ} \end{matrix}],

[\begin{matrix} O^{(i, ℓ + 1)} \\ {\bar{O}}^{(i, ℓ + 1)} \end{matrix}] = [\begin{matrix} O^{(i, ℓ)} \\ {\bar{O}}^{(i, ℓ)} \end{matrix}] - [\begin{matrix} Q_{ℓ}^{†} diag (\frac{| P |}{{| P |}_{max}}) Q_{ℓ} & 0 \\ 0 & Q_{ℓ}^{T} diag (\frac{| P |}{{| P |}_{max}}) Q_{ℓ} \end{matrix}] {(H_{cc, ℓ}^{A})}^{- 1} [\begin{matrix} \nabla_{O} f_{A, ℓ + 1} (O^{(i, ℓ)}) \\ \nabla_{\bar{O}} f_{A, ℓ + 1} ({\bar{O}}^{(i, ℓ)}) \end{matrix}],

[\begin{matrix} O^{(i + 1)} \\ {\bar{O}}^{(i + 1)} \end{matrix}] = [\begin{matrix} O^{(i)} \\ {\bar{O}}^{(i)} \end{matrix}] - α^{(i)} {(H_{cc})}^{- 1} [\begin{matrix} \nabla_{O} f (O^{(i)}) \\ \nabla_{\bar{O}} f (O^{(i)}) \end{matrix}] .

\begin{array}{l} g & = [\begin{matrix} g_{1} \\ ⋮ \\ g_{N_{img}} \end{matrix}] = [\begin{matrix} F^{- 1} & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & F^{- 1} \end{matrix}] [\begin{matrix} diag (P) & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & diag (P) \end{matrix}] [\begin{matrix} Q_{1} \\ ⋮ \\ Q_{N_{img}} \end{matrix}] O \\ = DO = [\begin{matrix} d_{1}^{†} \\ ⋮ \\ d_{N_{img} m^{2}}^{†} \end{matrix}] O, \end{array}

{| g |}^{2} = [\begin{matrix} O^{†} d_{1} d_{1}^{†} O \\ ⋮ \\ O^{†} d_{N_{img} m^{2}} d_{N_{img} m^{2}}^{†} O \end{matrix}] = [\begin{matrix} Tr (d_{1} d_{1}^{†} {OO}^{†}) \\ ⋮ \\ Tr (d_{N_{img} m^{2}} d_{N_{img} m^{2}}^{†} {OO}^{†}) \end{matrix}] = [\begin{matrix} Tr (D_{1} X) \\ ⋮ \\ Tr (D_{N_{img} m^{2}} X) \end{matrix}] = 𝒜 (X),

\begin{array}{l} f_{I} (X) = {(I - {| g |}^{2})}^{†} (I - {| g |}^{2}) \\ = {(I - 𝒜 (X))}^{†} (I - 𝒜 (X)) . \end{array}

min_{X} f_{I}^{'} (X) = min_{X} {(I - 𝒜 (X))}^{†} (I - 𝒜 (X)) + α Tr (X),

min_{R} f_{AL, I} (R) = min_{R} \frac{σ}{2} {(I - 𝒜 ({RR}^{†}))}^{†} (I - 𝒜 ({RR}^{†})) + y^{T} (I - 𝒜 ({RR}^{†})) + Tr ({RR}^{†}),

min_{O} f_{AL, I} (O) = min_{O} \frac{σ}{2} [\sum_{ℓ} {(I_{ℓ} - {| g_{ℓ} |}^{2} + \frac{2}{σ} y_{ℓ})}^{†} (I_{ℓ} - {| g_{ℓ} |}^{2})] + O^{†} O .

\nabla_{O} f_{AL, I} (O) = - σ \sum_{ℓ} Q_{ℓ}^{†} diag (\bar{P}) F diag (g_{ℓ}) (I_{ℓ} - {| g_{ℓ} |}^{2} + \frac{1}{σ} y_{ℓ}) + O .

Error = \frac{{‖ O_{recover} - O_{true} ‖}_{2}^{2}}{{‖ O_{true} ‖}_{2}^{2}},

\nabla_{P} f_{A, ℓ} (O, P) = - diag ({\bar{Q}}_{ℓ} \bar{O}) [F diag (\frac{\sqrt{I_{ℓ}}}{| g_{ℓ} |}) g_{ℓ} - diag (P) Q_{ℓ} O] .

\begin{array}{l} f_{A} (O) = \sum_{ℓ} f_{A ℓ}^{†} f_{A ℓ} \\ f_{I} (O) = \sum_{ℓ} f_{I ℓ}^{†} f_{I ℓ}, \end{array}

\nabla_{O} f_{A} (O) = {\sum_{ℓ} [\frac{\partial (f_{A ℓ}^{†} f_{A ℓ})}{\partial O}]}^{†} = \sum_{ℓ} {[\frac{\partial f_{A ℓ}^{†} f_{A ℓ}}{\partial f_{A ℓ}} \frac{\partial f_{A ℓ}}{\partial O}]}^{†} .

\begin{array}{l} \frac{\partial (f_{A ℓ}^{†} f_{A ℓ})}{\partial f_{A ℓ}} = 2 f_{A ℓ}^{†} \\ \frac{\partial f_{A ℓ}}{\partial O} = \frac{\partial {({| g_{ℓ} |}^{2})}^{1 / 2}}{\partial ({| g_{ℓ} |}^{2})} \frac{\partial (diag ({\bar{g}}_{ℓ}) g_{ℓ})}{\partial O} = - \frac{1}{2} diag (\frac{{\bar{g}}_{ℓ}}{| g_{ℓ} |}) F^{- 1} diag (P) Q_{ℓ}, \end{array}

\begin{array}{l} \nabla_{O} f_{A} (O) & = - \sum_{ℓ} Q_{ℓ}^{†} diag (\bar{P}) F diag (\frac{g_{ℓ}}{| g_{ℓ} |}) (\sqrt{I_{ℓ}} - | g_{ℓ} |) \\ = - \sum_{ℓ} Q_{ℓ}^{†} diag (\bar{P}) (F diag (\sqrt{\frac{I_{ℓ}}{| g_{ℓ} |}}) g_{ℓ} - diag (P) Q_{ℓ} O) . \end{array}

\frac{\partial f_{I ℓ}}{\partial O} = - \frac{\partial (diag ({\bar{g}}_{ℓ}) g_{ℓ})}{\partial O} = - diag ({\bar{g}}_{ℓ}) F^{- 1} diag (P) Q_{ℓ} .

\nabla_{O} f_{I} (O) = \sum_{ℓ} [\frac{\partial (f_{I ℓ}^{†} f_{I ℓ})}{\partial f_{I ℓ}} \frac{\partial f_{I ℓ}}{\partial O}] = - 2 \sum_{ℓ} Q_{ℓ}^{†} diag (\bar{P}) F diag (g_{ℓ}) (I_{ℓ} - {| g_{ℓ} |}^{2}) .

\begin{array}{l} \nabla_{O} ℒ_{Poisson} (O) & = {(\frac{\partial ℒ_{Poisson}}{\partial O})}^{†} = {(\sum_{ℓ} \sum_{j} [- \frac{I_{ℓ, j}}{a_{ℓ, j}^{†}} a_{ℓ, j}^{†} + a_{ℓ, j}^{†}])}^{†} \\ = - {(\sum_{ℓ} \sum_{j} [I_{ℓ, j} - a_{ℓ, j}^{†} O] \frac{1}{a_{ℓ, j}^{†} O} a_{ℓ, j}^{†})}^{†} \\ = - {(\sum_{ℓ} {(I_{ℓ} - {| g_{ℓ} |}^{2})}^{†} diag (\frac{1}{{| g_{ℓ} |}^{2}}) diag ({\bar{g}}_{ℓ}) F^{- 1} diag (P) Q_{ℓ})}^{†} \\ = - \sum_{ℓ} Q_{ℓ}^{†} diag (\bar{P}) F diag (\frac{g_{ℓ}}{{| g_{ℓ} |}^{2}}) (I_{ℓ} - {| g_{ℓ} |}^{2}) . \end{array}

O^{(i + 1)} = O^{(i)} - α^{(i)} \nabla_{O} f (O^{(i)}),

f (c) \approx f (c_{0}) + \nabla f {(c_{0})}^{†} (c - c_{0}) + \frac{1}{2} {(c - c_{0})}^{†} H_{cc} (c_{0}) (c - c_{0}),

H_{cc} = [\begin{matrix} H_{OO} & H_{\bar{O} O} \\ H_{O \bar{O}} & H_{\bar{O} \bar{O}} \end{matrix}],

\begin{array}{l} H_{OO} = \frac{\partial}{\partial O} {(\frac{\partial f}{\partial \bar{O}})}^{†}, H_{\bar{O} O} = \frac{\partial}{\partial \bar{O}} {(\frac{\partial f}{\partial O})}^{†} \\ H_{O \bar{O}} = \frac{\partial}{\partial O} {(\frac{\partial f}{\partial O})}^{†}, H_{\bar{O} \bar{O}} = \frac{\partial}{\partial \bar{O}} {(\frac{\partial f}{\partial \bar{O}})}^{†} . \end{array}

\begin{array}{l} H_{OO}^{A} = \sum_{ℓ} Q_{ℓ}^{†} diag (\bar{P}) F [1 - \frac{1}{2} diag (\frac{\sqrt{I_{ℓ}}}{| g_{ℓ} |})] F^{- 1} diag (P) Q_{ℓ} \\ H_{\bar{O} O}^{A} = \frac{1}{2} \sum_{ℓ} Q_{ℓ}^{†} diag (\bar{P}) F diag [(\frac{\sqrt{I_{ℓ}} g_{ℓ}^{2}}{{| g_{ℓ} |}^{3}})] {\bar{F}}^{- 1} diag (\bar{P}) {\bar{Q}}_{ℓ} \\ H_{O \bar{O}}^{A} = \frac{1}{2} \sum_{ℓ} Q_{ℓ}^{T} diag (P) \bar{F} diag [(\frac{\sqrt{I_{ℓ}} {\bar{g}}_{ℓ}^{2}}{{| g_{ℓ} |}^{3}})] F^{- 1} diag (P) Q_{ℓ} \\ H_{\bar{O} \bar{O}}^{A} = \sum_{ℓ} Q_{ℓ}^{T} diag (P) \bar{F} [1 - \frac{1}{2} diag (\frac{\sqrt{I_{ℓ}}}{| g_{ℓ} |})] {\bar{F}}^{- 1} diag (\bar{P}) {\bar{Q}}_{ℓ}, \end{array}

\begin{array}{l} H_{OO}^{I} = 2 \sum_{ℓ} Q_{ℓ}^{†} diag (\bar{P}) F diag (2 {| g_{ℓ} |}^{2} - I_{ℓ}) F^{- 1} diag (P) Q_{ℓ} \\ H_{\bar{O} O}^{I} = 2 \sum_{ℓ} Q_{ℓ}^{†} diag (\bar{P}) F diag (g_{ℓ}^{2}) {\bar{F}}^{- 1} diag (\bar{P}) {\bar{Q}}_{ℓ} \\ H_{O \bar{O}}^{I} = 2 \sum_{ℓ} Q_{ℓ}^{T} diag (P) \bar{F} diag ({\bar{g}}_{ℓ}^{2}) F^{- 1} diag (P) Q_{ℓ} \\ H_{\bar{O} \bar{O}}^{I} = 2 \sum_{ℓ} Q_{ℓ}^{T} diag (P) \bar{F} diag (2 {| g_{ℓ} |}^{2} - I_{ℓ}) {\bar{F}}^{- 1} diag (\bar{P}) {\bar{Q}}_{ℓ} . \end{array}

\begin{array}{l} H_{OO}^{P} = \sum_{ℓ} Q_{ℓ}^{†} diag ({| P |}^{2}) Q_{ℓ} \\ H_{\bar{O} O}^{P} = \sum_{ℓ} Q_{ℓ}^{†} diag (\bar{P}) F diag (\frac{I_{ℓ} g_{ℓ}^{2}}{{| g_{ℓ} |}^{4}}) {\bar{F}}^{- 1} diag (\bar{P}) {\bar{Q}}_{ℓ} \\ H_{O \bar{O}}^{P} = \sum_{ℓ} Q_{ℓ}^{T} diag (P) \bar{F} diag (\frac{I_{ℓ} {\bar{g}}_{ℓ}^{2}}{{| g_{ℓ} |}^{4}}) F^{- 1} diag (P) Q_{ℓ} \\ H_{\bar{O} \bar{O}}^{P} = \sum_{ℓ} Q_{ℓ}^{T} diag ({| P |}^{2}) {\bar{Q}}_{ℓ} . \end{array}

[\begin{matrix} O^{(i + 1)} \\ {\bar{O}}^{(i + 1)} \end{matrix}] = [\begin{matrix} O^{(i)} \\ {\bar{O}}^{(i)} \end{matrix}] - α^{(i)} H_{cc}^{- 1} [\begin{matrix} \nabla_{O} f (O^{(i)}) \\ \nabla_{\bar{O}} f (O^{(i)}) \end{matrix}] .

Gerchberg Saxton	Sequential Gauss-Newton	Amplitude Newton	Amplitude Wirtinger	Poisson Newton	Poisson Wirtinger
N/A	δ = 5	N/A	i₀ = 10θ_max = 0.05	N/A	i₀ = 10θ_max = 0.05
Intensity Wirtinger	PhaseLift	Intensity Newton
i₀ = 10θ_max = 1	σ = 10¹⁰	N/A

	Ideal data		Misaligned data
	Iteration number	Runtime (s)	Iteration number	Runtime (s)
Gerchberg Saxton	4	2.22	diverges	diverges
Sequential Gauss-Newton	23	12.97	83	46.8
Amplitude Newton	13	100.49	20	154.6
Amplitude Wirtinger	46	26.28	158	89.52
Poisson Newton	28	211.68	77	582.1
Poisson Wirtinger	96	54.46	153	87.36
Intensity Wirtinger	1481	651.64	diverges	diverges
PhaseLift	67	386.28	diverges	diverges
Intensity Newton	12	74.44	diverges	diverges

Gerchberg Saxton	Sequential Gauss-Newton	Amplitude Newton	Amplitude Wirtinger	Poisson Newton	Poisson Wirtinger
N/A	δ = 5	N/A	i₀ = 10θ_max = 0.05	N/A	i₀ = 10θ_max = 0.05
Intensity Wirtinger	PhaseLift	Intensity Newton
i₀ = 10θ_max = 1	σ = 10¹⁰	N/A

	Ideal data		Misaligned data
	Iteration number	Runtime (s)	Iteration number	Runtime (s)
Gerchberg Saxton	4	2.22	diverges	diverges
Sequential Gauss-Newton	23	12.97	83	46.8
Amplitude Newton	13	100.49	20	154.6
Amplitude Wirtinger	46	26.28	158	89.52
Poisson Newton	28	211.68	77	582.1
Poisson Wirtinger	96	54.46	153	87.36
Intensity Wirtinger	1481	651.64	diverges	diverges
PhaseLift	67	386.28	diverges	diverges
Intensity Newton	12	74.44	diverges	diverges