Abstract

The key limitations of digital inline holography (DIH) for particle tracking applications are poor longitudinal resolution, particle concentration limits, and case-specific processing. We utilize an inverse problem method with fused lasso regularization to perform full volumetric reconstructions of particle fields. By exploiting data sparsity in the solution and utilizing GPU processing, we dramatically reduce the computational cost usually associated with inverse reconstruction approaches. We demonstrate the accuracy of the proposed method using synthetic and experimental holograms. Finally, we present two practical applications (high concentration microorganism swimming and microfiber rotation) to extend the capabilities of DIH beyond what was possible using prior methods.

© 2019 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]
  50. A. Chengala, M. Hondzo, and J. Sheng, “Microalga propels along vorticity direction in a shear flow,” Phys. Rev. E 87, 052704 (2013).
    [Crossref]
  51. D. Vigolo, S. Radl, and H. A. Stone, “Unexpected trapping of particles at a T junction,” Proc. Natl. Acad. Sci. 111, 4770–4775 (2014).
    [Crossref] [PubMed]
  52. E. Katz, A. L. Yarin, W. Salalha, and E. Zussman, “Alignment and self-assembly of elongated micronsize rods in several flow fields,” J. Appl. Phys. 100, 034313 (2006).
    [Crossref]
  53. S. Parsa, J. S. Guasto, M. Kishore, N. T. Ouellette, J. P. Gollub, and G. A. Voth, “Rotation and alignment of rods in two-dimensional chaotic flow,” Phys. Fluids 23, 043302 (2011).
    [Crossref]
  54. S. Parsa, E. Calzavarini, F. Toschi, and G. A. Voth, “Rotation rate of rods in turbulent fluid flow,” Phys. Rev. Lett. 109, 134501 (2012).
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  55. G. G. Marcus, S. Parsa, S. Kramel, R. Ni, and G. A. Voth, “Measurements of the solid-body rotation of anisotropic particles in 3D turbulence,” New J. Phys. 16, 102001 (2014).
    [Crossref]

2019 (2)

M. Toloui, A. Abraham, and J. Hong, “Experimental investigation of turbulent flow over surfaces of rigid and flexible roughness,” Exp. Therm. Fluid Sci. 101, 263–275 (2019).
[Crossref]

A. Berdeu, O. Flasseur, L. Méès, L. Denis, F. Momey, T. Olivier, N. Grosjean, and C. Fournier, “Reconstruction of in-line holograms: combining model-based and regularized inversion,” Opt. Express 27, 14951(2019).
[Crossref]

2018 (2)

2017 (3)

Y.-C. Wu, A. Shiledar, Y.-C. Li, J. Wong, S. Feng, X. Chen, C. Chen, K. Jin, S. Janamian, Z. Yang, Z. S. Ballard, Z. Göröcs, A. Feizi, and A. Ozcan, “Air quality monitoring using mobile microscopy and machine learning,” Light. Sci. & Appl. 6, e17046 (2017).
[Crossref]

M. Toloui, K. Mallery, and J. Hong, “Improvements on digital inline holographic PTV for 3D wall-bounded turbulent flow measurements,” Meas. Sci. Technol. 28, 044009 (2017).
[Crossref]

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]

2016 (5)

N. Verrier, N. Grosjean, E. Dib, L. Méès, C. Fournier, and J.-L. Marié, “Improvement of the size estimation of 3D tracked droplets using digital in-line holography with joint estimation reconstruction,” Meas. Sci. Technol. 27, 045001 (2016).
[Crossref]

H. Ling, S. Srinivasan, K. Golovin, G. H. McKinley, A. Tuteja, and J. Katz, “High-resolution velocity measurement in the inner part of turbulent boundary layers over super-hydrophobic surfaces,” J. Fluid Mech. 801, 670–703 (2016).
[Crossref]

S. S. Kumar, Y. Sun, S. Zou, and J. Hong, “3D holographic observatory for long-term monitoring of complex behaviors in Drosophila,” Sci. Reports 6, 33001 (2016).
[Crossref]

C. A. Lindensmith, S. Rider, M. Bedrossian, J. K. Wallace, E. Serabyn, G. M. Showalter, J. W. Deming, and J. L. Nadeau, “A submersible, off-axis holographic microscope for detection of microbial motility and morphology in aqueous and icy environments,” Plos One 11, e0147700 (2016).
[Crossref] [PubMed]

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

2015 (2)

M. Toloui and J. Hong, “High fidelity digital inline holographic method for 3D flow measurements,” Opt. Express 23, 27159 (2015).
[Crossref] [PubMed]

M. J. Beals, J. P. Fugal, R. A. Shaw, J. Lu, S. M. Spuler, and J. L. Stith, “Holographic measurements of inhomogeneous cloud mixing at the centimeter scale,” Science 350, 87–90 (2015).
[Crossref] [PubMed]

2014 (7)

M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113, 1–6 (2014).
[Crossref]

X. Yu, J. Hong, C. Liu, and M. K. Kim, “Review of digital holographic microscopy for three-dimensional profiling and tracking,” Opt. Eng. 53, 112306 (2014).
[Crossref]

J. Kühn, B. Niraula, K. Liewer, J. Kent Wallace, E. Serabyn, E. Graff, C. Lindensmith, and J. L. Nadeau, “A Mach-Zender digital holographic microscope with sub-micrometer resolution for imaging and tracking of marine micro-organisms,” Rev. Sci. Instruments 85, 123113 (2014).
[Crossref]

N. Parikh and S. Boyd, “Proximal algorithms,” Foundations Trends Optim. 1, 127–239 (2014).
[Crossref]

D. Vigolo, S. Radl, and H. A. Stone, “Unexpected trapping of particles at a T junction,” Proc. Natl. Acad. Sci. 111, 4770–4775 (2014).
[Crossref] [PubMed]

T. Latychevskaia and H.-W. Fink, “Holographic time-resolved particle tracking by means of three-dimensional volumetric deconvolution,” Opt. Express 22, 20994 (2014).
[Crossref] [PubMed]

G. G. Marcus, S. Parsa, S. Kramel, R. Ni, and G. A. Voth, “Measurements of the solid-body rotation of anisotropic particles in 3D turbulence,” New J. Phys. 16, 102001 (2014).
[Crossref]

2013 (5)

D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-m. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
[Crossref] [PubMed]

M. Seifi, C. Fournier, N. Grosjean, L. Méès, J.-L. Marié, and L. Denis, “Accurate 3D tracking and size measurement of evaporating droplets using in-line digital holography and "inverse problems" reconstruction approach,” Opt. Express 21, 27964 (2013).
[Crossref]

A. Chengala, M. Hondzo, and J. Sheng, “Microalga propels along vorticity direction in a shear flow,” Phys. Rev. E 87, 052704 (2013).
[Crossref]

N. A. Buchmann, C. Atkinson, and J. Soria, “Ultra-high-speed tomographic digital holographic velocimetry in supersonic particle-laden jet flows,” Meas. Sci. Technol. 24, 024005 (2013).
[Crossref]

S. Talapatra and J. Katz, “Three-dimensional velocity measurements in a roughness sublayer using microscopic digital in-line holography and optical index matching,” Meas. Sci. Technol. 24, 024004 (2013).
[Crossref]

2012 (4)

H. Yu, K. Kanov, E. Perlman, J. Graham, E. Frederix, R. Burns, A. Szalay, G. Eyink, and C. Meneveau, “Studying Lagrangian dynamics of turbulence using on-demand fluid particle tracking in a public turbulence database,” J. Turbul. 13, 1–29 (2012).
[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. Wilson and R. Zhang, “3D localization of weak scatterers in digital holographic microscopy using Rayleigh-Sommerfeld back-propagation,” Opt. Express 20, 16735 (2012).
[Crossref]

S. Parsa, E. Calzavarini, F. Toschi, and G. A. Voth, “Rotation rate of rods in turbulent fluid flow,” Phys. Rev. Lett. 109, 134501 (2012).
[Crossref] [PubMed]

2011 (3)

S. Parsa, J. S. Guasto, M. Kishore, N. T. Ouellette, J. P. Gollub, and G. A. Voth, “Rotation and alignment of rods in two-dimensional chaotic flow,” Phys. Fluids 23, 043302 (2011).
[Crossref]

L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic deconvolution microscopy for high-resolution particle tracking,” Opt. Express 19, 16410 (2011).
[Crossref] [PubMed]

H. M. Amaro, A. C. Guedes, and F. X. Malcata, “Advances and perspectives in using microalgae to produce biodiesel,” Appl. Energy 88, 3402–3410 (2011).
[Crossref]

2010 (1)

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42, 531–555 (2010).
[Crossref]

2009 (6)

J. Sheng, E. Malkiel, and J. Katz, “Buffer layer structures associated with extreme wall stress events in a smooth wall turbulent boundary layer,” J. Fluid Mech. 633, 17–60 (2009).
[Crossref]

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm,” SIAM J. on Imaging Sci. 2, 183–202 (2009).
[Crossref]

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

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

M. Kempkes, E. Darakis, T. Khanam, A. Rajendran, V. Kariwala, M. Mazzotti, T. J. Naughton, and A. K. Asundi, “Three dimensional digital holographic profiling of micro-fibers,” Opt. Express 17, 2938–2943 (2009).
[Crossref] [PubMed]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[Crossref] [PubMed]

2008 (1)

Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eyink, “A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence,” J. Turbul. 9, N31 (2008).
[Crossref]

2007 (3)

2006 (1)

E. Katz, A. L. Yarin, W. Salalha, and E. Zussman, “Alignment and self-assembly of elongated micronsize rods in several flow fields,” J. Appl. Phys. 100, 034313 (2006).
[Crossref]

2005 (2)

R. Tibshirani, M. Saunders, S. Rosset, J. Zhu, and K. Knight, “Sparsity and smoothness via the fused lasso,” J. Royal Statiscical Soc. Ser. B 67, 91–108 (2005).
[Crossref]

L. Denis, C. Fournier, T. Fournel, and C. Ducottet, “Twin-image noise reduction by phase retrieval in in-line digital holography,” Proc. SPIE 5914, 59140J (2005).
[Crossref]

2004 (1)

2003 (1)

1999 (1)

V. Kebbel, M. Adams, H.-J. Hartmann, and W. Jüptner, “Digital holography as a versatile optical diagnostic method for microgravity experiments,” Meas. Sci. Technol. 10, 893–899 (1999).
[Crossref]

1996 (1)

J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
[Crossref]

Abraham, A.

M. Toloui, A. Abraham, and J. Hong, “Experimental investigation of turbulent flow over surfaces of rigid and flexible roughness,” Exp. Therm. Fluid Sci. 101, 263–275 (2019).
[Crossref]

Adams, M.

V. Kebbel, M. Adams, H.-J. Hartmann, and W. Jüptner, “Digital holography as a versatile optical diagnostic method for microgravity experiments,” Meas. Sci. Technol. 10, 893–899 (1999).
[Crossref]

Allano, D.

Amaro, H. M.

H. M. Amaro, A. C. Guedes, and F. X. Malcata, “Advances and perspectives in using microalgae to produce biodiesel,” Appl. Energy 88, 3402–3410 (2011).
[Crossref]

Asundi, A. K.

Atkinson, C.

N. A. Buchmann, C. Atkinson, and J. Soria, “Ultra-high-speed tomographic digital holographic velocimetry in supersonic particle-laden jet flows,” Meas. Sci. Technol. 24, 024005 (2013).
[Crossref]

Ballard, Z. S.

Y.-C. Wu, A. Shiledar, Y.-C. Li, J. Wong, S. Feng, X. Chen, C. Chen, K. Jin, S. Janamian, Z. Yang, Z. S. Ballard, Z. Göröcs, A. Feizi, and A. Ozcan, “Air quality monitoring using mobile microscopy and machine learning,” Light. Sci. & Appl. 6, e17046 (2017).
[Crossref]

Baraniuk, R.

T. Goldstein, C. Studer, and R. Baraniuk, “A field guide to forward-backward splitting with a FASTA implementation,” arXiv:1411.3406 p. 25 (2014).

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]

Barry, M.

M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113, 1–6 (2014).
[Crossref]

Beals, M. J.

M. J. Beals, J. P. Fugal, R. A. Shaw, J. Lu, S. M. Spuler, and J. L. Stith, “Holographic measurements of inhomogeneous cloud mixing at the centimeter scale,” Science 350, 87–90 (2015).
[Crossref] [PubMed]

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm,” SIAM J. on Imaging Sci. 2, 183–202 (2009).
[Crossref]

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M. J. Beals, J. P. Fugal, R. A. Shaw, J. Lu, S. M. Spuler, and J. L. Stith, “Holographic measurements of inhomogeneous cloud mixing at the centimeter scale,” Science 350, 87–90 (2015).
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H. M. Amaro, A. C. Guedes, and F. X. Malcata, “Advances and perspectives in using microalgae to produce biodiesel,” Appl. Energy 88, 3402–3410 (2011).
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J. Sheng, E. Malkiel, and J. Katz, “Buffer layer structures associated with extreme wall stress events in a smooth wall turbulent boundary layer,” J. Fluid Mech. 633, 17–60 (2009).
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M. Toloui, K. Mallery, and J. Hong, “Improvements on digital inline holographic PTV for 3D wall-bounded turbulent flow measurements,” Meas. Sci. Technol. 28, 044009 (2017).
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G. G. Marcus, S. Parsa, S. Kramel, R. Ni, and G. A. Voth, “Measurements of the solid-body rotation of anisotropic particles in 3D turbulence,” New J. Phys. 16, 102001 (2014).
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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).
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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).
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Marié, J.-L.

Marks, D. L.

Mazzotti, M.

McKinley, G. H.

H. Ling, S. Srinivasan, K. Golovin, G. H. McKinley, A. Tuteja, and J. Katz, “High-resolution velocity measurement in the inner part of turbulent boundary layers over super-hydrophobic surfaces,” J. Fluid Mech. 801, 670–703 (2016).
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A. Berdeu, O. Flasseur, L. Méès, L. Denis, F. Momey, T. Olivier, N. Grosjean, and C. Fournier, “Reconstruction of in-line holograms: combining model-based and regularized inversion,” Opt. Express 27, 14951(2019).
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F. Jolivet, F. Momey, L. Denis, L. Méès, N. Faure, N. Grosjean, F. Pinston, J.-L. Marié, and C. Fournier, “Regularized reconstruction of absorbing and phase objects from a single in-line hologram, application to fluid mechanics and micro-biology,” Opt. Express 26, 8923 (2018).
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N. Verrier, N. Grosjean, E. Dib, L. Méès, C. Fournier, and J.-L. Marié, “Improvement of the size estimation of 3D tracked droplets using digital in-line holography with joint estimation reconstruction,” Meas. Sci. Technol. 27, 045001 (2016).
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M. Seifi, C. Fournier, N. Grosjean, L. Méès, J.-L. Marié, and L. Denis, “Accurate 3D tracking and size measurement of evaporating droplets using in-line digital holography and "inverse problems" reconstruction approach,” Opt. Express 21, 27964 (2013).
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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).
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H. Yu, K. Kanov, E. Perlman, J. Graham, E. Frederix, R. Burns, A. Szalay, G. Eyink, and C. Meneveau, “Studying Lagrangian dynamics of turbulence using on-demand fluid particle tracking in a public turbulence database,” J. Turbul. 13, 1–29 (2012).
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Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eyink, “A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence,” J. Turbul. 9, N31 (2008).
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E. Perlman, R. Burns, Y. Li, and C. Meneveau, “Data exploration of turbulence simulations using a database cluster,” in Proceedings of the 2007 ACM/IEEE Conference on Supercomputing (SC ’07), (IEEE, 2007).

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Molaei, M.

M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113, 1–6 (2014).
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Nadeau, J. L.

C. A. Lindensmith, S. Rider, M. Bedrossian, J. K. Wallace, E. Serabyn, G. M. Showalter, J. W. Deming, and J. L. Nadeau, “A submersible, off-axis holographic microscope for detection of microbial motility and morphology in aqueous and icy environments,” Plos One 11, e0147700 (2016).
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J. Kühn, B. Niraula, K. Liewer, J. Kent Wallace, E. Serabyn, E. Graff, C. Lindensmith, and J. L. Nadeau, “A Mach-Zender digital holographic microscope with sub-micrometer resolution for imaging and tracking of marine micro-organisms,” Rev. Sci. Instruments 85, 123113 (2014).
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Ni, R.

G. G. Marcus, S. Parsa, S. Kramel, R. Ni, and G. A. Voth, “Measurements of the solid-body rotation of anisotropic particles in 3D turbulence,” New J. Phys. 16, 102001 (2014).
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Niraula, B.

J. Kühn, B. Niraula, K. Liewer, J. Kent Wallace, E. Serabyn, E. Graff, C. Lindensmith, and J. L. Nadeau, “A Mach-Zender digital holographic microscope with sub-micrometer resolution for imaging and tracking of marine micro-organisms,” Rev. Sci. Instruments 85, 123113 (2014).
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Ouellette, N. T.

S. Parsa, J. S. Guasto, M. Kishore, N. T. Ouellette, J. P. Gollub, and G. A. Voth, “Rotation and alignment of rods in two-dimensional chaotic flow,” Phys. Fluids 23, 043302 (2011).
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Y.-C. Wu, A. Shiledar, Y.-C. Li, J. Wong, S. Feng, X. Chen, C. Chen, K. Jin, S. Janamian, Z. Yang, Z. S. Ballard, Z. Göröcs, A. Feizi, and A. Ozcan, “Air quality monitoring using mobile microscopy and machine learning,” Light. Sci. & Appl. 6, e17046 (2017).
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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).
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Parikh, N.

N. Parikh and S. Boyd, “Proximal algorithms,” Foundations Trends Optim. 1, 127–239 (2014).
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G. G. Marcus, S. Parsa, S. Kramel, R. Ni, and G. A. Voth, “Measurements of the solid-body rotation of anisotropic particles in 3D turbulence,” New J. Phys. 16, 102001 (2014).
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S. Parsa, E. Calzavarini, F. Toschi, and G. A. Voth, “Rotation rate of rods in turbulent fluid flow,” Phys. Rev. Lett. 109, 134501 (2012).
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S. Parsa, J. S. Guasto, M. Kishore, N. T. Ouellette, J. P. Gollub, and G. A. Voth, “Rotation and alignment of rods in two-dimensional chaotic flow,” Phys. Fluids 23, 043302 (2011).
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H. Yu, K. Kanov, E. Perlman, J. Graham, E. Frederix, R. Burns, A. Szalay, G. Eyink, and C. Meneveau, “Studying Lagrangian dynamics of turbulence using on-demand fluid particle tracking in a public turbulence database,” J. Turbul. 13, 1–29 (2012).
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Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eyink, “A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence,” J. Turbul. 9, N31 (2008).
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E. Perlman, R. Burns, Y. Li, and C. Meneveau, “Data exploration of turbulence simulations using a database cluster,” in Proceedings of the 2007 ACM/IEEE Conference on Supercomputing (SC ’07), (IEEE, 2007).

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D. Vigolo, S. Radl, and H. A. Stone, “Unexpected trapping of particles at a T junction,” Proc. Natl. Acad. Sci. 111, 4770–4775 (2014).
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Rider, S.

C. A. Lindensmith, S. Rider, M. Bedrossian, J. K. Wallace, E. Serabyn, G. M. Showalter, J. W. Deming, and J. L. Nadeau, “A submersible, off-axis holographic microscope for detection of microbial motility and morphology in aqueous and icy environments,” Plos One 11, e0147700 (2016).
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R. Tibshirani, M. Saunders, S. Rosset, J. Zhu, and K. Knight, “Sparsity and smoothness via the fused lasso,” J. Royal Statiscical Soc. Ser. B 67, 91–108 (2005).
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E. Katz, A. L. Yarin, W. Salalha, and E. Zussman, “Alignment and self-assembly of elongated micronsize rods in several flow fields,” J. Appl. Phys. 100, 034313 (2006).
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R. Tibshirani, M. Saunders, S. Rosset, J. Zhu, and K. Knight, “Sparsity and smoothness via the fused lasso,” J. Royal Statiscical Soc. Ser. B 67, 91–108 (2005).
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Serabyn, E.

C. A. Lindensmith, S. Rider, M. Bedrossian, J. K. Wallace, E. Serabyn, G. M. Showalter, J. W. Deming, and J. L. Nadeau, “A submersible, off-axis holographic microscope for detection of microbial motility and morphology in aqueous and icy environments,” Plos One 11, e0147700 (2016).
[Crossref] [PubMed]

J. Kühn, B. Niraula, K. Liewer, J. Kent Wallace, E. Serabyn, E. Graff, C. Lindensmith, and J. L. Nadeau, “A Mach-Zender digital holographic microscope with sub-micrometer resolution for imaging and tracking of marine micro-organisms,” Rev. Sci. Instruments 85, 123113 (2014).
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M. J. Beals, J. P. Fugal, R. A. Shaw, J. Lu, S. M. Spuler, and J. L. Stith, “Holographic measurements of inhomogeneous cloud mixing at the centimeter scale,” Science 350, 87–90 (2015).
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M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113, 1–6 (2014).
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J. Sheng, E. Malkiel, and J. Katz, “Buffer layer structures associated with extreme wall stress events in a smooth wall turbulent boundary layer,” J. Fluid Mech. 633, 17–60 (2009).
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Y.-C. Wu, A. Shiledar, Y.-C. Li, J. Wong, S. Feng, X. Chen, C. Chen, K. Jin, S. Janamian, Z. Yang, Z. S. Ballard, Z. Göröcs, A. Feizi, and A. Ozcan, “Air quality monitoring using mobile microscopy and machine learning,” Light. Sci. & Appl. 6, e17046 (2017).
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Showalter, G. M.

C. A. Lindensmith, S. Rider, M. Bedrossian, J. K. Wallace, E. Serabyn, G. M. Showalter, J. W. Deming, and J. L. Nadeau, “A submersible, off-axis holographic microscope for detection of microbial motility and morphology in aqueous and icy environments,” Plos One 11, e0147700 (2016).
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N. A. Buchmann, C. Atkinson, and J. Soria, “Ultra-high-speed tomographic digital holographic velocimetry in supersonic particle-laden jet flows,” Meas. Sci. Technol. 24, 024005 (2013).
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Spuler, S. M.

M. J. Beals, J. P. Fugal, R. A. Shaw, J. Lu, S. M. Spuler, and J. L. Stith, “Holographic measurements of inhomogeneous cloud mixing at the centimeter scale,” Science 350, 87–90 (2015).
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Srinivasan, S.

H. Ling, S. Srinivasan, K. Golovin, G. H. McKinley, A. Tuteja, and J. Katz, “High-resolution velocity measurement in the inner part of turbulent boundary layers over super-hydrophobic surfaces,” J. Fluid Mech. 801, 670–703 (2016).
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M. J. Beals, J. P. Fugal, R. A. Shaw, J. Lu, S. M. Spuler, and J. L. Stith, “Holographic measurements of inhomogeneous cloud mixing at the centimeter scale,” Science 350, 87–90 (2015).
[Crossref] [PubMed]

Stocker, R.

M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113, 1–6 (2014).
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Stone, H. A.

D. Vigolo, S. Radl, and H. A. Stone, “Unexpected trapping of particles at a T junction,” Proc. Natl. Acad. Sci. 111, 4770–4775 (2014).
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S. S. Kumar, Y. Sun, S. Zou, and J. Hong, “3D holographic observatory for long-term monitoring of complex behaviors in Drosophila,” Sci. Reports 6, 33001 (2016).
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H. Yu, K. Kanov, E. Perlman, J. Graham, E. Frederix, R. Burns, A. Szalay, G. Eyink, and C. Meneveau, “Studying Lagrangian dynamics of turbulence using on-demand fluid particle tracking in a public turbulence database,” J. Turbul. 13, 1–29 (2012).
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Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eyink, “A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence,” J. Turbul. 9, N31 (2008).
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R. Tibshirani, M. Saunders, S. Rosset, J. Zhu, and K. Knight, “Sparsity and smoothness via the fused lasso,” J. Royal Statiscical Soc. Ser. B 67, 91–108 (2005).
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M. Toloui, A. Abraham, and J. Hong, “Experimental investigation of turbulent flow over surfaces of rigid and flexible roughness,” Exp. Therm. Fluid Sci. 101, 263–275 (2019).
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M. Toloui, K. Mallery, and J. Hong, “Improvements on digital inline holographic PTV for 3D wall-bounded turbulent flow measurements,” Meas. Sci. Technol. 28, 044009 (2017).
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M. Toloui and J. Hong, “High fidelity digital inline holographic method for 3D flow measurements,” Opt. Express 23, 27159 (2015).
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S. Parsa, E. Calzavarini, F. Toschi, and G. A. Voth, “Rotation rate of rods in turbulent fluid flow,” Phys. Rev. Lett. 109, 134501 (2012).
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L. Denis, D. Lorenz, E. Thiébaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. letters 34, 3475–3477 (2009).
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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).
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H. Ling, S. Srinivasan, K. Golovin, G. H. McKinley, A. Tuteja, and J. Katz, “High-resolution velocity measurement in the inner part of turbulent boundary layers over super-hydrophobic surfaces,” J. Fluid Mech. 801, 670–703 (2016).
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N. Verrier, N. Grosjean, E. Dib, L. Méès, C. Fournier, and J.-L. Marié, “Improvement of the size estimation of 3D tracked droplets using digital in-line holography with joint estimation reconstruction,” Meas. Sci. Technol. 27, 045001 (2016).
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D. Vigolo, S. Radl, and H. A. Stone, “Unexpected trapping of particles at a T junction,” Proc. Natl. Acad. Sci. 111, 4770–4775 (2014).
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Voth, G. A.

G. G. Marcus, S. Parsa, S. Kramel, R. Ni, and G. A. Voth, “Measurements of the solid-body rotation of anisotropic particles in 3D turbulence,” New J. Phys. 16, 102001 (2014).
[Crossref]

S. Parsa, E. Calzavarini, F. Toschi, and G. A. Voth, “Rotation rate of rods in turbulent fluid flow,” Phys. Rev. Lett. 109, 134501 (2012).
[Crossref] [PubMed]

S. Parsa, J. S. Guasto, M. Kishore, N. T. Ouellette, J. P. Gollub, and G. A. Voth, “Rotation and alignment of rods in two-dimensional chaotic flow,” Phys. Fluids 23, 043302 (2011).
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Wallace, J. K.

C. A. Lindensmith, S. Rider, M. Bedrossian, J. K. Wallace, E. Serabyn, G. M. Showalter, J. W. Deming, and J. L. Nadeau, “A submersible, off-axis holographic microscope for detection of microbial motility and morphology in aqueous and icy environments,” Plos One 11, e0147700 (2016).
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Walle, F.

Wan, M.

Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eyink, “A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence,” J. Turbul. 9, N31 (2008).
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Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eyink, “A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence,” J. Turbul. 9, N31 (2008).
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Y.-C. Wu, A. Shiledar, Y.-C. Li, J. Wong, S. Feng, X. Chen, C. Chen, K. Jin, S. Janamian, Z. Yang, Z. S. Ballard, Z. Göröcs, A. Feizi, and A. Ozcan, “Air quality monitoring using mobile microscopy and machine learning,” Light. Sci. & Appl. 6, e17046 (2017).
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E. Katz, A. L. Yarin, W. Salalha, and E. Zussman, “Alignment and self-assembly of elongated micronsize rods in several flow fields,” J. Appl. Phys. 100, 034313 (2006).
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H. Yu, K. Kanov, E. Perlman, J. Graham, E. Frederix, R. Burns, A. Szalay, G. Eyink, and C. Meneveau, “Studying Lagrangian dynamics of turbulence using on-demand fluid particle tracking in a public turbulence database,” J. Turbul. 13, 1–29 (2012).
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R. Tibshirani, M. Saunders, S. Rosset, J. Zhu, and K. Knight, “Sparsity and smoothness via the fused lasso,” J. Royal Statiscical Soc. Ser. B 67, 91–108 (2005).
[Crossref]

Zou, S.

S. S. Kumar, Y. Sun, S. Zou, and J. Hong, “3D holographic observatory for long-term monitoring of complex behaviors in Drosophila,” Sci. Reports 6, 33001 (2016).
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Zussman, E.

E. Katz, A. L. Yarin, W. Salalha, and E. Zussman, “Alignment and self-assembly of elongated micronsize rods in several flow fields,” J. Appl. Phys. 100, 034313 (2006).
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Annu. Rev. Fluid Mech. (1)

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42, 531–555 (2010).
[Crossref]

Appl. Energy (1)

H. M. Amaro, A. C. Guedes, and F. X. Malcata, “Advances and perspectives in using microalgae to produce biodiesel,” Appl. Energy 88, 3402–3410 (2011).
[Crossref]

Appl. Opt. (2)

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]

Exp. Therm. Fluid Sci. (1)

M. Toloui, A. Abraham, and J. Hong, “Experimental investigation of turbulent flow over surfaces of rigid and flexible roughness,” Exp. Therm. Fluid Sci. 101, 263–275 (2019).
[Crossref]

Foundations Trends Optim. (1)

N. Parikh and S. Boyd, “Proximal algorithms,” Foundations Trends Optim. 1, 127–239 (2014).
[Crossref]

IEEE Transactions on Image Process. (1)

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

J. Appl. Phys. (1)

E. Katz, A. L. Yarin, W. Salalha, and E. Zussman, “Alignment and self-assembly of elongated micronsize rods in several flow fields,” J. Appl. Phys. 100, 034313 (2006).
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J. Colloid Interface Sci. (1)

J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996).
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J. Fluid Mech. (2)

J. Sheng, E. Malkiel, and J. Katz, “Buffer layer structures associated with extreme wall stress events in a smooth wall turbulent boundary layer,” J. Fluid Mech. 633, 17–60 (2009).
[Crossref]

H. Ling, S. Srinivasan, K. Golovin, G. H. McKinley, A. Tuteja, and J. Katz, “High-resolution velocity measurement in the inner part of turbulent boundary layers over super-hydrophobic surfaces,” J. Fluid Mech. 801, 670–703 (2016).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Royal Statiscical Soc. Ser. B (1)

R. Tibshirani, M. Saunders, S. Rosset, J. Zhu, and K. Knight, “Sparsity and smoothness via the fused lasso,” J. Royal Statiscical Soc. Ser. B 67, 91–108 (2005).
[Crossref]

J. Turbul. (2)

H. Yu, K. Kanov, E. Perlman, J. Graham, E. Frederix, R. Burns, A. Szalay, G. Eyink, and C. Meneveau, “Studying Lagrangian dynamics of turbulence using on-demand fluid particle tracking in a public turbulence database,” J. Turbul. 13, 1–29 (2012).
[Crossref]

Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eyink, “A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence,” J. Turbul. 9, N31 (2008).
[Crossref]

Light. Sci. & Appl. (1)

Y.-C. Wu, A. Shiledar, Y.-C. Li, J. Wong, S. Feng, X. Chen, C. Chen, K. Jin, S. Janamian, Z. Yang, Z. S. Ballard, Z. Göröcs, A. Feizi, and A. Ozcan, “Air quality monitoring using mobile microscopy and machine learning,” Light. Sci. & Appl. 6, e17046 (2017).
[Crossref]

Meas. Sci. Technol. (5)

M. Toloui, K. Mallery, and J. Hong, “Improvements on digital inline holographic PTV for 3D wall-bounded turbulent flow measurements,” Meas. Sci. Technol. 28, 044009 (2017).
[Crossref]

V. Kebbel, M. Adams, H.-J. Hartmann, and W. Jüptner, “Digital holography as a versatile optical diagnostic method for microgravity experiments,” Meas. Sci. Technol. 10, 893–899 (1999).
[Crossref]

N. A. Buchmann, C. Atkinson, and J. Soria, “Ultra-high-speed tomographic digital holographic velocimetry in supersonic particle-laden jet flows,” Meas. Sci. Technol. 24, 024005 (2013).
[Crossref]

N. Verrier, N. Grosjean, E. Dib, L. Méès, C. Fournier, and J.-L. Marié, “Improvement of the size estimation of 3D tracked droplets using digital in-line holography with joint estimation reconstruction,” Meas. Sci. Technol. 27, 045001 (2016).
[Crossref]

S. Talapatra and J. Katz, “Three-dimensional velocity measurements in a roughness sublayer using microscopic digital in-line holography and optical index matching,” Meas. Sci. Technol. 24, 024004 (2013).
[Crossref]

New J. Phys. (2)

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]

G. G. Marcus, S. Parsa, S. Kramel, R. Ni, and G. A. Voth, “Measurements of the solid-body rotation of anisotropic particles in 3D turbulence,” New J. Phys. 16, 102001 (2014).
[Crossref]

Opt. Eng. (1)

X. Yu, J. Hong, C. Liu, and M. K. Kim, “Review of digital holographic microscopy for three-dimensional profiling and tracking,” Opt. Eng. 53, 112306 (2014).
[Crossref]

Opt. Express (12)

M. Seifi, C. Fournier, N. Grosjean, L. Méès, J.-L. Marié, and L. Denis, “Accurate 3D tracking and size measurement of evaporating droplets using in-line digital holography and "inverse problems" reconstruction approach,” Opt. Express 21, 27964 (2013).
[Crossref]

J. Gao and J. Katz, “Self-calibrated microscopic dual-view tomographic holography for 3D flow measurements,” Opt. Express 26, 16708–16725 (2018).
[Crossref] [PubMed]

M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12, 2270–2279 (2004).
[Crossref] [PubMed]

L. Wilson and R. Zhang, “3D localization of weak scatterers in digital holographic microscopy using Rayleigh-Sommerfeld back-propagation,” Opt. Express 20, 16735 (2012).
[Crossref]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[Crossref] [PubMed]

L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic deconvolution microscopy for high-resolution particle tracking,” Opt. Express 19, 16410 (2011).
[Crossref] [PubMed]

T. Latychevskaia and H.-W. Fink, “Holographic time-resolved particle tracking by means of three-dimensional volumetric deconvolution,” Opt. Express 22, 20994 (2014).
[Crossref] [PubMed]

M. Toloui and J. Hong, “High fidelity digital inline holographic method for 3D flow measurements,” Opt. Express 23, 27159 (2015).
[Crossref] [PubMed]

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

F. Jolivet, F. Momey, L. Denis, L. Méès, N. Faure, N. Grosjean, F. Pinston, J.-L. Marié, and C. Fournier, “Regularized reconstruction of absorbing and phase objects from a single in-line hologram, application to fluid mechanics and micro-biology,” Opt. Express 26, 8923 (2018).
[Crossref] [PubMed]

A. Berdeu, O. Flasseur, L. Méès, L. Denis, F. Momey, T. Olivier, N. Grosjean, and C. Fournier, “Reconstruction of in-line holograms: combining model-based and regularized inversion,” Opt. Express 27, 14951(2019).
[Crossref]

M. Kempkes, E. Darakis, T. Khanam, A. Rajendran, V. Kariwala, M. Mazzotti, T. J. Naughton, and A. K. Asundi, “Three dimensional digital holographic profiling of micro-fibers,” Opt. Express 17, 2938–2943 (2009).
[Crossref] [PubMed]

Opt. letters (1)

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

Phys. Fluids (1)

S. Parsa, J. S. Guasto, M. Kishore, N. T. Ouellette, J. P. Gollub, and G. A. Voth, “Rotation and alignment of rods in two-dimensional chaotic flow,” Phys. Fluids 23, 043302 (2011).
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Phys. Rev. E (1)

A. Chengala, M. Hondzo, and J. Sheng, “Microalga propels along vorticity direction in a shear flow,” Phys. Rev. E 87, 052704 (2013).
[Crossref]

Phys. Rev. Lett. (3)

S. Parsa, E. Calzavarini, F. Toschi, and G. A. Voth, “Rotation rate of rods in turbulent fluid flow,” Phys. Rev. Lett. 109, 134501 (2012).
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T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
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M. Molaei, M. Barry, R. Stocker, and J. Sheng, “Failed escape: solid surfaces prevent tumbling of Escherichia coli,” Phys. Rev. Lett. 113, 1–6 (2014).
[Crossref]

Plos One (1)

C. A. Lindensmith, S. Rider, M. Bedrossian, J. K. Wallace, E. Serabyn, G. M. Showalter, J. W. Deming, and J. L. Nadeau, “A submersible, off-axis holographic microscope for detection of microbial motility and morphology in aqueous and icy environments,” Plos One 11, e0147700 (2016).
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Proc. Natl. Acad. Sci. (1)

D. Vigolo, S. Radl, and H. A. Stone, “Unexpected trapping of particles at a T junction,” Proc. Natl. Acad. Sci. 111, 4770–4775 (2014).
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Proc. SPIE (1)

L. Denis, C. Fournier, T. Fournel, and C. Ducottet, “Twin-image noise reduction by phase retrieval in in-line digital holography,” Proc. SPIE 5914, 59140J (2005).
[Crossref]

Rev. Sci. Instruments (1)

J. Kühn, B. Niraula, K. Liewer, J. Kent Wallace, E. Serabyn, E. Graff, C. Lindensmith, and J. L. Nadeau, “A Mach-Zender digital holographic microscope with sub-micrometer resolution for imaging and tracking of marine micro-organisms,” Rev. Sci. Instruments 85, 123113 (2014).
[Crossref]

Sci. Reports (1)

S. S. Kumar, Y. Sun, S. Zou, and J. Hong, “3D holographic observatory for long-term monitoring of complex behaviors in Drosophila,” Sci. Reports 6, 33001 (2016).
[Crossref]

Science (1)

M. J. Beals, J. P. Fugal, R. A. Shaw, J. Lu, S. M. Spuler, and J. L. Stith, “Holographic measurements of inhomogeneous cloud mixing at the centimeter scale,” Science 350, 87–90 (2015).
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SIAM J. on Imaging Sci. (1)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm,” SIAM J. on Imaging Sci. 2, 183–202 (2009).
[Crossref]

Other (4)

T. Goldstein, C. Studer, and R. Baraniuk, “A field guide to forward-backward splitting with a FASTA implementation,” arXiv:1411.3406 p. 25 (2014).

T.-C. Poon and J.-P. Liu, Introduction to modern igital holography with MATLAB(Cambridge University, 2014).

N. Burns and J. Watson, “Data extraction from underwater holograms of marine organisms,” in OCEANS 2007 - Europe, (IEEE, 2007), pp. 1–6.

E. Perlman, R. Burns, Y. Li, and C. Meneveau, “Data exploration of turbulence simulations using a database cluster,” in Proceedings of the 2007 ACM/IEEE Conference on Supercomputing (SC ’07), (IEEE, 2007).

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Figures (7)

Fig. 1
Fig. 1 (a) Recorded holographic image of a 90 nm Ag nanowire. (b) Hologram after image enhancement. (c) Volumetric reconstruction of the sample using the deconvolution method. (d) Reconstruction using RIHVR with sparsity ( l 1) regularization. (e)Reconstruction using RIHVR with fused lasso (FL) regularization. For visualization, (c) uses the intensity as the transparency alpha value while (d) and (e) show all non-zero values at equal intensity.
Fig. 2
Fig. 2 Sample simulated hologram containing 1000 particles
Fig. 3
Fig. 3 (a) Smoothed 3D particle trajectories extracted from a synthetic hologram using RIHVR. (b) Localization error of tracked particles relative to their true locations. (c) Error in the RMS velocity components of the three test methods compared to ground truth.
Fig. 4
Fig. 4 (a) Extraction rate for each method for increasing particle concentrations. (b) The number of particles which can be accurately extracted is higher for RIHVR than the other methods. Solid line is 100% EP, dashed line is 50%.
Fig. 5
Fig. 5 (a) DIH imaging system, recorded hologram, XY projection of processed reconstruction. Scale bars are 100 μm. (b) Subset (25%) of tracked particles. (c) PDF of velocity fluctuations in each direction. The w distribution has longer tails and a sharper peak but is not substantially wider than the other two components.
Fig. 6
Fig. 6 2D view of the reconstructed cell trajectories showing complex behaviors Also illustrates the true cell concentration of processed volume. Scale bars are 100 μm.
Fig. 7
Fig. 7 (a) Visualization of the measured rod trajectories. Solid lines show the measured 3D orientation of the rods. Colors indicate individual particles. Vorticity isosurfaces (ω = 3000s1) are from the CFD simulation. (b) View of the experimental fiber orientations in the yz plane. (c) Contour map of the particle rotation rate (s1) expected from the simulation. (d) Measured 3D particle rotation rate.

Equations (11)

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

s d = n s L d 2
x ^ = arg min x { H x b 2 2 + λ R ( x ) f ( x ) + g ( x ) }
H x = ( 1 N z z = 1 N z e j k 0 z x z * h z )
F T { h z ( x , y ) } = H z ( m , n ) = exp [ j k 0 z 1 ( k 0 m 2 π M s ) 2 ( k 0 n 2 π N s ) 2 ]
x k = p r o x λ L ( x k 1 L f ( x k 1 ) )
p r o x L ( v ) = arg min x ( g ( x ) + 1 2 L x v 2 2 ) v L g ( x )
R ( x ) = x 1 = i = 1 N | x i |
R ( x ) = x T V = i = 1 N x j = 1 N y ( x i , j x i 1 , j ) 2 + ( x i , j x i , j 1 ) 2
g ( x ) = λ R ( x ) = λ l 1 x 1 + λ T V x T V
p r o x λ L ( v ) = ( 1 λ L | v | ) + s i g n ( v )
p ˙ i = Ω i j p j + ( S i j p j p i p j S j k p k )

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