## Uncertainty characterization of particle depth measurement using digital in-line holography and the hybrid method |

Optics Express, Vol. 21, Issue 22, pp. 26432-26449 (2013)

http://dx.doi.org/10.1364/OE.21.026432

Acrobat PDF (5849 KB)

### Abstract

In the detection of particles using digital in-line holography, measurement accuracy is substantially influenced by the hologram processing method. In particular, a number of methods have been proposed to determine the out-of-plane particle depth (*z* location). However, due to the lack of consistent uncertainty characterization, it has been unclear which method is best suited to a given measurement problem. In this work, depth determination accuracies of seven particle detection methods, including a recently proposed hybrid method, are systematically investigated in terms of relative depth measurement errors and uncertainties. Both synthetic and experimental holograms of particle fields are considered at conditions relevant to particle sizing and tracking. While all methods display a range of particle conditions where they are most accurate, in general the hybrid method is shown to be the most robust with depth uncertainty less than twice the particle diameter over a wide range of particle field conditions.

© 2013 OSA

## 1. Introduction

1. H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. **15**, 673 (2004). [CrossRef]

4. N. A. Buchmann, C. Atkinson, and J. Soria, “Ultra-high-speed tomographic digital holographic velocimetry in supersonic particle-laden jet flows,” Meas. Sci. Technol. **24**, 024005 (2013). [CrossRef]

5. J. Lee, K. A. Sallam, K. C. Lin, and C. D. Carter, “Spray structure in near-injector region of aerated jet in subsonic crossflow,” J. Propul. Power **25**, 258–266 (2009). [CrossRef]

8. J. Gao, D. R. Guildenbecher, P. L. Reu, V. Kulkarni, P. E. Sojka, and J. Chen, “Quantitative, three-dimensional diagnostics of multiphase drop fragmentation via digital in-line holography,” Opt. Lett. **38**, 1893–1895 (2013). [CrossRef] [PubMed]

9. 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. Nat. Acad. Sci. USA **104**, 17512–17517 (2007). [CrossRef] [PubMed]

10. 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**, 064004 (2011). [CrossRef]

11. L. Tian, N. Loomis, J. A. Domínguez-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] [PubMed]

12. D. Lebrun, D. Allano, L. Méès, F. Walle, F. Corbin, R. Boucheron, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt. **50**, H1–H9 (2011). [CrossRef] [PubMed]

13. J. P. Fugal, R. A. Shaw, E. W. Saw, and A. V. Sergeyev, “Airborne digital holographic system for cloud particle measurements,” Appl. Opt. **43**, 5987–5995 (2004). [CrossRef] [PubMed]

15. T. Khanam, M. N. Rahman, A. Rajendran, V. Kariwala, and A. K. Asundi, “Accurate size measurement of needle-shaped particles using digital holography,” Chem. Eng. Sci. **66**, 2699–2706 (2011). [CrossRef]

*z*location). Reconstruction at an accurate depth gives a well-focused particle image and ensures precise extraction of the particle’s size, shape and transverse location (

*x*–

*y*location). In addition to hardware issues (e.g., low pixel resolution of digital cameras) that affect the accuracy of particle detection, the methods (algorithms) to extract the depth also have a remarkable influence on the measurement accuracy.

26. G. Pan and H. Meng, “Digital holography of particle fields: Reconstruction by use of complex amplitude,” Appl. Opt. **42**, 827–833 (2003). [CrossRef] [PubMed]

27. W. Yang, A. B. Kostinski, and R. A. Shaw, “Phase signature for particle detection with digital in-line holography,” Opt. Lett. **31**, 1399–1401 (2006). [CrossRef] [PubMed]

28. F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express **14**, 5895–5908 (2006). [CrossRef] [PubMed]

29. C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul”, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Laser Eng. **33**, 409–421 (2000). [CrossRef]

30. S. Soontaranon, J. Widjaja, and T. Asakura, “Extraction of object position from in-line holograms by using single wavelet coefficient,” Opt. Commun. **281**, 1461–1467 (2008). [CrossRef]

31. F. Soulez, L. Denis, C. Fournier, Éric Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. A **24**, 1164–1171 (2007). [CrossRef]

7. Y. Yang and B. seon Kang, “Digital particle holographic system for measurements of spray field characteristics,” Opt. Laser Eng. **49**, 1254–1263 (2011). [CrossRef]

8. J. Gao, D. R. Guildenbecher, P. L. Reu, V. Kulkarni, P. E. Sojka, and J. Chen, “Quantitative, three-dimensional diagnostics of multiphase drop fragmentation via digital in-line holography,” Opt. Lett. **38**, 1893–1895 (2013). [CrossRef] [PubMed]

11. L. Tian, N. Loomis, J. A. Domínguez-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] [PubMed]

23. V. Palero, M. Arroyo, and J. Soria, “Digital holography for micro-droplet diagnostics,” Exp. Fluids **43**, 185–195 (2007). [CrossRef]

17. J. 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]

26. G. Pan and H. Meng, “Digital holography of particle fields: Reconstruction by use of complex amplitude,” Appl. Opt. **42**, 827–833 (2003). [CrossRef] [PubMed]

34. D. K. Singh and P. K. Panigrahi, “Improved digital holographic reconstruction algorithm for depth error reduction and elimination of out-of-focus particles,” Opt. Express **18**, 2426–2448 (2010). [CrossRef] [PubMed]

11. L. Tian, N. Loomis, J. A. Domínguez-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] [PubMed]

20. Y. Yang, G. Li, 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] [PubMed]

34. D. K. Singh and P. K. Panigrahi, “Improved digital holographic reconstruction algorithm for depth error reduction and elimination of out-of-focus particles,” Opt. Express **18**, 2426–2448 (2010). [CrossRef] [PubMed]

## 2. Introduction to DIH

*h*(

*m*,

*n*) by an imaging sensor (CCD or CMOS) and stored in a computer. In the reconstruction step, the Rayleigh-Sommerfeld diffraction equation is evaluated numerically to simulate the analog reconstruction process in which the reference wave illuminates the hologram and further propagates to a reconstruction plane forming the reconstructed image. The numerically reconstructed complex amplitude

*E*can be expressed as is the discrete analytical expression for the Fourier transform of the Rayleigh-Sommerfeld diffraction kernel [32]. The reconstructed amplitude is

_{r}*A*= |

_{r}*E*|, and the reconstructed intensity is

_{r}*I*= |

_{r}*E*|

_{r}^{2}. One advantage of numerical reconstruction is digital focusing, in which particles at different depths can be brought into focus by altering the value of

*z*during reconstruction using Eq. (1), as shown in Fig. 1(b). Accordingly, the depth at which a particle is in focus is determined as the

_{r}*z*coordinate of the particle. Further, the transverse (

*x*and

*y*) coordinates and in-plane size and shape of the particle can be evaluated from the focused image of the particle.

## 3. Hybrid method for particle field detection

22. D. R. Guildenbecher, J. Gao, P. L. Reu, and J. Chen, “Digital holography simulations and experiments to quantify the accuracy of 3D particle location and 2D sizing using a proposed hybrid method,” Appl. Opt. **52**, 3790–3801 (2013). [CrossRef] [PubMed]

*I*, maximum Tenengrad map,

_{min}*T*and its depth map,

_{max}*D*, as shown in Figs. 2(a)–2(d). These are expressed mathematically as

_{T}*T*(

*k*,

*l*,

*z*) is the sharpness of the reconstructed image, quantified by the Tenengrad operator, which can be expressed as where

_{r}*S*and

_{x}*S*are the horizontal and vertical Sobel kernels, respectively. The depth range (

_{y}*z*

_{min}≤

*z*≤

_{r}*z*

_{max}) for the volume reconstruction is established such that the particles of interest are enclosed. It is assumed that the interval between consecutive depths is small enough to neglect its influence on the measurement accuracy.

*I*using an automatically selected threshold. Symbol

_{min}*𝒯*{} denotes the thresholding operation using a threshold

_{t}*t*, which results in a binary image, as illustrated in Fig. 2(f). Symbol

*ℰ*{} denotes an operation that finds the exterior edge pixels (exterior contours) of binary segments, which can be realized by subtracting the original binary image from the morphologically dilated binary image. Figure 2(g) shows the

*ℰ*{} operation applied to the binary image in Fig. 2(f). An optimal threshold is selected from a sequence of tentative thresholds bounded by the minimum and maximum values in

*I*. The selection process is automated using the global sharpness

_{min}*S*(

*t*) at each tentative threshold

*t*, which is defined as where · denotes pointwise multiplication. The profile of

*S*(

*t*) achieved from Figs. 2(b) and 2(c) is shown in Fig. 2(e).

*t*is selected from the maximum value of

_{o}*S*(

*t*) and defines the optimal threshold, at which the particle segments (Fig. 2(f)) are separated from each other and easily identified.

*x*–

*y*) location information of the segments obtained in the second step, rectangular local windows are defined to enclose individual particles, as indicated by the red rectangles in Fig. 2. The typical size of a local window is twice that of the segment enclosed. Next, the procedures in the second step are applied locally in each window to find the optimal threshold for each particle. Specifically, the optimal threshold for a particle,

*t′*, is obtained by maximizing the local sharpness,

_{o}*S*(

_{W}*t*), expressed as where

*W*is the set of pixels that belong to the window. The edge pixels are identified by

*ℰ*{

*𝒯*

_{t′o}{

*I*}}. Assuming that the edge sharpness is maximized when the particle is in focus, the

_{min}*z*location is estimated by conditionally averaging the depths of the edge pixels (according to

*D*). To exclude potential outliers during the averaging, the edge pixels, whose sharpness values are less than half of the maximum sharpness value in the window (according to

_{T}*T*), are eliminated. An initial depth

_{max}*z′*is then determined by taking the mean of the depths of the rest of edge pixels.

_{d}*z′*, a new optimal threshold,

_{d}*t″*, is decided by replacing

_{o}*I*in Eq. (8) with the local intensity,

_{min}*I*(

_{r}*k*,

*l*,

*z′*). The final depth,

_{d}*z*, is determined for the particle, following the procedure in the third step. Since the local intensity, shown in the inset in Fig. 2, describes the particle better than

_{d}*I*, the refined particle binary image with smooth contours is obtained by thresholding the local intensity with

_{min}*t″*, as shown in Fig. 2(h). Further, the transverse location, in-plane size and shape of the particle can be measured from the refined binary image.

_{o}## 4. Alternative particle detection methods

### Laplacian (LAP) method

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

*z*direction by maximizing the sum of squared Laplacian values in an investigation window enclosing the particle. The LAP value can be expressed as where is the Laplacian kernel. Then, the particle depth

*z*is determined by maximizing

_{d}*LAP*(

*z*). In the present study, the size of the window is twice that of the particle enclosed.

_{r}### Correlation coefficient (CC) method

25. Y. Yang, B. seon Kang, and Y. jun Choo, “Application of the correlation coefficient method for determination of the focal plane to digital particle holography,” Appl. Opt. **47**, 817–824 (2008). [CrossRef] [PubMed]

*z*direction, which can be written as where

*C*(

*k*,

*l*,

*z*) =

_{r}*I*(

_{r}*k*,

*l*,

*z*) −

_{r}*Ī*(

_{W}*z*), and

_{r}*Ī*(

_{W}*z*) is the mean intensity in the window. Δ

_{r}*C*is the correlation interval, which is set to 1

_{z}*cm*[25

25. Y. Yang, B. seon Kang, and Y. jun Choo, “Application of the correlation coefficient method for determination of the focal plane to digital particle holography,” Appl. Opt. **47**, 817–824 (2008). [CrossRef] [PubMed]

### Variance (VAR) method

*z*location can be determined by minimizing the variance of the intensity of the particle [23

23. V. Palero, M. Arroyo, and J. Soria, “Digital holography for micro-droplet diagnostics,” Exp. Fluids **43**, 185–195 (2007). [CrossRef]

24. E. Darakis, T. Khanam, A. Rajendran, V. Kariwala, T. J. Naughton, and A. K. Asundi, “Microparticle characterization using digital holography,” Chem. Eng. Sci. **65**, 1037–1044 (2010). [CrossRef]

*P*is the set of pixels that belong to the particle,

*N*is the number of the particle pixels, and

_{P}*Ī*(

_{P}*z*) is mean intensity of the particle.

_{r}### Minimum intensity (MINI) method

15. T. Khanam, M. N. Rahman, A. Rajendran, V. Kariwala, and A. K. Asundi, “Accurate size measurement of needle-shaped particles using digital holography,” Chem. Eng. Sci. **66**, 2699–2706 (2011). [CrossRef]

### Minimum edge intensity (MINEI) method

**49**, 1549–1554 (2010). [CrossRef] [PubMed]

*I*(

_{min}*k*,

*l*).

*IE*is the set of interior-edge pixels, which is a subset of

*P*.

*N*is the number of interior-edge pixels.

_{IE}### Integrated gradient (IG) method

20. Y. Yang, G. Li, 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] [PubMed]

*z*direction. The first criterion IG

_{1}(

*z*) is defined as where

_{r}*EE*is the set of exterior-edge pixels, which are the background pixels that just encompasses the particle.

*N*is the number of exterior-edge pixels. The second criterion IG

_{EE}_{2}(

*z*) is defined as

_{r}## 5. Quantification of measurement accuracy by synthetic holograms

### 5.1. Accuracy in detecting a single particle

*D*

_{0}= 40, 70, 100, 130, 160, 200, 230 and 270

*μm*. Each of them is located at

*z*

_{0}= 0.05, 0.10, 0.15, 0.20, 0.25 and 0.30

*m*from the hologram plane. The diameter of a square is that of a circle with an equivalent area. Here, the approach used in [22

22. D. R. Guildenbecher, J. Gao, P. L. Reu, and J. Chen, “Digital holography simulations and experiments to quantify the accuracy of 3D particle location and 2D sizing using a proposed hybrid method,” Appl. Opt. **52**, 3790–3801 (2013). [CrossRef] [PubMed]

33. F. Slimani, G. Grehan, G. Gouesbet, and D. Allano, “Near-field Lorenz-Mie theory and its application to micro-holography,” Appl. Opt. **23**, 4140–4148 (1984). [CrossRef]

*nm*. The size of the hologram is 1024 × 1024 with 7.4 × 7.4

*μm*

^{2}pixels. A single particle is centered in the hologram. Some of the effects introduced by digital recording, such as digitization and readout noise, are also incorporated in the simulation. The intensity of each pixel obtained from the analytical expression is scaled linearly to the range [0, 9830] and then digitized by rounding to the nearest integer. Normally distributed random noise with a standard deviation of 164 is added to each pixel. This procedure is analogous to an experimental hologram recorded by a 14-bit digital camera with the brightest pixel having a grayscale value of 9830 (60% of the full scale) and the readout noise is 1% of the full scale. For each particle condition (Fresnel number), 10 such holograms are simulated and processed to investigate the effects of random noises on particle detection. Therefore, a total of 480 holograms are simulated for each shape.

*F*= 0.063. Figure 4(a) shows the simulated hologram, while Fig. 4(b) shows the radial intensity profile along the dotted line in Fig. 4(a). The inset in Fig. 4(a) shows the reconstructed intensity when the hologram is re-focused to the actual particle position,

*z*

_{0}, using Eq. (1), while Fig. 4(c) shows the radial intensity profile along the solid line in the re-focused image. In addition, Fig. 4(c) shows the intensity profiles when the hologram is re-focused to

*z*=

_{r}*z*

_{0}± 10

*D*

_{0}(green and blue lines). Finally, the bottom row (Figs. 4(d)–4(f)) is for a case of relatively low

*F*= 0.004 and is displayed in a similar manner.

*F*. As the bottom row in Fig. 4 reveals, at small

*F*the interference fringes have a larger radial extent. Combined with the limited hologram size, this effect results in a decrease of the number of higher-order lobes recorded by the sensor. For example, in Fig. 4(b) at

*F*= 0.063 higher-order lobes up to the fifth order are recorded; however, at

*F*= 0.004, only part of the second-order lobe is recorded. The loss of higher-order lobes results in less sharp or uniform particle images in reconstruction, as shown in the comparison between the insets in Figs. 4(a) and 4(d). Consequently, focus metrics which rely on edge sharpness (e.g., the LAP method) or particle image uniformity (e.g., the VAR method) tend to display higher uncertainties at lower

*F*.

*F*. This method relies on the assumption that pixels on the particle edge experience a minimum in intensity when reconstructed at the actual particle position,

*z*

_{0}. Figure 4(f) reveals that this is generally true at low

*F*(notice the intensity near the particle edge is minimum when

*z*=

_{r}*z*

_{0}, red line, in comparison to

*z*=

_{r}*z*

_{0}± 10

*D*

_{0}, green and blue lines). In contrast, at higher

*F*Fig. 4(c) reveals that rapid variation in intensity at the particle edge results in some pixels near the border which do not follow this trend. Combined with necessary discretization of the edge region as selected by the MINEI method, this effect tends to increase error in the measured particle depth at higher

*F*.

*F*, on the method accuracy. In practice it is well known that many other parameters can affect accuracy including the particle diameter/pixel size ratio, signal to noise ratio, particle morphology, position of the particle with respect to the detector edge, laser wavelength and coherence, particle number density, three-dimensionality of the particle fields, particle overlap within the field of view, etc. The results provided in Media 1 can be used to explore some of these effects (such as the particle diameter/pixel size ratio and the particle morphology). In addition, results presented in the proceeding sections use simulations and experiments of particle fields to provide some additional insight into the effects of particle number density, three-dimensionality of the particle field, and particle overlap. Nevertheless, due to the wide range of factors affecting measurement accuracy, caution should be applied before extending any of the results or conclusions presented here to conditions not explored in this work.

*x*–

*y*location and shape. In contrast, the HYBRID method has been specifically designed to automatically select thresholds for image segmentation, and therefore can also be used to measure particle

*x*–

*y*location and shape. Figure 5 shows the relative error and uncertainty of these quantities as measured by the HYBRID method for the range of conditions explored in this section. A particle can be located in the

*x*–

*y*plane with sub-pixel accuracy (less than 0.1 pixels). The relative error of size measurement is less than 5%, except at small Fresnel numbers where sizing becomes difficult due to decreased edge sharpness as displayed in the bottom row in Fig. 4. Similar results are given in [22

22. D. R. Guildenbecher, J. Gao, P. L. Reu, and J. Chen, “Digital holography simulations and experiments to quantify the accuracy of 3D particle location and 2D sizing using a proposed hybrid method,” Appl. Opt. **52**, 3790–3801 (2013). [CrossRef] [PubMed]

### 5.2. Accuracy in detecting a particle field

*mm*× 7.6

*mm*× 5

*cm*, where 5

*cm*is the dimension in the

*z*direction. Dimensions in the

*x*and

*y*directions are determined by the hologram size (1024 × 1024 with 7.4 × 7.4

*μm*

^{2}pixels). To exclude the effects of overlapping particles on particle detection accuracy, the particles are separated transversely from each other, such that the minimum in-plane distance between neighbor particle centers is 3

*D*

_{0}. Particle diameters are normally distributed with a standard deviation of 10

*μm*. The mean diameter

*D̄*

_{0}is 50

*μm*or 100

*μm*. The distance between the hologram plane and the nearest surface of the particle volume,

*L*, is 0.06

*m*or 0.12

*m*. The particle number density,

*ρ*, which is the average number of particles over the cross-section area of the volume, has two values: 3

_{n}*mm*

^{−2}and 6

*mm*

^{−2}. Eight different particle fields are generated by varying

*D̄*

_{0},

*L*and

*ρ*. To simulate the effects of particle displacement between sequentially recorded holograms, six holograms are calculated wherein the particle field is displaced 1

_{n}*mm*in the

*z*direction between each hologram. In total 48 holograms are generated. The wavelength is 532

*nm*.

*z*

_{0,i}>

*z*

_{0,i+1}. The complex amplitude immediately after impinging on the

*i*particle can be expressed as is the mask function of the

^{th}*i*particle with diameter

^{th}*D*

_{0,i}located at (

*x*

_{0,i},

*y*

_{0,i},

*z*

_{0,i}).

*i*− 1)

*particle, and*

^{th}*K*is the number of particles. To alleviate the space-domain aliasing introduced by the inverse FFT operation, the simulation is conducted within a much larger “frame”, and the final hologram is cropped from the center of the larger hologram. Specifically, a 4096 × 4096 hologram is first simulated, then the 1024 × 1024 hologram used for uncertainty characterization is cropped from the center. The effects of digital recording are also included in the same manner as in Section 5.1. Two sample synthetic holograms of particle fields are shown in Figs. 6(a) and 6(b). Again, the alternative methods are operated on the exact particle pixels and windows, while the HYBRID method is implemented without knowledge about the particle field. Due to the limited size of the hologram, some fringes are lost for particles close to the borders, leading to inaccurate particle detection. Therefore, for

*L*= 0.06

*m*, particles within 50 pixels of the borders are neglected; for

*L*= 0.12

*m*, particles within 100 pixels of the borders are neglected. An average Fresnel number

*F̄*can be calculated for each particle field using

*D̄*

_{0}and the distance between the hologram plane and the particle field center, as shown in the first row of Table 1.

*z*locations of the particles identified by the HYBRID method are determined using alternative methods by applying them directly to the exact particle pixels and windows. For each pair of particles in consecutive holograms, an individual displacement Δ

*z*can be obtained by subtraction of the first depth from the second depth. The standard deviation of displacements detected from all pairs of particles in the five translations is taken as the

_{d}*z*-displacement uncertainty, which is then divided by 2

^{1/2}to obtain the depth measurement uncertainty,

*δ*. To evaluate the depth error, the detected depth,

_{z}*z*, is compared with the particle actual depth,

_{d}*z*

_{0}, and the relative depth error is calculated by |

*z*−

_{d}*z*

_{0}|/

*D*

_{0}. For each particle field, the mean value of Δ

*z*is computed from all particle pairs in the five translations, and the mean value of

_{d}**|**

*z*−

_{d}*z*

_{0}|/

*D*

_{0}is computed from all detected particles in the six holograms, as listed in Table 1. The relative depth uncertainty is with respect to the mean particle diameter,

*D̄*

_{0}.

*F̄*, higher particle number density has negative effects on all particle detection methods, as is expected due to the increased noise introduced by neighboring particles. Similar to the results presented in Section 5.1, Table 1 indicates that the HYBRID, MINI, and IG methods all perform relatively well over the range of conditions explored here. Also, the VAR method is generally less accurate at low

*F̄*, while the MINEI method is less accurate at higher

*F̄*. However, in contrast to the results presented in Section 5.1, the LAP method shows relatively high uncertainty over the entire range of

*F̄*, and the CC method degrades severely at high

*F̄*. Both the LAP and CC methods are performed within windows that enclose target particles, thus noise due to light diffraction from adjacent particles are also included in the window, as shown in the background in Fig. 6(d). It is theorized that the focus metrics of the LAP and CC methods are severely degraded by this noise resulting in the high uncertainty shown in Table 1. Overall, the depth error and uncertainty of the HYBRID method are both less than one particle diameter, while the MINI and IG methods demonstrate superior performance compared to the HYBRID method in terms of smaller depth errors and uncertainties. One is reminded that knowledge of the exact binary images of synthetic particles contribute to their superior performances.

*D*−

_{d}*D*

_{0}|/

*D*

_{0}, |

*W*−

_{d}*D*

_{0}|/

*D*

_{0}and |

*H*−

_{d}*D*

_{0}|/

*D*

_{0}are calculated from all detected particles in the six holograms, as also listed in Table 2. The relative errors in measurement of the particle diameter, width and height are about 5%, which demonstrates the validity of the particle binary images extracted by the HYBRID method.

## 6. Quantification of measurement accuracy by experimental holograms

*x*,

*y*,

*z*position of each particle is not known. Therefore, only depth uncertainty,

*δ*, can be quantified by processing of sequential holograms of particle fields translated in the

_{z}*z*direction.

### 6.1. Accuracy in detecting a planar particle field

_{i}O

_{2}) particles on a thin flat glass. The surface of the glass, on which the particles are placed, is perpendicular to the

*z*direction. The glass is further fixed on a translation stage, so that the particle field can be translated to different

*z*positions. The expanded and collimated beam from a He-Ne laser (

*λ*= 632.8

*nm*) is used to illuminate the particle field, and the hologram is recorded by a CCD camera (Cooke pco.2000) with 2048 × 2048, 7.4 × 7.4

*μm*

^{2}pixels. Most of the particles reside in a region of 1500 × 700 pixels, which is then cropped from the original size as the hologram. The glass is translated to sequential

*z*positions with a positioning uncertainty of 12.7

*μm*. In an experiment, the glass slide is translated to a total of six different

*z*-positions with uniform spacing, Δ

*z*

_{0}. Two different values of Δ

*z*

_{0}are considered (127

*μm*and 635

*μm*), yielding a total of twelve holograms. Particle

*x*–

*y*positions and shapes are measured using the HYBRID method with the lower limit on detectable diameter set to 25

*μm*(3 pixels). The nominal mean diameter of the S

_{i}O

_{2}particles is around 40

*μm*. However, due to particle agglomeration and the size threshold, the measured average diameter,

*D̄*, is 86

_{d}*μm*. The average depth of particles is approximately 4.3

*cm*. Using these values, an average Fresnel number

*F̄*of 0.065 is found. The particle number density

*ρ*is 3.7

_{n}*mm*

^{−2}. Shown in Fig. 7 are a part of the hologram, a reconstructed intensity image at the average depth of all particles, the corresponding binary images extracted by the HYBRID method and the depth distribution of the particles measured from one hologram with a standard deviation of 174

*μm*(∼ 2

*D̄*).

_{d}*x*and

*y*directions. Following the accuracy quantification method used in Section 5.2, the mean displacement and depth uncertainty are quantified for each method, as shown in Table 3. The HYBRID, CC, MINI and MINEI methods demonstrate comparable performances. Interestingly, the CC method shows a considerable improvement, which may be attributed to the fact that the particles are on the same plane, and thus the CC values for each particle tend to reach their maximums at the same

*z*location. Unlike in the processing of synthetic holograms, the exact particle information in experimental holograms is not available to alternative methods. As a result, the MINI and IG methods downgrade from their ideal performances. The LAP and VAR methods display the highest uncertainties.

### 6.2. Accuracy in detecting a 3D particle field

*cm*× 5

*cm*× 5

*cm*. Due to the high viscosity (10000

*cSt*) of the oil, the particles settle very slowly and can be assumed to be stationary during the duration of an experiment [22

**52**, 3790–3801 (2013). [CrossRef] [PubMed]

*μm*. In these experiments Δ

*z*

_{0}= 2

*mm*, and the particle field is displaced seven times in the

*z*direction. The collimated laser beam from a DPSS laser (Coherent Verdi V6,

*λ*= 532

*nm*) illuminates the particle field, and the resulting hologram is recorded by a CCD camera (Redlake MegaPlus EC16000) with 4872 × 3248, 7.4 × 7.4

*μm*

^{2}pixels. To improve statistical convergence, the procedure is repeated after stirring the oil to create a new particle field and degassing in a vacuum to remove any bubbles. In total, fourteen holograms are recorded and processed by the HYBRID method. Particles that are within 200 pixels to the hologram borders are rejected to account for the degradation of accuracy due to loss of diffraction fringes. The particle size distribution is previously measured using a Malvern Mastersizer [22

**52**, 3790–3801 (2013). [CrossRef] [PubMed]

*μm*to minimize false particle detection in regions where noise results in local intensity gradients. The average diameter is measured to be 438

*μm*, close to that measured by the Mastersizer (450

*μm*). The average distance between the hologram plane and the cuvette center is about 19.5

*cm*. Therefore,

*F̄*= 0.452 with measured

*ρ*= 0.14

_{n}*mm*

^{−2}. Fig. 8 shows a photo of the cuvette filled with silicone oil and particles, a sample region of the hologram and the corresponding particle binary image extracted by the HYBRID method.

*z*and

_{d}*δ*are presented in Table 4. Overlapping particles, as circled in Fig. 8(b), are erroneously detected as single particles with incorrect size and position [22

_{z}/D̄_{d}**52**, 3790–3801 (2013). [CrossRef] [PubMed]

## 7. Conclusion

7. Y. Yang and B. seon Kang, “Digital particle holographic system for measurements of spray field characteristics,” Opt. Laser Eng. **49**, 1254–1263 (2011). [CrossRef]

8. J. Gao, D. R. Guildenbecher, P. L. Reu, V. Kulkarni, P. E. Sojka, and J. Chen, “Quantitative, three-dimensional diagnostics of multiphase drop fragmentation via digital in-line holography,” Opt. Lett. **38**, 1893–1895 (2013). [CrossRef] [PubMed]

**49**, 1549–1554 (2010). [CrossRef] [PubMed]

23. V. Palero, M. Arroyo, and J. Soria, “Digital holography for micro-droplet diagnostics,” Exp. Fluids **43**, 185–195 (2007). [CrossRef]

*x*–

*y*plane. Note, to allow for accurate sizing, the particle diameter is generally significantly larger than the pixel size for the conditions investigated here. Extension of these results to applications where the particle size is on the order of the pixel size (such as HPIV [17

17. J. 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]

26. G. Pan and H. Meng, “Digital holography of particle fields: Reconstruction by use of complex amplitude,” Appl. Opt. **42**, 827–833 (2003). [CrossRef] [PubMed]

## Acknowledgments

### Nomenclature

| reconstructed amplitude |

| actual particle diameter |

| measured particle diameter |

| depth map of I_{min} |

| depth map of T_{max} |

| reconstructed complex amplitude |

| set of exterior-edge pixels |

| Fresnel number |

| intensity distribution of a digital hologram |

| minimum intensity map |

| reconstructed intensity |

| set of interior-edge pixels |

| discrete coordinates in the image plane |

| Laplacian kernel |

| distance between the hologram plane and the nearest surface of a particle field |

| discrete coordinates in the hologram plane |

| discrete coordinates in the spatial frequency domain |

| number of pixels in the x and y directions |

| set of pixels that belong to a particle |

| quantified sharpness |

S_{y} | horizontal and vertical Sobel kernels |

| threshold |

| Tenengrad map |

| maximum Tenengrad map |

| set of pixels that belong to a local window |

| actual particle distance |

| measured particle distance |

| reconstruction distance |

| depth uncertainty |

Δ | correlation interval |

Δ | actual displacement |

Δ | measured displacement |

Δ | dimension of an individual pixel in the x and y directions |

| wavelength |

| particle number density |

| edge finding operation |

| fast Fourier transform |

^{1} | inverse fast Fourier transform |

| thresholding operation |

## References and links

1. | H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. |

2. | J. Sheng, E. Malkiel, and J. Katz, “Using digital holographic microscopy for simultaneous measurements of 3D near wall velocity and wall shear stress in a turbulent boundary layer,” Exp. Fluids |

3. | D. Chareyron, J. L. Marié, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography inverse method for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. |

4. | N. A. Buchmann, C. Atkinson, and J. Soria, “Ultra-high-speed tomographic digital holographic velocimetry in supersonic particle-laden jet flows,” Meas. Sci. Technol. |

5. | J. Lee, K. A. Sallam, K. C. Lin, and C. D. Carter, “Spray structure in near-injector region of aerated jet in subsonic crossflow,” J. Propul. Power |

6. | 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. |

7. | Y. Yang and B. seon Kang, “Digital particle holographic system for measurements of spray field characteristics,” Opt. Laser Eng. |

8. | J. Gao, D. R. Guildenbecher, P. L. Reu, V. Kulkarni, P. E. Sojka, and J. Chen, “Quantitative, three-dimensional diagnostics of multiphase drop fragmentation via digital in-line holography,” Opt. Lett. |

9. | 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. Nat. Acad. Sci. USA |

10. | 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. |

11. | L. Tian, N. Loomis, J. A. Domínguez-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. |

12. | D. Lebrun, D. Allano, L. Méès, F. Walle, F. Corbin, R. Boucheron, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt. |

13. | J. P. Fugal, R. A. Shaw, E. W. Saw, and A. V. Sergeyev, “Airborne digital holographic system for cloud particle measurements,” Appl. Opt. |

14. | Y.-S. Choi and S.-J. Lee, “Three-dimensional volumetric measurement of red blood cell motion using digital holographic microscopy,” Appl. Opt. |

15. | T. Khanam, M. N. Rahman, A. Rajendran, V. Kariwala, and A. K. Asundi, “Accurate size measurement of needle-shaped particles using digital holography,” Chem. Eng. Sci. |

16. | S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Laser Technol. |

17. | J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. |

18. | V. Ilchenko, T. Lex, and T. Sattelmayer, “Depth position detection of the particles in digital holographic particle image velocimetry (DHPIV),” Proc. SPIE |

19. | J. P. Fugal, T. J. Schulz, and R. A. Shaw, “Practical methods for automated reconstruction and characterization of particles in digital in-line holograms,” Meas. Sci. Technol. |

20. | Y. Yang, G. Li, 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. |

21. | Y. Wu, X. Wu, Z. Wang, L. Chen, and K. Cen, “Coal powder measurement by digital holography with expanded measurement area,” Appl. Opt. |

22. | D. R. Guildenbecher, J. Gao, P. L. Reu, and J. Chen, “Digital holography simulations and experiments to quantify the accuracy of 3D particle location and 2D sizing using a proposed hybrid method,” Appl. Opt. |

23. | V. Palero, M. Arroyo, and J. Soria, “Digital holography for micro-droplet diagnostics,” Exp. Fluids |

24. | E. Darakis, T. Khanam, A. Rajendran, V. Kariwala, T. J. Naughton, and A. K. Asundi, “Microparticle characterization using digital holography,” Chem. Eng. Sci. |

25. | Y. Yang, B. seon Kang, and Y. jun Choo, “Application of the correlation coefficient method for determination of the focal plane to digital particle holography,” Appl. Opt. |

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

27. | W. Yang, A. B. Kostinski, and R. A. Shaw, “Phase signature for particle detection with digital in-line holography,” Opt. Lett. |

28. | F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express |

29. | C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul”, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Laser Eng. |

30. | S. Soontaranon, J. Widjaja, and T. Asakura, “Extraction of object position from in-line holograms by using single wavelet coefficient,” Opt. Commun. |

31. | F. Soulez, L. Denis, C. Fournier, Éric Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. A |

32. | J. W. Goodman, |

33. | F. Slimani, G. Grehan, G. Gouesbet, and D. Allano, “Near-field Lorenz-Mie theory and its application to micro-holography,” Appl. Opt. |

34. | D. K. Singh and P. K. Panigrahi, “Improved digital holographic reconstruction algorithm for depth error reduction and elimination of out-of-focus particles,” Opt. Express |

**OCIS Codes**

(100.6890) Image processing : Three-dimensional image processing

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(350.4990) Other areas of optics : Particles

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: July 23, 2013

Revised Manuscript: October 6, 2013

Manuscript Accepted: October 11, 2013

Published: October 28, 2013

**Citation**

Jian Gao, Daniel R. Guildenbecher, Phillip L. Reu, and Jun Chen, "Uncertainty characterization of particle depth measurement using digital in-line holography and the hybrid method," Opt. Express **21**, 26432-26449 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-26432

Sort: Year | Journal | Reset

### References

- H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol.15, 673 (2004). [CrossRef]
- J. Sheng, E. Malkiel, and J. Katz, “Using digital holographic microscopy for simultaneous measurements of 3D near wall velocity and wall shear stress in a turbulent boundary layer,” Exp. Fluids45, 1023–1035 (2008). [CrossRef]
- D. Chareyron, J. L. Marié, C. Fournier, J. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography inverse method for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys.14, 043039 (2012). [CrossRef]
- N. A. Buchmann, C. Atkinson, and J. Soria, “Ultra-high-speed tomographic digital holographic velocimetry in supersonic particle-laden jet flows,” Meas. Sci. Technol.24, 024005 (2013). [CrossRef]
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- Y. Yang and B. seon Kang, “Digital particle holographic system for measurements of spray field characteristics,” Opt. Laser Eng.49, 1254–1263 (2011). [CrossRef]
- J. Gao, D. R. Guildenbecher, P. L. Reu, V. Kulkarni, P. E. Sojka, and J. Chen, “Quantitative, three-dimensional diagnostics of multiphase drop fragmentation via digital in-line holography,” Opt. Lett.38, 1893–1895 (2013). [CrossRef] [PubMed]
- 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. Nat. Acad. Sci. USA104, 17512–17517 (2007). [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, 064004 (2011). [CrossRef]
- L. Tian, N. Loomis, J. A. Domínguez-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] [PubMed]
- D. Lebrun, D. Allano, L. Méès, F. Walle, F. Corbin, R. Boucheron, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt.50, H1–H9 (2011). [CrossRef] [PubMed]
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