Abstract

We demonstrate 3D phase and absorption recovery from partially coherent intensity images captured with a programmable LED array source. Images are captured through-focus with four different illumination patterns. Using first Born and weak object approximations (WOA), a linear 3D differential phase contrast (DPC) model is derived. The partially coherent transfer functions relate the sample’s complex refractive index distribution to intensity measurements at varying defocus. Volumetric reconstruction is achieved by a global FFT-based method, without an intermediate 2D phase retrieval step. Because the illumination is spatially partially coherent, the transverse resolution of the reconstructed field achieves twice the NA of coherent systems and improved axial resolution.

© 2016 Optical Society of America

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References

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2016 (2)

2015 (7)

2014 (7)

2013 (5)

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nature Photon. 7, 113–117 (2013).
[Crossref]

S. Jones, M. King, and A. Ward, “Determining the unique refractive index properties of solid polystyrene aerosol using broadband Mie scattering from optically trapped beads,” Phys. Chem. 15, 20735–20741 (2013).

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

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

L. Tian, J. C. Petruccelli, Q. Miao, H. Kudrolli, V. Nagarkar, and G. Barbastathis, “Compressive X-ray phase tomography based on the transport of intensity equation,” Opt. Lett. 38, 3418–3421 (2013).
[Crossref] [PubMed]

2012 (2)

T. L. Jensen, J. H. Joergensen, P. C. Hansen, and S. H. Jensen, “Implementation of an Optimal First-Order Method for Strongly Convex Total Variation Regularization,” BIT Numer. Math. 52, 329–356 (2012). http://www.imm.dtu.dk/~pcha/TVReg/
[Crossref]

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nat. Methods 9, 1195–1197 (2012).
[Crossref] [PubMed]

2011 (4)

2009 (3)

2007 (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

2005 (1)

2004 (2)

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[Crossref] [PubMed]

C. J. R. Sheppard, “Defocused transfer function for a partially coherent microscope and application to phase retrieval,” J. Opt. Soc. Am. A 21, 828–831 (2004).
[Crossref]

2002 (1)

1999 (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

1998 (1)

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

1985 (1)

1984 (1)

D. Hamilton and C. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microsc. 133, 27–39 (1984).
[Crossref]

1977 (1)

J. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

1942 (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[Crossref]

Alieva, T.

Anastasio, M. A.

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[Crossref] [PubMed]

Babacan, S. D.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nature Photon. 8, 256–263 (2014).
[Crossref]

Badizadegan, K.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266–277 (2009).
[Crossref] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Barbastathis, G.

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999).
[Crossref]

Boss, D.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nature Photon. 7, 113–117 (2013).
[Crossref]

Brady, D.

Bronnikov, A.

Carney, P. S.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nature Photon. 8, 256–263 (2014).
[Crossref]

Chen, M.

Choi, K.

Choi, W.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266–277 (2009).
[Crossref] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Chu, K. K.

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nat. Methods 9, 1195–1197 (2012).
[Crossref] [PubMed]

Claus, R. A.

Cloetens, P.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

Cogswell, C. J.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[Crossref] [PubMed]

Cotte, Y.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nature Photon. 7, 113–117 (2013).
[Crossref]

Dasari, R. R.

Dauwels, J.

Depeursinge, C.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nature Photon. 7, 113–117 (2013).
[Crossref]

Ding, H.

Dyck, D. V.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

Edwards, C.

Fang-Yen, C.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266–277 (2009).
[Crossref] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Feld, M. S.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266–277 (2009).
[Crossref] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Ford, T. N.

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nat. Methods 9, 1195–1197 (2012).
[Crossref] [PubMed]

Gaylord, T. K.

Gbur, G.

Gillette, M. U.

Goddard, L. L.

T. H. Nguyen, C. Edwards, L. L. Goddard, and G. Popescu, “Quantitative phase imaging of weakly scattering objects using partially coherent illumination,” Opt. Express 24, 11683–11693 (2016).
[Crossref] [PubMed]

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nature Photon. 8, 256–263 (2014).
[Crossref]

Goy, A.

Guigay, J.

J. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Guigay, J. P.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

Gureyev, T. E.

Hamilton, D.

D. Hamilton and C. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microsc. 133, 27–39 (1984).
[Crossref]

Hansen, P. C.

T. L. Jensen, J. H. Joergensen, P. C. Hansen, and S. H. Jensen, “Implementation of an Optimal First-Order Method for Strongly Convex Total Variation Regularization,” BIT Numer. Math. 52, 329–356 (2012). http://www.imm.dtu.dk/~pcha/TVReg/
[Crossref]

Horisaki, R.

Horstmeyer, R.

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

Huang, Y.

Jenkins, M. H.

Jensen, S. H.

T. L. Jensen, J. H. Joergensen, P. C. Hansen, and S. H. Jensen, “Implementation of an Optimal First-Order Method for Strongly Convex Total Variation Regularization,” BIT Numer. Math. 52, 329–356 (2012). http://www.imm.dtu.dk/~pcha/TVReg/
[Crossref]

Jensen, T. L.

T. L. Jensen, J. H. Joergensen, P. C. Hansen, and S. H. Jensen, “Implementation of an Optimal First-Order Method for Strongly Convex Total Variation Regularization,” BIT Numer. Math. 52, 329–356 (2012). http://www.imm.dtu.dk/~pcha/TVReg/
[Crossref]

Jingshan, Z.

Joergensen, J. H.

T. L. Jensen, J. H. Joergensen, P. C. Hansen, and S. H. Jensen, “Implementation of an Optimal First-Order Method for Strongly Convex Total Variation Regularization,” BIT Numer. Math. 52, 329–356 (2012). http://www.imm.dtu.dk/~pcha/TVReg/
[Crossref]

Jones, S.

S. Jones, M. King, and A. Ward, “Determining the unique refractive index properties of solid polystyrene aerosol using broadband Mie scattering from optically trapped beads,” Phys. Chem. 15, 20735–20741 (2013).

Jourdain, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nature Photon. 7, 113–117 (2013).
[Crossref]

Kamilov, U. S.

Kim, T.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nature Photon. 8, 256–263 (2014).
[Crossref]

King, M.

S. Jones, M. King, and A. Ward, “Determining the unique refractive index properties of solid polystyrene aerosol using broadband Mie scattering from optically trapped beads,” Phys. Chem. 15, 20735–20741 (2013).

Kolner, C.

Kudrolli, H.

Landuyt, J. V.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

Lang, W.

W. Lang, Nomarski Differential Interference-Contrast Microscopy (Oberkochen, Carl Zeiss, 1982).

Larkin, K. G.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214, 7–12 (2004).
[Crossref] [PubMed]

Li, X.

Liu, S.

Z. Liu, L. Tian, S. Liu, and L. Waller, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19, 106002 (2014).
[Crossref] [PubMed]

Liu, Z.

L. Tian, Z. Liu, L.-H. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
[Crossref]

Z. Liu, L. Tian, S. Liu, and L. Waller, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19, 106002 (2014).
[Crossref] [PubMed]

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
[Crossref]

Lue, N.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[Crossref] [PubMed]

Magistretti, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nature Photon. 7, 113–117 (2013).
[Crossref]

Marks, D.

Marquet, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nature Photon. 7, 113–117 (2013).
[Crossref]

Mehta, S. B.

Mertz, J.

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nat. Methods 9, 1195–1197 (2012).
[Crossref] [PubMed]

Miao, Q.

Millet, L.

Mir, M.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nature Photon. 8, 256–263 (2014).
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Appl. Opt. (1)

Appl. Phys. Lett. (1)

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912–2914 (1999).
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Biomed. Opt. Express (1)

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D. Hamilton and C. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microsc. 133, 27–39 (1984).
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J. Opt. Soc. Am. A (7)

Nat. Methods (2)

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G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nature Photon. 7, 739–745 (2013).
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Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nature Photon. 7, 113–117 (2013).
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Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
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Opt. Express (11)

J. Zhong, L. Tian, J. Dauwels, and L. Waller, “Partially coherent phase imaging with simultaneous source recovery,” Opt. Express 6, 257–265 (2015).
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D. Brady, K. Choi, D. Marks, and R. Horisaki, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
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[Crossref] [PubMed]

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266–277 (2009).
[Crossref] [PubMed]

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[Crossref] [PubMed]

J. A. Rodrigo and T. Alieva, “Rapid quantitative phase imaging for partially coherent light microscopy,” Opt. Express 22, 13472–13483 (2014).
[Crossref] [PubMed]

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

X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
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T. H. Nguyen, C. Edwards, L. L. Goddard, and G. Popescu, “Quantitative phase imaging of weakly scattering objects using partially coherent illumination,” Opt. Express 24, 11683–11693 (2016).
[Crossref] [PubMed]

L. Tian and L. Waller, “Quantitative differential phase contrast imaging in an LED array microscope,” Opt. Express 23, 11394–11403 (2015).
[Crossref] [PubMed]

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Opt. Lett. (4)

Optica (3)

Optik (1)

J. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121–125 (1977).

Phys. Chem. (1)

S. Jones, M. King, and A. Ward, “Determining the unique refractive index properties of solid polystyrene aerosol using broadband Mie scattering from optically trapped beads,” Phys. Chem. 15, 20735–20741 (2013).

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]

Physica (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).
[Crossref]

Other (3)

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[Crossref]

Supplementary Material (1)

NameDescription
» Visualization 1: AVI (741 KB)      Through focus of Embryos

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Figures (6)

Fig. 1
Fig. 1

3D Differential Phase Contrast (DPC) microscopy. The setup is a microscope equipped with LED array illumination and an axial motion stage. Through-focus intensity stacks are captured using 4 source patterns (top, bottom, right, and left half-circles). The intensity data is related to the 3D refractive index distribution by illumination-dependent transfer functions, according to the Born approximation. A deconvolution algorithm then recovers the 3D complex refractive index.

Fig. 2
Fig. 2

(a) Absorption (HIm) and phase (HRe) 3D transfer functions for the brightfield and DPC stacks. The NA of both illumination and detection is 0.65. (b) Lateral and axial resolution (measured by Fourier bandwidth normalized to units of NA) improve as the NA increases. Gray circles indicate the case shown in (a).

Fig. 3
Fig. 3

3D refractive index reconstructions of a 10µm diameter polystyrene bead immersed in oil of refractive index 1.58 (40× 0.65 NA objective). (a) Cross-sectional views of 3D brightfield and DPC measurements. (b) Absorption and (c) phase reconstructions with 2 and TV regularization. (d) 1D cross-sections of recovered refractive index.

Fig. 4
Fig. 4

Comparison of recovered 3D refractive index (6 μm polystyrene beads) with 20 × 0.4 NA and 40 × 0.65 NA objectives. Larger NA provides better lateral and axial resolution.

Fig. 5
Fig. 5

Comparison between 2D phase reconstructions (TIE and 2D DPC) and 2D slices of our 3D DPC refractive index reconstruction for a human mammary epithelial MCF10A cell. (a) TIE and 2D DPC, next to three slices of the recovered refractive index from 3D DPC at three different depths. (b) 3D view of the recovered index’s 3D Fourier spectrum. (c) 3D rendering of the recovered refractive index distribution.

Fig. 6
Fig. 6

Reconstructed 3D refractive index of embryo cells. (Left) Full field-of-view at a single focus plane. (Right) 3 axial slices for regions in the blue dashed and orange boxes.

Equations (12)

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U ( r ) = U I ( r ) + U ( r ) V ( r ) G ( r r ) d 3 r ,
U ( r ) U I ( r ) + U I ( r ) V ( r ) G ( r r ) d 3 r .
I img ( x , z ) = S ( u ) | [ e i 2 π ( u x + η z ) + ( e i 2 π ( u x + η z ) V ( x , z ) ) G ( x , z ) ] h ( x ) | 2 d 2 u ,
I ˜ img ( u , η ) = S ( u ) [ P ( u ) ( δ ( u + u , η + η ) + V ˜ ( u + u , η + η ) G ˜ ( u , η ) ) ] [ P * ( u ) ( δ ( u u , η η ) + V ˜ * ( u u , η η ) G ˜ * ( u , η ) ) ] d 2 u ,
I ˜ img ( u , η ) = I ˜ o + I ˜ s s + S ( u ) [ P ( u ) V ˜ * ( u , η ) G ˜ * ( u u , η η ) P * ( u u ) + P * ( u ) V ˜ ( u , η ) G ˜ ( u + u , η + η ) P ( u + u ) ] d 2 u ,
I ˜ img = I ˜ o + H Re V ˜ Re + H Im V ˜ Im ,
H Re ( u , η ) = S ( u ) [ P * ( u ) G ˜ ( u + u , η + η ) P ( u + u ) + P ( u ) G ˜ * ( u u , η η ) P * ( u u ) ] d 2 u
H Im ( u , η ) = i S ( u ) [ P * ( u ) G ˜ ( u + u , η + η ) P ( u + u ) P ( u ) G ˜ * ( u u , η η ) P * ( u u ) ] d 2 u .
min V ˜ Re , V ˜ Im l I ˜ l H Re , l V ˜ Re H Im , l V ˜ Im 2 2 + α V ˜ Re 2 2 + β V ˜ Im 2 2 ,
I BF = I t o p + I b o t t o m + I r i g h t + I l e f t 2 , I BF = I BF | I ˜ o | | I ˜ o | , I DPC = I t o p / r i g h t I b o t t o m / l e f t | I ˜ o | .
V Re = F 1 { ( l | H Im , l | 2 + β ) l H Re , l * I ˜ l l H Re , l * H Im , l l H Im , l I ˜ ( l | H Re , l | 2 + α ) ( l | H Im , l | 2 + β ) l H Re , l H Im , l * l H Re , l * H Im , l } V Im = F 1 { ( l | H Re , l | 2 + α ) l H Im , l * I ˜ l l H Re , l H Im , l * l H Re , l * I ˜ ( l | H Re , l | 2 + α ) ( l | H Im , l | 2 + β ) l H Re , l H Im , l * l H Re , l * H Im , l } ,
min V ˜ Re , V ˜ Im l I ˜ l H Re , l V ˜ Re H Im , l V ˜ Im 2 2 + γ Re V Re TV + γ Im V Im TV , s . t . V Re , V Im 0 ,

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