Abstract

In this work, an optimum frequency combination (OFC) method is proposed to reconstruct high quality phase information of the complex light field, which is really valuable for many objects such as optical elements and cells. It is shown that the difference image between two symmetrical separated, larger defocused planes contains a lot of lower frequency components of the phase distribution and the higher frequency components can be easily observed in the difference image between two nearly focused planes. Based on the phase transfer function (PTF), our method combines different frequency components with high Signal-to-Noise Ratio (SNR) together to estimate a more accurate frequency spectrum of the object’s phase distribution without any complicated linear or nonlinear regression. Then, we can directly reconstruct a high-quality phase map through inverse Fourier transform. What’s more, in order to compensate the phase discrepancy resulted from strong absorption in the intensity, an iterative compensation algorithm is proposed. Both the simulation and experimental results demonstrate that our iterative OFC (IOFC) method can give a computationally efficient and noise-robust phase reconstruction for absorptive phase objects with higher accuracy and fewer defocus planes.

© 2015 Optical Society of America

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References

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2015 (1)

2014 (5)

2013 (4)

2012 (3)

2011 (1)

2010 (6)

2009 (1)

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]

2007 (2)

2005 (4)

2004 (4)

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

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

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

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

2002 (1)

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

2001 (1)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[Crossref]

2000 (2)

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[Crossref] [PubMed]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref] [PubMed]

1999 (1)

1998 (3)

1984 (1)

1983 (1)

1978 (1)

1971 (1)

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

1969 (1)

R. K. Crane, “Interference phase measurement,” Appl. Opt. 8, 538 (1969).

Acosta, E.

Allen, L. J.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[Crossref]

Allman, B.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54, 191–197 (2005).
[Crossref]

Allman, B. E.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref] [PubMed]

Anastasio, M. A.

Arif, M.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref] [PubMed]

Asundi, A.

Bai, X.

Bajt, S.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[Crossref] [PubMed]

Barbastathis, G.

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

Barty, A.

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

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

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[Crossref] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[Crossref]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[Crossref]

Baruchel, J.

S. Zabler, P. Cloetens, J. P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[Crossref]

Beleggia, M.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

Bie, R.

Blanchard, P. M.

Boistel, R.

Bunk, O.

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

Carney, P. S.

Carranza, J. M.

Chen, Q.

Claus, R. A.

Cloetens, P.

Colomb, T.

Crane, R. K.

R. K. Crane, “Interference phase measurement,” Appl. Opt. 8, 538 (1969).

Cuche, E.

Cui, L.

Dauwels, J.

David, C.

Depeursinge, C.

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]

Emery, Y.

Falaggis, K.

Faulkner, H. M.

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

Fienup, J. R.

Gaylord, T. K.

Gerchberg, R.

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

Gorthi, S. S.

Greenaway, A. H.

Guigay, J. P.

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32, 1617–1619 (2007).
[Crossref] [PubMed]

S. Zabler, P. Cloetens, J. P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[Crossref]

Horstmeyer, R.

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

Hu, Y.

Huang, L.

Ishizuka, K.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54, 191–197 (2005).
[Crossref]

Jacobson, D. L.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref] [PubMed]

Jenkins, M. H.

Kou, S. S.

Kozacki, T.

Kujawinska, M.

Kwon, O. Y.

Langer, M.

Long, J. M.

Magistretti, P. J.

Marquet, P.

McCartney, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[Crossref] [PubMed]

McMahon, P. J.

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

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

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref] [PubMed]

Menzel, A.

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

Nugent, K. A.

K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59, 1–99 (2010).
[Crossref]

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

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

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[Crossref] [PubMed]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[Crossref]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[Crossref]

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

Oxley, M. P.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[Crossref]

Paganin, D.

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

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

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[Crossref] [PubMed]

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
[Crossref] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[Crossref]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[Crossref]

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

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]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

Popescu, G.

G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw Hill Professional, 2011).

Qu, W.

Rappaz, B.

Roberts, A.

Rodenburg, J. M.

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

Saxton, W.

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

Schlenker, M.

S. Zabler, P. Cloetens, J. P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[Crossref]

Schofield, M. A.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

Schonbrun, E.

Schoonover, R. W.

Sheppard, C. J.

Sheppard, C. J. R.

Soto, M.

Stampanoni, M.

Sun, J.

Teague, M. R.

Thibault, P.

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.

Volkov, V. V.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref] [PubMed]

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S. Zabler, P. Cloetens, J. P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073705 (2005).
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Zhang, J.

Zhang, L.

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Zhong, J.

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Adv. Phys. (1)

K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59, 1–99 (2010).
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J. Opt. Soc. Am. (1)

Nat. Photonics (1)

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

Nature (1)

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408, 158–159 (2000).
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Opt. Commun. (1)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
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Opt. Express (12)

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
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L. Waller, S. S. Kou, C. J. Sheppard, and G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express 18, 22817–22825 (2010).
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C. Zuo, Q. Chen, W. Qu, and A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express 21, 24060–24075 (2013).
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C. Zuo, J. Sun, J. Zhang, Y. Hu, and Q. Chen, “Lensless phase microscopy and diffraction tomography with multi-angle and multi-wavelength illuminations using a LED matrix,” Opt. Express 23, 14314–14328 (2015).
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L. Waller, S. S. Kou, C. J. R. Sheppard, and G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express 18, 22817–22825 (2010).
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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, 12552–12561 (2010).
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R. Bie, X. H. Yuan, M. Zhao, and L. Zhang, “Method for estimating the axial intensity derivative in the TIE with higher order intensity derivatives and noise suppression,” Opt. Express 20, 8186–8191 (2012).
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C. Zuo, Q. Chen, Y. Yu, and A. Asundi, “Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter-theory and applications,” Opt. Express 21, 5346–5362 (2013).
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B. Xue, S. Zheng, L. Cui, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes,” Opt. Express 19, 20244–20250 (2011).
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S. Zheng, B. Xue, W. Xue, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express 20, 972–985 (2012).
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J. Zhong, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of Intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22, 10661–10674 (2014).
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C. Zuo, Q. Chen, L. Huang, and A. Asundi, “Phase discrepancy analysis and compensation for fast Fourier transform based solution of the transport of intensity equation,” Opt. Express 22, 17172–17186 (2014).
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Optik (1)

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Phys. Rev. Lett. (1)

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

Rev. Sci. Instrum. (1)

S. Zabler, P. Cloetens, J. P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[Crossref]

Ultramicroscopy (3)

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[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]

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[Crossref] [PubMed]

Other (1)

G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw Hill Professional, 2011).

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Figures (13)

Fig. 1
Fig. 1

Explanation for phase transfer function. (a) the one-dimension curve of G(u,v,z); (b) the 2D representation of |G(u,v,z)|; (c) presents the frequency spectrum of I ^ ( u , v , z ) I ^ ( u , v , z ) without noise; (d) shows the frequency spectrum of I ^ ( u , v , z ) I ^ ( u , v , z ) with additive white Gaussian noise.

Fig. 2
Fig. 2

Explanation for phase transfer function. (a) the one-dimension curve of OFCG(u,v); (b) the 2D representation of OFC phase transfer function OFCG(u,v); (c) presents the frequency spectrum of O F C I ^ ( u , v ) without noise; (d) shows the frequency spectrum of O F C I ^ ( u , v ) with additive white Gaussian noise.

Fig. 3
Fig. 3

Comparison of reconstructed phase error in different measurement situations. (a) - (d) respectively show true intensity distribution, true phase distribution, reconstructed phase distribution, and phase error map when the phase is small and the intensity image in focus is almost a constant; (e) - (h) respectively show true intensity distribution, true phase distribution, reconstructed phase distribution, and phase error map when the phase is large and the intensity image in focus is spatial varying, by using OFC method.

Fig. 4
Fig. 4

Block diagram of the IOFC-TIE method.

Fig. 5
Fig. 5

Comparison of RMSE in phase distributions for different measurement strategies with noise level increasing. Using exponential spacing scheme, OFC-TIE achieves the best performance with no more than 15 sampling planes.

Fig. 6
Fig. 6

Comparison of phase quality among simulation results utilizing different measurement strategies. OFC-TIE acquires the best phase quality with fewer defocus planes. (a) is the true phase distribution; (b) is the true intensity distribution; (c) is the reconstructed phase result by using OFS-TIE with Equally Spaced Stack 1; (d) is the reconstructed phase result by using OFS-TIE with Equally Spaced Stack 2; (e) is the reconstructed phase result by using OFS-TIE with Equally Spaced Stack 3; (f) is the phase result by using GP-TIE method, when Exponentially Spaced Stack is used; (g) is the phase result by using OFC-TIE method, when Exponentially Spaced Stack is used.

Fig. 7
Fig. 7

Comparison of phase quality among simulation results utilizing different measurement strategies for absorptive phase objects. OFS-TIE with 129 measurement images acquires the best phase quality. (a) is the true phase distribution; (b) is the true intensity distribution; (c), (d), (e) are the reconstructed phase result by using OFS-TIE with different Equally Spaced Stacks; (f) is the phase result by using GP-TIE method, when Exponentially Spaced Stack is used; (g) is the phase result by using OFC-TIE method without iterative compensation algorithm.

Fig. 8
Fig. 8

Comparison of IC-TIE algorithm and IOFC-TIE algorithm for their compensation efficiencies. After five iterations of compensation, IOFC-TIE algorithm acquires the better phase quality than IC-TIE algorithm does. (a) is the RMSE in reconstructed phase distributions for two different compensation methods with iteration number increasing; (b) is the phase result by using GP-TIE before compensation (RMSE 0.0490); (c) is the compensated phase result by using IC-TIE algorithm (RMSE 0.0175); (d) is the phase result by using OFC-TIE before compensation (RMSE 0.0358); (e) is the compensated phase result by using IOFC-TIE algorithm (RMSE 0.0099).

Fig. 9
Fig. 9

(a) The experimental test setup; (b) the DIC image of the tested phase sample.

Fig. 10
Fig. 10

The left column shows the less defocused image (left up), further defocused image (left down), and the difference image (right) with defocus values are (a) 40µm, (c) 120µm, (e) 600µm, (g) 1040µm, and (i)1280µm. The corresponding theoretical PTF and TIE PTFs are shown in the right column.

Fig. 11
Fig. 11

Comparison of phase quality among experimental results utilizing different measurement strategies. The phase object under test is a geometry pattern etched on PMMA substrate. OFC-TIE acquires the best phase quality with fewer defocus planes. (a) Equally Spaced Stack 1, each image has 256 × 256 pixels with effective size 4.65µm×4.65µm.; (b) is the phase result recovered by using OFS-TIE method, when all 129 equally spaced images are used; (c) is the phase result reconstructed by using Equally Spaced Stack 2, [−160µm to 160µm,dz = 20µm]; (d) is the phase result reconstructed by using Equally Spaced Stack 3, [−1280µm to 1280µm,dz = 160µm]; (e) is the phase recovered with exponentially spaced z steps, by using GP-TIE algorithm; (f) is the phase recovered with OFC-TIE algorithm by using Exponentially Spaced Stack.

Fig. 12
Fig. 12

Quantitative phase comparison of different methods. (a) The 3D topography characterization of the test sample by a confocal microscope shown in inverse height. (b) Phase profile obtained along the line. (e) Comparison of line profiles between OFS-TIE, GP-TIE, and OFC-TIE method with different data capturing strategies.

Fig. 13
Fig. 13

Comparison of phase quality among experimental results utilizing different measurement strategies for measuring SMCC−7721 human hepatocellular carcinoma cells. IOFC acquires the best phase quality with fewer defocus planes. (a) Equally Spaced Stack 1, each image has 616 × 616 pixels with effective size 0.34µm × 0.34µm.; (b) is the phase result by using OFS method, when all 129 equally spaced images are used; (c) is the phase result by using Equally Spaced Stack 2, [−8µm to 8µm,dz = 1µm]; (d) is the phase result by using Equally Spaced Stack 3, [−64µm to 64µm,dz = 8µm]; (e) is the phase recovered with exponentially spaced z steps, by using GP-TIE algorithm; (f) is the phase recovered with IOFC-TIE algorithm by using Exponentially Spaced Stack.

Equations (17)

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I ^ ( u , v , z ) = I 0 { δ ( u , v ) 2 cos [ π λ z ( u 2 + v 2 ) ] U ( u , v ) + 2 sin [ π λ z ( u 2 + v 2 ) ] Φ ( u , v ) } ,
I ^ ( u , v , z ) I ^ ( u , v , z ) 4 I 0 = sin [ π λ z ( u 2 + v 2 ) ] Φ ( u , v ) .
G ( u , v , z ) = sin [ π λ z ( u 2 + v 2 ) ] .
z n = β z n 1 .
O F C G ( u , v ) = k = 1 n { G ( u , v , z k ) } ,
O F C I ^ ( u , v ) = k = 1 n { I ^ ( u , v , z k ) I ^ ( u , v , z k ) } 4 I 0 .
Φ ( u , v ) = O F C I ^ ( u , v ) O F C G ( u , v ) .
w 0 = A e i φ = A cos ( φ ) + i A sin ( φ ) = A cos ( φ ) + i A cos ( φ ) tan ( φ ) = A + i A φ ,
W 0 = { w 0 } = { A } + i { A ϕ } ,
W z = W 0 e i k z 1 λ 2 ( u 2 + v 2 ) = W 0 e i k z e i k z ( 1 1 λ 2 ( u 2 + v 2 ) ) = W 0 e i k z e i ω = e i k z { [ { A } cos ( ω ) + { A φ } sin ( ω ) ] i [ { A } sin ( ω ) { A φ } cos ( ω ) ] } ,
I z = w z w z * = 1 { W z } 1 { W z } * = [ 1 { { A } cos ( ω ) + { A φ } sin ( ω ) } ] 2 + [ 1 { { A } sin ( ω ) { A φ } cos ( ω ) } ] 2 ,
I z I z = 4 [ 1 { { A } cos ( ω ) } 1 { { A φ } sin ( ω ) } 1 { { A } sin ( ω ) } 1 { { A φ } cos ( ω ) } ] .
I z I z 4 = ( A m ) 2 1 { sin ( ω ) Φ } + R ,
R = A m 1 { { A r φ } sin ( ω ) } + 1 { { A r } cos ( ω ) } 1 { { A φ } sin ( ω ) } 1 { { A r } sin ( ω ) } 1 { { A φ } cos ( ω ) } ,
I ^ z I ^ z 4 { R } 4 ( A m ) 2 = G ( u , v , z ) Φ .
O F C I ^ 4 { R } 4 ( A m ) 2 = O F C G Φ .
φ = arctan ( 1 { O F C I ^ 4 { R } 4 ( A m ) 2 O F C G } ) .

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