Abstract

In low light conditions, such as in astronomy and non-invasive bio-imaging applications, the imaging performance is mostly degraded due to noise. In this paper, we demonstrate a transport of intensity equation based technique that uses photon-counting phase imaging. To achieve the phase imaging in a photon starved condition, a method proposed by Paganin et al. [J. Micros. 214, 51 (2004) [CrossRef]  ] has been used. The method uses the fact that the magnitude of the wavefront curvature determines the quality of the recovered phase image for a given noise level and defocus distance. The effectiveness of the proposed method has been illustrated through simulations and experimental results using inexpensive partially coherent illumination. The study can find applications in non-invasive phase imaging.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  28. K. Yan, L. Lifei, D. Xuejie, Z. Tongyi, L. Dongjian, and Z. Wei, “Photon-limited depth and reflectivity imaging with sparsity regularization,” Opt. Commun. 392, 25–30 (2017).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2019 (1)

Y. Chen, Y. Zou, P. Huang, J. Qi, W. He, and Q. Chen, “Reconstruction algorithm of low-light integral imaging by electron-multiplying charge-coupled device,” Opt. Eng. 58(5), 1 (2019).
[Crossref]

2018 (3)

2017 (4)

K. Yan, L. Lifei, D. Xuejie, Z. Tongyi, L. Dongjian, and Z. Wei, “Photon-limited depth and reflectivity imaging with sparsity regularization,” Opt. Commun. 392, 25–30 (2017).
[Crossref]

X. Meng, H. Huang, K. Yan, X. Tian, W. Yu, H. Cui, Y. Kong, L. Xue, C. Liu, and S. Wang, “Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method,” Lab Chip 17(1), 104–109 (2017).
[Crossref]

I. Muniraj, C. Guo, R. Malallah, J. P. Ryle, J. J. Healy, B.-G. Lee, and J. T. Sheridan, “Low photon count based digital holography for quadratic phase cryptography,” Opt. Lett. 42(14), 2774–2777 (2017).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Optical asymmetric cryptosystem based on photon counting and phase-truncated Fresnel transforms,” J. Mod. Opt. 64(8), 878–886 (2017).
[Crossref]

2016 (2)

2015 (2)

C. Zuo, Q. Chen, L. Tian, L. Waller, and A. Asundi, “Transport of intensity phase retrieval and computational imaging of partially coherent fields: The phase space perspective,” Opt. Lasers Eng. 71, 20–32 (2015).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

2014 (2)

G. Pedrini, A. Faridian, A. K. Singh, and W. Osten, “Phase retrieval for optical metrology,” Proc. SPIE 9276, 927602 (2014).
[Crossref]

S. K. Rajput, D. Kumar, and N. K. Nishchal, “Photon counting imaging and polarized light encoding for secure image verification and hologram watermarking,” J. Opt. 16(12), 125406 (2014).
[Crossref]

2013 (5)

P. Latorre-Carmona, B. Javidi, F. Pla, and E. Tajahuerce, “Photon counting 3-D object recognition using digital holography,” IEEE Photonics J. 5(6), 6900309 (2013).
[Crossref]

I. Rasnik, T. French, K. Jacobson, and K. Berland, “Electronic cameras for low, light microscopy,” Methods Cell Biol. 114, 211–241 (2013).
[Crossref]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Noninterferometric single-shot quantitative phase microscopy,” J. Microsc. 38(18), 3538–3541 (2013).
[Crossref]

J. Petruccelli, L. Tien, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21(12), 14430–14441 (2013).
[Crossref]

J. Fienup, “Phase retrieval algorithms: a personal tour,” Appl. Opt. 52(1), 45–56 (2013).
[Crossref]

2012 (2)

T. J. Gould and J. Bewersdorf, “Nanoscopy at low light intensities shows its potential,” eLife 1, e00475 (2012).
[Crossref]

I. Testa, N. T. Urban, S. Jakobs, C. Eggling, K. L. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

2011 (1)

D. P. Kelly, T. Meinecke, N. Sabitov, S. Sinzinger, and J. T. Sheridan, “Digital holography and phase retrieval: a theoretical investigation,” Proc. SPIE 8074, 807401 (2011).
[Crossref]

2010 (1)

2009 (1)

2007 (1)

2005 (1)

2004 (3)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase amplitude microscopy III. The effect of noise,” J. Micros. 214(1), 51–61 (2004).
[Crossref]

G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 29(21), 2503–2505 (2004).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using digital holography,” Opt. Eng. 43(12), 2959–2966 (2004).
[Crossref]

2002 (1)

1998 (1)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

1997 (1)

T. E. Gureyevand K and A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[Crossref]

1992 (1)

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

1983 (1)

1982 (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21(5), 829–832 (1982).
[Crossref]

1955 (1)

F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955).
[Crossref]

Abeywickrema, U.

Arthur, K.

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” Phys. Rev. Lett. 121(24), 243902 (2018).
[Crossref]

Asundi, A.

C. Zuo, Q. Chen, L. Tian, L. Waller, and A. Asundi, “Transport of intensity phase retrieval and computational imaging of partially coherent fields: The phase space perspective,” Opt. Lasers Eng. 71, 20–32 (2015).
[Crossref]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Noninterferometric single-shot quantitative phase microscopy,” J. Microsc. 38(18), 3538–3541 (2013).
[Crossref]

Badizadegan, K.

Banerjee, P. P.

Barbastathis, G.

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase amplitude microscopy III. The effect of noise,” J. Micros. 214(1), 51–61 (2004).
[Crossref]

Basunia, M.

Berland, K.

I. Rasnik, T. French, K. Jacobson, and K. Berland, “Electronic cameras for low, light microscopy,” Methods Cell Biol. 114, 211–241 (2013).
[Crossref]

Bewersdorf, J.

T. J. Gould and J. Bewersdorf, “Nanoscopy at low light intensities shows its potential,” eLife 1, e00475 (2012).
[Crossref]

Brooker, G.

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: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Chen, Q.

Y. Chen, Y. Zou, P. Huang, J. Qi, W. He, and Q. Chen, “Reconstruction algorithm of low-light integral imaging by electron-multiplying charge-coupled device,” Opt. Eng. 58(5), 1 (2019).
[Crossref]

C. Zuo, Q. Chen, L. Tian, L. Waller, and A. Asundi, “Transport of intensity phase retrieval and computational imaging of partially coherent fields: The phase space perspective,” Opt. Lasers Eng. 71, 20–32 (2015).
[Crossref]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Noninterferometric single-shot quantitative phase microscopy,” J. Microsc. 38(18), 3538–3541 (2013).
[Crossref]

Chen, Y.

Y. Chen, Y. Zou, P. Huang, J. Qi, W. He, and Q. Chen, “Reconstruction algorithm of low-light integral imaging by electron-multiplying charge-coupled device,” Opt. Eng. 58(5), 1 (2019).
[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: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Cui, H.

X. Meng, H. Huang, K. Yan, X. Tian, W. Yu, H. Cui, Y. Kong, L. Xue, C. Liu, and S. Wang, “Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method,” Lab Chip 17(1), 104–109 (2017).
[Crossref]

Dasari, R. R.

Deflores, L. P.

Dongjian, L.

K. Yan, L. Lifei, D. Xuejie, Z. Tongyi, L. Dongjian, and Z. Wei, “Photon-limited depth and reflectivity imaging with sparsity regularization,” Opt. Commun. 392, 25–30 (2017).
[Crossref]

Durand, F.

S. W. Hasinoff, F. Durand, and W. T. Freeman, “Noise optimal capture for dynamic range photography,” Proc. IEEE Computer Society Confer. on Computer Vision and Pattern Recognition, San Francisco, USA, 553–560 (2010).

Eggling, C.

I. Testa, N. T. Urban, S. Jakobs, C. Eggling, K. L. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[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: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Faridian, A.

G. Pedrini, A. Faridian, A. K. Singh, and W. Osten, “Phase retrieval for optical metrology,” Proc. SPIE 9276, 927602 (2014).
[Crossref]

Feld, M. S.

Fienup, J.

Fienup, J. R.

Freeman, W. T.

S. W. Hasinoff, F. Durand, and W. T. Freeman, “Noise optimal capture for dynamic range photography,” Proc. IEEE Computer Society Confer. on Computer Vision and Pattern Recognition, San Francisco, USA, 553–560 (2010).

French, T.

I. Rasnik, T. French, K. Jacobson, and K. Berland, “Electronic cameras for low, light microscopy,” Methods Cell Biol. 114, 211–241 (2013).
[Crossref]

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21(5), 829–832 (1982).
[Crossref]

Gould, T. J.

T. J. Gould and J. Bewersdorf, “Nanoscopy at low light intensities shows its potential,” eLife 1, e00475 (2012).
[Crossref]

Goy, A.

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” Phys. Rev. Lett. 121(24), 243902 (2018).
[Crossref]

Guan, X.

Guo, C.

Gureyevand K, T. E.

T. E. Gureyevand K and A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[Crossref]

Hasinoff, S. W.

S. W. Hasinoff, F. Durand, and W. T. Freeman, “Noise optimal capture for dynamic range photography,” Proc. IEEE Computer Society Confer. on Computer Vision and Pattern Recognition, San Francisco, USA, 553–560 (2010).

Hayasaki, Y.

He, W.

Y. Chen, Y. Zou, P. Huang, J. Qi, W. He, and Q. Chen, “Reconstruction algorithm of low-light integral imaging by electron-multiplying charge-coupled device,” Opt. Eng. 58(5), 1 (2019).
[Crossref]

Healy, J. J.

Hell, S. W.

I. Testa, N. T. Urban, S. Jakobs, C. Eggling, K. L. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

Huang, H.

X. Meng, H. Huang, K. Yan, X. Tian, W. Yu, H. Cui, Y. Kong, L. Xue, C. Liu, and S. Wang, “Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method,” Lab Chip 17(1), 104–109 (2017).
[Crossref]

Huang, P.

Y. Chen, Y. Zou, P. Huang, J. Qi, W. He, and Q. Chen, “Reconstruction algorithm of low-light integral imaging by electron-multiplying charge-coupled device,” Opt. Eng. 58(5), 1 (2019).
[Crossref]

Iwai, H.

Jacobson, K.

I. Rasnik, T. French, K. Jacobson, and K. Berland, “Electronic cameras for low, light microscopy,” Methods Cell Biol. 114, 211–241 (2013).
[Crossref]

Jakobs, S.

I. Testa, N. T. Urban, S. Jakobs, C. Eggling, K. L. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

Javidi, B.

P. Latorre-Carmona, B. Javidi, F. Pla, and E. Tajahuerce, “Photon counting 3-D object recognition using digital holography,” IEEE Photonics J. 5(6), 6900309 (2013).
[Crossref]

S. Yeom, B. Javidi, and E. Watson, “Photon counting passive 3D image sensing for automatic target recognition,” Opt. Express 13(23), 9310–9330 (2005).
[Crossref]

Joseph, J.

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using digital holography,” Opt. Eng. 43(12), 2959–2966 (2004).
[Crossref]

Kelly, D. P.

D. P. Kelly, T. Meinecke, N. Sabitov, S. Sinzinger, and J. T. Sheridan, “Digital holography and phase retrieval: a theoretical investigation,” Proc. SPIE 8074, 807401 (2011).
[Crossref]

Komuro, K.

Kong, Y.

X. Meng, H. Huang, K. Yan, X. Tian, W. Yu, H. Cui, Y. Kong, L. Xue, C. Liu, and S. Wang, “Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method,” Lab Chip 17(1), 104–109 (2017).
[Crossref]

Kumar, D.

D. Kumar and N. K. Nishchal, “Synthesis and reconstruction of multi-plane phase-only Fresnel holograms,” Optik 127(24), 12069–12077 (2016).
[Crossref]

S. K. Rajput, D. Kumar, and N. K. Nishchal, “Photon counting imaging and polarized light encoding for secure image verification and hologram watermarking,” J. Opt. 16(12), 125406 (2014).
[Crossref]

Latorre-Carmona, P.

P. Latorre-Carmona, B. Javidi, F. Pla, and E. Tajahuerce, “Photon counting 3-D object recognition using digital holography,” IEEE Photonics J. 5(6), 6900309 (2013).
[Crossref]

Lee, B.-G.

Li, S.

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” Phys. Rev. Lett. 121(24), 243902 (2018).
[Crossref]

Lifei, L.

K. Yan, L. Lifei, D. Xuejie, Z. Tongyi, L. Dongjian, and Z. Wei, “Photon-limited depth and reflectivity imaging with sparsity regularization,” Opt. Commun. 392, 25–30 (2017).
[Crossref]

Liu, C.

X. Meng, H. Huang, K. Yan, X. Tian, W. Yu, H. Cui, Y. Kong, L. Xue, C. Liu, and S. Wang, “Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method,” Lab Chip 17(1), 104–109 (2017).
[Crossref]

Liu, F.

Lucke, R. L.

Malallah, R.

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase amplitude microscopy III. The effect of noise,” J. Micros. 214(1), 51–61 (2004).
[Crossref]

Meinecke, T.

D. P. Kelly, T. Meinecke, N. Sabitov, S. Sinzinger, and J. T. Sheridan, “Digital holography and phase retrieval: a theoretical investigation,” Proc. SPIE 8074, 807401 (2011).
[Crossref]

Meng, X.

X. Meng, H. Huang, K. Yan, X. Tian, W. Yu, H. Cui, Y. Kong, L. Xue, C. Liu, and S. Wang, “Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method,” Lab Chip 17(1), 104–109 (2017).
[Crossref]

Miao, J.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Muniraj, I.

Nishchal, N. K.

S. K. Rajput and N. K. Nishchal, “Optical asymmetric cryptosystem based on photon counting and phase-truncated Fresnel transforms,” J. Mod. Opt. 64(8), 878–886 (2017).
[Crossref]

D. Kumar and N. K. Nishchal, “Synthesis and reconstruction of multi-plane phase-only Fresnel holograms,” Optik 127(24), 12069–12077 (2016).
[Crossref]

S. K. Rajput, D. Kumar, and N. K. Nishchal, “Photon counting imaging and polarized light encoding for secure image verification and hologram watermarking,” J. Opt. 16(12), 125406 (2014).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using digital holography,” Opt. Eng. 43(12), 2959–2966 (2004).
[Crossref]

Nomura, T.

Nugent, A.

T. E. Gureyevand K and A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[Crossref]

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase amplitude microscopy III. The effect of noise,” J. Micros. 214(1), 51–61 (2004).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

Osten, W.

G. Pedrini, A. Faridian, A. K. Singh, and W. Osten, “Phase retrieval for optical metrology,” Proc. SPIE 9276, 927602 (2014).
[Crossref]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase amplitude microscopy III. The effect of noise,” J. Micros. 214(1), 51–61 (2004).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

Paxman, R. G.

Pedrini, G.

G. Pedrini, A. Faridian, A. K. Singh, and W. Osten, “Phase retrieval for optical metrology,” Proc. SPIE 9276, 927602 (2014).
[Crossref]

Petruccelli, J.

Pla, F.

P. Latorre-Carmona, B. Javidi, F. Pla, and E. Tajahuerce, “Photon counting 3-D object recognition using digital holography,” IEEE Photonics J. 5(6), 6900309 (2013).
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Popescu, G.

Qi, J.

Y. Chen, Y. Zou, P. Huang, J. Qi, W. He, and Q. Chen, “Reconstruction algorithm of low-light integral imaging by electron-multiplying charge-coupled device,” Opt. Eng. 58(5), 1 (2019).
[Crossref]

Qu, W.

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Noninterferometric single-shot quantitative phase microscopy,” J. Microsc. 38(18), 3538–3541 (2013).
[Crossref]

Rajput, S. K.

S. K. Rajput and N. K. Nishchal, “Optical asymmetric cryptosystem based on photon counting and phase-truncated Fresnel transforms,” J. Mod. Opt. 64(8), 878–886 (2017).
[Crossref]

S. K. Rajput, D. Kumar, and N. K. Nishchal, “Photon counting imaging and polarized light encoding for secure image verification and hologram watermarking,” J. Opt. 16(12), 125406 (2014).
[Crossref]

Rasnik, I.

I. Rasnik, T. French, K. Jacobson, and K. Berland, “Electronic cameras for low, light microscopy,” Methods Cell Biol. 114, 211–241 (2013).
[Crossref]

Rickard, L. J.

Rosen, J.

Ryle, J. P.

Sabitov, N.

D. P. Kelly, T. Meinecke, N. Sabitov, S. Sinzinger, and J. T. Sheridan, “Digital holography and phase retrieval: a theoretical investigation,” Proc. SPIE 8074, 807401 (2011).
[Crossref]

Schulz, T. J.

Segev, M.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Shechtman, Y.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
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Sheridan, J. T.

I. Muniraj, C. Guo, R. Malallah, J. P. Ryle, J. J. Healy, B.-G. Lee, and J. T. Sheridan, “Low photon count based digital holography for quadratic phase cryptography,” Opt. Lett. 42(14), 2774–2777 (2017).
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D. P. Kelly, T. Meinecke, N. Sabitov, S. Sinzinger, and J. T. Sheridan, “Digital holography and phase retrieval: a theoretical investigation,” Proc. SPIE 8074, 807401 (2011).
[Crossref]

Singh, A. K.

G. Pedrini, A. Faridian, A. K. Singh, and W. Osten, “Phase retrieval for optical metrology,” Proc. SPIE 9276, 927602 (2014).
[Crossref]

Singh, K.

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using digital holography,” Opt. Eng. 43(12), 2959–2966 (2004).
[Crossref]

Sinzinger, S.

D. P. Kelly, T. Meinecke, N. Sabitov, S. Sinzinger, and J. T. Sheridan, “Digital holography and phase retrieval: a theoretical investigation,” Proc. SPIE 8074, 807401 (2011).
[Crossref]

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

Tajahuerce, E.

P. Latorre-Carmona, B. Javidi, F. Pla, and E. Tajahuerce, “Photon counting 3-D object recognition using digital holography,” IEEE Photonics J. 5(6), 6900309 (2013).
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Testa, I.

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Tian, L.

C. Zuo, Q. Chen, L. Tian, L. Waller, and A. Asundi, “Transport of intensity phase retrieval and computational imaging of partially coherent fields: The phase space perspective,” Opt. Lasers Eng. 71, 20–32 (2015).
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L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010).
[Crossref]

Tian, X.

X. Meng, H. Huang, K. Yan, X. Tian, W. Yu, H. Cui, Y. Kong, L. Xue, C. Liu, and S. Wang, “Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method,” Lab Chip 17(1), 104–109 (2017).
[Crossref]

Tien, L.

Tongyi, Z.

K. Yan, L. Lifei, D. Xuejie, Z. Tongyi, L. Dongjian, and Z. Wei, “Photon-limited depth and reflectivity imaging with sparsity regularization,” Opt. Commun. 392, 25–30 (2017).
[Crossref]

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I. Testa, N. T. Urban, S. Jakobs, C. Eggling, K. L. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

Vaughan, J. C.

Waller, L.

C. Zuo, Q. Chen, L. Tian, L. Waller, and A. Asundi, “Transport of intensity phase retrieval and computational imaging of partially coherent fields: The phase space perspective,” Opt. Lasers Eng. 71, 20–32 (2015).
[Crossref]

L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010).
[Crossref]

Wang, S.

X. Meng, H. Huang, K. Yan, X. Tian, W. Yu, H. Cui, Y. Kong, L. Xue, C. Liu, and S. Wang, “Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method,” Lab Chip 17(1), 104–109 (2017).
[Crossref]

Watson, E.

Wei, Z.

K. Yan, L. Lifei, D. Xuejie, Z. Tongyi, L. Dongjian, and Z. Wei, “Photon-limited depth and reflectivity imaging with sparsity regularization,” Opt. Commun. 392, 25–30 (2017).
[Crossref]

Willig, K. L.

I. Testa, N. T. Urban, S. Jakobs, C. Eggling, K. L. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

Xue, L.

X. Meng, H. Huang, K. Yan, X. Tian, W. Yu, H. Cui, Y. Kong, L. Xue, C. Liu, and S. Wang, “Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method,” Lab Chip 17(1), 104–109 (2017).
[Crossref]

Xuejie, D.

K. Yan, L. Lifei, D. Xuejie, Z. Tongyi, L. Dongjian, and Z. Wei, “Photon-limited depth and reflectivity imaging with sparsity regularization,” Opt. Commun. 392, 25–30 (2017).
[Crossref]

Yamamoto, H.

Yamamoto, M.

Yamazaki, Y.

Yan, K.

K. Yan, L. Lifei, D. Xuejie, Z. Tongyi, L. Dongjian, and Z. Wei, “Photon-limited depth and reflectivity imaging with sparsity regularization,” Opt. Commun. 392, 25–30 (2017).
[Crossref]

X. Meng, H. Huang, K. Yan, X. Tian, W. Yu, H. Cui, Y. Kong, L. Xue, C. Liu, and S. Wang, “Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method,” Lab Chip 17(1), 104–109 (2017).
[Crossref]

Yeom, S.

Yu, W.

X. Meng, H. Huang, K. Yan, X. Tian, W. Yu, H. Cui, Y. Kong, L. Xue, C. Liu, and S. Wang, “Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method,” Lab Chip 17(1), 104–109 (2017).
[Crossref]

Yu, Y.

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955).
[Crossref]

Zhang, H.

Zhou, W.-J.

Zou, Y.

Y. Chen, Y. Zou, P. Huang, J. Qi, W. He, and Q. Chen, “Reconstruction algorithm of low-light integral imaging by electron-multiplying charge-coupled device,” Opt. Eng. 58(5), 1 (2019).
[Crossref]

Zuo, C.

C. Zuo, Q. Chen, L. Tian, L. Waller, and A. Asundi, “Transport of intensity phase retrieval and computational imaging of partially coherent fields: The phase space perspective,” Opt. Lasers Eng. 71, 20–32 (2015).
[Crossref]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Noninterferometric single-shot quantitative phase microscopy,” J. Microsc. 38(18), 3538–3541 (2013).
[Crossref]

Appl. Opt. (5)

eLife (1)

T. J. Gould and J. Bewersdorf, “Nanoscopy at low light intensities shows its potential,” eLife 1, e00475 (2012).
[Crossref]

IEEE Photonics J. (1)

P. Latorre-Carmona, B. Javidi, F. Pla, and E. Tajahuerce, “Photon counting 3-D object recognition using digital holography,” IEEE Photonics J. 5(6), 6900309 (2013).
[Crossref]

IEEE Signal Process. Mag. (1)

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: A contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

J. Micros. (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase amplitude microscopy III. The effect of noise,” J. Micros. 214(1), 51–61 (2004).
[Crossref]

J. Microsc. (1)

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Noninterferometric single-shot quantitative phase microscopy,” J. Microsc. 38(18), 3538–3541 (2013).
[Crossref]

J. Mod. Opt. (1)

S. K. Rajput and N. K. Nishchal, “Optical asymmetric cryptosystem based on photon counting and phase-truncated Fresnel transforms,” J. Mod. Opt. 64(8), 878–886 (2017).
[Crossref]

J. Opt. (1)

S. K. Rajput, D. Kumar, and N. K. Nishchal, “Photon counting imaging and polarized light encoding for secure image verification and hologram watermarking,” J. Opt. 16(12), 125406 (2014).
[Crossref]

J. Opt. Soc. Am. (1)

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

Lab Chip (1)

X. Meng, H. Huang, K. Yan, X. Tian, W. Yu, H. Cui, Y. Kong, L. Xue, C. Liu, and S. Wang, “Smartphone based hand-held quantitative phase microscope using the transport of intensity equation method,” Lab Chip 17(1), 104–109 (2017).
[Crossref]

Methods Cell Biol. (1)

I. Rasnik, T. French, K. Jacobson, and K. Berland, “Electronic cameras for low, light microscopy,” Methods Cell Biol. 114, 211–241 (2013).
[Crossref]

Neuron (1)

I. Testa, N. T. Urban, S. Jakobs, C. Eggling, K. L. Willig, and S. W. Hell, “Nanoscopy of living brain slices with low light levels,” Neuron 75(6), 992–1000 (2012).
[Crossref]

Opt. Commun. (3)

K. Yan, L. Lifei, D. Xuejie, Z. Tongyi, L. Dongjian, and Z. Wei, “Photon-limited depth and reflectivity imaging with sparsity regularization,” Opt. Commun. 392, 25–30 (2017).
[Crossref]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

T. E. Gureyevand K and A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[Crossref]

Opt. Eng. (3)

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using digital holography,” Opt. Eng. 43(12), 2959–2966 (2004).
[Crossref]

Y. Chen, Y. Zou, P. Huang, J. Qi, W. He, and Q. Chen, “Reconstruction algorithm of low-light integral imaging by electron-multiplying charge-coupled device,” Opt. Eng. 58(5), 1 (2019).
[Crossref]

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21(5), 829–832 (1982).
[Crossref]

Opt. Express (3)

Opt. Lasers Eng. (1)

C. Zuo, Q. Chen, L. Tian, L. Waller, and A. Asundi, “Transport of intensity phase retrieval and computational imaging of partially coherent fields: The phase space perspective,” Opt. Lasers Eng. 71, 20–32 (2015).
[Crossref]

Opt. Lett. (4)

Optik (1)

D. Kumar and N. K. Nishchal, “Synthesis and reconstruction of multi-plane phase-only Fresnel holograms,” Optik 127(24), 12069–12077 (2016).
[Crossref]

Phys. Rev. Lett. (2)

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” Phys. Rev. Lett. 121(24), 243902 (2018).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

Proc. SPIE (2)

G. Pedrini, A. Faridian, A. K. Singh, and W. Osten, “Phase retrieval for optical metrology,” Proc. SPIE 9276, 927602 (2014).
[Crossref]

D. P. Kelly, T. Meinecke, N. Sabitov, S. Sinzinger, and J. T. Sheridan, “Digital holography and phase retrieval: a theoretical investigation,” Proc. SPIE 8074, 807401 (2011).
[Crossref]

Science (1)

F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955).
[Crossref]

Other (1)

S. W. Hasinoff, F. Durand, and W. T. Freeman, “Noise optimal capture for dynamic range photography,” Proc. IEEE Computer Society Confer. on Computer Vision and Pattern Recognition, San Francisco, USA, 553–560 (2010).

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Figures (6)

Fig. 1.
Fig. 1. Focussed intensity image of micro-lens (a) without photon-limiting, (b) photon-limited with Np = 105, (c) photon-limited with Np = 104, (d), (e), and (f) show their retrieved phase, respectively.
Fig. 2.
Fig. 2. Focussed intensity image of micro-lens array (a) without photon-limiting, (b) photon-limited with Np = 105, (c) photon-limited with Np = 104,(d), (e), and (f) show their retrieved phase, respectively.
Fig. 3.
Fig. 3. (Color online) Schematic diagram of optical setup(L1, condenser lens; NDF, variable neutral density filter; O, phase object; MO, microscopic objective; CL, collimating lens; IP, image plane; L2-L3, 4f imaging relay system; CCD, camera).
Fig. 4.
Fig. 4. Micro-lens: (a), (b), and (c) focussed images recorded on camera in varying photon-limiting conditions, viz., no photon limiting, SNR 7.2 and SNR 6.7, respectively; (d), (e), and (f) retrieved phase of (a), (b), and (c), respectively.
Fig. 5.
Fig. 5. Micro-lens array: (a), (b), and (c) focussed images recorded on camera in varying photon-limiting conditions, viz., no photon limiting, SNR 7.2 and SNR 6.7, respectively; (d), (e), and (f) retrieved phase of (a), (b), and (c), respectively.
Fig. 6.
Fig. 6. Onion’s peel: (a), (b), and (c) focussed images recorded on camera in varying photon-limiting conditions, viz., no photon limiting, SNR 7.2 and SNR 6.7, respectively; (d), (e), and (f) retrieved phase of (a), (b), and (c), respectively.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

[ I z ( x , y ) ϕ z ( x , y ) ] = 2 π λ I z
2 π λ I z = I z ( x , y ) ϕ z ( x , y , 0 ) + I z ( x , y ) 2 ϕ z ( x , y , 0 ) I 0 2 ϕ z ( x , y , 0 )
2 ϕ 0 ( x , y ) = 2 π λ I 0 I ( x , y ) z
4 π 2 ( k x 2 + k y 2 ) Φ 0 ( k x , k y ) = 2 π λ I 0 [ I ( x , y ) z ]
ϕ 0 ( x , y ) = 2 π λ I 0 1 [ 1 4 π 2 ( k x 2 + k y 2 ) [ I ( x , y ) z ] ]
I z I d z ( x , y ) I d z ( x , y ) 2 d z
P ( n ; r , t ) = [ a ( r ) t ] n exp [ a ( r ) t ] n ! , n = 1 ,   2 ,   3 ,  
n p ( r i ) = N p I p ( r i ) j = 1 N T I p ( r j )
I d z ( x , y ) = I d z i d e a l ( x , y ) + σ d z ( x , y ) I d z ( x , y ) = I d z i d e a l ( x , y ) + σ d z ( x , y )
I d z ( x , y ) I d z ( x , y ) = [ I d z i d e a l ( x , y ) I d z i d e a l ( x , y ) ] + [ σ d z ( x , y ) σ d z ( x , y ) ]
I d z ( x , y ) I d z ( x , y ) 2 d z = I i d e a l ( x , y , 0 ) d z + d z 2 3 ! 3 I i d e a l ( x , y , 0 ) d z 3 + σ ( x , y ) 2 d z
2 π λ I 0 σ 2 d z 2 ϕ z ( x , y ) R M S
k 6 I 0 3 I i d e a l ( x , y , 0 ) z 3 R M S d z 2 2 ϕ z ( x , y ) R M S
S N R = 20 log | | A i d e a l | | F | | A i d e a l A n o i s y | | F