Abstract

We propose an asymmetric wavelet method to reconstruct a particle from a hologram illuminated by an elliptical, astigmatic Gaussian beam. The particle can be reconstructed by a convolution of the asymmetric wavelet and hologram. The reconstructed images have the same size and resolution as the recorded hologram; therefore, the reconstructed 3D field is convenient for automatic particle locating and sizing. The asymmetric wavelet method is validated by both simulated holograms of spherical particles and experimental holograms of opaque, nonspherical coal particles.

© 2013 Optical Society of America

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  1. M. Adams, T. M. Kreis, and W. P. O. Jueptner, “Particle size and position measurement with digital holography,” Proc. SPIE 3098, 234–240 (1997).
    [CrossRef]
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    [CrossRef]
  3. Y. Yang and B.-S. Kang, “Experimental validation for the determination of particle positions by the correlation coefficient method in digital particle holography,” Appl. Opt. 47, 5953–5960 (2008).
    [CrossRef]
  4. L. Tian, N. Loomis, J. A. Dominguez-Caballero, and G. Barbastathis, “Quantitative measurement of size and three-dimensional position of fast-moving bubbles in air-water mixture flows using digital holography,” Appl. Opt. 49, 1549–1554 (2010).
    [CrossRef]
  5. S.-I. Satake, Y. Yonemoto, T. Kikuchi, and T. Kunugi, “Detection of microbubble position by a digital hologram,” Appl. Opt. 50, 5999–6005 (2011).
    [CrossRef]
  6. D. K. Singh and P. K. Panigrahi, “Automatic threshold technique for holographic particle field characterization,” Appl. Opt. 51, 3874–3887 (2012).
    [CrossRef]
  7. Y. Yang, G. Y. Li, L. L. Tang, and L. Huang, “Integrated gray-level gradient method applied for the extraction of three-dimensional velocity fields of sprays in in-line digital holography,” Appl. Opt. 51, 255–267 (2012).
    [CrossRef]
  8. S.-I. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12, 442–444 (2005).
    [CrossRef]
  9. L. Cao, G. Pan, J. de Jong, S. Woodward, and H. Meng, “Hybrid digital holographic imaging system for three-dimensional dense particle field measurement,” Appl. Opt. 47, 4501–4508 (2008).
    [CrossRef]
  10. Q. Lü, Y. Chen, R. Yuan, B. Ge, Y. Gao, and Y. Zhang, “Trajectory and velocity measurement of a particle in spray by digital holography,” Appl. Opt. 48, 7000–7007 (2009).
    [CrossRef]
  11. S. Kim and S. J. Lee, “Measurement of dean flow in a curved micro-tube using micro digital holographic particle tracking velocimetry,” Exp. Fluids 46, 255–264 (2009).
    [CrossRef]
  12. Y. Wu, X. Wu, Z. Wang, L. Chen, and K. Cen, “Coal powder measurement by digital holography with expanded measurement area,” Appl. Opt. 50, H22–H29 (2011).
    [CrossRef]
  13. Y. Wu, X. Wu, Z. Wang, G. Gréhan, L. Chen, and K. Cen, “Measurement of microchannel flow with digital holographic microscopy by integrated nearest neighbor and cross-correlation particle pairing,” Appl. Opt. 50, H297–H305 (2011).
    [CrossRef]
  14. J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42, 531–555 (2010).
    [CrossRef]
  15. D. Lebrun, S. Belad, and C. Zkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
    [CrossRef]
  16. M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
    [CrossRef]
  21. 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, 1137–1148 (2012).
    [CrossRef]
  22. J. Crane, P. Dunn, B. J. Thompson, J. Knapp, and J. Zeiss, “Far-field holography of ampule contaminants,” Appl. Opt. 21, 2548–2553 (1982).
    [CrossRef]
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    [CrossRef]
  24. S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. 26, 974–976 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  29. M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
    [CrossRef]
  30. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, 2011).
  31. Y. Yuan, K. Ren, S. Coëtmellec, and D. Lebrun, “Rigorous description of holograms of particles illuminated by an astigmatic elliptical Gaussian beam,” J. Phys. 147, 012052 (2009).
    [CrossRef]

2012 (3)

2011 (3)

2010 (3)

2009 (3)

Y. Yuan, K. Ren, S. Coëtmellec, and D. Lebrun, “Rigorous description of holograms of particles illuminated by an astigmatic elliptical Gaussian beam,” J. Phys. 147, 012052 (2009).
[CrossRef]

Q. Lü, Y. Chen, R. Yuan, B. Ge, Y. Gao, and Y. Zhang, “Trajectory and velocity measurement of a particle in spray by digital holography,” Appl. Opt. 48, 7000–7007 (2009).
[CrossRef]

S. Kim and S. J. Lee, “Measurement of dean flow in a curved micro-tube using micro digital holographic particle tracking velocimetry,” Exp. Fluids 46, 255–264 (2009).
[CrossRef]

2008 (4)

2005 (2)

S.-I. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12, 442–444 (2005).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

2003 (2)

M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
[CrossRef]

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

2002 (1)

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

2001 (2)

1999 (1)

1997 (1)

M. Adams, T. M. Kreis, and W. P. O. Jueptner, “Particle size and position measurement with digital holography,” Proc. SPIE 3098, 234–240 (1997).
[CrossRef]

1988 (1)

J. Wen and M. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

1986 (1)

C. Vikram and M. Billet, “Fraunhofer holography in cylindrical tunnels: neutralizing window curvature effects,” Opt. Eng. 25, 251189 (1986).
[CrossRef]

1982 (1)

1970 (1)

Adams, M.

M. Adams, T. M. Kreis, and W. P. O. Jueptner, “Particle size and position measurement with digital holography,” Proc. SPIE 3098, 234–240 (1997).
[CrossRef]

Allano, D.

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
[CrossRef]

Barbastathis, G.

Belad, S.

Billet, M.

C. Vikram and M. Billet, “Fraunhofer holography in cylindrical tunnels: neutralizing window curvature effects,” Opt. Eng. 25, 251189 (1986).
[CrossRef]

Blu, T.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

Breazeale, M.

J. Wen and M. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Brunel, M.

Cao, L.

Cen, K.

Chen, L.

Chen, Y.

Coëtmellec, S.

Collins, J.

Crane, J.

de Jong, J.

De Nicola, S.

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. 26, 974–976 (2001).
[CrossRef]

Dirksen, D.

Dominguez-Caballero, J. A.

Dunn, P.

Ferraro, P.

Finizio, A.

Gao, Y.

Ge, B.

Gouesbet, G.

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, 2011).

Gréhan, G.

Grilli, S.

Huang, L.

Ito, T.

S.-I. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12, 442–444 (2005).
[CrossRef]

Janssen, A. J.

Jueptner, W. P. O.

M. Adams, T. M. Kreis, and W. P. O. Jueptner, “Particle size and position measurement with digital holography,” Proc. SPIE 3098, 234–240 (1997).
[CrossRef]

Kang, B.-S.

Katz, J.

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

Kemper, B.

Kikuchi, T.

Kim, S.

S. Kim and S. J. Lee, “Measurement of dean flow in a curved micro-tube using micro digital holographic particle tracking velocimetry,” Exp. Fluids 46, 255–264 (2009).
[CrossRef]

Knapp, J.

Kreis, T. M.

M. Adams, T. M. Kreis, and W. P. O. Jueptner, “Particle size and position measurement with digital holography,” Proc. SPIE 3098, 234–240 (1997).
[CrossRef]

Kunugi, T.

S.-I. Satake, Y. Yonemoto, T. Kikuchi, and T. Kunugi, “Detection of microbubble position by a digital hologram,” Appl. Opt. 50, 5999–6005 (2011).
[CrossRef]

S.-I. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12, 442–444 (2005).
[CrossRef]

Langehanenberg, P.

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, 1137–1148 (2012).
[CrossRef]

N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
[CrossRef]

Y. Yuan, K. Ren, S. Coëtmellec, and D. Lebrun, “Rigorous description of holograms of particles illuminated by an astigmatic elliptical Gaussian beam,” J. Phys. 147, 012052 (2009).
[CrossRef]

N. Verrier, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography in thick optical systems: application to visualization in pipes,” Appl. Opt. 47, 4147–4157 (2008).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
[CrossRef]

D. Lebrun, S. Belad, and C. Zkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
[CrossRef]

Lee, S. J.

S. Kim and S. J. Lee, “Measurement of dean flow in a curved micro-tube using micro digital holographic particle tracking velocimetry,” Exp. Fluids 46, 255–264 (2009).
[CrossRef]

Li, G. Y.

Liebling, M.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

Loomis, N.

Lü, Q.

Malek, M.

M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
[CrossRef]

Meng, H.

Nicola, S. D.

Nicolas, F.

Pan, G.

Panigrahi, P. K.

Pierattini, G.

Remacha, C.

Ren, K.

Y. Yuan, K. Ren, S. Coëtmellec, and D. Lebrun, “Rigorous description of holograms of particles illuminated by an astigmatic elliptical Gaussian beam,” J. Phys. 147, 012052 (2009).
[CrossRef]

Satake, S.-I.

S.-I. Satake, Y. Yonemoto, T. Kikuchi, and T. Kunugi, “Detection of microbubble position by a digital hologram,” Appl. Opt. 50, 5999–6005 (2011).
[CrossRef]

S.-I. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12, 442–444 (2005).
[CrossRef]

Sato, K.

S.-I. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12, 442–444 (2005).
[CrossRef]

Shen, H.

Sheng, J.

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

Singh, D. K.

Stuart, A.

Tang, L. L.

Taniguchi, J.

S.-I. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12, 442–444 (2005).
[CrossRef]

Thompson, B. J.

Tian, L.

Unser, M.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

Verrier, N.

Vikram, C.

C. Vikram and M. Billet, “Fraunhofer holography in cylindrical tunnels: neutralizing window curvature effects,” Opt. Eng. 25, 251189 (1986).
[CrossRef]

Vikram, C. S.

C. S. Vikram and B. J. Thompson, Particle Field Holography (Cambridge University, 2005), Vol. 11.

von Bally, G.

Wang, Z.

Wen, J.

J. Wen and M. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Woodward, S.

Wu, X.

Wu, Y.

Yang, Y.

Yonemoto, Y.

Yuan, R.

Yuan, Y.

Y. Yuan, K. Ren, S. Coëtmellec, and D. Lebrun, “Rigorous description of holograms of particles illuminated by an astigmatic elliptical Gaussian beam,” J. Phys. 147, 012052 (2009).
[CrossRef]

Zeiss, J.

Zhang, Y.

Zkul, C.

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. Opt. (14)

D. Lebrun, S. Belad, and C. Zkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
[CrossRef]

N. Verrier, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography in thick optical systems: application to visualization in pipes,” Appl. Opt. 47, 4147–4157 (2008).
[CrossRef]

P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, “Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging,” Appl. Opt. 47, D176–D182 (2008).
[CrossRef]

Y. Yang and B.-S. Kang, “Experimental validation for the determination of particle positions by the correlation coefficient method in digital particle holography,” Appl. Opt. 47, 5953–5960 (2008).
[CrossRef]

L. Tian, N. Loomis, J. A. Dominguez-Caballero, and G. Barbastathis, “Quantitative measurement of size and three-dimensional position of fast-moving bubbles in air-water mixture flows using digital holography,” Appl. Opt. 49, 1549–1554 (2010).
[CrossRef]

S.-I. Satake, Y. Yonemoto, T. Kikuchi, and T. Kunugi, “Detection of microbubble position by a digital hologram,” Appl. Opt. 50, 5999–6005 (2011).
[CrossRef]

D. K. Singh and P. K. Panigrahi, “Automatic threshold technique for holographic particle field characterization,” Appl. Opt. 51, 3874–3887 (2012).
[CrossRef]

Y. Yang, G. Y. Li, L. L. Tang, and L. Huang, “Integrated gray-level gradient method applied for the extraction of three-dimensional velocity fields of sprays in in-line digital holography,” Appl. Opt. 51, 255–267 (2012).
[CrossRef]

L. Cao, G. Pan, J. de Jong, S. Woodward, and H. Meng, “Hybrid digital holographic imaging system for three-dimensional dense particle field measurement,” Appl. Opt. 47, 4501–4508 (2008).
[CrossRef]

Q. Lü, Y. Chen, R. Yuan, B. Ge, Y. Gao, and Y. Zhang, “Trajectory and velocity measurement of a particle in spray by digital holography,” Appl. Opt. 48, 7000–7007 (2009).
[CrossRef]

Y. Wu, X. Wu, Z. Wang, L. Chen, and K. Cen, “Coal powder measurement by digital holography with expanded measurement area,” Appl. Opt. 50, H22–H29 (2011).
[CrossRef]

Y. Wu, X. Wu, Z. Wang, G. Gréhan, L. Chen, and K. Cen, “Measurement of microchannel flow with digital holographic microscopy by integrated nearest neighbor and cross-correlation particle pairing,” Appl. Opt. 50, H297–H305 (2011).
[CrossRef]

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, 1137–1148 (2012).
[CrossRef]

J. Crane, P. Dunn, B. J. Thompson, J. Knapp, and J. Zeiss, “Far-field holography of ampule contaminants,” Appl. Opt. 21, 2548–2553 (1982).
[CrossRef]

Exp. Fluids (1)

S. Kim and S. J. Lee, “Measurement of dean flow in a curved micro-tube using micro digital holographic particle tracking velocimetry,” Exp. Fluids 46, 255–264 (2009).
[CrossRef]

IEEE Trans. Image Process. (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. Wen and M. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. (1)

Y. Yuan, K. Ren, S. Coëtmellec, and D. Lebrun, “Rigorous description of holograms of particles illuminated by an astigmatic elliptical Gaussian beam,” J. Phys. 147, 012052 (2009).
[CrossRef]

Opt. Commun. (1)

M. Malek, S. Coëtmellec, D. Allano, and D. Lebrun, “Formulation of in-line holography process by a linear shift invariant system: application to the measurement of fiber diameter,” Opt. Commun. 223, 263–271 (2003).
[CrossRef]

Opt. Eng. (1)

C. Vikram and M. Billet, “Fraunhofer holography in cylindrical tunnels: neutralizing window curvature effects,” Opt. Eng. 25, 251189 (1986).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (1)

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331–340 (2002).
[CrossRef]

Opt. Lett. (1)

Opt. Rev. (1)

S.-I. Satake, T. Kunugi, K. Sato, T. Ito, and J. Taniguchi, “Three-dimensional flow tracking in a micro channel with high time resolution using micro digital-holographic particle-tracking velocimetry,” Opt. Rev. 12, 442–444 (2005).
[CrossRef]

Proc. SPIE (1)

M. Adams, T. M. Kreis, and W. P. O. Jueptner, “Particle size and position measurement with digital holography,” Proc. SPIE 3098, 234–240 (1997).
[CrossRef]

Other (2)

C. S. Vikram and B. J. Thompson, Particle Field Holography (Cambridge University, 2005), Vol. 11.

G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, 2011).

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Figures (4)

Fig. 1.
Fig. 1.

Illustration of digital particle holography with elliptical Gaussian beam illumination.

Fig. 2.
Fig. 2.

Simulated hologram and reconstructed symmetrical wavelet method. (a) Simulated hologram using GLMT, with λ=532nm, ωx0=2μm, ωy0=3mm, particle diameter d=140μm, Zp=2.7cm, and ZCCD=5.0cm; (b) fringe comparison with elliptical Gaussian beam and plane wave illumination; (c) reconstructed image at z=2.3cm with wavelet method; (d) reconstructed image at z=4.26cm with wavelet method.

Fig. 3.
Fig. 3.

Validation with simulated hologram in Fig. 2(a). (a) Reconstructed image using modified Fresnel approximation reconstruction with zx=4.26cm and zy=2.3cm; (b) reconstructed image with wavelet method, where z=2.3cm; (c) reconstructed image with FRFT, where αx=0.601π/2 and αy=0.408π/2; (d) rescaled image with FRFT reconstruction.

Fig. 4.
Fig. 4.

Experimental hologram validation. (a) Hologram of a coal particle illuminated by a elliptical Gaussian beam; (b) reconstructed coal particle image with asymmetric wavelet method, where z=9.85cm; (c) rescaled image with modified Fresnel approximation reconstruction; (d) rescaled image with FRFT reconstruction, where αx=0.801π/2 and αy=0.530π/2.

Equations (12)

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

U(u,v)=G1(x,y)[1T(x,y)]exp[iπλBx(Axu22ux+Dxx2)]exp[iπλBy(Ayv22vy+Dyy2)]dx(dy),
G1(x,y)=exp(x2ωx12y2ωy12)exp[iπλ(x2Rx1+y2Ry1)]
R(u,v)=G1(x,y)exp[iπλBx(Axu22ux+Dxx2)]exp[iπλBy(Ayv22vy+Dyy2)]dxdy=Crefexp(iπDxx2λBx+iπDyy2λBy),exp[πx2λBx(λBxπωx12+iBxRx1iAx)+πy2λBy(λByπωy12+iByRy1iAy)],
Cref=π/((λBx/πωx12)+(iBx/Rx1)iAx)((λBy/πωy12)+(iBy/Ry1)iAy).
O(u,v)=G1(x,y)T(x,y)exp[iπλBx(Axu22ux+Dxx2)]exp[iπλBy(Ayv22vy+Dyy2)]dxdy=Cobjexp(iπDxx2λBx+iπDyy2λBy)k=110{Akexp[πx2λBx(λBxπωx12+iBxRx1iAxλBxBkπr2)](λBxπωx12+iBxRx1iAxλBxBkπr2)exp[πy2λBy(λByπωy12+iByRy1iAyλByBkπr2)](λByπωy12+iByRy1iAyλByBkπr2)},
Iholo(u,v)=U(u,v)·U(u,v)¯=R·R¯+O·O¯+R·O¯+O·R¯.
arg(Iholo)=arg(O·R¯+R·O¯)=I{πx2λBx(λBxπωx12+iBxRx1iAx)+πy2λBy(λByπωy12+iByRy1iAy)}=πx2λBxRx+πy2λByRy=πx2λzBxRxz+πy2λzByRyz=πx2λzx,eq+πy2λzy,eq,
ψCa,Cb(x,y)=[sin(x2Ca2+y2Cb2)Mψ(σa,σb)]exp(x2Ca2σa2y2Cb2σb2).
Ca2=λBxRxπ=λzx,eqπ,Cb2=λByRyπ=λzy,eqπ.
ψCa=1,Cb=1(x,y)dxdy=0.
Mψ(σa,σb)=i2σaσb(11+iσa21+iσb211iσa21iσb2),
Ire(x,y)=Iholo(x,y)ψCa,Cb(x,y)=F1[F(Iholo)·F(ψCa,Cb)]

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