Abstract

A matched filter method is provided for obtaining improved particle size estimates from digital in-line holograms. This improvement is relative to conventional reconstruction and pixel counting methods for particle size estimation, which is greatly limited by the CCD camera pixel size. The proposed method is based on iterative application of a sign matched filter in the Fourier domain, with sign meaning the matched filter takes values of ±1 depending on the sign of the angular spectrum of the particle aperture function. Using simulated data the method is demonstrated to work for particle diameters several times the pixel size. Holograms of piezoelectrically generated water droplets taken in the laboratory show greatly improved particle size measurements. The method is robust to additive noise and can be applied to real holograms over a wide range of matched-filter particle sizes.

© 2012 OSA

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References

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  1. J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech.42, 531–555 (2010).
    [CrossRef]
  2. J. P. Fugal and R. A. Shaw, “Cloud particle size distributions measured with an airborne digital in-line holographic instrument,” Atmos. Meas. Tech.2, 259–271 (2009).
    [CrossRef]
  3. S. Soontaranon, J. Widjaja, and T. Asakura, “Improved holographic particle sizing by using absolute values of the wavelet transform,” Opt. Commun.240, 253–260 (2004).
    [CrossRef]
  4. L. Denis, C. Fournier, T. Fournel, C. Ducottet, and D. Jeulin, “Direct extraction of the mean particle size from a digital hologram,” Appl. Opt.45, 944–952 (2006).
    [CrossRef] [PubMed]
  5. J. Lu, J. P. Fugal, H. Nordsiek, E. W. Saw, R. A. Shaw, and W. Yang, “Lagrangian particle tracking in three dimensions via single-camera in-line digital holography,” New J. Phys.10, 125013 (2008).
    [CrossRef]
  6. G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt.42, 827–833 (2003).
    [CrossRef] [PubMed]
  7. W. Yang, A. B. Kostinski, and R. A. Shaw, “Depth-of-focus reduction for digital in-line holography of particle fields,” Opt. Lett.30, 1303–1305 (2005).
    [CrossRef] [PubMed]
  8. 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. A24, 1164–1171 (2007).
    [CrossRef]
  9. J. Gire, L. Denis, C. Fournier, E. Thiébaut, F. Soulez, and C. Ducottet, “Digital holography of particles: benefits of the ‘inverse problem’ approach,” Meas. Sci. Technol.19, 074005 (2008).
    [CrossRef]
  10. L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic deconvolution microscopy for high-resolution particle tracking,” Opt. Express19, 16410–16417 (2011).
    [CrossRef] [PubMed]
  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. J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Boston, Mass., 1996).
  13. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  14. J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt.23, 812–816 (1984).
    [CrossRef] [PubMed]
  15. C. Fournier, L. Denis, and T. Fournel, “On the single point resolution of on-axis digital holography,” J. Opt. Soc. Am. A27, 1856–1862 (2010).
    [CrossRef]

2011

2010

C. Fournier, L. Denis, and T. Fournel, “On the single point resolution of on-axis digital holography,” J. Opt. Soc. Am. A27, 1856–1862 (2010).
[CrossRef]

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

2009

J. P. Fugal and R. A. Shaw, “Cloud particle size distributions measured with an airborne digital in-line holographic instrument,” Atmos. Meas. Tech.2, 259–271 (2009).
[CrossRef]

2008

J. Lu, J. P. Fugal, H. Nordsiek, E. W. Saw, R. A. Shaw, and W. Yang, “Lagrangian particle tracking in three dimensions via single-camera in-line digital holography,” New J. Phys.10, 125013 (2008).
[CrossRef]

J. Gire, L. Denis, C. Fournier, E. Thiébaut, F. Soulez, and C. Ducottet, “Digital holography of particles: benefits of the ‘inverse problem’ approach,” Meas. Sci. Technol.19, 074005 (2008).
[CrossRef]

2007

2006

2005

2004

S. Soontaranon, J. Widjaja, and T. Asakura, “Improved holographic particle sizing by using absolute values of the wavelet transform,” Opt. Commun.240, 253–260 (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

1984

Asakura, T.

S. Soontaranon, J. Widjaja, and T. Asakura, “Improved holographic particle sizing by using absolute values of the wavelet transform,” Opt. Commun.240, 253–260 (2004).
[CrossRef]

Cheong, F. C.

Denis, L.

Dixon, L.

Ducottet, C.

J. Gire, L. Denis, C. Fournier, E. Thiébaut, F. Soulez, and C. Ducottet, “Digital holography of particles: benefits of the ‘inverse problem’ approach,” Meas. Sci. Technol.19, 074005 (2008).
[CrossRef]

L. Denis, C. Fournier, T. Fournel, C. Ducottet, and D. Jeulin, “Direct extraction of the mean particle size from a digital hologram,” Appl. Opt.45, 944–952 (2006).
[CrossRef] [PubMed]

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]

Fournel, T.

Fournier, C.

C. Fournier, L. Denis, and T. Fournel, “On the single point resolution of on-axis digital holography,” J. Opt. Soc. Am. A27, 1856–1862 (2010).
[CrossRef]

J. Gire, L. Denis, C. Fournier, E. Thiébaut, F. Soulez, and C. Ducottet, “Digital holography of particles: benefits of the ‘inverse problem’ approach,” Meas. Sci. Technol.19, 074005 (2008).
[CrossRef]

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. A24, 1164–1171 (2007).
[CrossRef]

L. Denis, C. Fournier, T. Fournel, C. Ducottet, and D. Jeulin, “Direct extraction of the mean particle size from a digital hologram,” Appl. Opt.45, 944–952 (2006).
[CrossRef] [PubMed]

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]

Fugal, J. P.

J. P. Fugal and R. A. Shaw, “Cloud particle size distributions measured with an airborne digital in-line holographic instrument,” Atmos. Meas. Tech.2, 259–271 (2009).
[CrossRef]

J. Lu, J. P. Fugal, H. Nordsiek, E. W. Saw, R. A. Shaw, and W. Yang, “Lagrangian particle tracking in three dimensions via single-camera in-line digital holography,” New J. Phys.10, 125013 (2008).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Gianino, P. D.

Gire, J.

J. Gire, L. Denis, C. Fournier, E. Thiébaut, F. Soulez, and C. Ducottet, “Digital holography of particles: benefits of the ‘inverse problem’ approach,” Meas. Sci. Technol.19, 074005 (2008).
[CrossRef]

Goepfert, C.

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Boston, Mass., 1996).

Grier, D. G.

Horner, J. L.

Jeulin, D.

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]

Kostinski, A. B.

Lu, J.

J. Lu, J. P. Fugal, H. Nordsiek, E. W. Saw, R. A. Shaw, and W. Yang, “Lagrangian particle tracking in three dimensions via single-camera in-line digital holography,” New J. Phys.10, 125013 (2008).
[CrossRef]

Meng, H.

Nordsiek, H.

J. Lu, J. P. Fugal, H. Nordsiek, E. W. Saw, R. A. Shaw, and W. Yang, “Lagrangian particle tracking in three dimensions via single-camera in-line digital holography,” New J. Phys.10, 125013 (2008).
[CrossRef]

Pan, G.

Saw, E. W.

J. Lu, J. P. Fugal, H. Nordsiek, E. W. Saw, R. A. Shaw, and W. Yang, “Lagrangian particle tracking in three dimensions via single-camera in-line digital holography,” New J. Phys.10, 125013 (2008).
[CrossRef]

Shaw, R. A.

J. P. Fugal and R. A. Shaw, “Cloud particle size distributions measured with an airborne digital in-line holographic instrument,” Atmos. Meas. Tech.2, 259–271 (2009).
[CrossRef]

J. Lu, J. P. Fugal, H. Nordsiek, E. W. Saw, R. A. Shaw, and W. Yang, “Lagrangian particle tracking in three dimensions via single-camera in-line digital holography,” New J. Phys.10, 125013 (2008).
[CrossRef]

W. Yang, A. B. Kostinski, and R. A. Shaw, “Depth-of-focus reduction for digital in-line holography of particle fields,” Opt. Lett.30, 1303–1305 (2005).
[CrossRef] [PubMed]

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]

Soontaranon, S.

S. Soontaranon, J. Widjaja, and T. Asakura, “Improved holographic particle sizing by using absolute values of the wavelet transform,” Opt. Commun.240, 253–260 (2004).
[CrossRef]

Soulez, F.

J. Gire, L. Denis, C. Fournier, E. Thiébaut, F. Soulez, and C. Ducottet, “Digital holography of particles: benefits of the ‘inverse problem’ approach,” Meas. Sci. Technol.19, 074005 (2008).
[CrossRef]

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. A24, 1164–1171 (2007).
[CrossRef]

Thiébaut, E.

J. Gire, L. Denis, C. Fournier, E. Thiébaut, F. Soulez, and C. Ducottet, “Digital holography of particles: benefits of the ‘inverse problem’ approach,” Meas. Sci. Technol.19, 074005 (2008).
[CrossRef]

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. A24, 1164–1171 (2007).
[CrossRef]

Widjaja, J.

S. Soontaranon, J. Widjaja, and T. Asakura, “Improved holographic particle sizing by using absolute values of the wavelet transform,” Opt. Commun.240, 253–260 (2004).
[CrossRef]

Yang, W.

J. Lu, J. P. Fugal, H. Nordsiek, E. W. Saw, R. A. Shaw, and W. Yang, “Lagrangian particle tracking in three dimensions via single-camera in-line digital holography,” New J. Phys.10, 125013 (2008).
[CrossRef]

W. Yang, A. B. Kostinski, and R. A. Shaw, “Depth-of-focus reduction for digital in-line holography of particle fields,” Opt. Lett.30, 1303–1305 (2005).
[CrossRef] [PubMed]

Annu. Rev. Fluid Mech.

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.

Atmos. Meas. Tech.

J. P. Fugal and R. A. Shaw, “Cloud particle size distributions measured with an airborne digital in-line holographic instrument,” Atmos. Meas. Tech.2, 259–271 (2009).
[CrossRef]

J. Opt. Soc. Am. A

Meas. Sci. Technol.

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]

J. Gire, L. Denis, C. Fournier, E. Thiébaut, F. Soulez, and C. Ducottet, “Digital holography of particles: benefits of the ‘inverse problem’ approach,” Meas. Sci. Technol.19, 074005 (2008).
[CrossRef]

New J. Phys.

J. Lu, J. P. Fugal, H. Nordsiek, E. W. Saw, R. A. Shaw, and W. Yang, “Lagrangian particle tracking in three dimensions via single-camera in-line digital holography,” New J. Phys.10, 125013 (2008).
[CrossRef]

Opt. Commun.

S. Soontaranon, J. Widjaja, and T. Asakura, “Improved holographic particle sizing by using absolute values of the wavelet transform,” Opt. Commun.240, 253–260 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Other

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Boston, Mass., 1996).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

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Figures (6)

Fig. 1
Fig. 1

The filter-modified aperture (on the reconstructed field at focus z = z0) when a′ matches with the detected particle size a. The black curve is the modified aperture function, exhibiting greatly intensified center values. In contrast, the aperture function before applying the filter is shown as the blue curve. This computation assumes a circular aperture of 250 pixels in radius on a computational array of 2048 × 2048.

Fig. 2
Fig. 2

Axial intensity profiles verses depth along the optical axis. Panel a shows the conventional, unfiltered result and panel b shows the axial profile after filtering with P′(η,ξ) = sgn[P(η,ξ)], i.e., for filter diameter equal to the particle diameter of 56 μm. The bottom two images show the axial intensity profiles for matched filters corresponding to the actual diameter minus and plus ten percent (50 and 62 μm respectively). The intensity scales for panels b, c, and d are normalized by the same factor, such that the best matched filter results in a maximum intensity of unity. The plots are made using 10 μm steps along the z-axis.

Fig. 3
Fig. 3

The peak-intensity versus filter-size, showing a maximum when filter size is equal to the true particle size.

Fig. 4
Fig. 4

Axial intensity profiles verses depth along the optical axis (left) and peak-intensity versus filter-size (right) for three different noise levels. The top panels are for additive Gaussian white noise with SNR of 1; the middle panels are for the same but with SNR of 1/10. The bottom panels are for ‘noise’ resulting from 1000 particles within the roughly cube-shaped volume.

Fig. 5
Fig. 5

The peak-intensity versus filter-size for a single hologram recorded in the lab. The filter size varies from 44 μm to 68 μm in 0.2 μm steps. The maximum agrees with the actual particle diameter to within experimental uncertainty.

Fig. 6
Fig. 6

Distributions of estimated particle diameters from 136 holograms. The blue histogram, with d̄ = 55.7 μm and σd = 2.8 μm, is for the the conventional pixel counting method based on a global intensity threshold. The red histogram, with d̄ = 56.0 μm and σd = 0.23 μm, is for the same holograms analyzed with the sign matched filter method.

Equations (3)

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e z ( u , v ) = F 1 { P ( η , ξ ) H z 0 z ( η , ξ ) } .
e z ( u , v ) = F 1 { P ( η , ξ ) × P ( η , ξ ) H z 0 z ( η , ξ ) } .
P ( η , ξ ) = { 1 for all ( η , ξ ) that satisfy J 1 ( a η 2 + ξ 2 ) / ( a η 2 + ξ 2 ) 0 1 for all ( η , ξ ) that satisfy J 1 ( a η 2 + ξ 2 ) / ( a η 2 + ξ 2 ) < 0 .

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