Abstract

Fourier ptychographic microscopy (FPM) is a recently developed computational microscopy technique for wide-field-of-view and super-resolution complex imaging. Wirtinger-flow-based methods can effectively suppress noise and reduce data acquisition time, but they are time-consuming during the phase reconstruction. In this paper, we present a Wirtinger-flow-based reconstruction method for FPM, which combines the Poisson maximum likelihood objective function, improved truncated Wirtinger criteria, and improved adaptive momentum method. Both the simulation and experimental results demonstrate that the proposed method runs faster and the reconstruction quality is similar to or better than other state-of-the-art Wirtinger-flow-based methods.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013).
    [Crossref] [PubMed]
  2. G. Zheng, X. Ou, R. Horstmeyer, J. Chung, and C. Yang, “Fourier ptychographic microscopy: a gigapixel superscope for biomedicine,” Opt. Photonics News 25(4), 26–33 (2014).
    [Crossref]
  3. C. Zuo, J. Sun, and Q. Chen, “Adaptive step-size strategy for noise-robust Fourier ptychographic microscopy,” Opt. Express 24(18), 20724–20744 (2016).
    [Crossref] [PubMed]
  4. T. R. Hillman, T. Gutzler, S. A. Alexandrov, and D. D. Sampson, “High-resolution, wide-field object reconstruction with synthetic aperture Fourier holographic optical microscopy,” Opt. Express 17(10), 7873–7892 (2009).
    [Crossref] [PubMed]
  5. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
    [Crossref] [PubMed]
  6. 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(10), 904–911 (2015).
    [Crossref]
  7. T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
    [Crossref]
  8. L. Granero, V. Micó, Z. Zalevsky, and J. García, “Synthetic aperture superresolved microscopy in digital lensless Fourier holography by time and angular multiplexing of the object information,” Appl. Opt. 49(5), 845–857 (2010).
    [Crossref] [PubMed]
  9. T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
    [Crossref] [PubMed]
  10. Y. Choi, M. Kim, C. Yoon, T. D. Yang, K. J. Lee, and W. Choi, “Synthetic aperture microscopy for high resolution imaging through a turbid medium,” Opt. Lett. 36(21), 4263–4265 (2011).
    [Crossref] [PubMed]
  11. C. Guo, Y. Zhao, J. Tan, S. Liu, and Z. Liu, “Adaptive lens-free computational coherent imaging using autofocusing quantification with speckle illumination,” Opt. Express 26(11), 14407–14420 (2018).
    [Crossref] [PubMed]
  12. J. Rosenblatt, “Phase retrieval,” Commun. Math. Phys. 95(3), 317–343 (1984).
    [Crossref]
  13. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
    [Crossref] [PubMed]
  14. X. Pan, C. Liu, and J. Zhu, “Single shot ptychographical iterative engine based on multi-beam illumination,” Appl. Phys. Lett. 103(17), 171105 (2013).
    [Crossref]
  15. A. Maiden, D. Johnson, and P. Li, “Further improvements to the ptychographical iterative engine,” Optica 4(7), 736–745 (2017).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  19. J. Liu, Y. Li, W. Wang, H. Zhang, Y. Wang, J. Tan, and C. Liu, “Stable and robust frequency domain position compensation strategy for Fourier ptychographic microscopy,” Opt. Express 25(23), 28053–28067 (2017).
    [Crossref]
  20. L. Bian, J. Suo, G. Situ, G. Zheng, F. Chen, and Q. Dai, “Content adaptive illumination for Fourier ptychography,” Opt. Lett. 39(23), 6648–6651 (2014).
    [Crossref] [PubMed]
  21. P. C. Konda, J. M. Taylor, and A. R. Harvey, “High-resolution microscopy with low-resolution objectives: correcting phase aberrations in Fourier ptychography,” Proc. SPIE 9630, 96300X (2015).
    [Crossref]
  22. P. C. Konda, J. M. Taylor, and A. R. Harvey, “Calibration and aberration correction in multi-aperture Fourier ptychography,” in Imaging and Applied Optics 2016, OSA Technical Digest (online) (Optical Society of America, 2016), paper CT2D.2.
  23. L. Bian, G. Zheng, K. Guo, J. Suo, C. Yang, F. Chen, and Q. Dai, “Motion-corrected Fourier ptychography,” Biomed. Opt. Express 7(11), 4543–4553 (2016).
    [Crossref] [PubMed]
  24. L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Wirtinger flow optimization,” Opt. Express 23(4), 4856–4866 (2015).
    [Crossref] [PubMed]
  25. L. Bian, J. Suo, J. Chung, X. Ou, C. Yang, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Poisson maximum likelihood and truncated Wirtinger gradient,” Sci. Rep. 6(1), 27384 (2016).
    [Crossref] [PubMed]
  26. Y. Zhang, P. Song, and Q. Dai, “Fourier ptychographic microscopy using a generalized Anscombe transform approximation of the mixed Poisson-Gaussian likelihood,” Opt. Express 25(1), 168–179 (2017).
    [Crossref] [PubMed]
  27. E. Bostan, M. Soltanolkotabi, M. D. Ren, and L. Waller. “Accelerated Wirtinger flow for multiplexed Fourier ptychographic microscopy,” arXiv preprint arXiv:1803.03714 (2018).
  28. G. Wang, G. B. Giannakis, and Y. C. Eldar, “Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow,” IEEE Trans. Inf. Theory 64(2), 773–794 (2018).
    [Crossref]
  29. G. Wang, L. Zhang, G. B. Giannakis, M. Akcakaya, and J. Chen, “Sparse phase retrieval via truncated amplitude flow,” IEEE Trans. Signal Process. 66(2), 479–491 (2018).
    [Crossref]
  30. B. T. Polyak, “Some methods of speeding up the convergence of iteration methods,” USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964).
    [Crossref]
  31. X. Jiang, S. Rajan, and X. Liu, “Wirtinger flow method with optimal stepsize for phase petrieval,” IEEE Signal Process. Lett. 23(11), 1627–1631 (2016).
    [Crossref]
  32. Y. Nesterov, “A method of solving a convex programming problem with convergence rate O(1/k2),” Soviet Mathematics Doklady 2(27), 372–376 (1983).

2018 (3)

C. Guo, Y. Zhao, J. Tan, S. Liu, and Z. Liu, “Adaptive lens-free computational coherent imaging using autofocusing quantification with speckle illumination,” Opt. Express 26(11), 14407–14420 (2018).
[Crossref] [PubMed]

G. Wang, G. B. Giannakis, and Y. C. Eldar, “Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow,” IEEE Trans. Inf. Theory 64(2), 773–794 (2018).
[Crossref]

G. Wang, L. Zhang, G. B. Giannakis, M. Akcakaya, and J. Chen, “Sparse phase retrieval via truncated amplitude flow,” IEEE Trans. Signal Process. 66(2), 479–491 (2018).
[Crossref]

2017 (3)

2016 (5)

2015 (4)

2014 (2)

L. Bian, J. Suo, G. Situ, G. Zheng, F. Chen, and Q. Dai, “Content adaptive illumination for Fourier ptychography,” Opt. Lett. 39(23), 6648–6651 (2014).
[Crossref] [PubMed]

G. Zheng, X. Ou, R. Horstmeyer, J. Chung, and C. Yang, “Fourier ptychographic microscopy: a gigapixel superscope for biomedicine,” Opt. Photonics News 25(4), 26–33 (2014).
[Crossref]

2013 (3)

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013).
[Crossref] [PubMed]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref] [PubMed]

X. Pan, C. Liu, and J. Zhu, “Single shot ptychographical iterative engine based on multi-beam illumination,” Appl. Phys. Lett. 103(17), 171105 (2013).
[Crossref]

2011 (1)

2010 (1)

2009 (1)

2007 (1)

T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

1995 (1)

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
[Crossref]

1984 (1)

J. Rosenblatt, “Phase retrieval,” Commun. Math. Phys. 95(3), 317–343 (1984).
[Crossref]

1983 (1)

Y. Nesterov, “A method of solving a convex programming problem with convergence rate O(1/k2),” Soviet Mathematics Doklady 2(27), 372–376 (1983).

1982 (1)

1964 (1)

B. T. Polyak, “Some methods of speeding up the convergence of iteration methods,” USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964).
[Crossref]

Akcakaya, M.

G. Wang, L. Zhang, G. B. Giannakis, M. Akcakaya, and J. Chen, “Sparse phase retrieval via truncated amplitude flow,” IEEE Trans. Signal Process. 66(2), 479–491 (2018).
[Crossref]

Alexandrov, S. A.

Bian, L.

Boppart, S. A.

T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Chen, F.

Chen, J.

G. Wang, L. Zhang, G. B. Giannakis, M. Akcakaya, and J. Chen, “Sparse phase retrieval via truncated amplitude flow,” IEEE Trans. Signal Process. 66(2), 479–491 (2018).
[Crossref]

Chen, M.

Chen, Q.

Choi, W.

Choi, Y.

Chung, J.

L. Bian, J. Suo, J. Chung, X. Ou, C. Yang, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Poisson maximum likelihood and truncated Wirtinger gradient,” Sci. Rep. 6(1), 27384 (2016).
[Crossref] [PubMed]

G. Zheng, X. Ou, R. Horstmeyer, J. Chung, and C. Yang, “Fourier ptychographic microscopy: a gigapixel superscope for biomedicine,” Opt. Photonics News 25(4), 26–33 (2014).
[Crossref]

Dai, Q.

Dong, J.

Eldar, Y. C.

G. Wang, G. B. Giannakis, and Y. C. Eldar, “Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow,” IEEE Trans. Inf. Theory 64(2), 773–794 (2018).
[Crossref]

Fienup, J. R.

García, J.

Gesell, L. H.

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
[Crossref]

Giannakis, G. B.

G. Wang, G. B. Giannakis, and Y. C. Eldar, “Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow,” IEEE Trans. Inf. Theory 64(2), 773–794 (2018).
[Crossref]

G. Wang, L. Zhang, G. B. Giannakis, M. Akcakaya, and J. Chen, “Sparse phase retrieval via truncated amplitude flow,” IEEE Trans. Signal Process. 66(2), 479–491 (2018).
[Crossref]

Granero, L.

Guo, C.

Guo, K.

Gutzler, T.

Harvey, A. R.

P. C. Konda, J. M. Taylor, and A. R. Harvey, “High-resolution microscopy with low-resolution objectives: correcting phase aberrations in Fourier ptychography,” Proc. SPIE 9630, 96300X (2015).
[Crossref]

Hillman, T. R.

Horstmeyer, R.

G. Zheng, X. Ou, R. Horstmeyer, J. Chung, and C. Yang, “Fourier ptychographic microscopy: a gigapixel superscope for biomedicine,” Opt. Photonics News 25(4), 26–33 (2014).
[Crossref]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref] [PubMed]

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013).
[Crossref] [PubMed]

Jiang, X.

X. Jiang, S. Rajan, and X. Liu, “Wirtinger flow method with optimal stepsize for phase petrieval,” IEEE Signal Process. Lett. 23(11), 1627–1631 (2016).
[Crossref]

Johnson, D.

Kim, M.

Konda, P. C.

P. C. Konda, J. M. Taylor, and A. R. Harvey, “High-resolution microscopy with low-resolution objectives: correcting phase aberrations in Fourier ptychography,” Proc. SPIE 9630, 96300X (2015).
[Crossref]

Lapides, J.

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
[Crossref]

Lee, K. J.

Li, P.

Li, Y.

Liu, C.

Liu, J.

Liu, S.

Liu, X.

X. Jiang, S. Rajan, and X. Liu, “Wirtinger flow method with optimal stepsize for phase petrieval,” IEEE Signal Process. Lett. 23(11), 1627–1631 (2016).
[Crossref]

Liu, Z.

Maiden, A.

Marks, D. L.

T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Micó, V.

Nesterov, Y.

Y. Nesterov, “A method of solving a convex programming problem with convergence rate O(1/k2),” Soviet Mathematics Doklady 2(27), 372–376 (1983).

Ou, X.

L. Bian, J. Suo, J. Chung, X. Ou, C. Yang, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Poisson maximum likelihood and truncated Wirtinger gradient,” Sci. Rep. 6(1), 27384 (2016).
[Crossref] [PubMed]

G. Zheng, X. Ou, R. Horstmeyer, J. Chung, and C. Yang, “Fourier ptychographic microscopy: a gigapixel superscope for biomedicine,” Opt. Photonics News 25(4), 26–33 (2014).
[Crossref]

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013).
[Crossref] [PubMed]

Pan, X.

X. Pan, C. Liu, and J. Zhu, “Single shot ptychographical iterative engine based on multi-beam illumination,” Appl. Phys. Lett. 103(17), 171105 (2013).
[Crossref]

Polyak, B. T.

B. T. Polyak, “Some methods of speeding up the convergence of iteration methods,” USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964).
[Crossref]

Price, C. H.

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
[Crossref]

Rajan, S.

X. Jiang, S. Rajan, and X. Liu, “Wirtinger flow method with optimal stepsize for phase petrieval,” IEEE Signal Process. Lett. 23(11), 1627–1631 (2016).
[Crossref]

Ralston, T. S.

T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Rosenblatt, J.

J. Rosenblatt, “Phase retrieval,” Commun. Math. Phys. 95(3), 317–343 (1984).
[Crossref]

Sampson, D. D.

Scott Carney, P.

T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Situ, G.

Soltanolkotabi, M.

Song, P.

Sun, J.

Suo, J.

Tan, J.

Tang, G.

Taylor, J. M.

P. C. Konda, J. M. Taylor, and A. R. Harvey, “High-resolution microscopy with low-resolution objectives: correcting phase aberrations in Fourier ptychography,” Proc. SPIE 9630, 96300X (2015).
[Crossref]

Tian, L.

Turpin, T. M.

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
[Crossref]

Waller, L.

Wang, G.

G. Wang, L. Zhang, G. B. Giannakis, M. Akcakaya, and J. Chen, “Sparse phase retrieval via truncated amplitude flow,” IEEE Trans. Signal Process. 66(2), 479–491 (2018).
[Crossref]

G. Wang, G. B. Giannakis, and Y. C. Eldar, “Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow,” IEEE Trans. Inf. Theory 64(2), 773–794 (2018).
[Crossref]

Wang, W.

Wang, Y.

Yang, C.

L. Bian, G. Zheng, K. Guo, J. Suo, C. Yang, F. Chen, and Q. Dai, “Motion-corrected Fourier ptychography,” Biomed. Opt. Express 7(11), 4543–4553 (2016).
[Crossref] [PubMed]

L. Bian, J. Suo, J. Chung, X. Ou, C. Yang, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Poisson maximum likelihood and truncated Wirtinger gradient,” Sci. Rep. 6(1), 27384 (2016).
[Crossref] [PubMed]

G. Zheng, X. Ou, R. Horstmeyer, J. Chung, and C. Yang, “Fourier ptychographic microscopy: a gigapixel superscope for biomedicine,” Opt. Photonics News 25(4), 26–33 (2014).
[Crossref]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref] [PubMed]

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013).
[Crossref] [PubMed]

Yang, T. D.

Yeh, L.-H.

Yoon, C.

Zalevsky, Z.

Zhang, H.

Zhang, L.

G. Wang, L. Zhang, G. B. Giannakis, M. Akcakaya, and J. Chen, “Sparse phase retrieval via truncated amplitude flow,” IEEE Trans. Signal Process. 66(2), 479–491 (2018).
[Crossref]

Zhang, Y.

Zhao, Y.

Zheng, G.

Zhong, J.

Zhu, J.

X. Pan, C. Liu, and J. Zhu, “Single shot ptychographical iterative engine based on multi-beam illumination,” Appl. Phys. Lett. 103(17), 171105 (2013).
[Crossref]

Zuo, C.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

X. Pan, C. Liu, and J. Zhu, “Single shot ptychographical iterative engine based on multi-beam illumination,” Appl. Phys. Lett. 103(17), 171105 (2013).
[Crossref]

Biomed. Opt. Express (2)

Commun. Math. Phys. (1)

J. Rosenblatt, “Phase retrieval,” Commun. Math. Phys. 95(3), 317–343 (1984).
[Crossref]

IEEE Signal Process. Lett. (1)

X. Jiang, S. Rajan, and X. Liu, “Wirtinger flow method with optimal stepsize for phase petrieval,” IEEE Signal Process. Lett. 23(11), 1627–1631 (2016).
[Crossref]

IEEE Trans. Inf. Theory (1)

G. Wang, G. B. Giannakis, and Y. C. Eldar, “Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow,” IEEE Trans. Inf. Theory 64(2), 773–794 (2018).
[Crossref]

IEEE Trans. Signal Process. (1)

G. Wang, L. Zhang, G. B. Giannakis, M. Akcakaya, and J. Chen, “Sparse phase retrieval via truncated amplitude flow,” IEEE Trans. Signal Process. 66(2), 479–491 (2018).
[Crossref]

Nat. Photonics (1)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref] [PubMed]

Nat. Phys. (1)

T. S. Ralston, D. L. Marks, P. Scott Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Opt. Express (7)

C. Zuo, J. Sun, and Q. Chen, “Adaptive step-size strategy for noise-robust Fourier ptychographic microscopy,” Opt. Express 24(18), 20724–20744 (2016).
[Crossref] [PubMed]

T. R. Hillman, T. Gutzler, S. A. Alexandrov, and D. D. Sampson, “High-resolution, wide-field object reconstruction with synthetic aperture Fourier holographic optical microscopy,” Opt. Express 17(10), 7873–7892 (2009).
[Crossref] [PubMed]

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23(26), 33214–33240 (2015).
[Crossref] [PubMed]

J. Liu, Y. Li, W. Wang, H. Zhang, Y. Wang, J. Tan, and C. Liu, “Stable and robust frequency domain position compensation strategy for Fourier ptychographic microscopy,” Opt. Express 25(23), 28053–28067 (2017).
[Crossref]

Y. Zhang, P. Song, and Q. Dai, “Fourier ptychographic microscopy using a generalized Anscombe transform approximation of the mixed Poisson-Gaussian likelihood,” Opt. Express 25(1), 168–179 (2017).
[Crossref] [PubMed]

C. Guo, Y. Zhao, J. Tan, S. Liu, and Z. Liu, “Adaptive lens-free computational coherent imaging using autofocusing quantification with speckle illumination,” Opt. Express 26(11), 14407–14420 (2018).
[Crossref] [PubMed]

L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Wirtinger flow optimization,” Opt. Express 23(4), 4856–4866 (2015).
[Crossref] [PubMed]

Opt. Lett. (3)

Opt. Photonics News (1)

G. Zheng, X. Ou, R. Horstmeyer, J. Chung, and C. Yang, “Fourier ptychographic microscopy: a gigapixel superscope for biomedicine,” Opt. Photonics News 25(4), 26–33 (2014).
[Crossref]

Optica (2)

Proc. SPIE (2)

P. C. Konda, J. M. Taylor, and A. R. Harvey, “High-resolution microscopy with low-resolution objectives: correcting phase aberrations in Fourier ptychography,” Proc. SPIE 9630, 96300X (2015).
[Crossref]

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
[Crossref]

Sci. Rep. (1)

L. Bian, J. Suo, J. Chung, X. Ou, C. Yang, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Poisson maximum likelihood and truncated Wirtinger gradient,” Sci. Rep. 6(1), 27384 (2016).
[Crossref] [PubMed]

Soviet Mathematics Doklady (1)

Y. Nesterov, “A method of solving a convex programming problem with convergence rate O(1/k2),” Soviet Mathematics Doklady 2(27), 372–376 (1983).

USSR Comput. Math. Math. Phys. (1)

B. T. Polyak, “Some methods of speeding up the convergence of iteration methods,” USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964).
[Crossref]

Other (3)

E. Bostan, M. Soltanolkotabi, M. D. Ren, and L. Waller. “Accelerated Wirtinger flow for multiplexed Fourier ptychographic microscopy,” arXiv preprint arXiv:1803.03714 (2018).

P. C. Konda, J. M. Taylor, and A. R. Harvey, “Calibration and aberration correction in multi-aperture Fourier ptychography,” in Imaging and Applied Optics 2016, OSA Technical Digest (online) (Optical Society of America, 2016), paper CT2D.2.

J. Fu, P. Li, R. Tan, and L. Chen, “Performance evaluation of ptychographical iterative engine algorithm for coherent diffractive imaging,” in 2012 7th International Conference on System of Systems Engineering (IEEE, 2012), pp. 111–114.

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Figures (2)

Fig. 1
Fig. 1 Comparison of the reconstruction results by the three state-of-the-arts and the proposed DTWFP. The upper left corner of each amplitude map is the RE at the end of 50 iterations.
Fig. 2
Fig. 2 Comparison of the reconstruction results by the three state-of-the-arts and the proposed DTWFP under real data.

Tables (1)

Tables Icon

Table 1 Comparison of running time between the three state-of-the-arts and the proposed DTWFP.

Equations (16)

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I j = | F 1 { P ( k ) F ( s ( r ) e i 2 π k j r ) } | 2 = | F 1 { P ( k ) S ( k k j ) } | 2 j = 1... N ,
b j = | F 1 { P ( k ) S ( k k j ) } | 2 = | A j S | 2 ,
[ I ˜ n ] i ~ P o i s s o n ( [ b n ] i ) ,
p ( [ I ˜ n ] i | [ b n ] i ) = [ b n ] i [ I ˜ n ] i [ I ˜ n ] i ! e [ b n ] i ,
min f ( S ) = n = 1 N i = 1 m | [ A n ] i S | 2 n = 1 N i = 1 m [ I ˜ n ] i log ( | [ A n ] i S | 2 ) ,
f ( S ) = d f ( S ) d S * = d { n = 1 N i = 1 m | [ A n S ] i | 2 n = 1 N i = 1 m [ I ˜ n ] i log ( | [ A n S ] i | 2 ) } d S * = n = 1 N i = 1 m d [ | [ A n S ] i | 2 [ I ˜ n ] i log ( | [ A n S ] i | 2 ) ] d S * = n = 1 N i = 1 m d [ ( [ A n S ] i ) * ( [ A n S ] i ) [ I ˜ n ] i log ( ( [ A n S ] i ) * ( [ A n S ] i ) ) ] d S * = n = 1 N i = 1 m ( [ A n ] i H [ A n S ] i [ I ˜ n ] i | [ A n S ] i | 2 [ A n ] i H [ A n S ] i ) ,
ζ = { 1 i m · N | | [ I ˜ n ] i | [ A n S ] i | 2 | a h [ I ˜ ] | [ A S ] | 2 1 m · N · | [ A n S ] i | 2 S | [ A n S ] i | 2 [ I ˜ n ] i 1 ( 1 + γ ) 2 } ,
ζ f ( S ) = n = 1 N i = 1 m { ( [ A n ] i H [ A n S ] i [ I ˜ n ] i | [ A n S ] i | 2 [ A n ] i H [ A n S ] i ) } ζ ,
ζ f ( S ) = n = 1 N { P * ( k + k n ) F { F 1 { P ( k + k n ) S ( k ) } I ˜ n | F 1 { P ( k + k n ) S ( k ) } | 2 F 1 { P ( k + k n ) S ( k ) } } } ζ ζ f ( P ) = n = 1 N { S * ( k k n ) F { F 1 { P ( k ) S ( k k n ) } I ˜ n | F 1 { P ( k ) S ( k k n ) } | 2 F 1 { P ( k ) S ( k k n ) } } } ζ ,
v t = β t * v t 1 + μ t * ζ f ( S ) ,
μ t = min ( 1 e t / t 0 , μ max ) m · N ,
β t = 0.9 0.9 / ( 1 + ( 3 · t / 2 · T ) ^ 8 ) ,
S t + 1 = S t + 0.9 v t ,
μ t = arg min f ( S t μ f ( S t ) ) ,
d f ( S t μ f ( S t ) ) d μ = n i { μ 3 | h t | 2 | h t | 2 3 μ 2 Re ( y t * h t ) | h t | 2 + μ [ | y t | 2 | h t | 2 ... + 2 Re ( y t * h t ) Re ( y t * h t ) [ I ˜ n ] i | h t | 2 ] + [ I ˜ n ] i Re ( y t * h t ) ... | y t | 2 Re ( y t * h t ) } , y t = [ A n ] i S t , h t = [ A n ] i f ( S t ) ,
R E = min φ [ 0 , 2 π ) e j φ S S r 2 S r 2 ,

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