Abstract

We propose a microparticle detection scheme in digital holography. In our inverse problem approach, we estimate the optimal particles set that best models the observed hologram image. Such a method can deal with data that have missing pixels. By considering the camera as a truncated version of a wider sensor, it becomes possible to detect particles even out of the camera field of view. We tested the performance of our algorithm against simulated and experimental data for diluted particle conditions. With real data, our algorithm can detect particles far from the detector edges in a working area as large as 16 times the camera field of view. A study based on simulated data shows that, compared with classical methods, our algorithm greatly improves the precision of the estimated particle positions and radii. This precision does not depend on the particle’s size or location (i.e., whether inside or outside the detector field of view).

© 2007 Optical Society of America

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References

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  1. T. M. Kreis, M. Adams, and W. Jüptner, "Methods of digital holography: a comparison," Proc. SPIE 3098, 224-233 (1997).
    [CrossRef]
  2. S. Murata and N. Yasuda, "Potential of digital holography in particle measurement," Opt. Laser Technol. 32, 567-574 (2000).
    [CrossRef]
  3. G. Pan and H. Meng, "Digital holography of particle fields: reconstruction by use of complex amplitude," Appl. Opt. 42, 827-833 (2003).
    [CrossRef] [PubMed]
  4. K. D. Hinsch and S. F. Herrmann, eds., "Special issue: holographic particle image velocimetry," Meas. Sci. Technol. 15, 601-769 (2004).
    [CrossRef]
  5. F. Sheng, E. Malkiel, and J. Katz, "Digital holographic microscope for measuring three-dimensional particle distributions and motions," Appl. Opt. 45, 3893-3901 (2006).
    [CrossRef] [PubMed]
  6. T. Ooms, W. Koek, J. Braat, and J. Westerweel, "Optimizing Fourier filtering for digital holographic particle image velocimetry," Meas. Sci. Technol. 17, 304-312 (2006).
    [CrossRef]
  7. Y. Zhang, G. Pedrini, W. Osten, and H. Tiziani, "Applications of fractional transforms to object reconstruction from in-line holograms," Opt. Lett. 29, 1793-1795 (2004).
    [CrossRef] [PubMed]
  8. 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, 409-421 (2000).
    [CrossRef]
  9. M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Process. 12, 29-43 (2003).
    [CrossRef]
  10. H. Royer, "Holographic velocimetry of submicron particles," Opt. Commun. 20, 73-75 (1977).
    [CrossRef]
  11. 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, 686-693 (2004).
    [CrossRef]
  12. F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, "Inverse problem approach for particle digital holography: accurate location based on local optimisation," J. Opt. Soc. Am. A 24, 1164-1171 (2007).
    [CrossRef]
  13. J. W. Goodman, Introduction to Fourier Optics (Mc Graw-Hill, 1996).
  14. G. A. Tayler and B. J. Thompson, "Fraunhofer holography applied to particle size analysis: a reassessment," Opt. Acta 23, 261-304 (1976).
  15. L. Denis, C. Fournier, T. Fournel, C. Ducottet, and D. Jeulin, "Direct extraction of mean particle size from a digital hologram," Appl. Opt. 45, 944-952 (2006).
    [CrossRef] [PubMed]

2007 (1)

2006 (3)

2004 (3)

Y. Zhang, G. Pedrini, W. Osten, and H. Tiziani, "Applications of fractional transforms to object reconstruction from in-line holograms," Opt. Lett. 29, 1793-1795 (2004).
[CrossRef] [PubMed]

K. D. Hinsch and S. F. Herrmann, eds., "Special issue: holographic particle image velocimetry," Meas. Sci. Technol. 15, 601-769 (2004).
[CrossRef]

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, 686-693 (2004).
[CrossRef]

2003 (2)

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

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

2000 (2)

S. Murata and N. Yasuda, "Potential of digital holography in particle measurement," Opt. Laser Technol. 32, 567-574 (2000).
[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, 409-421 (2000).
[CrossRef]

1997 (1)

T. M. Kreis, M. Adams, and W. Jüptner, "Methods of digital holography: a comparison," Proc. SPIE 3098, 224-233 (1997).
[CrossRef]

1977 (1)

H. Royer, "Holographic velocimetry of submicron particles," Opt. Commun. 20, 73-75 (1977).
[CrossRef]

1976 (1)

G. A. Tayler and B. J. Thompson, "Fraunhofer holography applied to particle size analysis: a reassessment," Opt. Acta 23, 261-304 (1976).

Appl. Opt. (3)

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. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (3)

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, 686-693 (2004).
[CrossRef]

T. Ooms, W. Koek, J. Braat, and J. Westerweel, "Optimizing Fourier filtering for digital holographic particle image velocimetry," Meas. Sci. Technol. 17, 304-312 (2006).
[CrossRef]

K. D. Hinsch and S. F. Herrmann, eds., "Special issue: holographic particle image velocimetry," Meas. Sci. Technol. 15, 601-769 (2004).
[CrossRef]

Opt. Acta (1)

G. A. Tayler and B. J. Thompson, "Fraunhofer holography applied to particle size analysis: a reassessment," Opt. Acta 23, 261-304 (1976).

Opt. Commun. (1)

H. Royer, "Holographic velocimetry of submicron particles," Opt. Commun. 20, 73-75 (1977).
[CrossRef]

Opt. Laser Technol. (1)

S. Murata and N. Yasuda, "Potential of digital holography in particle measurement," Opt. Laser Technol. 32, 567-574 (2000).
[CrossRef]

Opt. Lasers Eng. (1)

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, 409-421 (2000).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

T. M. Kreis, M. Adams, and W. Jüptner, "Methods of digital holography: a comparison," Proc. SPIE 3098, 224-233 (1997).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (Mc Graw-Hill, 1996).

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Figures (11)

Fig. 1
Fig. 1

Synopsis of the method.

Fig. 2
Fig. 2

Notation used in the hologram model. The parameters p j = { x j , y j , z j , r j } are the position and radius of the jth particle, ( x k , y k ) is the location of the kth pixel, and ρ j , k is the distance between the projection of the jth particle on the detector and the kth pixel.

Fig. 3
Fig. 3

Computing region R and working area T compared with the CCD size ( W × H ) .

Fig. 4
Fig. 4

Comparison of lateral errors (on x and y) for classical and inverse problem approaches. Pixel size, 6.7 μ m .

Fig. 5
Fig. 5

Comparison of depth error (parameter z) for classical and inverse problem approaches. Pixel size, 6.7 μ m .

Fig. 6
Fig. 6

Error on radius (parameter r) for the inverse problem approach. Pixel size, 6.7 μ m .

Fig. 7
Fig. 7

In-line holography setup.

Fig. 8
Fig. 8

Superimposition of an experimental hologram (in the box) on a synthesized hologram of 18 reconstructed particles.

Fig. 9
Fig. 9

Montage of slices of the criterion Q ̊ computed on an experimental hologram as a function of the lateral position ( x , y ) of the sought particle. The montage is composed of 13 maps numbered in order of their computation from 1 to 13. Each map is computed at optimal depth z for the particle detected at that stage and after the detection and removal of preceding particles. Detected particles are indicated by arrows. To improve readability, the color scale has been inverted: maxima of Q ̊ appear as dark spots.

Fig. 10
Fig. 10

3D reconstruction of the droplet jet materialized by the positions of the detected droplets. Gray area, camera field of view.

Fig. 11
Fig. 11

Histogram of measured particle diameters.

Equations (33)

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I ( x , y ) = I src + I bg 2 I src j = 1 n η j Re ( f j ̱ ( x , y ) ) + I src i = 1 n j = 1 n η i f i ̱ ( x , y ) η j f j ̱ ( x , y ) ,
f j ̱ ( x , y ) = r j 2 i ρ j ( x , y ) J 1 ( 2 π r j ρ j ( x , y ) λ z j ) exp ( i π ρ j 2 ( x , y ) λ z j ) ,
I ( x , y ) I 0 j = 1 n α j g j ( x x j , y y j ) ,
g j ( Δ x , Δ y ) = r j 2 ρ ( Δ x , Δ y ) J 1 ( 2 π r j ρ ( Δ x , Δ y ) λ z j ) × sin ( π ρ 2 ( Δ x , Δ y ) λ z j ) ,
P n = k w ( x k , y k ) [ m n ( x k , y k ) d ( x k , y k ) ] 2 ,
w ( x k , y k ) = { 1 Var ( d ( x k , y k ) ) if k th pixel is measured 0 otherwise ,
w ( x k , y k ) = { 1 if k th pixel is measured 0 otherwise .
m n ( x k , y k ) = I 0 j = 1 n α j g j ( x j x k , x j y k ) .
r n 1 ( x k , y k ) = d ( x k , y k ) + j = 1 n 1 α j g j ( x j x k , y j y k ) c n 1 ,
c n 1 = 1 σ w k w ( x k , y k ) [ d ( x k , y k ) + j = 1 n 1 α j g j ( x j x k , y j y k ) ] ,
σ w r def k w ( x k , y k ) r n 1 ( y k , y k ) = 0 ,
P n = k w ( x k , y k ) [ I n r n 1 ( x k , y k ) α n g n ( x n x k , x n y k ) ] 2
P n = σ w r 2 + 2 α n σ w r g 2 α n I n σ w g + I n 2 σ w + α n 2 σ w g 2 ,
σ w = k w ( x k , y k ) ,
σ w r 2 = k w ( x k , y k ) r n 1 2 ( x k , y k ) ,
σ w r g = k w ( x k , y k ) r n 1 ( x k , y k ) g n ( x n x k , y n y k ) ,
σ w g = k w ( x k , y k ) g n ( x n x k , y n y k ) ,
σ w g 2 = k w ( x k , y k ) g n 2 ( x n x k , y n y k ) .
P n I n I n = I n + , α n = α n + = 0 ,
P n α n I n = I n + , α n = α n + = 0 ,
I n + = σ w g σ w r g σ w σ w g 2 σ w g 2 = α n + σ w g σ w ,
α n + = σ w σ w r g σ w σ w g 2 σ w g 2 .
P n + def P n I n = I n + , α n = α n + = σ w r 2 σ w ( σ w r g ) 2 σ w σ w g 2 σ w g 2 .
Q n def σ w ( σ w r g ) 2 σ w σ w g 2 σ w g 2
α n + = σ w σ w r g σ w σ w g 2 σ w g 2 > 0 .
g ̊ k def g n ( x k , y k ) .
r ̊ k def { r n 1 ( x k , y k ) d k is measured 0 otherwise ,
w ̊ k def { Var ( d k ) 1 d k is measured 0 otherwise ( mainly outside the field of view ) .
a ̊ def F 1 diag ( F w ̊ ) F g ̊ ,
[ diag ( u ) v ] k = u k v k .
b ̊ def F 1 diag ( F diag ( w ̊ ) r ̊ ) F g ̊ ,
c ̊ def F 1 diag ( F w ̊ ) F diag ( g ̊ ) g ̊ .
Q k Q ̊ k = σ w b ̊ k σ w c ̊ k å k 2 ,

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