We have used a digital in-line holography system with numerical reconstruction for 3D particle field extraction. In this system the diffraction patterns (holograms) are directly recorded on a charge-coupled device (CCD) camera. The numerical reconstruction is based on the wavelet transformation method. A sample volume is reconstructed by computing the wavelet components for different scale parameters. These parameters are related to the axial distance between a particle and the CCD camera. The particle images are identified and localized by analyzing the maximum of the wavelet transform modulus and the equivalent diameter of the particle image. The general process for the 3D particle location and data processing method are presented. As in classical holography we found that the signal to noise ratio depends only on the shadow density. Nevertheless, we show that both the volume depth and the shadow density affect the percentage of extracted particles.
©2004 Optical Society of America
In in-line digital holography, the diffraction pattern is directly recorded on a charge-coupled device (CCD) camera and the reconstruction process is performed numerically. Several numerical methods have been proposed to determine the characteristics of micro-objects from their diffraction patterns. Reconstruction methods based on the Fresnel integral , the complex amplitude  and the wavelet transformation (WT) [3–6] have been investigated. More recently, the use of the Wigner distribution function associated to the fractional order Fourier transformation for the 3D location of particles and fibers has been proposed [7, 8]. Digital holographic techniques have also been applied to three-dimensional-three component (3D-3C) or three-dimensional-two component (3D-2C) particle image velocimetry [9–14] and to other fields such as biology .
In classical holography, the diffraction pattern produced by a particle field is recorded on a holographic plate or film with a collimated laser beam. The reconstruction of the 3D particle images is performed by illuminating developed hologram with the reference beam. Thereafter, a CCD camera is used to scan mechanically, plane by plane, a region of the sample volume, and the particle locations are computed from the reconstructed volume by an automated process. Recently, this technique has been used by Pu and Meng to develop a new, successful, holographic particle image velocimetry (HPIV) system .
In Gabor holography it is well-known that the characteristics of the recording medium, i.e. particle number density and the depth of sample volume may significantly influence the signal-to-noise ratio (SNR) of the reconstructed images [16, 17]. Even so, to our knowledge, no quantitative study to determine the percentage of the extracted particles has been achieved. In this work we show that this parameter is very important in determining whether holographic particle tracking velocimetry (HPTV) can be successfully implemented in the case of digital holography.
The structure of this paper is as follows: Section 2 presents the background of digital in-line holography and the theoretical basis of the hologram reconstruction using WT. In Section 3, we present the method used for measuring 3D particle position. This method is based on the analysis of the maximum of the wavelet transform modulus (WTMM) and the equivalent diameter (Deq) of the reconstructed particle image. This analysis is performed for different axial positions (z). In the same section, we show how, for small particles, this method can be used for an automated particle field extraction process. In Section 4, the results of this particle extraction process are presented from simulated in-line holograms. Moreover, we show that both volume depth and shadow density hamper the percentage of extracted particles. In addition, the dependence of the signal-to-noise ratio on the shadow density is established. Concluding remarks are reported in Section 5.
2. Theoretical background
Let us recall the digital reconstruction process proposed by Buraga-Lefevre et al . Consider an opaque object O(ξ,η) of diameter d illuminated by a monochromatic plane wave (see Fig. 1). The intensity distribution on a plane (x,y) located at a distance z 0 from an object can be approximated by:
where ** denotes the two-dimensional convolution operation over spatial variables x and y.
Note that the intermodulation term has been dropped. This simplification is valid in the general case of far-field in-line holography (i.e. d 2/λz 0 ≪ 1).
The daughter wavelet functions are defined for the scale parameter a by :
The scale parameter a 0 of the wavelet is related to the distance z 0 as follows :
In the same way, this approach can be used for hologram reconstruction. The reconstructed image at a given distance zr can be seen as the WT of the intensity distribution recorded by the photosensitive plane. When ar = a 0 = a (i.e. the interrogation plane located at a distance zr corresponds to the object plane located at a distance z 0), it can be shown that, by dropping a multiplicative constant we have:
The object function O(x,y) can easily be reconstructed. This method can be extended to the case of several particles of different diameters provided that the far-field condition is maintained.
In fact, the function given by Eq. (3) is not really a wavelet and must be modified in order to check the admissibility conditions required for a wavelet function (zero mean and localization). It is shown  that the following function can be used:
where σ is a parameter that depends on the frame grabber characteristics and M Ψ(σ) is adjusted in order to have a zero mean value of ΨGa(x,y).
3. General process for extracting 3D particle locations
Numerous metrology applications involving particle fields (such as velocity field computation from pairs of images and particle tracking in 3D turbulence flow) require a robust and accurate method for extracting particle coordinates from reconstructed 3D images.
Firstly, let us consider a sample volume of thickness L. This volume corresponds to a set of reconstructed z-planes WTIz (a,x,y) such that the inequalities zr - L/2 ≤ z ≤ zr + L/2 hold. Figure 2(a) shows an example of the simulated diffraction pattern of a 30μm particle located at z 0 = 50mm. The reconstructed particle image is shown in Fig. 2(b).
In our 3D particle field extraction process, we use both the WTMM and the equivalent diameter (Deq) variations versus zr. The WTMM method consists of searching for the particle position that gives the maximum of WTMM(z). An example of the WTMM versus z-position is shown in Fig. 3(a) by the line marked with square. The details of this method are given by Buraga-Lefebvre et al .
In a previous paper , we have shown that from a WT point of view, the recording-reconstruction process can be seen as a linear shift invariant system where the point-spread function does not depend on the axial distance (z 0). This result is used for particle or fiber diameter measurement by elaborating a unique calibration curve, regardless of the recording distance. Furthermore, it was observed that the image width is minimal when the position of the interrogation plane corresponds to the position of the particle (i.e. zr = z 0). From this result, we have established an additional criterion based on the evolution of the equivalent image diameter Deq with zr. Deq is defined by the following expression :
where Seq is the equivalent area of the reconstructed particle image at a given axial distance z:
An example of the equivalent diameter variations with zr is shown in Fig. 3(a) by line marked with circle.
The general process for determining 3D particle locations within the studied volume is outlined in Fig. 4.
Before selecting the supposed particle images by a thresholding operation, a background subtraction is performed (step 2). The background is estimated for each z-plane by using a morphological opening operator. The size of the structuring element of this operator must be larger than the particle diameter. The objective of this operation is to improve the image signal-to-noise ratio by accentuating the contrast of the reconstructed 2D object images. This operation is followed by a 2D-labelling operation (step 3). Thus, the x-y coordinates of the labelled objects in each plane are available.
In step 4, we define a 3D-labelled object by clustering the 2D-labelled objects located on the same z-axis. During this operation, we retain only the 3D-labelled objects that are contained in at least three consecutive planes. The use of this rejection principle is justified by the great depth of field. Indeed, in in-line holography, a given particle image is inevitably detected in several adjacent z-planes (see Fig. 3(a)). In addition, this operation enables the speckle and other noises to be reduced in the considered volume.
Finally (i.e. in step 5), the result of step 4 is used to recognize the particles among all the 3D-labelled objects in the volume, using both the WTMM variations and the Deq variations with zr.
Let us consider Z 1 and Z 2 as the axial coordinates corresponding respectively to the maximum of WTMM(z) and the minimum of Deq(z). The joint exploitation of these extrema proved efficient in the elimination of ambiguities between the 3D particle images and other 3D objects which are present in several planes. In this volume, only the 3D-labelled objects presenting the parabolic shapes of the WTMM and Deq curves (as shown in Fig. 3(a)) and those checking Z 1 ≈ Z 2 are considered as 3D particle images. The estimated z-location of the considered particle is given by (Z 1 + Z 2)/2. At the end of this extraction process, the 3D particle positions corresponding to a given volume are obtained.
4. Results and discussion
In order to evaluate the potential of the proposed method, we have used synthesized in-line holograms consisting of 1024 × 1024 pixels of 9 μm. Particles are randomly distributed in a sample volume of depth L. An example of a simulated particle field hologram with particle diameter d = 30μm is shown in Fig. 5(a). This pattern is generated with particle number density ns = 15 mm-3 and L = 3mm . The numerical reconstruction at zr = 51mm is shown in Fig 5(b). The whole volume (200 z-planes) is reconstructed by using the method described in Section 2. Then, this volume is processed according to the procedure discussed in Section 3.
We have presented in Fig. 6 the histogram of the measured depth-error δz. δz is evaluated for each particle image by comparing the expected and estimated axial locations. For this example, the mean square error computed on the depth estimation is equal to 120 μm.
The reliability of the extraction method is very important for implementing holographic particle tracking velocimetry (HPTV). Indeed, the efficiency of the HPTV depends on the number of calculated velocity vectors. The number of these vectors is, of course, related to the percentage of particle coordinates extracted from the reconstructed images. For this reason, we have studied the percentage of extracted particles (Ep) and the signal-to-noise ratio (SNR) in several cases likely to be encountered. These two parameters have been measured for different particle number densities (ns) and for different depths (L) of the sample volume. We have also tested this particle extraction process in the case of Gaussian size distribution.
¿From the reconstructed particle images, the SNR is given by the ratio of the average intensity of the extracted particle images (Isg) to the standard deviation of the background noise (σbn):
The results are plotted versus the non-dimensional parameter sd = nsLd 2 used by Royer . This parameter, called shadow density is said to describe hologram quality.
Figure 7 illustrates the dependence of Ep on sd. From this figure, it is clear that the number of extracted particles decreases with increased shadow density. Nonetheless, as shown by the different cases for which the Ep is computed, the same shadow density sd may lead to different values of Ep. Therefore, shadow density alone is not a good prediction for Ep. Considering the depth of the sample volume, we show that L has a much stronger influence compared to ns. For example, in both situations : [L = 3mm, ns = 2mm-3, d = 30μm] and [L = 20mm, ns = 0.3mm-3, d = 30μm], the same shadow density is obtained (sd = 0.54%), but we extract respectively 93.12% and 81.73% of particles.
Note that in practical cases of HPTV experiments, the depth of the sample volume is often greater than 10 mm. In such cases, a high percentage (Ep> 70%) of extracted particles is needed to obtain a significant number of paired particles. Thus, the particle number density must be reduced to less than 1 mm-3, approximately.
However, the measured SNR leads to a satisfactory estimation of the shadow density as demonstrated by Meng et al . This idea is confirmed in the case of digital in-line holography as shown in Fig. 8 by the curves representing the situation [L = 3mm, d = 30μm, ns=(2, 5, 10, 15, 20, 25, 30)mm-3] and [L = 20mm, d = 30μm, ns=(0.3, 0.7, 2.5, 9)mm-3]. Note that in all the simulations the measured SNR is greater than 6dB.
The case corresponding to a Gaussian size distribution with L = 3mm and ns = (1, 2, 4, 6, 8)mm-3 deserves a particular interpretation. Indeed, for this case, the extraction process fails for very small particles. Thus, the measured SNR is clearly greater than predicted because it is measured only from the reconstructed particle images whose diameters are greater than 2×δccd, where δccd is the pixel size of the CCD camera. The intensity of very small particle images cannot be differentiated from the background noise. This overestimation of the SNR is well shown in Fig. 8 by the star-marked line.
The above remarks prove that the SNR is not the only indicator for the percentage of particles to be extracted. For a given SNR, this percentage depends mainly on the spatial distribution of particles inside the sample volume.
4.2. Experimental results
The above extraction method has been tested using an RS-3 standard reticle (Malvern Equipment) which is approximately parallel to the recording plane. This reticle is an optical glass plate with a pattern of small particles photographically deposited on the surface. The light source used is a low-power laser diode. The wavelength of the generated laser beam is λ = 635 nm. The diffraction patterns are recorded by means of a CCD camera with 1008×1018 pixels of 9 μm.
Figure 9 shows an example of a hologram recorded at approximately z 0 = 45mm from the reticle. The information contained in this diffraction pattern leads to a satisfactory numerical reconstruction as shown in Fig. 9(b). To extract the particle field, we have built a sample volume corresponding to a set of reconstructed z-planes with 44.5 ≤ zr ≤ 46.5 mm. Then, this volume has been processed according to the procedure discussed in Section 3. We have represented in Fig. 10 the histogram of the deviations between the measured axial locations and the best fit plane obtained by a 2D polynomial regression on the 3D particle coordinates. It was found that the standard deviation is equal to 180μm. The shadow density, measured by a common imaging system, has been evaluated to sd = 10%. The SNR measured from the reconstructed particle images is equal to 7.54dB (see “Experiment” point in Fig. 8).
Knowing the total number of particles in the reticle and by counting the number of extracted particles, we have found that Ep≃ 21%. This result is in good agreement with the simulations presented in Fig. 7 (see “Experiment” point in Fig. 7).
In this study, we have tested the applicability of digital in-line holography to particle field extraction. The 3D coordinates are measured by analyzing both the maximum of the wavelet transform modulus and the equivalent diameter versus the axial reconstruction distance zr.
As expected, the reliability of the extraction process depends on the SNR of the reconstruction fields.
Here, we have confirmed that the SNR depends on the shadow density alone. This result, well-established in the case of classical holography, has been extended here to digital in-line holography.
Nevertheless, we have demonstrated by simulated holograms, that the percentage of extracted particles is not only a function of the shadow density, but is also dependent on the spatial distribution of particles. In particular, we have shown that the depth of the particle sample volume acts as an essential parameter. Note that the experimental point obtained by using a calibrated reticle is in good agreement with the simulations.
This result provides criteria for estimating the success of the 3D HPTV method. Indeed, this study shows that particle number density should be drastically reduced for successful diagnostics in thick volumes. However, digital in-line holography may bring additional information to conventional existing methods for particle field investigation.
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