Abstract

Reconstruction of phase objects is a central problem in digital holography, whose various applications include microscopy, biomedical imaging, and fluid mechanics. Starting from a single in-line hologram, there is no direct way to recover the phase of the diffracted wave in the hologram plane. The reconstruction of absorbing and phase objects therefore requires the inversion of the non-linear hologram formation model. We propose a regularized reconstruction method that includes several physically-grounded constraints such as bounds on transmittance values, maximum/minimum phase, spatial smoothness or the absence of any object in parts of the field of view. To solve the non-convex and non-smooth optimization problem induced by our modeling, a variable splitting strategy is applied and the closed-form solution of the sub-problem (the so-called proximal operator) is derived. The resulting algorithm is efficient and is shown to lead to quantitative phase estimation on reconstructions of accurate simulations of in-line holograms based on the Mie theory. As our approach is adaptable to several in-line digital holography configurations, we present and discuss the promising results of reconstructions from experimental in-line holograms obtained in two different applications: the tracking of an evaporating droplet (size ∼ 100μm) and the microscopic imaging of bacteria (size ∼ 1μm).

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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2017 (7)

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. C. Öztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

C. Fournier, F. Jolivet, L. Denis, N. Verrier, E. Thiebaut, C. Allier, and T. Fournel, “Pixel super-resolution in digital holography by regularized reconstruction,” Appl. Opt. 56, 69–77 (2017).
[Crossref]

D. Ryu, Z. Wang, K. He, G. Zheng, R. Horstmeyer, and O. Cossairt, “Subsampled phase retrieval for temporal resolution enhancement in lensless on-chip holographic video,” Biomed. Opt. Express 8, 1981–1995 (2017).
[Crossref] [PubMed]

A. Berdeu, F. Momey, B. Laperrousaz, T. Bordy, X. Gidrol, J.-M. Dinten, N. Picollet-D’hahan, and C. Allier, “Comparative study of fully three-dimensional reconstruction algorithms for lens-free microscopy,” Appl. Opt. 56, 3939–3951 (2017).
[Crossref] [PubMed]

C. Schretter, D. Blinder, S. Bettens, H. Ottevaere, and P. Schelkens, “Regularized non-convex image reconstruction in digital holographic microscopy,” Opt. Express 25, 16491–16508 (2017).
[Crossref] [PubMed]

S. Bettens, H. Yan, D. Blinder, H. Ottevaere, C. Schretter, and P. Schelkens, “Studies on the sparsifying operator in compressive digital holography,” Opt. Express 25, 18656–18676 (2017).
[Crossref] [PubMed]

A. Sinha, J. Lee, S. Li, and G. Barbastathis, “Lensless computational imaging through deep learning,” Optica 4, 1117–1125 (2017).
[Crossref]

2016 (4)

Y. Endo, T. Shimobaba, T. Kakue, and T. Ito, “GPU-accelerated compressive holography,” Opt. Express 24, 8437–8445 (2016).
[Crossref] [PubMed]

F. Soulez, É. Thiébaut, A. Schutz, A. Ferrari, F. Courbin, and M. Unser, “Proximity operators for phase retrieval,” Appl. Opt. 55, 7412–7421 (2016).
[Crossref] [PubMed]

Y. Rivenson, Y. Wu, H. Wang, Y. Zhang, A. Feizi, and A. Ozcan, “Sparsity-based multi-height phase recovery in holographic microscopy,” Sci. Rep. 6, 37862 (2016).
[Crossref] [PubMed]

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681(2016).

2014 (3)

N. Parikh and S. Boyd, Proximal Algorithms, Foundations and Trends® in Optimization 1, 127–239 (2014).
[Crossref]

C. Fournier, L. Denis, M. Seifi, and T. Fournel, “Digital hologram processing in on-axis holography,” Multi-Dimensional Imaging 0, 51–73 (2014).
[Crossref]

R. Horisaki, Y. Ogura, M. Aino, and J. Tanida, “Single-shot phase imaging with a coded aperture,” Opt. Lett. 39, 6466–6469 (2014).
[Crossref] [PubMed]

2013 (2)

2012 (3)

F. Eilenberger, S. Minardi, D. Pliakis, and T. Pertsch, “Digital holography from shadowgraphic phase estimates,” Opt. Lett. 37, 509–511 (2012).
[Crossref] [PubMed]

M. Jericho, H. Kreuzer, M. Kanka, and R. Riesenberg, “Quantitative phase and refractive index measurements with point-source digital in-line holographic microscopy,” Appl. Opt. 51, 1503–1515 (2012).
[Crossref] [PubMed]

D. Chareyron, J.-L. Marié, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography ‘inverse method’ for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[Crossref]

2011 (1)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends® in Machine Learning 3, 1–122 (2011).
[Crossref]

2009 (2)

2007 (2)

2004 (2)

2002 (1)

E. Thiébaut, “Optimization issues in blind deconvolution algorithms,” Proc. SPIE 4847, pp. 174–183 (2002).
[Crossref]

2001 (1)

1996 (1)

G. Toker and J. Stricker, “Holographic study of suspended vaporizing volatile liquid droplets in still air,” Int. J. Heat Mass Tran. 39, 3475–3482 (1996). .
[Crossref]

1995 (1)

1994 (1)

1982 (1)

1980 (1)

J. Nocedal, “Updating quasi-Newton matrices with limited storage,” Math. Comput. 35, 773–782 (1980).
[Crossref]

1972 (1)

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

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref] [PubMed]

Aino, M.

Allier, C.

Barbastathis, G.

Barbier, B.

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. C. Öztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

Becker, J.-M.

R. Mourya, L. Denis, J.-M. Becker, and E. Thiebaut, “Augmented lagrangian without alternating directions: Practical algorithms for inverse problems in imaging,” in Proc. of 2015 IEEE International Conference on Image Processing (ICIP), (IEEE, 2015), pp. 1205–1209.
[Crossref]

Berdeu, A.

Bettens, S.

Blinder, D.

Blu, T.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley, 1983).

Bordy, T.

Bostan, E.

Bourquard, A.

Boyd, S.

N. Parikh and S. Boyd, Proximal Algorithms, Foundations and Trends® in Optimization 1, 127–239 (2014).
[Crossref]

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends® in Machine Learning 3, 1–122 (2011).
[Crossref]

Chareyron, D.

L. Méès, N. Grosjean, D. Chareyron, J.-L. Marié, M. Seifi, and C. Fournier, “Evaporating droplet hologram simulation for digital in-line holography setup with divergent beam,” J. Opt. Soc. Am. A 30, 2021–2028 (2013).
[Crossref]

D. Chareyron, J.-L. Marié, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography ‘inverse method’ for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[Crossref]

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends® in Machine Learning 3, 1–122 (2011).
[Crossref]

Combettes, P. L.

P. L. Combettes and J.-C. Pesquet, “Proximal Splitting Methods in Signal Processing,” in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, (Springer, 2011), pp. 185–212.
[Crossref]

Cossairt, O.

Cossairt, O.S.

Z. Wang, Q. Dai, D. Ryu, K. He, R. Horstmeyer, A. Katsaggelos, and O.S. Cossairt, “Dictionary-based phase retrieval for space-time super resolution using lens-free on-chip holographic video,” in Imaging and Applied Optics 2017 (3D, AIO, COSI, IS, MATH, pcAOP), OSA Technical Digest (online) (Optical Society of America, 2017), paper CTu2B.3.
[Crossref]

Courbin, F.

Dai, Q.

Z. Wang, Q. Dai, D. Ryu, K. He, R. Horstmeyer, A. Katsaggelos, and O.S. Cossairt, “Dictionary-based phase retrieval for space-time super resolution using lens-free on-chip holographic video,” in Imaging and Applied Optics 2017 (3D, AIO, COSI, IS, MATH, pcAOP), OSA Technical Digest (online) (Optical Society of America, 2017), paper CTu2B.3.
[Crossref]

Denis, L.

C. Fournier, F. Jolivet, L. Denis, N. Verrier, E. Thiebaut, C. Allier, and T. Fournel, “Pixel super-resolution in digital holography by regularized reconstruction,” Appl. Opt. 56, 69–77 (2017).
[Crossref]

C. Fournier, L. Denis, M. Seifi, and T. Fournel, “Digital hologram processing in on-axis holography,” Multi-Dimensional Imaging 0, 51–73 (2014).
[Crossref]

D. Chareyron, J.-L. Marié, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography ‘inverse method’ for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[Crossref]

L. Denis, D. Lorenz, E. Thiébaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. 34, 3475–3477 (2009).
[Crossref] [PubMed]

F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in particle digital holography: out-of-field particle detection made possible,” J. Opt. Soc. Am. A 24, 3708–3716 (2007).
[Crossref]

F. Soulez, L. Denis, C. Fournier, É. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. A 24, 1164–1171 (2007).
[Crossref]

R. Mourya, L. Denis, J.-M. Becker, and E. Thiebaut, “Augmented lagrangian without alternating directions: Practical algorithms for inverse problems in imaging,” in Proc. of 2015 IEEE International Conference on Image Processing (ICIP), (IEEE, 2015), pp. 1205–1209.
[Crossref]

Depeursinge, C.

Dinten, J.-M.

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends® in Machine Learning 3, 1–122 (2011).
[Crossref]

Eilenberger, F.

Endo, Y.

Feizi, A.

Y. Rivenson, Y. Wu, H. Wang, Y. Zhang, A. Feizi, and A. Ozcan, “Sparsity-based multi-height phase recovery in holographic microscopy,” Sci. Rep. 6, 37862 (2016).
[Crossref] [PubMed]

Ferrari, A.

Fessler, J. A.

Fienup, J. R.

Fournel, T.

C. Fournier, F. Jolivet, L. Denis, N. Verrier, E. Thiebaut, C. Allier, and T. Fournel, “Pixel super-resolution in digital holography by regularized reconstruction,” Appl. Opt. 56, 69–77 (2017).
[Crossref]

C. Fournier, L. Denis, M. Seifi, and T. Fournel, “Digital hologram processing in on-axis holography,” Multi-Dimensional Imaging 0, 51–73 (2014).
[Crossref]

Fournier, C.

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. C. Öztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

C. Fournier, F. Jolivet, L. Denis, N. Verrier, E. Thiebaut, C. Allier, and T. Fournel, “Pixel super-resolution in digital holography by regularized reconstruction,” Appl. Opt. 56, 69–77 (2017).
[Crossref]

C. Fournier, L. Denis, M. Seifi, and T. Fournel, “Digital hologram processing in on-axis holography,” Multi-Dimensional Imaging 0, 51–73 (2014).
[Crossref]

L. Méès, N. Grosjean, D. Chareyron, J.-L. Marié, M. Seifi, and C. Fournier, “Evaporating droplet hologram simulation for digital in-line holography setup with divergent beam,” J. Opt. Soc. Am. A 30, 2021–2028 (2013).
[Crossref]

D. Chareyron, J.-L. Marié, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography ‘inverse method’ for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[Crossref]

L. Denis, D. Lorenz, E. Thiébaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. 34, 3475–3477 (2009).
[Crossref] [PubMed]

F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in particle digital holography: out-of-field particle detection made possible,” J. Opt. Soc. Am. A 24, 3708–3716 (2007).
[Crossref]

F. Soulez, L. Denis, C. Fournier, É. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. A 24, 1164–1171 (2007).
[Crossref]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref] [PubMed]

Gerchberg, R. W.

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

Gidrol, X.

Gire, J.

D. Chareyron, J.-L. Marié, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography ‘inverse method’ for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[Crossref]

Goepfert, C.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, (Roberts and Company Publishers, 2005).

Gouesbet, G.

Gréhan, G.

Grosjean, N.

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. C. Öztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

L. Méès, N. Grosjean, D. Chareyron, J.-L. Marié, M. Seifi, and C. Fournier, “Evaporating droplet hologram simulation for digital in-line holography setup with divergent beam,” J. Opt. Soc. Am. A 30, 2021–2028 (2013).
[Crossref]

D. Chareyron, J.-L. Marié, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography ‘inverse method’ for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[Crossref]

Günaydin, H.

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” arXiv preprint arXiv:1705.04286 (2018).

Hare, W.

W. Hare and C. Sagastizábal, “Computing proximal points of nonconvex functions,” Math. Program. 116, 221–258 (2009).
[Crossref]

He, K.

D. Ryu, Z. Wang, K. He, G. Zheng, R. Horstmeyer, and O. Cossairt, “Subsampled phase retrieval for temporal resolution enhancement in lensless on-chip holographic video,” Biomed. Opt. Express 8, 1981–1995 (2017).
[Crossref] [PubMed]

Z. Wang, Q. Dai, D. Ryu, K. He, R. Horstmeyer, A. Katsaggelos, and O.S. Cossairt, “Dictionary-based phase retrieval for space-time super resolution using lens-free on-chip holographic video,” in Imaging and Applied Optics 2017 (3D, AIO, COSI, IS, MATH, pcAOP), OSA Technical Digest (online) (Optical Society of America, 2017), paper CTu2B.3.
[Crossref]

Horisaki, R.

Horstmeyer, R.

D. Ryu, Z. Wang, K. He, G. Zheng, R. Horstmeyer, and O. Cossairt, “Subsampled phase retrieval for temporal resolution enhancement in lensless on-chip holographic video,” Biomed. Opt. Express 8, 1981–1995 (2017).
[Crossref] [PubMed]

Z. Wang, Q. Dai, D. Ryu, K. He, R. Horstmeyer, A. Katsaggelos, and O.S. Cossairt, “Dictionary-based phase retrieval for space-time super resolution using lens-free on-chip holographic video,” in Imaging and Applied Optics 2017 (3D, AIO, COSI, IS, MATH, pcAOP), OSA Technical Digest (online) (Optical Society of America, 2017), paper CTu2B.3.
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley, 1983).

Im, H.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681(2016).

Ito, T.

Iwamoto, Y.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681(2016).

Jeong, S.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681(2016).

Jericho, M.

Jolivet, F.

Kai, L.

Kakue, T.

Kanka, M.

Kato, J.-i.

Katsaggelos, A.

Z. Wang, Q. Dai, D. Ryu, K. He, R. Horstmeyer, A. Katsaggelos, and O.S. Cossairt, “Dictionary-based phase retrieval for space-time super resolution using lens-free on-chip holographic video,” in Imaging and Applied Optics 2017 (3D, AIO, COSI, IS, MATH, pcAOP), OSA Technical Digest (online) (Optical Society of America, 2017), paper CTu2B.3.
[Crossref]

Kreuzer, H.

Lance, M.

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. C. Öztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

D. Chareyron, J.-L. Marié, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography ‘inverse method’ for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[Crossref]

Laperrousaz, B.

Lee, H.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681(2016).

Lee, J.

Li, S.

Liebling, M.

Lorenz, D.

Marié, J.-L.

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. C. Öztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

L. Méès, N. Grosjean, D. Chareyron, J.-L. Marié, M. Seifi, and C. Fournier, “Evaporating droplet hologram simulation for digital in-line holography setup with divergent beam,” J. Opt. Soc. Am. A 30, 2021–2028 (2013).
[Crossref]

D. Chareyron, J.-L. Marié, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography ‘inverse method’ for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[Crossref]

Massoli, P.

Méès, L.

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. C. Öztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

L. Méès, N. Grosjean, D. Chareyron, J.-L. Marié, M. Seifi, and C. Fournier, “Evaporating droplet hologram simulation for digital in-line holography setup with divergent beam,” J. Opt. Soc. Am. A 30, 2021–2028 (2013).
[Crossref]

D. Chareyron, J.-L. Marié, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography ‘inverse method’ for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[Crossref]

Minardi, S.

Mizuno, J.

Momey, F.

Mourya, R.

R. Mourya, L. Denis, J.-M. Becker, and E. Thiebaut, “Augmented lagrangian without alternating directions: Practical algorithms for inverse problems in imaging,” in Proc. of 2015 IEEE International Conference on Image Processing (ICIP), (IEEE, 2015), pp. 1205–1209.
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J. Nocedal, “Updating quasi-Newton matrices with limited storage,” Math. Comput. 35, 773–782 (1980).
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Ohta, S.

Onofri, F.

Ottevaere, H.

Ozcan, A.

Y. Rivenson, Y. Wu, H. Wang, Y. Zhang, A. Feizi, and A. Ozcan, “Sparsity-based multi-height phase recovery in holographic microscopy,” Sci. Rep. 6, 37862 (2016).
[Crossref] [PubMed]

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” arXiv preprint arXiv:1705.04286 (2018).

Öztürk, O. C.

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. C. Öztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

Parikh, N.

N. Parikh and S. Boyd, Proximal Algorithms, Foundations and Trends® in Optimization 1, 127–239 (2014).
[Crossref]

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends® in Machine Learning 3, 1–122 (2011).
[Crossref]

Pathania, D.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681(2016).

Pavillon, N.

Peleato, B.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends® in Machine Learning 3, 1–122 (2011).
[Crossref]

Pertsch, T.

Pesquet, J.-C.

P. L. Combettes and J.-C. Pesquet, “Proximal Splitting Methods in Signal Processing,” in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, (Springer, 2011), pp. 185–212.
[Crossref]

Picollet-D’hahan, N.

Pivovarov, M.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681(2016).

Pliakis, D.

Riesenberg, R.

Rivenson, Y.

Y. Rivenson, Y. Wu, H. Wang, Y. Zhang, A. Feizi, and A. Ozcan, “Sparsity-based multi-height phase recovery in holographic microscopy,” Sci. Rep. 6, 37862 (2016).
[Crossref] [PubMed]

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” arXiv preprint arXiv:1705.04286 (2018).

Ryu, D.

D. Ryu, Z. Wang, K. He, G. Zheng, R. Horstmeyer, and O. Cossairt, “Subsampled phase retrieval for temporal resolution enhancement in lensless on-chip holographic video,” Biomed. Opt. Express 8, 1981–1995 (2017).
[Crossref] [PubMed]

Z. Wang, Q. Dai, D. Ryu, K. He, R. Horstmeyer, A. Katsaggelos, and O.S. Cossairt, “Dictionary-based phase retrieval for space-time super resolution using lens-free on-chip holographic video,” in Imaging and Applied Optics 2017 (3D, AIO, COSI, IS, MATH, pcAOP), OSA Technical Digest (online) (Optical Society of America, 2017), paper CTu2B.3.
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R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237 (1972).

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Schretter, C.

Schutz, A.

Seifi, M.

Shimobaba, T.

Sinha, A.

Song, J.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681(2016).

Sotthivirat, S.

Soulez, F.

Stricker, J.

G. Toker and J. Stricker, “Holographic study of suspended vaporizing volatile liquid droplets in still air,” Int. J. Heat Mass Tran. 39, 3475–3482 (1996). .
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Swisher, C. L.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681(2016).

Tanida, J.

Teng, D.

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” arXiv preprint arXiv:1705.04286 (2018).

Thiebaut, E.

C. Fournier, F. Jolivet, L. Denis, N. Verrier, E. Thiebaut, C. Allier, and T. Fournel, “Pixel super-resolution in digital holography by regularized reconstruction,” Appl. Opt. 56, 69–77 (2017).
[Crossref]

R. Mourya, L. Denis, J.-M. Becker, and E. Thiebaut, “Augmented lagrangian without alternating directions: Practical algorithms for inverse problems in imaging,” in Proc. of 2015 IEEE International Conference on Image Processing (ICIP), (IEEE, 2015), pp. 1205–1209.
[Crossref]

Thiébaut, E.

Thiébaut, É.

Toker, G.

G. Toker and J. Stricker, “Holographic study of suspended vaporizing volatile liquid droplets in still air,” Int. J. Heat Mass Tran. 39, 3475–3482 (1996). .
[Crossref]

Trede, D.

Tronchin, T.

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. C. Öztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

Unser, M.

Verrier, N.

Wang, H.

Y. Rivenson, Y. Wu, H. Wang, Y. Zhang, A. Feizi, and A. Ozcan, “Sparsity-based multi-height phase recovery in holographic microscopy,” Sci. Rep. 6, 37862 (2016).
[Crossref] [PubMed]

Wang, Z.

D. Ryu, Z. Wang, K. He, G. Zheng, R. Horstmeyer, and O. Cossairt, “Subsampled phase retrieval for temporal resolution enhancement in lensless on-chip holographic video,” Biomed. Opt. Express 8, 1981–1995 (2017).
[Crossref] [PubMed]

Z. Wang, Q. Dai, D. Ryu, K. He, R. Horstmeyer, A. Katsaggelos, and O.S. Cossairt, “Dictionary-based phase retrieval for space-time super resolution using lens-free on-chip holographic video,” in Imaging and Applied Optics 2017 (3D, AIO, COSI, IS, MATH, pcAOP), OSA Technical Digest (online) (Optical Society of America, 2017), paper CTu2B.3.
[Crossref]

Weissleder, R.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681(2016).

Wu, Y.

Y. Rivenson, Y. Wu, H. Wang, Y. Zhang, A. Feizi, and A. Ozcan, “Sparsity-based multi-height phase recovery in holographic microscopy,” Sci. Rep. 6, 37862 (2016).
[Crossref] [PubMed]

Yamaguchi, I.

Yan, H.

Zhang, Y.

Y. Rivenson, Y. Wu, H. Wang, Y. Zhang, A. Feizi, and A. Ozcan, “Sparsity-based multi-height phase recovery in holographic microscopy,” Sci. Rep. 6, 37862 (2016).
[Crossref] [PubMed]

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” arXiv preprint arXiv:1705.04286 (2018).

Zheng, G.

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C. Fournier, F. Jolivet, L. Denis, N. Verrier, E. Thiebaut, C. Allier, and T. Fournel, “Pixel super-resolution in digital holography by regularized reconstruction,” Appl. Opt. 56, 69–77 (2017).
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Biomed. Opt. Express (1)

Exp. Fluids (1)

J.-L. Marié, T. Tronchin, N. Grosjean, L. Méès, O. C. Öztürk, C. Fournier, B. Barbier, and M. Lance, “Digital holographic measurement of the lagrangian evaporation rate of droplets dispersing in a homogeneous isotropic turbulence,” Exp. Fluids 58, 11 (2017).
[Crossref]

Foundations and Trends® in Machine Learning (1)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends® in Machine Learning 3, 1–122 (2011).
[Crossref]

Foundations and Trends® in Optimization (1)

N. Parikh and S. Boyd, Proximal Algorithms, Foundations and Trends® in Optimization 1, 127–239 (2014).
[Crossref]

Int. J. Heat Mass Tran. (1)

G. Toker and J. Stricker, “Holographic study of suspended vaporizing volatile liquid droplets in still air,” Int. J. Heat Mass Tran. 39, 3475–3482 (1996). .
[Crossref]

J. Opt. Soc. Am. A (5)

Math. Comput. (1)

J. Nocedal, “Updating quasi-Newton matrices with limited storage,” Math. Comput. 35, 773–782 (1980).
[Crossref]

Math. Program. (1)

W. Hare and C. Sagastizábal, “Computing proximal points of nonconvex functions,” Math. Program. 116, 221–258 (2009).
[Crossref]

Multi-Dimensional Imaging (1)

C. Fournier, L. Denis, M. Seifi, and T. Fournel, “Digital hologram processing in on-axis holography,” Multi-Dimensional Imaging 0, 51–73 (2014).
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Nature (1)

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New J. Phys. (1)

D. Chareyron, J.-L. Marié, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography ‘inverse method’ for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Optica (1)

Optik (1)

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

Proc. SPIE (1)

E. Thiébaut, “Optimization issues in blind deconvolution algorithms,” Proc. SPIE 4847, pp. 174–183 (2002).
[Crossref]

Sci. Rep. (2)

Y. Rivenson, Y. Wu, H. Wang, Y. Zhang, A. Feizi, and A. Ozcan, “Sparsity-based multi-height phase recovery in holographic microscopy,” Sci. Rep. 6, 37862 (2016).
[Crossref] [PubMed]

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681(2016).

Other (7)

Z. Wang, Q. Dai, D. Ryu, K. He, R. Horstmeyer, A. Katsaggelos, and O.S. Cossairt, “Dictionary-based phase retrieval for space-time super resolution using lens-free on-chip holographic video,” in Imaging and Applied Optics 2017 (3D, AIO, COSI, IS, MATH, pcAOP), OSA Technical Digest (online) (Optical Society of America, 2017), paper CTu2B.3.
[Crossref]

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” arXiv preprint arXiv:1705.04286 (2018).

R. Mourya, L. Denis, J.-M. Becker, and E. Thiebaut, “Augmented lagrangian without alternating directions: Practical algorithms for inverse problems in imaging,” in Proc. of 2015 IEEE International Conference on Image Processing (ICIP), (IEEE, 2015), pp. 1205–1209.
[Crossref]

J. W. Goodman, Introduction to Fourier Optics, (Roberts and Company Publishers, 2005).

P. L. Combettes and J.-C. Pesquet, “Proximal Splitting Methods in Signal Processing,” in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, (Springer, 2011), pp. 185–212.
[Crossref]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley, 1983).

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories, (Springer, 2017).
[Crossref]

Supplementary Material (2)

NameDescription
» Visualization 1       Acquisition sequence of in-line holograms of an evaporating ether droplet in fluid mechanics experiment at Laboratoire de Mécanique de Fluides et Acoustique (LMFA UMR CNRS 5509 - Ecole Centrale de Lyon).
» Visualization 2       Phase reconstruction with the proposed method from the sequence of in-line holograms of an evaporating ether droplet in fluid mechanics experiment at Laboratoire de Mécanique de Fluides et Acoustique (LMFA UMR CNRS 5509 - Ecole Centrale de Lyon).

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Figures (11)

Fig. 1
Fig. 1 a) Experimental setup for observing evaporating droplets dedicated to fluid mechanics studies. b) Experimental hologram. The red rectangle corresponds to the field of view of interest in the reconstructions shown in Fig. 4 in section 4.1.3.
Fig. 2
Fig. 2 Simulation of an evaporating droplet of index nd = 1.35 and radius rd = 50 μm, with an isotropic vapor cloud modeled by a refractive index decay nv(r) − 1 = (ns − 1) exp (−(rrd)/σ), illuminated by a plane wave of wavelength λ = 532 nm, and recorded at a distance z = 0.5 m. a) Synthetic hologram of the phase-shifting and absorbing object. b) Ground truth phase images obtained by back-propagation of the corresponding complex holograms (i.e., including the phase in the hologram plane). The field of view corresponds to the red square shown on the holograms.
Fig. 3
Fig. 3 Reconstructions of simulation of the phase-shifting and absorbing object. a) Reconstruction phase for a data SNR of 100. b) Absolute value of the phase difference (error map) between the ground truth phase of Fig. 2(b) and the reconstruction, for data SNR = 100. Normalized maximum error EMAX ≃ 15.18% and normalized root mean squared error RMSE ≃ 1.53%. c) Reconstruction phase for data SNR = 50. d) Absolute value of the phase difference (error map) between the ground truth of Fig. 2(b) and the reconstruction, for the data SNR = 50. Normalized maximum error EMAX ≃ 14.91% and normalized root mean squared error RMSE ≃ 1.68%. e) Reconstruction phase for data SNR = 100 with the Error-Reduction Fienup’s algorithm with a known-support constraint. f) Absolute value of the phase difference (error map) between the ground truth of Fig. 2(b) and the reconstruction, for the data SNR = 100. Normalized maximum error EMAX ≃ 49.41% and normalized root mean squared error RMSE ≃ 10.67%.
Fig. 4
Fig. 4 Reconstructions of a time sequence of experimental holograms of evaporating droplets (See Visualization 1) with the proposed method (See Visualization 2). a) Data at 3 different time steps of the acquisition. The red rectangles corresponds to the field of view shown in the reconstructions on the rows below. b) Reconstructions of the phase at the same 3 stages. The evaporating cloud is clearly visible as a purely phase information. c) Reconstructions of the modulus at the same 3 stages. Only the droplet is visible as a purely absorbing object.
Fig. 5
Fig. 5 Qualitative comparison between a) the experimental hologram and b) the hologram model c* () obtained from the repropagation of the reconstruction.
Fig. 6
Fig. 6 Reconstructed phase at different optimization steps. a) 2 dual updates with 30 iterations (calculation time ∼ 1 minute). b) 4 dual updates with 30 iterations (calculation time ∼ 3 minutes). c) 16 dual updates with 50 iterations (strategy used for the reconstructions shown in Fig. 4) (calculation time ∼ 15 minutes). Calculation times are given for reconstructions performed on a CPU Intel Core i7-3630QM (2,40 GHz).
Fig. 7
Fig. 7 a) Inline holographic microscopic setup dedicated to the imaging of bacteria. b) Defocused hologram (z = 32 μm) of Gram stained bacteria at illumination wavelength λ = 610 nm.
Fig. 8
Fig. 8 Intensity images of the biological sample acquired at the focal plane of the objective (experimental ground truth images). A, B and C are 3 regions of interest (see Fig. 10). a) White-light illumination. b) Monochromatic illumination at λ = 610 nm.
Fig. 9
Fig. 9 Reconstructions from the defocused hologram (z = 32 μm) of Fig. 7(b) at illumination wavelength λ = 610 nm. A, B and C are 3 regions of interest (see Fig. 10).
Fig. 10
Fig. 10 Zooms on the 3 regions of interest A, B and C, of the ground truth images at the focal plane of Fig. 8, and of the reconstructions of Fig. 9.
Fig. 11
Fig. 11 Absorbance spectra of the chromophores used for standard Gram stain of bacteria.

Tables (1)

Tables Icon

Table 1 Physical constraints considered by our reconstruction method.

Equations (30)

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

A z _ ( x , y ) = ( h z _ * ( A 0 _ t _ ) ) ( x , y ) ,
h z _ ( x , y ) = z i _ λ exp ( i _ k r ) r 2 .
h z _ ( x , y ) = 1 i _ λ z exp ( i _ π ( x 2 + y 2 ) λ z ) .
I ( x , y ) = | A _ | 2 m ( x , y ) with m ( x , y ) = | ( h z _ * t _ ) ( x , y ) | 2 .
[ m ( t _ ) ] k = | [ H _ t _ ] k | 2 ,
[ d ] k = c [ m ( t _ ) ] k + [ n ] k = c | [ H _ t _ ] k | 2 + [ n ] k
t ^ _ = arg min t _ Λ { min c 𝒟 ( t _ , c ) } + ( t _ ) ,
𝒟 ( t _ , c ) = c m ( t _ ) d W 2 ,
c * ( t _ ) = arg min c c m ( t _ ) d W 2 .
c * ( t _ ) = m ( t _ ) T Wd m ( t _ ) T Wm ( t _ ) .
𝒟 * ( t _ ) = c * ( t _ ) m ( t _ ) d W 2 .
t _ = | t _ | . e i _ φ ( t _ ) .
TV ( t _ ) = k ( Δ k x 𝔢 ( t _ ) ) 2 + ( Δ k y 𝔢 ( t _ ) ) 2 + ( Δ k x 𝔪 ( t _ ) ) 2 + ( Δ k y 𝔪 ( t _ ) ) 2 + 2 .
t ^ _ = argmin t _ Ω Ψ 𝒟 * ( t _ ) + α 1 1 ( t _ ) + α 2 2 ( t _ ) + α 3 TV ( 𝔢 ( t _ ) , 𝔪 ( t _ ) )
{ t ^ _ = argmin t _ , a , b f α 3 ( t _ ) + g α 1 , α 2 ( a , b ) s . t . e ( t _ ) = a , m ( t _ ) = b , and ( a + i _ b ) Ω Ψ ,
β ( t _ , a , b , u _ ) = f α 3 ( t _ ) + g α 1 , α 2 ( a , b ) + β t _ ( a + i _ b ) + u _ 2 2
{ t _ ( j + 1 ) , a ( j + 1 ) , b ( j + 1 ) } = arg min t _ , ( a + i _ b ) Ω Ψ β ( t _ , a , b , u _ ( j ) )
dual update : u _ ( j + 1 ) = u _ ( j ) + t _ ( j + 1 ) ( a ( j + 1 ) + i _ b ( j + 1 ) )
t _ ( j + 1 ) = argmin t _ f α 3 ( t _ ) + g α 1 , α 2 ( a * ( t _ ) , b * ( t _ ) ) + β t _ ( a * ( t _ ) + i _ b * ( t _ ) ) + u _ ( j ) 2 2
with { a * ( t _ ) , b * ( t _ ) } = arg min ( a + i _ b ) Ω Ψ g α 1 , α 2 ( a , b ) + β t _ ( a + i _ b ) + u _ ( j ) 2 2
dual update : u _ ( j + 1 ) = u _ ( j ) + t _ ( j + 1 ) ( a * ( t _ ( j + 1 ) ) + i _ b * ( t _ ( j + 1 ) ) )
prox β 1 g α 1 , α 2 ( p , q ) = { arg min a , b β ( p a ) 2 + β ( q b ) 2 + g α 1 , α 2 ( a , b ) s . t . a 2 + b 2 1 , and φ min φ ( a + i _ b ) φ max
arg min a , b β ( p a ) 2 + β ( q b ) 2 + g α 1 , α 2 ( a , b )
( a * , b * ) = ( p + α 1 2 β p | p + i _ S α 2 / 2 β ( q ) | , S α 2 / 2 β ( q ) + α 1 2 β S α 2 / 2 β ( q ) | p + i _ S α 2 / 2 β ( q ) | )
S α 2 / 2 β ( q ) { q α 2 2 β if q α 2 2 β 0 if q [ α 2 2 β , α 2 2 β ] q + α 2 2 β if q α 2 2 β
( a * , b * ) = ( α 1 2 β , 0 )
( a * , b * ) = ( p | p + i _ S α 2 / 2 β ( q ) | , S α 2 / 2 β ( q ) | p + i _ S α 2 / 2 β ( q ) | )
| a * + i _ b * | = { 1 if ρ * ( φ * ) 1 ρ * ( φ * ) if ρ * ( φ * ) [ 0 , 1 ] 0 if ρ * ( φ * ) 0
ρ * ( φ ) = p cos ( φ ) + q sin ( φ ) + α 1 2 β α 2 2 β | sin ( φ ) |
φ * = arg min φ { φ min , φ max } β ( p ρ * ( φ ) cos ( φ ) ) 2 + β ( q ρ * ( φ ) sin ( φ ) ) 2 + α 1 ( 1 ρ * ( φ ) ) + α 2 ρ * ( φ ) | sin ( φ ) | .

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