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.

© 2015 Optical Society of America

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  1. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photonics 7, 739–745 (2013).
    [Crossref]
  2. J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Applied Physics Letters 85, 4795–4797 (2004).
    [Crossref]
  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]
  4. 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]
  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]
  6. 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).
  7. 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]
  8. 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]
  9. R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).
  10. J. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Optics Letters 327–29 (1978).
    [Crossref] [PubMed]
  11. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  12. V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
    [Crossref]
  13. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
    [Crossref]
  14. 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]
  15. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
    [Crossref] [PubMed]
  16. 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]
  17. C. Yang, J. Qian, A. Schirotzek, F. Maia, and S. Marchesini, “Iterative Algorithms for Ptychographic Phase Retrieval,” arXiv:1105.5628v1 (2011).
  18. X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
    [Crossref] [PubMed]
  19. 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]
  20. E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: Theory and algorithms,” arXiv:1407.1065 (2014).
  21. 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]
  22. J. Nocedal and S. Wright, “Numerical Optimization,” Springer Science & Business Media (2006).
  23. 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]
  24. 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]
  25. 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]
  26. S. Burer and R. Monteiro, “A Nonlinear Programming Algorithm for Solving Semidefinite Programs via Low-Rank Factorization,” Mathematical Programming 95, 329–357 (2003).
    [Crossref]
  27. 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]
  28. E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” arXiv:1310.3240 (2013).
  29. P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New Journal of Physics 14, 063004 (2012).
    [Crossref]
  30. K. Kreutz-Delgado, “The Complex Gradient Operator and the CR-Calculus,” arXiv:0906.4835v1 (2009).
  31. J. Nocedal and S. J. Wright, “Conjugate gradient methods,” Springer (2006).
  32. 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]
  33. 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]
  34. 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]
  35. 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]
  36. 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. 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]

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]

2014 (5)

2013 (4)

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

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]

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)

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

1982 (1)

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.

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]

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.

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]

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, 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]

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.

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.

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, 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.

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]

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.

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]

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.

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.

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]

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]

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]

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.

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]

Kraus, B.

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]

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.

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).

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

Liu, Z.

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, 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]

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.

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]

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, 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.

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]

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.

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. 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]

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.

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]

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.

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]

Suo, J.

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]

Thibault, P.

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

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]

Tian, L.

Tripathi, A.

Tropp, J. A.

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]

Vila-Comamala, J.

Voroninski, V.

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]

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]

Waller, L.

Willems, P.

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]

Williams, A.

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]

Wright, S.

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

Wright, S. J.

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

Yang, C.

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]

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]

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

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

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

Yeh, L.-H.

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).
[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]

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, 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]

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

Zhong, J.

Appl. Opt. (1)

Applied Physics Letters (1)

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

Biomed. Opt. Express (1)

Communications on Pure and Applied Math (1)

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]

Computerized Medical Imaging and Graphics (1)

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. Opt. Soc. Am. A (1)

Journal of biomedical optics (1)

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]

Mathematical Programming (1)

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

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)

Optica (2)

Optics express (1)

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]

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]

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]

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]

Optics Letters (1)

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

Optik (1)

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

Rev. Sci. Instrum. (1)

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

SIAM Journal on Imaging Sciences (1)

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]

SIAM Review (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]

Ultramicroscopy (3)

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]

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]

Other (8)

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

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).

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 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).

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Figures (11)

Fig. 1
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
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
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
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
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
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
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
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
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
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
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)

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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 ) ∑ ℓ ∑ r | I ℓ ( r ) − | ℱ − 1 { P ( u ) O ( u − u ℓ ) } | | 2 .
p [ I ℓ ( r ) | O ( u ) ] = 1 2 π σ w 2 exp [ − ( I ℓ ( r ) − I ^ ℓ ( r ) ) 2 2 σ w 2 ] ,
ℒ Gaussian ( O ( u ) ) = − log ∏ ℓ ∏ r p [ I ℓ ( r ) | O ( u ) ]                 = ∑ ℓ ∑ r [ 1 2 log ( 2 π σ w 2 ) + ( I ℓ ( r ) − I ^ ℓ ( r ) ) 2 2 σ w 2 ] .
min O ( u ) f I ( O ( u ) ) = min O ( u ) ∑ ℓ ∑ r | I ℓ ( r ) − | ℱ − 1 { P ( u ) O ( u − u ℓ ) } | 2 | 2 .
p [ I ℓ ( r ) | O ( u ) ] = [ I ^ ℓ ( r ) ] I ℓ ( r ) exp [ − I ^ ℓ ( r ) ] I ℓ ( r ) ! ≈ 1 2 π σ ℓ , r 2 exp [ − ( I ℓ ( r ) − I ^ ℓ ( r ) ) 2 2 σ ℓ , r 2 ] .
min O ( u ) ℒ Poisson ( O ( u ) ) = min O ( u ) ∑ ℓ ∑ r ( − I ℓ ( r ) log [ I ^ ℓ ( r ) ] + I ^ ℓ ( r ) + log [ I ℓ ( r ) ! ] )                               ≈ min O ( u ) ∑ ℓ ∑ r ( I ℓ ( r ) − I ^ ℓ ( r ) ) 2 2 σ ℓ , r 2 .
I ^ ℓ = | g ℓ | 2 = | F − 1 diag ( P ) Q ℓ O | 2 .
min O f A ( O ) = min O ∑ ℓ ( I ℓ − | g ℓ | ) † ( I ℓ − | g ℓ | ) ,
min O f I ( O ) = min O ∑ ℓ ( I ℓ − | g ℓ | 2 ) † ( I ℓ − | g ℓ | 2 ) .
| g ℓ | 2 = diag ( g ¯ ℓ ) F − 1 diag ( P ) Q ℓ O = A ℓ O = [ a ℓ , 1 † ⋮ a ℓ , m 2 † ] O ,
min O ℒ Poisson ( O ) = ∑ ℓ ∑ j [ − I ℓ , j log ( a ℓ , j † O ) a ℓ , j † O + log ( I ℓ , j ) ! ] .
f A , ℓ ( O ) = ( I ℓ − | g ℓ | ) † ( I ℓ − | g ℓ | ) ,
∇ O f A , ℓ ( O ) = − Q ℓ † diag ( P ¯ )     [ F diag ( I ℓ | g ℓ | ) g ℓ − diag ( P ) Q ℓ O ] .
O ( i , ℓ + 1 ) = O ( i , ℓ ) − 1 | P | max 2 ∇ O f A , ℓ + 1 ( O ( i , ℓ ) ) ,
O ( i + 1 ) = O ( i ) − α ( i ) ∇ O f I ( O ( i ) ) ,
α ( i ) = min ( 1 − e − i / i 0 , θ max ) ( O ( 0 ) ) † O ( 0 ) ,
H cc , ℓ A ≈ ( ∂ f A ℓ ∂ c ) † ( ∂ f A ℓ ∂ c ) = [ 1 2 Q ℓ † diag ( | P | 2 ) Q ℓ Q ℓ † diag ( P ¯ ) F diag ( g ℓ 2 | g ℓ | 2 ) F ¯ − 1 diag ( P ¯ ) Q ¯ ℓ Q ℓ T diag ( P ) F ¯ diag ( g ¯ ℓ 2 | g ℓ | 2 ) F − 1 diag ( P ) Q ℓ 1 2 Q ℓ T diag ( | P | 2 ) Q ¯ ℓ ] ,
( H cc , ℓ A ) − 1 ≈ [ 2 Q ℓ † diag ( 1 | P | 2 + Δ ) Q ℓ 0 0 2 Q ℓ T diag ( 1 | P | 2 + Δ ) Q ¯ ℓ ] ,
[ O ( i , ℓ + 1 ) O ¯ ( i , ℓ + 1 ) ] = [ O ( i , ℓ ) O ¯ ( i , ℓ ) ] − [ Q ℓ † diag ( | P | | P | max ) Q ℓ 0 0 Q ℓ T diag ( | P | | P | max ) Q ℓ ]     ( H cc , ℓ A ) − 1 [ ∇ O f A , ℓ + 1 ( O ( i , ℓ ) ) ∇ O ¯ f A , ℓ + 1 ( O ¯ ( i , ℓ ) ) ] ,
[ O ( i + 1 ) O ¯ ( i + 1 ) ] = [ O ( i ) O ¯ ( i ) ] − α ( i ) ( H cc ) − 1 [ ∇ O f ( O ( i ) ) ∇ O ¯ f ( O ( i ) ) ] .
g = [ g 1 ⋮ g N img ] = [ F − 1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ F − 1 ]     [ diag ( P ) ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ diag ( P ) ]     [ Q 1 ⋮ Q N img ] O = DO = [ d 1 † ⋮ d N img m 2 † ] O ,
| g | 2 = [ O † d 1 d 1 † O ⋮ O † d N img m 2 d N img m 2 † O ] = [ Tr ( d 1 d 1 † OO † ) ⋮ Tr ( d N img m 2 d N img m 2 † OO † ) ] = [ Tr ( D 1 X ) ⋮ Tr ( D N img m 2 X ) ] = 𝒜 ( X ) ,
f I ( X ) = ( I − | g | 2 ) † ( I − | g | 2 )                       = ( I − 𝒜 ( X ) ) † ( I − 𝒜 ( X ) ) .
min X f I ′ ( X ) = min X ( I − 𝒜 ( X ) ) † ( I − 𝒜 ( X ) ) + α Tr ( X ) ,
min R f AL , I ( R ) = min R σ 2 ( I − 𝒜 ( RR † ) ) † ( I − 𝒜 ( RR † ) ) + y T ( I − 𝒜 ( RR † ) ) + Tr ( RR † ) ,
min O f AL , I ( O ) = min O σ 2 [ ∑ ℓ ( I ℓ − | g ℓ | 2 + 2 σ y ℓ ) † ( I ℓ − | g ℓ | 2 ) ] + O † O .
∇ O f AL , I ( O ) = − σ ∑ ℓ Q ℓ † diag ( P ¯ ) F diag ( g ℓ )         ( I ℓ − | g ℓ | 2 + 1 σ y ℓ ) + O .
Error = ‖ O recover − O true ‖ 2 2 ‖ O true ‖ 2 2 ,
∇ P f A , ℓ ( O , P ) = − diag ( Q ¯ ℓ O ¯ )     [ F diag ( I ℓ | g ℓ | ) g ℓ − diag ( P ) Q ℓ O ] .
f A ( O ) = ∑ ℓ f A ℓ † f A ℓ f I ( O ) = ∑ ℓ f I ℓ † f I ℓ ,
∇ O f A ( O ) = ∑ ℓ [ ∂ ( f A ℓ † f A ℓ ) ∂ O ] † = ∑ ℓ [ ∂ f A ℓ † f A ℓ ∂ f A ℓ ∂ f A ℓ ∂ O ] † .
∂ ( f A ℓ † f A ℓ ) ∂ f A ℓ = 2 f A ℓ † ∂ f A ℓ ∂ O = ∂ ( | g ℓ | 2 ) 1 / 2 ∂ ( | g ℓ | 2 ) ∂ ( diag ( g ¯ ℓ ) g ℓ ) ∂ O = − 1 2 diag ( g ¯ ℓ | g ℓ | ) F − 1 diag ( P ) Q ℓ ,
∇ O f A ( O ) = − ∑ ℓ Q ℓ † diag ( P ¯ ) F diag ( g ℓ | g ℓ | )     ( I ℓ − | g ℓ | ) = − ∑ ℓ Q ℓ † diag ( P ¯ )     ( F diag ( I ℓ | g ℓ | ) g ℓ − diag ( P ) Q ℓ O ) .
∂ f I ℓ ∂ O = − ∂ ( diag ( g ¯ ℓ ) g ℓ ) ∂ O = − diag ( g ¯ ℓ ) F − 1 diag ( P ) Q ℓ .
∇ O f I ( O ) = ∑ ℓ [ ∂ ( f I ℓ † f I ℓ ) ∂ f I ℓ ∂ f I ℓ ∂ O ] = − 2 ∑ ℓ Q ℓ † diag ( P ¯ ) F diag ( g ℓ )   ( I ℓ − | g ℓ | 2 ) .
∇ O ℒ Poisson ( O ) = ( ∂ ℒ Poisson ∂ O ) † = ( ∑ ℓ ∑ j [ − I ℓ , j a ℓ , j † a ℓ , j † + a ℓ , j † ] ) † = − ( ∑ ℓ ∑ j [ I ℓ , j − a ℓ , j † O ] 1 a ℓ , j † O a ℓ , j † ) † = − ( ∑ ℓ ( I ℓ − | g ℓ | 2 ) † diag ( 1 | g ℓ | 2 ) diag ( g ¯ ℓ ) F − 1 diag ( P ) Q ℓ ) † = − ∑ ℓ Q ℓ † diag ( P ¯ ) F diag ( g ℓ | g ℓ | 2 )     ( I ℓ − | g ℓ | 2 ) .
O ( i + 1 ) = O ( i ) − α ( i ) ∇ O f ( O ( i ) ) ,
f ( c ) ≈ f ( c 0 ) + ∇ f ( c 0 ) † ( c − c 0 ) + 1 2 ( c − c 0 ) † H cc ( c 0 ) ( c − c 0 ) ,
H cc = [ H OO H O ¯ O H O O ¯ H O ¯ O ¯ ] ,
H OO = ∂ ∂ O ( ∂ f ∂ O ¯ ) † ,     H O ¯ O = ∂ ∂ O ¯ ( ∂ f ∂ O ) † H O O ¯ = ∂ ∂ O ( ∂ f ∂ O ) † ,     H O ¯ O ¯ = ∂ ∂ O ¯ ( ∂ f ∂ O ¯ ) † .
H OO A = ∑ ℓ Q ℓ † diag ( P ¯ ) F     [ 1 − 1 2 diag ( I ℓ | g ℓ | ) ]     F − 1 diag ( P ) Q ℓ H O ¯ O A = 1 2 ∑ ℓ Q ℓ † diag ( P ¯ ) F   diag   [ ( I ℓ g ℓ 2 | g ℓ | 3 ) ]     F ¯ − 1 diag ( P ¯ ) Q ¯ ℓ H O O ¯ A = 1 2 ∑ ℓ Q ℓ T diag ( P ) F ¯   diag   [ ( I ℓ g ¯ ℓ 2 | g ℓ | 3 ) ]     F − 1 diag ( P ) Q ℓ H O ¯ O ¯ A = ∑ ℓ Q ℓ T diag ( P ) F ¯     [ 1 −   1 2 diag ( I ℓ | g ℓ | ) ]     F ¯ − 1 diag ( P ¯ ) Q ¯ ℓ ,
H OO I = 2 ∑ ℓ Q ℓ † diag ( P ¯ ) F diag     ( 2 | g ℓ | 2 − I ℓ )     F − 1 diag ( P ) Q ℓ H O ¯ O I = 2 ∑ ℓ Q ℓ † diag ( P ¯ ) F   diag ( g ℓ 2 )     F ¯ − 1 diag ( P ¯ ) Q ¯ ℓ H O O ¯ I = 2 ∑ ℓ Q ℓ T diag ( P ) F ¯   diag ( g ¯ ℓ 2 )     F − 1 diag ( P ) Q ℓ H O ¯ O ¯ I = 2 ∑ ℓ Q ℓ T diag ( P ) F ¯   diag ( 2 | g ℓ | 2 − I ℓ )     F ¯ − 1 diag ( P ¯ ) Q ¯ ℓ .
H OO P = ∑ ℓ Q ℓ † diag ( | P | 2 ) Q ℓ H O ¯ O P = ∑ ℓ Q ℓ † diag ( P ¯ ) F   diag ( I ℓ g ℓ 2 | g ℓ | 4 )     F ¯ − 1 diag ( P ¯ ) Q ¯ ℓ H O O ¯ P = ∑ ℓ Q ℓ T diag ( P ) F ¯   diag ( I ℓ g ¯ ℓ 2 | g ℓ | 4 )     F − 1 diag ( P ) Q ℓ H O ¯ O ¯ P = ∑ ℓ Q ℓ T diag ( | P | 2 ) Q ¯ ℓ .
[ O ( i + 1 ) O ¯ ( i + 1 ) ] = [ O ( i ) O ¯ ( i ) ] − α ( i ) H cc − 1 [ ∇ O f ( O ( i ) ) ∇ O ¯ f ( O ( i ) ) ] .

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