Abstract

One advantage of digital in-line holography is the ability for a user to know the 3-D location of a moving particle recorded at a given time. When the time exposure is much larger than the time required for grabbing the particle image at a given location, the diffraction pattern is spread along the trajectory of this particle. This can be seen as a convolution between the diffraction pattern and a blurring function resulting from the motion of the particle during the camera exposure. This article shows that the reconstruction of holograms recorded under such conditions exhibit traces that could be processed for extracting 3D trajectories.

© 2013 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  26. F. Slimani, G. Gréhan, G. Gouesbet, and D. Allano, “Near-field Lorenz-Mie theory and its application to microholography,” Appl. Opt.23(22), 4140–4148 (1984).
    [CrossRef] [PubMed]
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    [CrossRef]

2013

2012

2011

2009

M. Heydt, P. Divós, M. Grunze, and A. Rosenhahn, “Analysis of holographic microscopy data to quantitatively investigate three-dimensional settlement dynamics of algal zoospores in the vicinity of surfaces,” Eur Phys J E Soft Matter30(2), 141–148 (2009).
[CrossRef] [PubMed]

2008

N. Salah, G. Godard, D. Lebrun, P. Paranthoen, D. Allano, and S. Coetmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard-von Karman vortex flow,” Meas. Sci. Technol.19(7), 074001 (2008).
[CrossRef]

2007

2006

2005

S. Pu, D. Lebrun, D. Allano, B. Patte-Rouland, M. Malek, and C. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids39(1), 1–9 (2005).
[CrossRef]

2004

H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol.15(4), 673–685 (2004).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol.15(4), 699–705 (2004).
[CrossRef]

2003

2002

2000

C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: three dimensional location of particles,” Opt. Lasers Eng.33(6), 409–421 (2000).
[CrossRef]

1996

1994

1993

1984

1967

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett.11(3), 77–79 (1967).
[CrossRef]

1958

G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat.29(2), 610–611 (1958).
[CrossRef]

Allano, D.

D. Lebrun, D. Allano, L. Méès, F. Walle, F. Corbin, R. Boucheron, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt.50(34), H1–H9 (2011).
[CrossRef] [PubMed]

N. Salah, G. Godard, D. Lebrun, P. Paranthoen, D. Allano, and S. Coetmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard-von Karman vortex flow,” Meas. Sci. Technol.19(7), 074001 (2008).
[CrossRef]

S. Pu, D. Lebrun, D. Allano, B. Patte-Rouland, M. Malek, and C. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids39(1), 1–9 (2005).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol.15(4), 699–705 (2004).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, and D. Allano, “Near-field Lorenz-Mie theory and its application to microholography,” Appl. Opt.23(22), 4140–4148 (1984).
[CrossRef] [PubMed]

Benkouider, A. M.

Boucheron, R.

Box, G. E. P.

G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat.29(2), 610–611 (1958).
[CrossRef]

Bruel, L.

F. Lamadie, L. Bruel, and M. Himbert, “Digital holographic measurements of liquid-liquid two-phase flows,” Opt. Lasers Eng.50(12), 1716–1725 (2012).
[CrossRef]

Brunel, M.

Buraga-Lefebvre, C.

C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: three dimensional location of particles,” Opt. Lasers Eng.33(6), 409–421 (2000).
[CrossRef]

Callens, N.

Cen, C.

S. Pu, D. Lebrun, D. Allano, B. Patte-Rouland, M. Malek, and C. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids39(1), 1–9 (2005).
[CrossRef]

Cheong, F. C.

Coetmellec, S.

N. Salah, G. Godard, D. Lebrun, P. Paranthoen, D. Allano, and S. Coetmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard-von Karman vortex flow,” Meas. Sci. Technol.19(7), 074001 (2008).
[CrossRef]

Coëtmellec, S.

M. Brunel, H. Shen, S. Coëtmellec, and D. Lebrun, “Extended ABCD matrix formalism for the description of femtosecond diffraction patterns; application to femtosecond Digital In-line Holography with anamorphic optical systems,” Appl. Opt.51(8), 1137–1148 (2012).
[CrossRef] [PubMed]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital In-line holography with a sub-picosecond laser beam,” Opt. Commun.268(1), 27–33 (2006).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol.15(4), 699–705 (2004).
[CrossRef]

D. Lebrun, A. M. Benkouider, S. Coëtmellec, and M. Malek, “Particle field digital holographic reconstruction in arbitrary tilted planes,” Opt. Express11(3), 224–229 (2003).
[CrossRef] [PubMed]

S. Coëtmellec, D. Lebrun, and C. Özkul, “Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier Transform,” Appl. Opt.41(2), 312–319 (2002).
[CrossRef] [PubMed]

C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: three dimensional location of particles,” Opt. Lasers Eng.33(6), 409–421 (2000).
[CrossRef]

Corbin, F.

Denis, L.

Divós, P.

M. Heydt, P. Divós, M. Grunze, and A. Rosenhahn, “Analysis of holographic microscopy data to quantitatively investigate three-dimensional settlement dynamics of algal zoospores in the vicinity of surfaces,” Eur Phys J E Soft Matter30(2), 141–148 (2009).
[CrossRef] [PubMed]

Dixon, L.

Dubois, F.

El Mallahi, A.

Fournier, C.

Fréchou, D.

Garcia-Sucerquia, J.

Godard, G.

N. Salah, G. Godard, D. Lebrun, P. Paranthoen, D. Allano, and S. Coetmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard-von Karman vortex flow,” Meas. Sci. Technol.19(7), 074001 (2008).
[CrossRef]

Goepfert, C.

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett.11(3), 77–79 (1967).
[CrossRef]

Gouesbet, G.

Gréhan, G.

Grier, D. G.

Grunze, M.

M. Heydt, P. Divós, M. Grunze, and A. Rosenhahn, “Analysis of holographic microscopy data to quantitatively investigate three-dimensional settlement dynamics of algal zoospores in the vicinity of surfaces,” Eur Phys J E Soft Matter30(2), 141–148 (2009).
[CrossRef] [PubMed]

Heydt, M.

M. Heydt, P. Divós, M. Grunze, and A. Rosenhahn, “Analysis of holographic microscopy data to quantitatively investigate three-dimensional settlement dynamics of algal zoospores in the vicinity of surfaces,” Eur Phys J E Soft Matter30(2), 141–148 (2009).
[CrossRef] [PubMed]

Himbert, M.

F. Lamadie, L. Bruel, and M. Himbert, “Digital holographic measurements of liquid-liquid two-phase flows,” Opt. Lasers Eng.50(12), 1716–1725 (2012).
[CrossRef]

Hoyos, M.

Jericho, M. H.

Jericho, S. K.

Jüptner, W.

Klages, P.

Kreuzer, H. J.

Kurowski, P.

Lamadie, F.

F. Lamadie, L. Bruel, and M. Himbert, “Digital holographic measurements of liquid-liquid two-phase flows,” Opt. Lasers Eng.50(12), 1716–1725 (2012).
[CrossRef]

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett.11(3), 77–79 (1967).
[CrossRef]

Lebrun, D.

M. Brunel, H. Shen, S. Coëtmellec, and D. Lebrun, “Extended ABCD matrix formalism for the description of femtosecond diffraction patterns; application to femtosecond Digital In-line Holography with anamorphic optical systems,” Appl. Opt.51(8), 1137–1148 (2012).
[CrossRef] [PubMed]

D. Lebrun, D. Allano, L. Méès, F. Walle, F. Corbin, R. Boucheron, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt.50(34), H1–H9 (2011).
[CrossRef] [PubMed]

N. Salah, G. Godard, D. Lebrun, P. Paranthoen, D. Allano, and S. Coetmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard-von Karman vortex flow,” Meas. Sci. Technol.19(7), 074001 (2008).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital In-line holography with a sub-picosecond laser beam,” Opt. Commun.268(1), 27–33 (2006).
[CrossRef]

S. Pu, D. Lebrun, D. Allano, B. Patte-Rouland, M. Malek, and C. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids39(1), 1–9 (2005).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol.15(4), 699–705 (2004).
[CrossRef]

D. Lebrun, A. M. Benkouider, S. Coëtmellec, and M. Malek, “Particle field digital holographic reconstruction in arbitrary tilted planes,” Opt. Express11(3), 224–229 (2003).
[CrossRef] [PubMed]

S. Coëtmellec, D. Lebrun, and C. Özkul, “Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier Transform,” Appl. Opt.41(2), 312–319 (2002).
[CrossRef] [PubMed]

C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: three dimensional location of particles,” Opt. Lasers Eng.33(6), 409–421 (2000).
[CrossRef]

D. Lebrun, C. E. Touil, and C. Özkul, “Methods for the deconvolution of defocused-image pairs recorded separately on two CCD cameras: application to particle sizing,” Appl. Opt.35(32), 6375–6381 (1996).
[CrossRef] [PubMed]

Malek, M.

S. Pu, D. Lebrun, D. Allano, B. Patte-Rouland, M. Malek, and C. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids39(1), 1–9 (2005).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol.15(4), 699–705 (2004).
[CrossRef]

D. Lebrun, A. M. Benkouider, S. Coëtmellec, and M. Malek, “Particle field digital holographic reconstruction in arbitrary tilted planes,” Opt. Express11(3), 224–229 (2003).
[CrossRef] [PubMed]

Méès, L.

Meinertzhagen, I. A.

Meng, H.

H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol.15(4), 673–685 (2004).
[CrossRef]

G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt.42(5), 827–833 (2003).
[CrossRef] [PubMed]

Minetti, C.

Monnom, O.

Muller, M. E.

G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat.29(2), 610–611 (1958).
[CrossRef]

Nicolas, F.

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital In-line holography with a sub-picosecond laser beam,” Opt. Commun.268(1), 27–33 (2006).
[CrossRef]

Onural, L.

Özkul, C.

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol.15(4), 699–705 (2004).
[CrossRef]

S. Coëtmellec, D. Lebrun, and C. Özkul, “Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier Transform,” Appl. Opt.41(2), 312–319 (2002).
[CrossRef] [PubMed]

C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: three dimensional location of particles,” Opt. Lasers Eng.33(6), 409–421 (2000).
[CrossRef]

D. Lebrun, C. E. Touil, and C. Özkul, “Methods for the deconvolution of defocused-image pairs recorded separately on two CCD cameras: application to particle sizing,” Appl. Opt.35(32), 6375–6381 (1996).
[CrossRef] [PubMed]

Pan, G.

H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol.15(4), 673–685 (2004).
[CrossRef]

G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt.42(5), 827–833 (2003).
[CrossRef] [PubMed]

Paranthoen, P.

N. Salah, G. Godard, D. Lebrun, P. Paranthoen, D. Allano, and S. Coetmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard-von Karman vortex flow,” Meas. Sci. Technol.19(7), 074001 (2008).
[CrossRef]

Patte-Rouland, B.

S. Pu, D. Lebrun, D. Allano, B. Patte-Rouland, M. Malek, and C. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids39(1), 1–9 (2005).
[CrossRef]

Pu, S.

S. Pu, D. Lebrun, D. Allano, B. Patte-Rouland, M. Malek, and C. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids39(1), 1–9 (2005).
[CrossRef]

Pu, Y.

H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol.15(4), 673–685 (2004).
[CrossRef]

Restrepo, J. F.

Rosenhahn, A.

M. Heydt, P. Divós, M. Grunze, and A. Rosenhahn, “Analysis of holographic microscopy data to quantitatively investigate three-dimensional settlement dynamics of algal zoospores in the vicinity of surfaces,” Eur Phys J E Soft Matter30(2), 141–148 (2009).
[CrossRef] [PubMed]

Salah, N.

N. Salah, G. Godard, D. Lebrun, P. Paranthoen, D. Allano, and S. Coetmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard-von Karman vortex flow,” Meas. Sci. Technol.19(7), 074001 (2008).
[CrossRef]

Schnars, U.

Shen, H.

Slimani, F.

Soulez, F.

Thiébaut, E.

Touil, C. E.

Walle, F.

Woodward, S. H.

H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol.15(4), 673–685 (2004).
[CrossRef]

Xu, W.

Yourassowsky, C.

Ann. Math. Stat.

G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat.29(2), 610–611 (1958).
[CrossRef]

Appl. Opt.

F. Slimani, G. Gréhan, G. Gouesbet, and D. Allano, “Near-field Lorenz-Mie theory and its application to microholography,” Appl. Opt.23(22), 4140–4148 (1984).
[CrossRef] [PubMed]

U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt.33(2), 179–181 (1994).
[CrossRef] [PubMed]

D. Lebrun, C. E. Touil, and C. Özkul, “Methods for the deconvolution of defocused-image pairs recorded separately on two CCD cameras: application to particle sizing,” Appl. Opt.35(32), 6375–6381 (1996).
[CrossRef] [PubMed]

S. Coëtmellec, D. Lebrun, and C. Özkul, “Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier Transform,” Appl. Opt.41(2), 312–319 (2002).
[CrossRef] [PubMed]

G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt.42(5), 827–833 (2003).
[CrossRef] [PubMed]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt.45(5), 836–850 (2006).
[CrossRef] [PubMed]

F. Dubois, N. Callens, C. Yourassowsky, M. Hoyos, P. Kurowski, and O. Monnom, “Digital holographic microscopy with reduced spatial coherence for three-dimensional particle flow analysis,” Appl. Opt.45(5), 864–871 (2006).
[CrossRef] [PubMed]

D. Lebrun, D. Allano, L. Méès, F. Walle, F. Corbin, R. Boucheron, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt.50(34), H1–H9 (2011).
[CrossRef] [PubMed]

M. Brunel, H. Shen, S. Coëtmellec, and D. Lebrun, “Extended ABCD matrix formalism for the description of femtosecond diffraction patterns; application to femtosecond Digital In-line Holography with anamorphic optical systems,” Appl. Opt.51(8), 1137–1148 (2012).
[CrossRef] [PubMed]

A. El Mallahi, C. Minetti, and F. Dubois, “Automated three-dimensional detection and classification of living organisms using digital holographic microscopy with partial spatial coherent source: application to the monitoring of drinking water resources,” Appl. Opt.52(1), A68–A80 (2013).
[CrossRef] [PubMed]

Appl. Phys. Lett.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett.11(3), 77–79 (1967).
[CrossRef]

Eur Phys J E Soft Matter

M. Heydt, P. Divós, M. Grunze, and A. Rosenhahn, “Analysis of holographic microscopy data to quantitatively investigate three-dimensional settlement dynamics of algal zoospores in the vicinity of surfaces,” Eur Phys J E Soft Matter30(2), 141–148 (2009).
[CrossRef] [PubMed]

Exp. Fluids

S. Pu, D. Lebrun, D. Allano, B. Patte-Rouland, M. Malek, and C. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids39(1), 1–9 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Meas. Sci. Technol.

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol.15(4), 699–705 (2004).
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H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol.15(4), 673–685 (2004).
[CrossRef]

N. Salah, G. Godard, D. Lebrun, P. Paranthoen, D. Allano, and S. Coetmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard-von Karman vortex flow,” Meas. Sci. Technol.19(7), 074001 (2008).
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Opt. Commun.

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital In-line holography with a sub-picosecond laser beam,” Opt. Commun.268(1), 27–33 (2006).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

F. Lamadie, L. Bruel, and M. Himbert, “Digital holographic measurements of liquid-liquid two-phase flows,” Opt. Lasers Eng.50(12), 1716–1725 (2012).
[CrossRef]

C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: three dimensional location of particles,” Opt. Lasers Eng.33(6), 409–421 (2000).
[CrossRef]

Opt. Lett.

Other

P. Picart and J. Li, Digital holography (John Wiley & Sons Ed. 2012)

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Figures (7)

Fig. 1
Fig. 1

Optical configuration of recording in-line holograms ( ξ,η ) : object plane ( x,y ) : 2D detector plane.

Fig. 2
Fig. 2

Simulation of a long time exposure hologram recorded by a 1024 × 1024 camera with 6.7 × 6.7 µm pixels, d = 60 μm, ze = 258 mm, τ=1000µs . V 0.5m. s 1

Fig. 3
Fig. 3

Simulation of reconstruction of long time exposure holograms: d = 60 μm, ze = 258 mm, τ=1000µs . (a) V 3m. s 1 , (b) V 2m. s 1 , and (c) V 0.1m. s 1

Fig. 4
Fig. 4

Variations of SNR measured from simulated reconstructed hologram versus object velocity and for 3 different time exposures. d = 60 μm, ze = 258 mm

Fig. 5
Fig. 5

Recording system of digital holograms with a long time exposure τ. LD: Laser diode, F. Optical fiber, SV: Sample volume.

Fig. 6
Fig. 6

Reconstruction of experimental of long time exposure holograms d 60 μm, (a) τ=1000µs, V 3m. s 1 , SNR = 5.2 dB, (b) τ=1000µs, V 2.4m. s 1 SNR = 5.3 dB, (c) τ=200µs, V 2.8m. s 1 ,SNR = 11.8 dB, (d) τ=200µs, V 2.7m. s 1 SNR = 7.0 dB,(e) τ=200µs, V 3.3m. s 1 SNR = 6.2 dB,(f) τ=100µs, V 1.7m. s 1 SNR = 10.9 dB

Fig. 7
Fig. 7

Evolution of the SNR of reconstructed holographic images versus time exposure. Parameters used for the simulations: d = 60 µm, V 3m. s 1 , ze = 258 mm.

Equations (13)

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A z e (x,y)= [ ( 1T )** h z e ] (x,y)
h z e (x,y)= 1 iλ z e exp[ i π λ z e ( x 2 + y 2 ) ].
I z e (x,y, z e )=1[ T**( h z e + h ¯ z e ) ](x,y)
E V (x,y, z e )= τ/2 τ/2 [ I(x V x t,y V y t, z e )+N(x,y,t) ] dt,
E V (x',y', z e )= τ/2 τ/2 [ I(x' V t,y', z e )+N(x',y',t) ] dt.
E V (x',y', z e )=τ[ I(x',y', z e )**Δ(x',y')+ N(x',y',t) ],
Δ(x,y)= 1 V τ rect( xcos(θ)+ysin(θ) V τ )δ( xsinθ+ycosθ ).
E V (x,y, z e )=τ[ I Δ (x,y, z e )+ N(x,y,t) ],
where I Δ (x,y, z e )=I(x,y, z e )**Δ(x,y).
E V (x,y)=τ{ 1 T Δ **( h z e + h ¯ z e )(x,y)+ N(x,y,t) },
where T Δ (x,y)=( T**Δ )(x,y).
R Δ (x,y)= [ I Δ **( h z r + h ¯ z r ) ] (x,y)+ N(x,y,t) **( h z r + h ¯ z r ).
R Δ (x,y)=( R**Δ )(x,y)+ N(x,y,t) **( h z r + h ¯ z r ),

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