Abstract

We propose an accurate and computationally efficient 3D scattering model, multi-layer Born (MLB), and use it to recover the 3D refractive index (RI) of thick biological samples. For inverse problems recovering the complex field of thick samples, weak scattering models (e.g., first Born) may fail or underestimate the RI, especially with a large index contrast. Multi-slice (MS) beam propagation methods model multiple scattering to provide more realistic reconstructions; however, MS does not properly account for highly oblique scattering, nor does it model backward scattering. Our proposed MLB model uses a first Born model at each of many slices, accurately capturing the oblique scattering effects and estimating the backward scattering process. When used in conjunction with an inverse solver, the model provides more accurate RI reconstructions for high-resolution phase tomography. Importantly, MLB retains a reasonable computation time that is critical for practical implementation with iterative inverse algorithms.

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

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Corrections

27 May 2020: A typographical correction was made to Eq. (6).

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2019 (2)

2018 (5)

2017 (5)

2016 (4)

2015 (9)

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).
[Crossref]

J. Lim, K. Lee, K. H. Jin, S. Shin, S. Lee, Y. Park, and J. C. Ye, “Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography,” Opt. Express 23, 16933–16948 (2015).
[Crossref]

M. H. Jenkins and T. K. Gaylord, “Three-dimensional quantitative phase imaging via tomographic deconvolution phase microscopy,” Appl. Opt. 54, 9213–9227 (2015).
[Crossref]

L. Tian and L. Waller, “3D intensity and phase imaging from light field measurements in an LED array microscope,” Optica 2, 104–111 (2015).
[Crossref]

U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, “Learning approach to optical tomography,” Optica 2, 517–522 (2015).
[Crossref]

S. Shin, K. Kim, J. Yoon, and Y. Park, “Active illumination using a digital micromirror device for quantitative phase imaging,” Opt. Lett. 40, 5407–5410 (2015).
[Crossref]

M. Habaza, B. Giboa, Y. Roichman, and N. T. Shaked, “Tomographic phase microscopy with 180° rotation of live cells in suspension by holographic optical tweezers,” Opt. Lett. 40, 1881–1884 (2015).
[Crossref]

B. O’Donoghue and E. Candes, “Adaptive restart for accelerated gradient schemes,” Found. Comput. Math. 15, 715–732 (2015).
[Crossref]

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

2014 (3)

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]

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,” Nat. Photonics. 8, 256–263 (2014).
[Crossref]

K. Kim, Z. Yaqoob, K. Lee, J. W. Kang, Y. Choi, P. Hosseini, P. T. C. So, and Y. Park, “Diffraction optical tomography using a quantitative phase imaging unit,” Opt. Lett. 39, 6935–6938 (2014).
[Crossref]

2013 (2)

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

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

2012 (2)

E. Mudry, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inverse Probl. 28, 065007 (2012).
[Crossref]

P. Bon, B. Wattellier, and S. Monneret, “Modeling quantitative phase image formation under tilted illuminations,” Opt. Lett. 37, 1718–1720 (2012).
[Crossref]

2010 (1)

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref]

2009 (5)

2006 (1)

2005 (1)

A. Dubois, K. Belkebir, and M. Saillard, “Retrieval of inhomogeneous targets from experimental frequency diversity data,” Inverse Probl. 21, S65–S79 (2005).
[Crossref]

2002 (1)

1999 (1)

H. Erdogan and J. A. Fessler, “Ordered subsets algorithms for transmission tomography,” Phys. Med. Biol. 44, 2835–2851 (1999).
[Crossref]

1983 (1)

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

1969 (1)

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

Alieva, T.

Allain, M.

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,” Nat. Photonics. 8, 256–263 (2014).
[Crossref]

Badizadegan, K.

Barty, A.

Beck, A.

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18, 2419–2434 (2009).
[Crossref]

Beetz, T.

Belkebir, K.

T. Zhang, C. Godavarthi, P. C. Chaumet, G. Maire, H. Giovannini, A. Talneau, M. Allain, K. Belkebir, and A. Sentenac, “Far-field diffraction microscopy at λ/10 resolution,” Optica 3, 609–612 (2016).
[Crossref]

E. Mudry, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inverse Probl. 28, 065007 (2012).
[Crossref]

P. C. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Probl. 25, 024003 (2009).
[Crossref]

A. Dubois, K. Belkebir, and M. Saillard, “Retrieval of inhomogeneous targets from experimental frequency diversity data,” Inverse Probl. 21, S65–S79 (2005).
[Crossref]

Best-Popescu, C.

Bon, P.

Born, M.

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

Boss, D.

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

Boufounos, P. T.

H.-Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: sparsity-driven image reconstruction under multiple scattering,” IEEE Trans. Comput. Imag. 4, 73–86 (2018).
[Crossref]

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23, 1052–1056 (2016).
[Crossref]

Y. Ma, H. Mansour, D. Liu, P. T. Boufounos, and U. S. Kamilov, “Accelerated image reconstruction for nonlinear diffractive imaging,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2018), p. 6473–6477.

Bunk, O.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref]

Candes, E.

B. O’Donoghue and E. Candes, “Adaptive restart for accelerated gradient schemes,” Found. Comput. Math. 15, 715–732 (2015).
[Crossref]

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,” Nat. Photonics. 8, 256–263 (2014).
[Crossref]

Chapman, H. N.

Chaumet, P. C.

T. Zhang, C. Godavarthi, P. C. Chaumet, G. Maire, H. Giovannini, A. Talneau, M. Allain, K. Belkebir, and A. Sentenac, “Far-field diffraction microscopy at λ/10 resolution,” Optica 3, 609–612 (2016).
[Crossref]

E. Mudry, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inverse Probl. 28, 065007 (2012).
[Crossref]

P. C. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Probl. 25, 024003 (2009).
[Crossref]

Chen, M.

Chen, Q.

Choi, W.

Choi, Y.

Chowdhury, S.

Chung, J.

Cotte, Y.

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

Cui, C.

Dasari, R. R.

Depeursinge, C.

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

Dierolf, M.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref]

Dubois, A.

A. Dubois, K. Belkebir, and M. Saillard, “Retrieval of inhomogeneous targets from experimental frequency diversity data,” Inverse Probl. 21, S65–S79 (2005).
[Crossref]

Eckert, R.

Eldridge, W. J.

Erdogan, H.

H. Erdogan and J. A. Fessler, “Ordered subsets algorithms for transmission tomography,” Phys. Med. Biol. 44, 2835–2851 (1999).
[Crossref]

Fang-Yen, C.

Fanous, M.

Feld, M. S.

Fessler, J. A.

H. Erdogan and J. A. Fessler, “Ordered subsets algorithms for transmission tomography,” Phys. Med. Biol. 44, 2835–2851 (1999).
[Crossref]

Gaylord, T. K.

Gbur, G.

Giboa, B.

Giovannini, H.

Godavarthi, C.

Goddard, L. L.

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,” Nat. Photonics. 8, 256–263 (2014).
[Crossref]

Goy, A.

Habaza, M.

Hau-Riege, S. P.

He, H.

Horstmeyer, R.

R. Horstmeyer, J. Chung, X. Ou, G. Zheng, and C. Yang, “Diffraction tomography with Fourier ptychography,” Optica 3, 827–835 (2016).
[Crossref]

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

Hosseini, P.

Howells, M. R.

Hu, Y.

Izatt, J.

Jacobsen, C.

Jenkins, M. H.

Jin, D.

Jin, K. H.

Jourdain, P.

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

Kamilov, U. S.

W. Tahir, U. S. Kamilov, and L. Tian, “Holographic particle localization under multiple scattering,” Adv. Photon. 1, 1–12 (2019).
[Crossref]

H.-Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: sparsity-driven image reconstruction under multiple scattering,” IEEE Trans. Comput. Imag. 4, 73–86 (2018).
[Crossref]

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23, 1052–1056 (2016).
[Crossref]

U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, “Learning approach to optical tomography,” Optica 2, 517–522 (2015).
[Crossref]

Y. Ma, H. Mansour, D. Liu, P. T. Boufounos, and U. S. Kamilov, “Accelerated image reconstruction for nonlinear diffractive imaging,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2018), p. 6473–6477.

Kandel, M. E.

Kang, J. W.

Kewish, C. M.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref]

Kim, G.

Kim, K.

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,” Nat. Photonics. 8, 256–263 (2014).
[Crossref]

Kou, S. S.

Lee, K.

Lee, S.

Lim, J.

Liu, D.

H.-Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: sparsity-driven image reconstruction under multiple scattering,” IEEE Trans. Comput. Imag. 4, 73–86 (2018).
[Crossref]

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23, 1052–1056 (2016).
[Crossref]

Y. Ma, H. Mansour, D. Liu, P. T. Boufounos, and U. S. Kamilov, “Accelerated image reconstruction for nonlinear diffractive imaging,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2018), p. 6473–6477.

Liu, H.-Y.

H.-Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: sparsity-driven image reconstruction under multiple scattering,” IEEE Trans. Comput. Imag. 4, 73–86 (2018).
[Crossref]

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]

Liu, Z.

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]

Ma, X.

Ma, Y.

Y. Ma, H. Mansour, D. Liu, P. T. Boufounos, and U. S. Kamilov, “Accelerated image reconstruction for nonlinear diffractive imaging,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2018), p. 6473–6477.

Magistretti, P.

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

Maire, G.

Mansour, H.

H.-Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: sparsity-driven image reconstruction under multiple scattering,” IEEE Trans. Comput. Imag. 4, 73–86 (2018).
[Crossref]

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23, 1052–1056 (2016).
[Crossref]

Y. Ma, H. Mansour, D. Liu, P. T. Boufounos, and U. S. Kamilov, “Accelerated image reconstruction for nonlinear diffractive imaging,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2018), p. 6473–6477.

Marchesini, S.

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R. Horstmeyer, J. Chung, X. Ou, G. Zheng, and C. Yang, “Diffraction tomography with Fourier ptychography,” Optica 3, 827–835 (2016).
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G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
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D. Jin, R. Zhou, Z. Yaqoob, and P. T. C. So, “Dynamic spatial filtering using a digital micromirror device for high-speed optical diffraction tomography,” Opt. Express 26, 428–437 (2018).
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Adv. Photon. (1)

W. Tahir, U. S. Kamilov, and L. Tian, “Holographic particle localization under multiple scattering,” Adv. Photon. 1, 1–12 (2019).
[Crossref]

Appl. Opt. (3)

Biomed. Opt. Express (2)

Found. Comput. Math. (1)

B. O’Donoghue and E. Candes, “Adaptive restart for accelerated gradient schemes,” Found. Comput. Math. 15, 715–732 (2015).
[Crossref]

IEEE Signal Process. Lett. (1)

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23, 1052–1056 (2016).
[Crossref]

IEEE Trans. Comput. Imag. (1)

H.-Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: sparsity-driven image reconstruction under multiple scattering,” IEEE Trans. Comput. Imag. 4, 73–86 (2018).
[Crossref]

IEEE Trans. Image Process. (1)

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18, 2419–2434 (2009).
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A. Dubois, K. Belkebir, and M. Saillard, “Retrieval of inhomogeneous targets from experimental frequency diversity data,” Inverse Probl. 21, S65–S79 (2005).
[Crossref]

P. C. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Probl. 25, 024003 (2009).
[Crossref]

E. Mudry, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inverse Probl. 28, 065007 (2012).
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J. Biomed. Opt. (1)

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]

J. Opt. Soc. Am. A (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]

Nat. Photonics. (2)

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

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,” Nat. Photonics. 8, 256–263 (2014).
[Crossref]

Nature (1)

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref]

Opt. Commun. (1)

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

Opt. Express (9)

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]

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

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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J. Lim, K. Lee, K. H. Jin, S. Shin, S. Lee, Y. Park, and J. C. Ye, “Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography,” Opt. Express 23, 16933–16948 (2015).
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T.-A. Pham, E. Soubies, A. Goy, J. Lim, F. Soulez, D. Psaltis, and M. Unser, “Versatile reconstruction framework for diffraction tomography with intensity measurements and multiple scattering,” Opt. Express 26, 2749–2763 (2018).
[Crossref]

J. M. Soto, J. A. Rodrigo, and T. Alieva, “Label-free quantitative 3D tomographic imaging for partially coherent light microscopy,” Opt. Express 25, 15699–15712 (2017).
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E. Soubies, T.-A. Pham, and M. Unser, “Efficient inversion of multiple-scattering model for optical diffraction tomography,” Opt. Express 25, 21786–21800 (2017).
[Crossref]

X. Ma, W. Xiao, and F. Pan, “Optical tomographic reconstruction based on multi-slice wave propagation method,” Opt. Express 25, 22595–22607 (2017).
[Crossref]

D. Jin, R. Zhou, Z. Yaqoob, and P. T. C. So, “Dynamic spatial filtering using a digital micromirror device for high-speed optical diffraction tomography,” Opt. Express 26, 428–437 (2018).
[Crossref]

Opt. Lett. (7)

Optica (6)

Phys. Med. Biol. (1)

H. Erdogan and J. A. Fessler, “Ordered subsets algorithms for transmission tomography,” Phys. Med. Biol. 44, 2835–2851 (1999).
[Crossref]

Sov. Math. Doklady (1)

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

Other (2)

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

Y. Ma, H. Mansour, D. Liu, P. T. Boufounos, and U. S. Kamilov, “Accelerated image reconstruction for nonlinear diffractive imaging,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2018), p. 6473–6477.

Supplementary Material (4)

NameDescription
» Supplement 1       Supplemental document
» Visualization 1       3D refractive index reconstruction of a 3T3 cell under an LED array microscope with a 60X 0.8NA objective lens (Nikon, CFI Plan Achromat).
» Visualization 2       3D refractive index reconstructions of an entire adult hermaphrodite C. elegans using the first Born approximation and a multiple-scattering forward model (MLB) with intensity images collected from a custom 1.2NA oil immersion microscopy.
» Visualization 3       3D refractive index reconstruction of a 100 micron 4-cell stage mouse embryo under an LED array microscope with a 20X 0.5NA objective lens (Nikon, CFI Plan Fluor).

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Figures (6)

Fig. 1.
Fig. 1. Our multi-layer Born (MLB) forward scattering model is validated as part of a 3D phase imaging framework. Intensity measurements with spatially coherent illumination from different angles are captured on an optical microscope and fed in as inputs to our 3D phase tomography algorithm. By solving a nonlinear optimization problem with our MLB scattering forward model, the 3D refractive index of a multiple-scattering object is recovered.
Fig. 2.
Fig. 2. Comparison of forward model accuracy for a 3D cell phantom using the first Born, Rytov, multi-slice (MS), and MLB scattering models. (a) Ground truth (SEAGLE simulations) of the forward and backward scattered amplitude with on-axis and off-axis illumination, along with the error maps for all other forward models. (b) Accuracy of each model as the maximum phase within each layer increases.
Fig. 3.
Fig. 3. Simulations comparing the 3D RI reconstructions using the first Born, Rytov, MS, and MLB forward scattering models with our inverse solver (Algorithm 3). (Top) Simulated measurements for five angles of incidence. (Bottom) $ x - y $ and $ x - z $ cross sections of the 3D refractive index distributions along with their mean squared error (MSE).
Fig. 4.
Fig. 4. Experimental results for a single polystyrene bead in low and high index contrast media. (a) Examples of the measured normalized transmitted amplitude. (b), (c) Orthogonal slices of 3D refractive index reconstructions of low/high contrast polystyrene beads from different scattering models. 1D cross sections along the black dotted lines are plotted on the right. The weakly scattering approximation is no longer valid when the refractive index contrast is large, so the multiple-scattering models (MS and MLB) perform better.
Fig. 5.
Fig. 5. 3D refractive index reconstruction of a 3T3 cell using the MLB scattering model. (a) Orthonormal views of the recovered 3T3 cell. (b) 3D rendering. (c) Zoomed-in comparison among differential phase contrast (DPC), quantitative phase, and a slice of 3D refractive index of the 3T3 cell at two different depths within green and blue boxes in (a). The 3D slices provide both good depth sectioning and quantitative information with high contrast.
Fig. 6.
Fig. 6. 3D refractive index reconstructions of an entire adult hermaphrodite C. elegans worm from the dataset in Ref. [17], using our MLB model. The insets show zoomed-in comparison between orthonormal cross-sections using the first Born, multi-slice (MS), and MLB methods, for the white box region that includes the mouth and pharynx of the C. elegans. Visualization 2 is a video showing different regions across the whole worm.

Tables (3)

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Algorithm 1. Multi-Layer Born Forward Scattering

Tables Icon

Algorithm 2. Gradient Computation

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Algorithm 3. 3D Intensity-Based Phase Tomography with MLB

Equations (11)

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V ( x , y , z ) = k o 2 ( n b 2 n 2 ( x , y , z ) ) ,
U tot ( x , y , z ) U in  ( x , y , z ) + G ( x x , y y , z z ) × U in ( x , y , z ) V ( x , y , z ) d x d y d z .
U n + 1 ( ρ , ( n + 1 ) Δ z ) = U n ( ρ , ( n + 1 ) Δ z ) + U s n ( ρ , ( n + 1 ) Δ z ) ,
U s n ( ρ , ( n + 1 ) Δ z ) = λ 2 Δ z 2 G ( ρ ρ , Δ z ζ ) × U n ( ρ ; n Δ z + ζ ) V ( ρ ; n Δ z + ζ ) d 2 ρ d ζ .
U n ( ρ , ( n + 1 ) Δ z ) = F 1 { P ( u , Δ z ) U ~ n ( u ) } .
G ~ ( u , z ) = G ( ρ , z ) e i 2 π ( u , ρ ) d 2 ρ = i e i 2 π ( n b / λ ) 2 | | u | | 2 | z | 4 π ( n b / λ ) 2 u 2 = i P ( u , | z | ) 4 π Γ ( u ) ,
U ~ s n ( u ) = Δ 2 Δ z 2 i P ( u , | Δ z ζ | ) 4 π Γ ( u ) P ( u , ζ ) U ~ n ( u ) × V ~ n ( u u ) d 2 u d ζ ,
Δ z 2 Δ z 2 P ( u , | Δ z ζ | ) P ( u , ζ ) d ζ = P ( u , Δ z ) sinc ( ( Γ ( u ) Γ ( u ) ) Δ z ) Δ z ,
U ~ s n ( u ) = G ~ ( u , Δ z ) sinc ( ( Γ ( u ) Γ ( u ) ) Δ z ) U ~ n ( u ) × V ~ n ( u u ) Δ z d 2 u .
min V L ( V ) = j = 1 N LED | U image , j ( V ) | I measure , j 2 + τ R ( V ) .
pro x τ R ( V ) = argmin x τ R ( x ) + 1 2 x V 2 .

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