Abstract

We propose a method using a focus function and its cross-correlation to measure depth-position and precise depth-displacement. The focus function provides acceptable results in the determination of depth-position of a transparent particle, an opaque particle, and a red blood cell. However, positional errors and a short time interval can cause unreliable results in identifying depth-displacement (Δz) and depth-directional velocity in digital holographic particle tracking velocimetry (DHPTV). To minimize the errors in Δz, we propose a method that directly obtains depth displacement from the cross-correlation of focus values between consecutive holograms. The feasibility of this method is demonstrated by quantitatively visualizing a 3D flow using HPTV.

© 2014 Optical Society of America

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References

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2014 (1)

K. W. Seo, H. J. Byeon, H. K. Huh, and S. J. Lee, “Particle migration and single-line particle focusing in microscale pipe flow of viscoelastic fluids,” RSC Adv. 4(7), 3512–3520 (2014).
[Crossref]

2013 (1)

2012 (2)

K. W. Seo, Y. S. Choi, and S. J. Lee, “Dean-coupled inertial migration and transient focusing of particles in a curved microscale pipe flow,” Exp. Fluids 53(6), 1867–1877 (2012).
[Crossref]

Y. S. Choi, K. W. Seo, M. H. Sohn, and S. J. Lee, “Advances in digital holographic micro-PTV for analyzing microscale flows,” Opt. Lasers Eng. 50(1), 39–45 (2012).
[Crossref]

2011 (4)

M. H. Sohn, K. W. Seo, Y. S. Choi, S. J. Lee, Y. S. Kang, and Y. S. Kang, “Determination of the swimming trajectory and speed of chain-forming dinoflagellate Cochlodinium polykrikoides with digital holographic particle tracking velocimetry,” Mar. Biol. 158(3), 561–570 (2011).
[Crossref]

Y. S. Choi and S. J. Lee, “High-accuracy three-dimensional position measurement of tens of micrometers size transparent microspheres using digital in-line holographic microscopy,” Opt. Lett. 36(21), 4167–4169 (2011).
[Crossref] [PubMed]

S. J. Lee, K. W. Seo, Y. S. Choi, and M. H. Sohn, “Three-dimensional motion measurements of free-swimming microorganisms using digital holographic microscopy,” Meas. Sci. Technol. 22(6), 064004 (2011).
[Crossref]

Y. S. Choi, K. W. Seo, and S. J. Lee, “Lateral and cross-lateral focusing of spherical particles in a square microchannel,” Lab Chip 11(3), 460–465 (2011).
[Crossref] [PubMed]

2009 (2)

Y. S. Choi and S. J. Lee, “Three-dimensional volumetric measurement of red blood cell motion using digital holographic microscopy,” Appl. Opt. 48(16), 2983–2990 (2009).
[Crossref] [PubMed]

T. A. Ooms, R. Lindken, and J. Westerweel, “Digital holographic microscopy applied to measurement of a flow in a T-shaped micromixer,” Exp. Fluids 47(6), 941–955 (2009).
[Crossref]

2008 (1)

S. Kim and S. J. Lee, “Effect of particle number density in in-line digital holographic particle velocimetry,” Exp. Fluids 44(4), 623–631 (2008).
[Crossref]

2007 (2)

S. Kim and S. J. Lee, “Measurement of 3D laminar flow inside a micro tube using micro digital holographic particle tracking velocimetry,” J. Micromech. Microeng. 17(10), 2157–2162 (2007).
[Crossref]

J. Sheng, E. Malkiel, J. Katz, J. Adolf, R. Belas, and A. R. Place, “Digital holographic microscopy reveals prey-induced changes in swimming behavior of predatory dinoflagellates,” Proc. Natl. Acad. Sci. U.S.A. 104(44), 17512–17517 (2007).
[Crossref] [PubMed]

2006 (2)

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[Crossref]

J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. 45(16), 3893–3901 (2006).
[Crossref] [PubMed]

2005 (3)

2004 (1)

C. Fournier, C. Ducottet, and T. Fournel, “Digital in-line holography: influence of the reconstruction function on the axial profile of a reconstructed particle image,” Meas. Sci. Technol. 15(4), 686–693 (2004).
[Crossref]

2003 (2)

2002 (1)

1996 (1)

S. J. Baek and S. J. Lee, “A new two-frame particle tracking algorithm using match probability,” Exp. Fluids 22(1), 23–32 (1996).
[Crossref]

1995 (1)

Adolf, J.

J. Sheng, E. Malkiel, J. Katz, J. Adolf, R. Belas, and A. R. Place, “Digital holographic microscopy reveals prey-induced changes in swimming behavior of predatory dinoflagellates,” Proc. Natl. Acad. Sci. U.S.A. 104(44), 17512–17517 (2007).
[Crossref] [PubMed]

Baek, S. J.

S. J. Baek and S. J. Lee, “A new two-frame particle tracking algorithm using match probability,” Exp. Fluids 22(1), 23–32 (1996).
[Crossref]

Belas, R.

J. Sheng, E. Malkiel, J. Katz, J. Adolf, R. Belas, and A. R. Place, “Digital holographic microscopy reveals prey-induced changes in swimming behavior of predatory dinoflagellates,” Proc. Natl. Acad. Sci. U.S.A. 104(44), 17512–17517 (2007).
[Crossref] [PubMed]

Byeon, H. J.

K. W. Seo, H. J. Byeon, H. K. Huh, and S. J. Lee, “Particle migration and single-line particle focusing in microscale pipe flow of viscoelastic fluids,” RSC Adv. 4(7), 3512–3520 (2014).
[Crossref]

Choi, Y. S.

Y. S. Choi, K. W. Seo, M. H. Sohn, and S. J. Lee, “Advances in digital holographic micro-PTV for analyzing microscale flows,” Opt. Lasers Eng. 50(1), 39–45 (2012).
[Crossref]

K. W. Seo, Y. S. Choi, and S. J. Lee, “Dean-coupled inertial migration and transient focusing of particles in a curved microscale pipe flow,” Exp. Fluids 53(6), 1867–1877 (2012).
[Crossref]

Y. S. Choi, K. W. Seo, and S. J. Lee, “Lateral and cross-lateral focusing of spherical particles in a square microchannel,” Lab Chip 11(3), 460–465 (2011).
[Crossref] [PubMed]

M. H. Sohn, K. W. Seo, Y. S. Choi, S. J. Lee, Y. S. Kang, and Y. S. Kang, “Determination of the swimming trajectory and speed of chain-forming dinoflagellate Cochlodinium polykrikoides with digital holographic particle tracking velocimetry,” Mar. Biol. 158(3), 561–570 (2011).
[Crossref]

S. J. Lee, K. W. Seo, Y. S. Choi, and M. H. Sohn, “Three-dimensional motion measurements of free-swimming microorganisms using digital holographic microscopy,” Meas. Sci. Technol. 22(6), 064004 (2011).
[Crossref]

Y. S. Choi and S. J. Lee, “High-accuracy three-dimensional position measurement of tens of micrometers size transparent microspheres using digital in-line holographic microscopy,” Opt. Lett. 36(21), 4167–4169 (2011).
[Crossref] [PubMed]

Y. S. Choi and S. J. Lee, “Three-dimensional volumetric measurement of red blood cell motion using digital holographic microscopy,” Appl. Opt. 48(16), 2983–2990 (2009).
[Crossref] [PubMed]

Ducottet, C.

C. Fournier, C. Ducottet, and T. Fournel, “Digital in-line holography: influence of the reconstruction function on the axial profile of a reconstructed particle image,” Meas. Sci. Technol. 15(4), 686–693 (2004).
[Crossref]

Fournel, T.

C. Fournier, C. Ducottet, and T. Fournel, “Digital in-line holography: influence of the reconstruction function on the axial profile of a reconstructed particle image,” Meas. Sci. Technol. 15(4), 686–693 (2004).
[Crossref]

Fournier, C.

C. Fournier, C. Ducottet, and T. Fournel, “Digital in-line holography: influence of the reconstruction function on the axial profile of a reconstructed particle image,” Meas. Sci. Technol. 15(4), 686–693 (2004).
[Crossref]

Grasser, T.

Guildenbecher, D. R.

Huh, H. K.

K. W. Seo, H. J. Byeon, H. K. Huh, and S. J. Lee, “Particle migration and single-line particle focusing in microscale pipe flow of viscoelastic fluids,” RSC Adv. 4(7), 3512–3520 (2014).
[Crossref]

Hussain, F.

Ito, T.

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[Crossref]

S. 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(6), 442–444 (2005).
[Crossref]

Jericho, M. H.

Kanamori, H.

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[Crossref]

Kang, Y. S.

M. H. Sohn, K. W. Seo, Y. S. Choi, S. J. Lee, Y. S. Kang, and Y. S. Kang, “Determination of the swimming trajectory and speed of chain-forming dinoflagellate Cochlodinium polykrikoides with digital holographic particle tracking velocimetry,” Mar. Biol. 158(3), 561–570 (2011).
[Crossref]

M. H. Sohn, K. W. Seo, Y. S. Choi, S. J. Lee, Y. S. Kang, and Y. S. Kang, “Determination of the swimming trajectory and speed of chain-forming dinoflagellate Cochlodinium polykrikoides with digital holographic particle tracking velocimetry,” Mar. Biol. 158(3), 561–570 (2011).
[Crossref]

Katz, J.

Kim, M. K.

Kim, S.

S. Kim and S. J. Lee, “Effect of particle number density in in-line digital holographic particle velocimetry,” Exp. Fluids 44(4), 623–631 (2008).
[Crossref]

S. Kim and S. J. Lee, “Measurement of 3D laminar flow inside a micro tube using micro digital holographic particle tracking velocimetry,” J. Micromech. Microeng. 17(10), 2157–2162 (2007).
[Crossref]

Kostinski, A. B.

Kreuzer, H. J.

Kunugi, T.

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[Crossref]

S. 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(6), 442–444 (2005).
[Crossref]

Lee, S. J.

K. W. Seo, H. J. Byeon, H. K. Huh, and S. J. Lee, “Particle migration and single-line particle focusing in microscale pipe flow of viscoelastic fluids,” RSC Adv. 4(7), 3512–3520 (2014).
[Crossref]

K. W. Seo, Y. S. Choi, and S. J. Lee, “Dean-coupled inertial migration and transient focusing of particles in a curved microscale pipe flow,” Exp. Fluids 53(6), 1867–1877 (2012).
[Crossref]

Y. S. Choi, K. W. Seo, M. H. Sohn, and S. J. Lee, “Advances in digital holographic micro-PTV for analyzing microscale flows,” Opt. Lasers Eng. 50(1), 39–45 (2012).
[Crossref]

S. J. Lee, K. W. Seo, Y. S. Choi, and M. H. Sohn, “Three-dimensional motion measurements of free-swimming microorganisms using digital holographic microscopy,” Meas. Sci. Technol. 22(6), 064004 (2011).
[Crossref]

M. H. Sohn, K. W. Seo, Y. S. Choi, S. J. Lee, Y. S. Kang, and Y. S. Kang, “Determination of the swimming trajectory and speed of chain-forming dinoflagellate Cochlodinium polykrikoides with digital holographic particle tracking velocimetry,” Mar. Biol. 158(3), 561–570 (2011).
[Crossref]

Y. S. Choi, K. W. Seo, and S. J. Lee, “Lateral and cross-lateral focusing of spherical particles in a square microchannel,” Lab Chip 11(3), 460–465 (2011).
[Crossref] [PubMed]

Y. S. Choi and S. J. Lee, “High-accuracy three-dimensional position measurement of tens of micrometers size transparent microspheres using digital in-line holographic microscopy,” Opt. Lett. 36(21), 4167–4169 (2011).
[Crossref] [PubMed]

Y. S. Choi and S. J. Lee, “Three-dimensional volumetric measurement of red blood cell motion using digital holographic microscopy,” Appl. Opt. 48(16), 2983–2990 (2009).
[Crossref] [PubMed]

S. Kim and S. J. Lee, “Effect of particle number density in in-line digital holographic particle velocimetry,” Exp. Fluids 44(4), 623–631 (2008).
[Crossref]

S. Kim and S. J. Lee, “Measurement of 3D laminar flow inside a micro tube using micro digital holographic particle tracking velocimetry,” J. Micromech. Microeng. 17(10), 2157–2162 (2007).
[Crossref]

S. J. Baek and S. J. Lee, “A new two-frame particle tracking algorithm using match probability,” Exp. Fluids 22(1), 23–32 (1996).
[Crossref]

Lindken, R.

T. A. Ooms, R. Lindken, and J. Westerweel, “Digital holographic microscopy applied to measurement of a flow in a T-shaped micromixer,” Exp. Fluids 47(6), 941–955 (2009).
[Crossref]

Malkiel, E.

Meinertzhagen, I. A.

Meng, H.

Ooms, T. A.

T. A. Ooms, R. Lindken, and J. Westerweel, “Digital holographic microscopy applied to measurement of a flow in a T-shaped micromixer,” Exp. Fluids 47(6), 941–955 (2009).
[Crossref]

Pan, G.

Place, A. R.

J. Sheng, E. Malkiel, J. Katz, J. Adolf, R. Belas, and A. R. Place, “Digital holographic microscopy reveals prey-induced changes in swimming behavior of predatory dinoflagellates,” Proc. Natl. Acad. Sci. U.S.A. 104(44), 17512–17517 (2007).
[Crossref] [PubMed]

Reu, P. L.

Satake, S.

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[Crossref]

S. 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(6), 442–444 (2005).
[Crossref]

Sato, K.

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[Crossref]

S. 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(6), 442–444 (2005).
[Crossref]

Seo, K. W.

K. W. Seo, H. J. Byeon, H. K. Huh, and S. J. Lee, “Particle migration and single-line particle focusing in microscale pipe flow of viscoelastic fluids,” RSC Adv. 4(7), 3512–3520 (2014).
[Crossref]

Y. S. Choi, K. W. Seo, M. H. Sohn, and S. J. Lee, “Advances in digital holographic micro-PTV for analyzing microscale flows,” Opt. Lasers Eng. 50(1), 39–45 (2012).
[Crossref]

K. W. Seo, Y. S. Choi, and S. J. Lee, “Dean-coupled inertial migration and transient focusing of particles in a curved microscale pipe flow,” Exp. Fluids 53(6), 1867–1877 (2012).
[Crossref]

Y. S. Choi, K. W. Seo, and S. J. Lee, “Lateral and cross-lateral focusing of spherical particles in a square microchannel,” Lab Chip 11(3), 460–465 (2011).
[Crossref] [PubMed]

S. J. Lee, K. W. Seo, Y. S. Choi, and M. H. Sohn, “Three-dimensional motion measurements of free-swimming microorganisms using digital holographic microscopy,” Meas. Sci. Technol. 22(6), 064004 (2011).
[Crossref]

M. H. Sohn, K. W. Seo, Y. S. Choi, S. J. Lee, Y. S. Kang, and Y. S. Kang, “Determination of the swimming trajectory and speed of chain-forming dinoflagellate Cochlodinium polykrikoides with digital holographic particle tracking velocimetry,” Mar. Biol. 158(3), 561–570 (2011).
[Crossref]

Shaw, R. A.

Sheng, J.

Sohn, M. H.

Y. S. Choi, K. W. Seo, M. H. Sohn, and S. J. Lee, “Advances in digital holographic micro-PTV for analyzing microscale flows,” Opt. Lasers Eng. 50(1), 39–45 (2012).
[Crossref]

M. H. Sohn, K. W. Seo, Y. S. Choi, S. J. Lee, Y. S. Kang, and Y. S. Kang, “Determination of the swimming trajectory and speed of chain-forming dinoflagellate Cochlodinium polykrikoides with digital holographic particle tracking velocimetry,” Mar. Biol. 158(3), 561–570 (2011).
[Crossref]

S. J. Lee, K. W. Seo, Y. S. Choi, and M. H. Sohn, “Three-dimensional motion measurements of free-swimming microorganisms using digital holographic microscopy,” Meas. Sci. Technol. 22(6), 064004 (2011).
[Crossref]

Stuaffacher, H. L.

Taniguchi, J.

S. Satake, T. Kunugi, K. Sato, T. Ito, H. Kanamori, and J. Taniguchi, “Measurements of 3D flow in a micro-pipe via micro digital holographic particle tracking velocimetry,” Meas. Sci. Technol. 17(7), 1647–1651 (2006).
[Crossref]

S. 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(6), 442–444 (2005).
[Crossref]

Westerweel, J.

T. A. Ooms, R. Lindken, and J. Westerweel, “Digital holographic microscopy applied to measurement of a flow in a T-shaped micromixer,” Exp. Fluids 47(6), 941–955 (2009).
[Crossref]

Xu, W.

Yang, W. D.

Yu, L. F.

Appl. Opt. (6)

Exp. Fluids (4)

S. J. Baek and S. J. Lee, “A new two-frame particle tracking algorithm using match probability,” Exp. Fluids 22(1), 23–32 (1996).
[Crossref]

S. Kim and S. J. Lee, “Effect of particle number density in in-line digital holographic particle velocimetry,” Exp. Fluids 44(4), 623–631 (2008).
[Crossref]

T. A. Ooms, R. Lindken, and J. Westerweel, “Digital holographic microscopy applied to measurement of a flow in a T-shaped micromixer,” Exp. Fluids 47(6), 941–955 (2009).
[Crossref]

K. W. Seo, Y. S. Choi, and S. J. Lee, “Dean-coupled inertial migration and transient focusing of particles in a curved microscale pipe flow,” Exp. Fluids 53(6), 1867–1877 (2012).
[Crossref]

J. Micromech. Microeng. (1)

S. Kim and S. J. Lee, “Measurement of 3D laminar flow inside a micro tube using micro digital holographic particle tracking velocimetry,” J. Micromech. Microeng. 17(10), 2157–2162 (2007).
[Crossref]

Lab Chip (1)

Y. S. Choi, K. W. Seo, and S. J. Lee, “Lateral and cross-lateral focusing of spherical particles in a square microchannel,” Lab Chip 11(3), 460–465 (2011).
[Crossref] [PubMed]

Mar. Biol. (1)

M. H. Sohn, K. W. Seo, Y. S. Choi, S. J. Lee, Y. S. Kang, and Y. S. Kang, “Determination of the swimming trajectory and speed of chain-forming dinoflagellate Cochlodinium polykrikoides with digital holographic particle tracking velocimetry,” Mar. Biol. 158(3), 561–570 (2011).
[Crossref]

Meas. Sci. Technol. (3)

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Opt. Lasers Eng. (1)

Y. S. Choi, K. W. Seo, M. H. Sohn, and S. J. Lee, “Advances in digital holographic micro-PTV for analyzing microscale flows,” Opt. Lasers Eng. 50(1), 39–45 (2012).
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S. 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(6), 442–444 (2005).
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K. W. Seo, H. J. Byeon, H. K. Huh, and S. J. Lee, “Particle migration and single-line particle focusing in microscale pipe flow of viscoelastic fluids,” RSC Adv. 4(7), 3512–3520 (2014).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 (a) Schematic diagram of in-line holographic microscopy to measure micro-scale fluid flows. (b) Typical holograms of a transparent particle, an opaque particle, and a red blood cell.
Fig. 2
Fig. 2 Reconstructed images for each hologram are projected into one image (projection image) to find the in-plane positions of particles. Red circles represent the centers of particles, which are determined by using a peak-searching algorithm. Red squares indicate the segmented regions whose focus value requires evaluation by scanning the reconstructed images of each particle.
Fig. 3
Fig. 3 (a) Holograms and reconstructed images of a transparent particle. In the first frame of hologram, the particle is positioned approximately 100 μm apart from the focal plane. The second frame of hologram captures the particle precisely positioned 100 μm apart from the first hologram along the z-direction. The red squared images represent optimally focused reconstructed images of a particle. The degree of focusing state is evaluated by applying the focus functions (b) LAP and (c) VAR. The peak represents the z-position of the particle, at which the reconstructed image is optimally focused. (d) Holograms and reconstructed images of an opaque particle. The actual displacement of particle between the two holograms is 100 μm. (e) Typical focus value profiles obtained by adopting LAP. The focus value profiles of an opaque particle do not exhibit a sharp peak such as that of a transparent particle. However, the two focus value profiles of the first and second holograms have a strong similarity.
Fig. 4
Fig. 4 (a) Reconstructed images of RBC at z = 50, 100, and 150 μm. The z-displacement of the RBC in consecutive holograms are precisely controled to 50 μm. The red squared images indicate optimally focused reconstructed images of the RBC. (b) The degree of focusing state is evaluated by adopting the focus function VAR in the reconstructed images. The peak represents the z-position of the RBC, at which the reconstructed image is optimally focused. The focus value profiles of the RBC show more complex curves than those of a sphere particle. However, the focus value profiles well-reflect the degree of focusing state and the curves are highly correlated.
Fig. 5
Fig. 5 Cross-correlation curves of the focus values depicted in Figs. 3 and 4. The depth-displacements of (a) a transparent particle, (b) an opaque particle, and (c) RBC are obtained by the cross-correlation of focus values. The distance from the center to the peak represents the depth-displacement of a particle or a biological sample. The distance of a particle between two consecutive holograms is precisely controlled to 100 μm apart. RBCs are positioned with intervals of 50 μm along the z-direction in consecutive holograms. The depth-displacements obtained by the cross-correlation of focus value profiles well-matched with their actual displacements.
Fig. 6
Fig. 6 Statistical analysis of depth-displacement of transparent particles, opaque particles, and RBCs. Depth-displacement of more than 100 particles or of RBCs are obtained by employing a conventional method ( Δ z = z 2 z 1 ) and a cross-correlation method ( Δ z = F 2 F 1 ).
Fig. 7
Fig. 7 Schematic diagram of the evaporation-induced flow inside a confined droplet.
Fig. 8
Fig. 8 (a) A typical hologram of tracer particles seeded inside the confined droplet. (b) Side view of the confined droplet. (c) Particle trajectories and (d) velocity field inside the confined droplet as seen at the side view.

Equations (6)

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A( k x , k y ;0)= E 0 ( x 0 , y 0 ,0)exp[i( k x x 0 + k y y 0 )]d x 0 d y 0 ,
E(x,y,z)= 1 {{ E 0 }exp[i k z z]}, k z = k 2 k x 2 k y 2 ,
I(m,n)=Re [E(x,y;z)] 2 +Im [E(x,y;z)] 2 .
LAP(z)= x,y { 2 I(x,y;z)} 2
VAR(z)= 1 N x N y x,y {I(x,y;z) I ¯ (z)} 2
F 1 F 2 = F 1 (z) F 2 (z+Δz)dz .

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