Abstract

Recently Fourier Ptychography (FP) has attracted great attention, due to its marked effectiveness in leveraging snapshot numbers for spatial resolution in large field-of-view imaging. To acquire high signal-to-noise-ratio (SNR) images under angularly varying illuminations for subsequent reconstruction, FP requires long exposure time, which largely limits its practical applications. In this paper, based on the recently reported Wirtinger flow algorithm, we propose an iterative optimization framework incorporating phase retrieval and noise relaxation together, to realize FP reconstruction using low SNR images captured under short exposure time. Experiments on both synthetic and real captured data validate the effectiveness of the proposed reconstruction method. Specifically, the proposed technique could save ~ 80% exposure time to achieve similar retrieval accuracy compared to the conventional FP. Besides, we have released our source code for non-commercial use.

© 2015 Optical Society of America

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References

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    [Crossref]
  2. G. Zheng, “Breakthroughs in photonics 2013: Fourier ptychographic imaging,” IEEE Photonics J. 6, 0701207 (2014).
  3. G. Zheng, X. Ou, R. Horstmeyer, J. Chung, and C. Yang, “Fourier ptychographic microscopy: A gigapixel superscope for biomedicine,” Opt. Photonics News 25, 26–33 (2014).
    [Crossref]
  4. X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via fourier ptychographic microscopy,” Opt. Lett. 38, 4845–4848 (2013).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  14. E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imag. Sci. 6, 199–225 (2013).
    [Crossref]
  15. E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
    [Crossref]
  16. I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. Ser. A 45, 1–35 (2012).
  17. E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” arXiv preprint arXiv:13103240 (2013).
  18. R. Remmert, Theory of Complex Functions (Springer, 1991).
    [Crossref]
  19. R. F. Fischer, Precoding and Signal Shaping for Digital Transmission (John Wiley & Sons, 2005).
  20. E. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” arXiv preprint arXiv:14071065 (2014).
  21. D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Academic, 1982).
  22. Y. Deng, Q. Dai, and Z. Zhang, “An overview of computational sparse models and their applications in artificial intelligence,”in Artif. Intell. Evol. Comput. and Metaheuristics, (2013), pp. 345–369.
    [Crossref]
  23. K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE T. Image Process. 16, 2080–2095 (2007).
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  24. P. Milanfar, “A tour of modern image filtering: new insights and methods, both practical and theoretical,” IEEE Signal Proc. Mag. 30, 106–128 (2013).
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  25. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE T. Image Process. 13, 600–612 (2004).
    [Crossref]
  26. A. G. Weber, “The usc-sipi image database version 5,” USC-SIPI Rep. 315, 1–24 (1997).
  27. L. Bian, J. Suo, G. Situ, G. Zheng, F. Chen, and Q. Dai, “Content adaptive illumination for fourier ptychography,” Opt. Lett. 39, 6648–6651 (2014).
    [PubMed]
  28. X. Li and V. Voroninski, “Sparse signal recovery from quadratic measurements via convex programming,” SIAM J. Math. Anal. 45, 3019–3033 (2013).
    [Crossref]
  29. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D. 60, 259–268 (1992).
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  30. A. Buades, B. Coll, and J.-M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Sim. 4, 490–530 (2005).
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  33. 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]

2014 (9)

G. Zheng, “Breakthroughs in photonics 2013: Fourier ptychographic imaging,” IEEE Photonics J. 6, 0701207 (2014).

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

S. Dong, P. Nanda, R. Shiradkar, K. Guo, and G. Zheng, “High-resolution fluorescence imaging via patternilluminated fourier ptychography,” Opt. Express 22, 20856–20870 (2014).
[Crossref] [PubMed]

S. Dong, K. Guo, P. Nanda, R. Shiradkar, and G. Zheng, “FPscope: a field-portable high-resolution microscope using a cellphone lens,” Biomed. Opt. Express 5, 3305–3310 (2014).
[Crossref] [PubMed]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” arXiv preprint arXiv:14027350 (2014).

E. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” arXiv preprint arXiv:14071065 (2014).

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

S. Dong, R. Shiradkar, P. Nanda, and G. Zheng, “Spectral multiplexing and coherent-state decomposition in fourier ptychographic imaging,” Biomed. Opt. Express 5, 1757–1767 (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]

2013 (7)

X. Li and V. Voroninski, “Sparse signal recovery from quadratic measurements via convex programming,” SIAM J. Math. Anal. 45, 3019–3033 (2013).
[Crossref]

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” arXiv preprint arXiv:13103240 (2013).

P. Milanfar, “A tour of modern image filtering: new insights and methods, both practical and theoretical,” IEEE Signal Proc. Mag. 30, 106–128 (2013).
[Crossref]

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imag. Sci. 6, 199–225 (2013).
[Crossref]

E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

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

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

2012 (1)

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. Ser. A 45, 1–35 (2012).

2010 (1)

2007 (1)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE T. Image Process. 16, 2080–2095 (2007).
[Crossref]

2005 (1)

A. Buades, B. Coll, and J.-M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Sim. 4, 490–530 (2005).
[Crossref]

2004 (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE T. Image Process. 13, 600–612 (2004).
[Crossref]

1997 (1)

A. G. Weber, “The usc-sipi image database version 5,” USC-SIPI Rep. 315, 1–24 (1997).

1996 (1)

L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Rev. 38, 49–95 (1996).
[Crossref]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D. 60, 259–268 (1992).
[Crossref]

1987 (1)

1982 (1)

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

1978 (1)

1972 (1)

R. W. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237 (1972).

Bertsekas, D. P.

D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Academic, 1982).

Bian, L.

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE T. Image Process. 13, 600–612 (2004).
[Crossref]

Boyd, S.

L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Rev. 38, 49–95 (1996).
[Crossref]

Buades, A.

A. Buades, B. Coll, and J.-M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Sim. 4, 490–530 (2005).
[Crossref]

Candes, E.

E. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” arXiv preprint arXiv:14071065 (2014).

Candes, E. J.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” arXiv preprint arXiv:13103240 (2013).

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imag. Sci. 6, 199–225 (2013).
[Crossref]

E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” arXiv preprint arXiv:14027350 (2014).

Chen, F.

Chung, J.

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

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” arXiv preprint arXiv:14027350 (2014).

Coll, B.

A. Buades, B. Coll, and J.-M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Sim. 4, 490–530 (2005).
[Crossref]

d’Aspremont, A.

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. Ser. A 45, 1–35 (2012).

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE T. Image Process. 16, 2080–2095 (2007).
[Crossref]

Dai, Q.

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

Y. Deng, Q. Dai, and Z. Zhang, “An overview of computational sparse models and their applications in artificial intelligence,”in Artif. Intell. Evol. Comput. and Metaheuristics, (2013), pp. 345–369.
[Crossref]

Deng, Y.

Y. Deng, Q. Dai, and Z. Zhang, “An overview of computational sparse models and their applications in artificial intelligence,”in Artif. Intell. Evol. Comput. and Metaheuristics, (2013), pp. 345–369.
[Crossref]

Dong, S.

Egiazarian, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE T. Image Process. 16, 2080–2095 (2007).
[Crossref]

Eldar, Y. C.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” arXiv preprint arXiv:14027350 (2014).

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imag. Sci. 6, 199–225 (2013).
[Crossref]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D. 60, 259–268 (1992).
[Crossref]

Fienup, J. R.

Fischer, R. F.

R. F. Fischer, Precoding and Signal Shaping for Digital Transmission (John Wiley & Sons, 2005).

Foi, A.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE T. Image Process. 16, 2080–2095 (2007).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237 (1972).

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-hill, 2008).

Guo, K.

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, 26–33 (2014).
[Crossref]

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

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

Humphry, M. J.

Katkovnik, V.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE T. Image Process. 16, 2080–2095 (2007).
[Crossref]

Li, X.

E. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” arXiv preprint arXiv:14071065 (2014).

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. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” arXiv preprint arXiv:13103240 (2013).

X. Li and V. Voroninski, “Sparse signal recovery from quadratic measurements via convex programming,” SIAM J. Math. Anal. 45, 3019–3033 (2013).
[Crossref]

Maiden, A. M.

Mallat, S.

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. Ser. A 45, 1–35 (2012).

Miao, J.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” arXiv preprint arXiv:14027350 (2014).

Milanfar, P.

P. Milanfar, “A tour of modern image filtering: new insights and methods, both practical and theoretical,” IEEE Signal Proc. Mag. 30, 106–128 (2013).
[Crossref]

Morel, J.-M.

A. Buades, B. Coll, and J.-M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Sim. 4, 490–530 (2005).
[Crossref]

Nanda, P.

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D. 60, 259–268 (1992).
[Crossref]

Ou, X.

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

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

Ramchandran, K.

Remmert, R.

R. Remmert, Theory of Complex Functions (Springer, 1991).
[Crossref]

Rodenburg, J. M.

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D. 60, 259–268 (1992).
[Crossref]

Segev, M.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” arXiv preprint arXiv:14027350 (2014).

Shechtman, Y.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” arXiv preprint arXiv:14027350 (2014).

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE T. Image Process. 13, 600–612 (2004).
[Crossref]

Shiradkar, R.

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE T. Image Process. 13, 600–612 (2004).
[Crossref]

Situ, G.

Soltanolkotabi, M.

E. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” arXiv preprint arXiv:14071065 (2014).

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” arXiv preprint arXiv:13103240 (2013).

Strohmer, T.

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imag. Sci. 6, 199–225 (2013).
[Crossref]

E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

Suo, J.

Tian, L.

Vandenberghe, L.

L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Rev. 38, 49–95 (1996).
[Crossref]

Voroninski, V.

E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

E. J. Candes, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imag. Sci. 6, 199–225 (2013).
[Crossref]

X. Li and V. Voroninski, “Sparse signal recovery from quadratic measurements via convex programming,” SIAM J. Math. Anal. 45, 3019–3033 (2013).
[Crossref]

Waldspurger, I.

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. Ser. A 45, 1–35 (2012).

Waller, L.

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE T. Image Process. 13, 600–612 (2004).
[Crossref]

Weber, A. G.

A. G. Weber, “The usc-sipi image database version 5,” USC-SIPI Rep. 315, 1–24 (1997).

Yang, C.

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

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

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

Zhang, Z.

Y. Deng, Q. Dai, and Z. Zhang, “An overview of computational sparse models and their applications in artificial intelligence,”in Artif. Intell. Evol. Comput. and Metaheuristics, (2013), pp. 345–369.
[Crossref]

Zheng, G.

Appl. Optics (1)

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

arXiv preprint (3)

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” arXiv preprint arXiv:14027350 (2014).

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” arXiv preprint arXiv:13103240 (2013).

E. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” arXiv preprint arXiv:14071065 (2014).

Biomed. Opt. Express (3)

Commun. Pure Appl. Math. (1)

E. J. Candes, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

IEEE Photonics J. (1)

G. Zheng, “Breakthroughs in photonics 2013: Fourier ptychographic imaging,” IEEE Photonics J. 6, 0701207 (2014).

IEEE Signal Proc. Mag. (1)

P. Milanfar, “A tour of modern image filtering: new insights and methods, both practical and theoretical,” IEEE Signal Proc. Mag. 30, 106–128 (2013).
[Crossref]

IEEE T. Image Process. (2)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE T. Image Process. 13, 600–612 (2004).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE T. Image Process. 16, 2080–2095 (2007).
[Crossref]

J. Opt. Soc. Am. A (1)

Math. Program. Ser. A (1)

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. Ser. A 45, 1–35 (2012).

Multiscale Model. Sim. (1)

A. Buades, B. Coll, and J.-M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Model. Sim. 4, 490–530 (2005).
[Crossref]

Nat. Photonics (1)

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

Opt. Express (1)

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, 26–33 (2014).
[Crossref]

Optik (1)

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Reconstruction results by WFP on different noise-level simulated data. The top row shows the intensity images, while the bottom row shows the phase images.

Fig. 2
Fig. 2

Reconstruction results of different methods on the same dataset.

Fig. 3
Fig. 3

Quantitative comparison between reconstruction results of WFP and AP on different scenes.

Fig. 4
Fig. 4

Reconstruction comparison between AP and WFP on real captured data.

Fig. 5
Fig. 5

Resolution demonstration of AP and WFP under different exposure time. (a) shows the amplitude recovery results by WFP with inputs captured under 1 ms exposure time. (b) and (c) respectively show the results by AP with inputs taken under 4 ms and 5 ms.

Tables (4)

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Algorithm 1: Wirtinger flow algorithm

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Algorithm 2: WFP algorithm

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Table 1 Quantitative comparison among the recovery of WFP at different noise levels.

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Table 2 Quantitative comparison among different reconstruction methods.

Equations (12)

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min f ( x ) = 1 2 ( Ax ) Ax b F 2 , x n .
x ( k + 1 ) = x ( k ) Δ f x | x = x ( k ) .
f x = 1 2 ( Ax ) Ax b F 2 x = A H [ ( | Ax | . 2 b ) ( Ax ) ] .
b = | Ax | 2 + n ,
| n | 3 σ .
n n 9 σ 2 + ε ε = 0 .
min f ( x ) = 1 2 ( Ax ) Ax + n b F 2 s . t . n n 9 σ 2 + ε ε = 0 .
min f ( x ) = 1 2 ( Ax ) Ax + n b F 2 + μ 2 n n 9 σ 2 + ε ε F 2 .
x ( k + 1 ) = x ( k ) Δ x f x | x = x ( k ) = x ( k ) Δ x A H [ ( | Ax | . 2 + n b ) ( Ax ) ] | x = x ( k ) ,
n ( k + 1 ) = n ( k ) Δ n f n | n = n ( k ) = n ( k ) Δ n [ ( | Az | . 2 + n b ) + μ ( n n 9 σ 2 + ε ε ) 2 n ] | n = n ( k ) .
f ε | ε = ε ( k + 1 ) = [ μ ( n n 9 σ 2 + ε ε ) 2 ε ] | ε = ε ( k + 1 ) = 0 ε ( k + 1 ) = max ( 9 σ 2 n n , 0 ) .
η = m 2 k 2 [ ( 1 ξ ) m ( k 1 ) + m ] 2 1 ( 1 ξ ) 2 .

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