## Improved particle size estimation in digital holography via sign matched filtering |

Optics Express, Vol. 20, Issue 12, pp. 12666-12674 (2012)

http://dx.doi.org/10.1364/OE.20.012666

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### 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

## 1. Introduction

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]

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]

10. L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic deconvolution microscopy for high-resolution particle tracking,” Opt. Express **19**, 16410–16417 (2011). [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. A **24**, 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]

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. A **24**, 1164–1171 (2007). [CrossRef]

## 2. Principle of operation

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]

*s,t*) is

*i*(

*s,t*) = 1 −

*o*(

*s,t*) −

*o*

^{*}(

*s,t*) +

*|o*(

*s,t*)

*|*

^{2}, where

^{*}denote the complex conjugate. After omitting the contribution from the higher-order (diffraction) term, the Fourier transform of

*i*(

*s,t*) can be approximated as

*P*(

*η*,

*ξ*) is the Fourier transform of the particle aperture function

*p*(

*x,y*) in the particle object plane (

*x,y*) at position

*z*

_{0}along the optical axis, and

*H*(

_{z}*η*,

*ξ*) = exp[−

*iπλz*(

*η*

^{2}+

*ξ*

^{2})] [12]. For a spherical particle

*p*(

*x,y*) can be assumed to be a circular aperture with the diameter of the particle. Then

*P*(

*η*,

*ξ*) is proportional to

*a*, where

*J*

_{1}is the Bessel function of the first kind. Omitting the effect of the twin image, and denoting

**F**

^{−1}{} as the inverse Fourier transform, the reconstructed field in an arbitrary plane (

*u,v*) at distance −

*z*from the hologram is

*P*′(

*η*,

*ξ*) with the Fourier spectrum of the recorded hologram, such that the reconstructed field becomes When

*P*′ is the

*z*-dependent Gaussian function or the inverse of the angular spectrum of detected particles, the axial profile of the reconstructed field has a maximum intensity value at

*z*=

*z*

_{0}, as described by Fournier et al. [11

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]

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]

*P*′(

*η*,

*ξ*) as the sign function of the angular spectrum of a circular aperture with radius

*a*′:

*a*′ of a preselected circular aperture function, and the resulting reconstructed field is equivalent to the diffraction field from a modified aperture

*p*(

_{m,a}*u, v*) =

**F**

^{−1}{

*P*′(

*η*,

*ξ*) ×

*P*(

*η*,

*ξ*)} (i.e., at

*z*=

*z*

_{0}the propagation kernel is

*H*

_{z0−z}(

*η*,

*ξ*) = 1 and therefore the reconstructed field has the simple form

*e*

_{z0}(

*u, v*) =

*p*(

_{m,a}*u, v*), which is the form of a matched correlation function.)

*p*will be real and circularly symmetric. Most importantly, when radius

_{m,a}*a*′ matches the detected particle radius

*a*, the modified aperture function, or the reconstructed field at the focus

*z*=

*z*

_{0}, is sharply peaked as illustrated in Fig. 1. This feature can be understood through Fourier theory [13]: when the filter size is matched, all spectrum values in the Fourier domain are positive and the center value of the modified aperture is the sum of all these positive values; in contrast, off-center values of the modified aperture are the sum of these positive values, but they are modulated by different harmonic waveforms that contain positive and negative values. Therefore we can expect that the modified aperture is a circular symmetric function that is center peaked. As seen in the figure, the modified aperture function allows some negative values and does not have a clear aperture edge; however, it has a significantly intensified field value at the center. The intensified center values effectively reduce the size of aperture, thereby resulting in a greatly reduced depth of focus [6

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]

**30**, 1303–1305 (2005). [CrossRef] [PubMed]

*a*′ of the filter does not match with the size of the detected particle we expect the modified aperture still to be a circularly symmetric real function, but with much lower center values and an overall wider aperture compared to the matching case. This is because when not matching, the sign filter would not be able to turn all the spectrum values to positive values, thus the center value of the modified aperture would be smaller than the the center value when size matches. Therefore a much wider depth of focus and lowered axial field peak can also be expected.

14. J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. **23**, 812–816 (1984). [CrossRef] [PubMed]

*J*

_{1}(

*x*)/

*x*), which contains no explicit phase angle variables, however if we confine the amplitude to be positive only, the positive or negative sign of the Jinc function can be substituted by exp(

*i*·

*π*) or exp(−

*i*·

*π*). In effect, the sign filter captures the binary phase of the aperture angular spectrum. As such, we can expect that the sign matched filter should also have the properties of low sensitivity to noise and high sensitivity to features of the detected object as exist for the phase-only matched filter [14

14. J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. **23**, 812–816 (1984). [CrossRef] [PubMed]

**30**, 1303–1305 (2005). [CrossRef] [PubMed]

*P*(

*η*,

*ξ*). In that case the result is identical to the field generated by a point source, having an infinite intensity peak at

*z*=

*z*

_{0}. In practice, however, the multiplication by

*P*′(

*η*,

*ξ*) as described in this paper avoids the noise problems that arise from division by very small values or zeros in

*P*′(

*η*,

*ξ*). As a consequence this method is much more robust to additive noise and can be applied for

*P*′(

*η*,

*ξ*) over a range of particle sizes.

*P*′(

*η*,

*ξ*) and then reconstruct the hologram as in Eq. (2). We can then select the maximum value from each axial intensity profile (i.e., for each

*P*′(

*η*,

*ξ*)) and use the maximum values to construct a plot of maximum intensity versus particle size

*a*′ for the applied filter. The highest intensity corresponds to the sharpest axial profile, that for which

*a*′ =

*a*, the radius of the actual particle.

## 3. Numerical results

*λ*= 0.527

*μ*m and the hologram is a 4008 × 2672 array with a pixel size of 9

*μ*m. The particle diameter is

*d*= 56

*μ*m. Panel (a) of Fig. 2 shows the axial (i.e., along the center of the particle) intensity profiles without imposing the matched filtering, analogous to Eq. (1). Panel (b) shows the axial intensity profiles for a matched filter corresponding to the actual particle size,

*P*′(

*η*,

*ξ*) = sgn[

*P*(

*η*,

*ξ*)]. This demonstrates how the peak intensity profile changes to a single peak through the multiplication of

*P*′(

*η*,

*ξ*), which greatly improves the depth of focus [7

**30**, 1303–1305 (2005). [CrossRef] [PubMed]

*P*′(

*η*,

*ξ*) directly from the Bessel function, as in Eq. (3) instead of a direct pixel-generated method. Finally, panels (c) and (d) show axial intensity profiles for matched filters with diameters 10% less than and 10% greater than the actual particle diameter, respectively. The intensity scale for panels (b), (c), and (d) are all normalized to the peak intensity for

*P*′(

*η*,

*ξ*) = sgn[

*P*(

*η*,

*ξ*)], so it can be seen that the single, central peak in the axial profile is maintained, but that it becomes less pronounced as the filter diameter changes from the actual particle diameter. An additional check, the results of which are not shown in detail here, was to vary the lateral position of the particle: up to the maximum tested distance of 1/40 of the hologram width from the hologram edge, there was no significant degradation in the peak quality.

*p*(

_{m,a}*u, v*) =

**F**

^{−1}{

*P*′(

*η*,

*ξ*) ×

*P*(

*η*,

*ξ*)}, with

*P*′ varying with the filter diameter

*a*′ and the peak appearing when

*a*′ =

*a*.

*μ*/

*σ*, where

*μ*is the mean of the hologram ‘signal’ and

*σ*is the gaussian standard deviation. Results for two representative values, SNR of 1 and 1/10, are shown in Fig. 4: plots in the left column are profiles of intensity along the optical axis, and plots in the right column are profiles of peak height versus filter diameter. Even for the very low SNR case the peak is quite evident. It is noted that the width of the curve in the peak height versus filter diameter profiles does not undergo any significant change with increasing noise. By taking 30 independent realizations of single particle holograms with SNR of 1 a diameter mean of 55.94

*μ*m and standard deviation of 0.11

*μ*m are obtained. In practice each size estimate is accomplished via a parabolic fitting to obtain the position of the peak. The second noise test considers the influence of particle density in the hologram. The third row in Fig. 4 shows results of depth and diameter profiles for a realization of a hologram containing 1000 particles. The 1000 particles are randomly placed within a 4008 × 9

*μ*m by 2672 × 9

*μ*m by 2672 × 9

*μ*m box. By taking 30 independent realizations of particle positions a diameter mean of 55.97

*μ*m and a standard deviation of 0.10

*μ*m are obtained. These initial characterizations of additive and multiple-particle noise indicate that the proposed filter method is robust under a variety of realistic conditions for implementation. This is further borne out in the following section with experimental results. A more fundamental analysis of the Cramer-Rao bound [15

15. C. Fournier, L. Denis, and T. Fournel, “On the single point resolution of on-axis digital holography,” J. Opt. Soc. Am. A **27**, 1856–1862 (2010). [CrossRef]

## 4. Experimental results

*λ*= 0.527

*μ*m). The beam is passed through a beam expander with the addition of a pinhole for spatial filtering. The camera is an Illunis XMV-11000 with 4008 × 2672 with 9

*μ*m pixels and 12-bit output. Water droplets are injected by a piezoelectric injector that generates monodisperse droplets. The mean droplet diameter is

*d*̄ = 56.1

*μ*m with an upper bound on the size variability of Δ

*d*= 0.5

*μ*m, as determined from a high resolution, telecentric microscope with a spatial resolution of 0.9

*μ*m (the diameter can be determined to better than the spatial resolution by considering the full area of the image). The experimental data consists of 136 holograms taken at

*z*

_{0}= 531 mm, with one particle taken from each hologram. We note that the particle is taken at the same location in the hologram to avoid any possible evaporation effects. First, we use conventional reconstruction and pixel counting to obtain the coarse particle size and location. Then each particle in the hologram was processed using the sign matched filtering method described here. We then repeat the reconstruction with filters

*P*′(

*η*,

*ξ*) corresponding to particle diameters in 0.2

*μ*m increments, obtaining the peak axial-profile intensity as a function of the particle filter size, as shown in Fig. 5. The peak is around 56

*μ*m as expected.

*d*̄ = 55.7

*μ*m and the standard deviation is

*σ*= 2.8

_{d}*μ*m. This standard deviation is consistent with the uncertainty scaling with square root of pixel size [5

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]

*d*̄= 56.0

*μ*m and the standard deviation is

*σ*= 0.23

_{d}*μ*m. This standard deviation is consistent with the expected variability from the piezoelectric droplet generator. This clearly demonstrates the ability of the new method to greatly improve the precision of particle size estimation.

## 5. Summary and conclusions

## Acknowledgments

## References and links

1. | J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. |

2. | J. P. Fugal and R. A. Shaw, “Cloud particle size distributions measured with an airborne digital in-line holographic instrument,” Atmos. Meas. Tech. |

3. | S. Soontaranon, J. Widjaja, and T. Asakura, “Improved holographic particle sizing by using absolute values of the wavelet transform,” Opt. Commun. |

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. |

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. |

6. | G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. |

7. | W. Yang, A. B. Kostinski, and R. A. Shaw, “Depth-of-focus reduction for digital in-line holography of particle fields,” Opt. Lett. |

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. A |

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. |

10. | L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic deconvolution microscopy for high-resolution particle tracking,” Opt. Express |

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. |

12. | J. Goodman, |

13. | J. D. Gaskill, |

14. | J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. |

15. | C. Fournier, L. Denis, and T. Fournel, “On the single point resolution of on-axis digital holography,” J. Opt. Soc. Am. A |

**OCIS Codes**

(070.4550) Fourier optics and signal processing : Correlators

(100.5090) Image processing : Phase-only filters

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: March 1, 2012

Revised Manuscript: May 8, 2012

Manuscript Accepted: May 12, 2012

Published: May 21, 2012

**Citation**

Jiang Lu, Raymond A. Shaw, and Weidong Yang, "Improved particle size estimation in digital holography via sign matched filtering," Opt. Express **20**, 12666-12674 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-12666

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### References

- J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech.42, 531–555 (2010). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt.42, 827–833 (2003). [CrossRef] [PubMed]
- 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]
- 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]
- 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. Dixon, F. C. Cheong, and D. G. Grier, “Holographic deconvolution microscopy for high-resolution particle tracking,” Opt. Express19, 16410–16417 (2011). [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]
- 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).
- J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt.23, 812–816 (1984). [CrossRef] [PubMed]
- 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]

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