1. Introduction

Microparticle measurements are used in fluid mechanics, microfluidics, and remote sensing. Digital holography (DH) is suitable for these applications, as it offers three-dimensional recovery of particle sizes and positions with data acquisition [1–3]. Examples of the applications are flow phenomena and holographic particle image velocimetry [4–9]. The prediction and control of particulate flows play an important role in practical engineering and environmental fields. In the optical setup to analyze such flow in DH, a wide range of observation in the transversal direction is required and a collimated wave illuminates particles and there is no lens such as a microscope objective between the particles and a bare image sensor. Since the particles appear elongated when optical setups with low numerical apertures (∼0.1) are used for DH, considering the minimum intensity of a reconstructed particle image as the particle center would be insufficient. Instead of using the minimum intensity as a metric, metrics such as edge sharpness and intensity distribution are applied to characterize the axial position and size of the particles [10,11]. However, the theoretical intensity distribution is not accurate and the estimated axial position that uses the intensity does not coincide with the true axial position because the Mie theory is not used in its calculation [12,13]. The deconvolution method models the observed blur in 3D reconstruction as the convolution of an object and a point-spread function [14]. However, the centroid axial position exhibits a similar offset. The offset depends on the size of the particle and increases with increasing particle radius in this method compared with the calculation using the Mie theory [15].

To determine the accurate axial particle position, a method using the phase signature has been proposed [16]. Moreover, the phase curvature has been used as a metric to determine the position of a particle with a 10-µm diameter using the Mie theory [17]. However, for a particle with a 60-µm diameter, which is a common particle size for turbulent flow visualization [9], the phase or curvature using a phase unwrapping algorithm near the particle center is unstable because the phase gradients are too large near the center. Thus, it is difficult to use this metric for the positioning of a particle center for a particle with a 60-µm diameter. Moreover, methods involving DH to obtain not only the correct axial particle positions but also particle sizes and refractive indices (RIs) are required. A direct hologram analysis method has been developed for the characterization and positioning of particles (from sub to a few µm diameters) in high-NA DH microscopy (DHM) [13]. In DHM, a collimated wave illuminates particles and a wave scattered by a particle propagates to the focal plane of a high-NA microscope objective. The characteristics and axial position of a particle are obtained by fitting experimental high-NA hologram patterns to theoretical ones using the Mie theory.

We previously proposed a method to obtain not only the axial position ${z_p}$ of a particle but also its radius ${r_p}$ and RI ${n_p}$, which is applicable to low-NA optical setup in Fig. 1 [18]. The model particle used in this study was a transparent sphere of a 60-µm diameter (about 100 wavelengths) with sparse particle fields, where individual particles could be isolated. We concentrated on the distinctive slope-shaped phase curvature pattern that the particle reconstruction offers, where the curvature decreases upon approaching the particle and changes the sign at the center (Fig. 2). As a result, we used the whole phase curvature, except the curvature value in an unstable region near the particle center, as the metric, and simulated particle parameters were optimized by minimizing the error function between the curvatures of the simulated and measured phases. Consequently, we could obtain not only the axial position but also the radius and RI of a particle. Here, although we use off-axis holography to remove the twin image and zero-order term in the experiment of our previous paper and this study, it is possible to use inline holography if the twin image problem is resolved.

 

Fig. 1. Optical setup for the single-shot phase-shifting method. z denotes the distance between the particle center z_p and the CMOS sensor surface. A beam splitter is horizontally inclined to satisfy the appropriate incident angle of the reference wave.

Download Full Size | PDF

 

Fig. 2. The pipeline of curvature calculation for experimental data. A reconstructed wave is obtained from the angular spectrum method using a hologram. Experimental phase curvatures were obtained from nine points.

Download Full Size | PDF

The problems of the above method are a computational burden due to nonlinear optimization algorithms using the Mie theory of a large diameter (60 µm) particle and stagnation in local minima due to inappropriate initial guesses. Contrary to sub–1 µm diameter particles, the calculation of 60-µm diameter particles with iteration is complex. Although all-infinite series in the calculation using Mie theory can be terminated after n_max terms [19,20], n_max for each series in 60 µm diameter particle is 27 times bigger than that in 1 µm diameter one, which indicates a much harder computational cost. For faster calculation, we turned to machine learning (ML). The computational cost for trained data using the Mie theory is relatively high. In high-NA DHM, the support vector machine [21] and convolutional neural network (CNN) [22] have been employed for fast computing. Recently, as direct hologram analysis using a collimated wave illuminating particles and a lens-less image sensor, the CNN-based method using the relation between the hologram pattern and parameter space has been proposed [23]. However, a trained hologram model of a particle was instead created using an opaque disk in place of the Mie theory, and the simulated axial positional error with a diameter range of 20–100 µm is as large as 0.25 mm [23]. Besides, RI was not obtained. As an alternative approach, scattering pattern from particles is used to determine the size and RI of a subwavelength particle using CNN in high-NA DHM [24]. However, the axial position of a particle could not be obtained.

For our approaching to ML, we use this phase curvature pattern as input data and three or four parameters as multioutput parameters. We examined some ML techniques, and overfitting is a severe problem in our applications. We chose the random forest (RF) method, a well-known and robust ML technique, with several fast multicore central processing unit (CPU) implementations [25,26]. We showed that the ML regression method could be used instead of our nonlinear optimization method with a factor of 10,000 faster calculations.

The aim of this study is the feasibility of this ML method instead of our earlier work using optimization algorithms. We had proposed this method in a previous study [27]. We extensively analyze and demonstrate the proposed method in this study. In Section 2, we describe the estimation of three parameters $({z_p},{r_p},{n_p})$ using the whole phase curvature with ML and propose another parameter δz, the position error of a particle center. In Section 3, we examine the experimental results of the proposed method using well-characterized particles and the method is validated for characterizing the distribution of three parameters $({z_p},{r_p},{n_p})$.

1.1 Optical setup and phase curvature

For an optical setup, we use a collimated wave as both the reference and object waves in a Mach–Zehnder interferometer (Fig. 1). We use a linearly polarized green laser (Cobolt Samba, 50 mW, wavelength λ = 0.532 µm) as the light source and a CMOS camera (Imaging Source, DMK 72BUC02), which has 2,592 × 1,944 pixels and pixel size d of 2.2 µm. The CMOS camera has a bare image sensor with no lens. The polarized direction of the laser is vertical (x-direction) in Fig. 1. To solve the problem of twin images, we use a tilted reference wave for a single-shot phase-shifting method [18,28–30]. This method uses a real-time phase shift [31] between three adjacent pixels on the camera, adjusting the oblique reference wave, and the tilted angle of the reference wave is sin⁻¹(λ / 3d) = 4.62°. RI of the medium surrounding the particle is 1.3337. The recovery image is obtained from the phase-shifting procedure. In this study, the image recovery method is not restricted to the phase-shifting method; other recovery methods, including inline hologram are also available. For an experiment, a small water droplet containing particles was introduced, using a syringe, into the water tank—an aluminum chamber (50 × 50 × 5 mm³) assembled using two cover glasses (1.0-mm thick). In the experiment, we captured a hologram movie, which allowed extracting 8-bit grayscale still images from each frame. The frame speed was 6 frames/sec. In addition, a 1,296 × 972-pixel hologram was used for reconstruction.

For trained data in ML, we employed curvature values using the Mie theory at several points along the optical axis [18], and three target parameters were estimated $({z_p},{r_p},{n_p})$. To calculate curvature, we simulated a scattering wave from a particle using the Mie theory. Using a hologram recorded scattering wave $u({x^{\prime},y^{\prime}} )$, the reconstructed image $U({x,y,z} )$ was obtained using the angular spectrum method [3], as given by

The image phase is expressed as follows:

The coefficient $a_2^{}(z )$ corresponding to the quadratic term ${T_2}(\eta )= 2{\eta ^2} - 1$ denotes the phase curvature of a phase including a constant. Thus, we used an input vector corresponding to the curvature values at z points for every three parameters $({z_p},{r_p},{n_p})$. Curvature patterns in an unstable region near the particle center were excluded. We supposed that the whole curvature pattern is symmetric at the particle center; thus, we calculated several curvatures at the half-region before the particle center [18].

To extract experimental phase curvature from a particle, the reconstructed image was calculated using the angular spectrum method [3], and we extracted the centroid (x_p,y_p) of the transversal intensity of the reconstructed particle. In this study, the positional error of the centroid was estimated to be 5 µm, which corresponds to 2 pixels of an image sensor. Because a z_p guess is necessary for ML, we found the point of minimal intensity of the centroid along the z-direction and calculated the zero-crossing point of the curvature pattern around the point of minimal intensity. Consequently, we chose this value as a z_p guess (Fig. 2) and calculated the curvatures using phase φ(x_p,y,z) at several designated points, except z_p.

2. Particle positioning and characterization using ML

To overcome the computational burden of nonlinear optimization algorithms, we turn to ML. We choose the RF method, which can avoid the overfitting problem, because the RF method is an ensemble technique capable of performing regression tasks using multiple decision trees (DTs) and a technique called Bootstrap aggregation [25,26], commonly known as bagging. Bagging involves training each DT on a different data sample, where sampling is performed with replacement. The core principle is to combine multiple DTs to determine the final output rather than relying on individual DTs. Therefore, it is expected to prevent overfitting. Among many high-quality CPU implementations of RFs, we employed scikit-learn [26] and included this estimator in the MultiOutputRegressor class.

2.1 ML using three parameters

In ML, a model must be trained with data that span the parameter range of interest at the desired resolution. A wider range of training parameters and resolutions enables a wider application of ML. However, there is a tradeoff between trained data and computational costs. The computation cost in creating the training data is high because of the calculation of the large-diameter particle model using the Mie theory. The main objective of this study is to compare ML results with those of the nonlinear optimization method in our previous study [18]. A global minimum optimization method is desired for the accurate nonlinear optimization. This method uses the approach to use many starting points and then call local minimum function from these points to find the corresponding local minima. The global minimum is then chosen to be the point that has the lowest objective value. However, the computational cost of this method is rather high, and we choose only local minimum optimization. We use MATLAB's function fmincon as the local minimum method. fmincon uses gradient based local optimization method with inequality constrained conditions and iterates from a starting point towards a local minimum. Here, we use sqp as implemented optimization techniques in fmincon. Therefore, a starting guess is required to be close to the true solution and the modest constrained parameter range is needed. The range of parameters $({z_p},{r_p},{n_p})$ for ML was set to be the same as the constrained three-parameter range used in the nonlinear optimization, whereas the resolution in ML was far less than that in the optimization method. In addition, we should ensure that the trained dataset is of a reasonable size, including the resolution. In the optimization method, the range of z_p was 157.5 mm ≤ z_p ≤ 162.5 mm, which is the thickness of our experimental water tank. Although the 5-mm range length in this case seems rather small, it is reasonable, given that the field of view (FOV) is restricted by the spatial bandwidth product of the DH system because of the limited pixel pitch and pixel number of the image sensor. The length of 5 mm satisfied that the modulation transfer function in the spatial frequency of 40 lp/mm is more than 0.93, which result was obtained from our experiment using the method in Ref. [30]. As the modest remained constrained parameter range in the local minimum method, we set the range of r_p to 27–32 µm and that of n_p to 1.58–1.60. We chose a rather high resolution in three parameters because the nonlinear optimization uses continuous resolution. Thus, trained parameters for the computed curvatures were evenly distributed, spanned by 157.5 mm ≤ z_p ≤ 162.5 mm, 27 µm ≤ r_p ≤ 32 µm, and 1.58 ≤ n_p ≤ 1.60 at resolutions of 10 µm in z_p, 1 µm in r_p, and 0.01 in n_p.

For the computational cost, we used simulated curvature data at nine points along z using the Mie theory, apart from the particle center z_p. The range of the nine points was 1 mm ≤ z ≤ 5 mm apart from z_p, and the sampling distance was 0.5 mm (circles in Fig. 3). These nine points were used in the optimization method. However, it is insufficient to use nine points as the input data in ML; hence, the input data were captured via the data interpolation of the nine points, and zero was set elsewhere. The region z for input curvature data was set in the range 152.5 mm ≤ z ≤ 162.5 mm (Fig. 3), and the input data contained curvatures at 1,001 points with a 0.01-mm sampling period. We employed a 9,018-member training set; examples of the trained data are shown in Fig. 4. We passed the input and output variable values of the dataset to the method proposed for the RF regression model. To validate our estimated result and parameter tuning, we used R² (the proportion of change in the response variable, which can be explained by the model) as a performance metric, enabling us to quickly and easily compare the models’ predictive capacity. If R² = 0, the model has the same predictive power as the mean of the test set, whereas R² = 1 shows that the model has a perfect prediction.

 

Fig. 3. Region z for the input curvature data. Nine simulated points using the Mie theory (red circles) are used for data interpolation, the range is 1 mm ≤ z ≤ 5 mm apart from a particle center, and the sampling distance is 0.5 mm.

Download Full Size | PDF

First, we tested the trained data. We split the 9,018-member simulated dataset into the training dataset (8,116) and test dataset (902) and validated the data. The codes are available in Ref. [33]. In this RF, we used the default parameter settings, i.e., n_estimator of 100, min_samples_split of 2, min_samples_leaf of 1, max_leaf_nodes of None, which means an unlimited number of leaf nodes, max_depth of None, which means that nodes are expanded until all leaves are pure or until all leaves contain less than min_samples_split samples. As the only setting parameter, we used max_features of “log2” in Section 3 for fast computing. After training, the estimated parameters using the test data and the corresponding test parameters were compared. Using almost the default settings, R² of the training and test data were as high as 0.99 due to the bagging algorithm of RF; deviations from each estimated parameter are shown in Table 1. The mean signed deviation (MSD) and root mean squared error (RMSE) of estimated parameters is minute, although the MSD and RMSE of the particle positions z_p were not small compared with those of r_p and n_p, attributable to the axial z resolution being 10 µm, and the maximal estimated error is likely 10 µm.

 

Fig. 4. Examples of curvature patterns for trained data. The calculated image area is 1,296 × 972 pixels, d is 2.2 µm, the laser wavelength is 0.532 µm, and n is 1.3337.

Download Full Size | PDF

Second, we simulated 1,000 models with three randomly distributed parameters within the above range and performed RF. Because the values of random radii, RIs, and positions are known, the tested and estimated values for each particle could be compared. Consequently, we found that the MSD and RMSE of estimated parameters from the true ones were minute (Table 1), whereas the MSD and RMSE for the particle positions z_p were larger than those of r_p and n_p, attributable to the axial resolution of 10 µm.

Table 1. Simulated MSD and RMSE of estimated parameters. For testing, the test input set is 902, whereas for estimation using an input dataset with random parameters, the number of input sets is 1,000.

View Table | View all tables in this article

Third, we investigated how noise affects our method using numerical simulations. Synthetic images were designed to mimic experimental holograms; in the images, three particles were randomly located at the calculated image area (1,296 × 972 pixels) in the transversal direction, and the distance ${z_p}$ was 160 ± 2.5 mm in the axial direction. We calculated phase curvatures for 600 hologram parameters $({z_p},{r_p},{n_p})$. The simulation was repeated for noise levels of 0%, 2%, 4%, and 6% of the maximum bit depth in hologram images. The MSD and RMSE from the true parameters are shown in Table 2. The MSD and RMSE of the parameters were minute. The small axial positional shift in the mean of z_p was the same as that in the optimization method [18]. Even when noise was added, the MSD of z_p was almost the same as that in the case of 0%. Therefore, this small shift is caused by the change in phase curvature due to multiple particle scattering comprising three particles. Although the RMSE of z_p is relatively small with a noise level of 0%–4%, the RMSE of z_p is slightly large with a noise level of 6%. Hence, low noise recording is required to accurately estimate the position z_p.

Table 2. Simulated MSD and RMSE of estimated parameters for some peak-to-peak noise ratios. The number of holograms is 600; the noise ratio is normalized by the maximal bit depth in the hologram.

View Table | View all tables in this article

2.2 More precise ML using four parameters

We describe the experimental performance for estimating z_p, r_p, and n_p in Section 3. The experimental input curvature data for estimation are obtained by not interpolation but third-order polynomial fitting with measured nine points because experimental data contains noise. If the estimation using three parameters results in residual estimated parameter errors for experimental data, we add another parameter for more precise estimation. In practice, the experimental curvature using the phase unwrapping algorithm is unstable near the particle center (Fig. 5), and there may be a few zero-crossing points for curvature values. In this case, we manually generate a guess using a curvature pattern as a guide. Therefore, z_p is an approximate value. We focused on the position error δz, the deviation from true z_p because the fitness of the curvature pattern for each z to the trained pattern as the metric is used in the ML method.

 

Fig. 5. An example of measured curvature using well-characterized polystyrene spheres. The curvature using the phase unwrapping algorithm is unstable near the particle center because the phase gradients are too large near the particle center.

Download Full Size | PDF

Figure 6 shows the geometry of the measured data with a position error of z_p. The blue curve indicates measured data without error, and the red curve indicates data with position error δz. In the red curve, the sampled end point z_n+ δz is (1 − δz) mm, apart from true z_p. However, if these curvature data are used for three-parameter estimation in ML, a particle center is supposed to be located at z_p+ δz. To avoid this inaccurate estimation, the particle center of the trained red curve should be regarded as the true z_p. This condition is satisfied by adding the parameter δz to the three parameters $({z_p},{r_p},{n_p})$ and regarding the shifted data by δz (red curve), which is defined at z = z₀+ δz,……,z_n+ δz as the corresponding trained data. Hence, trained data corresponding to each δz with three parameters $({z_p},{r_p},{n_p})$ are added.

 

Fig. 6. The geometry of measured sampling data with the particle center position error. The sampling end point z_n of measured curvature without position error is 1 mm apart from the particle center z_p (blue curve). The red curve indicates the measured data with position error δz. In this case, the sampling end point z_n + δz is (1 − δz) mm apart from a z_p. However, a particle center is supposed to be located at z_p + δz in the ML of three parameters.

Download Full Size | PDF

In practice, as trained data for four parameters (z_p,r_p,n_p,δz), we reused the trained data of three parameters. The trained data for three parameters were defined at z = z₀,……,z_n [ Fig. 7(a)]. First, the range δz_max of δz was determined. Next, for a given δz_max, the nominal trained data for (z_p,r_p,n_p,δz = 0) were defined at z = z₀ + δz_max,……,z_n − δz_max. Therefore, the length of the trained data for (z_p,r_p,n_p,δz) was shortened by 2δz_max/δz_res, in which δz_res is the resolution of δz. For each δz, the trained data for (z_p,r_p,n_p,δz) were defined at z = z₀ + δz_max + δz,……,z_n − δz_max + δz. Consequently, the amount of training data increased because it included the amount of trained data for three parameters multiplied by (2δz_max / δz_res + 1). Further, the experimental data for four parameters is defined at z = z₀ + δz_max,……,z_n − δz_max. Figures 7(b)–(d) show the example of δz = −0.2, 0, and 0.2 mm of trained data for four parameters (z_p,r_p,n_p,δz), respectively. Here, δz_max = 0.2 mm in all cases.

 

Fig. 7. Geometry of trained data for four parameters (z_p,r_p,n_p,δz). (a) Trained data using data of three parameters (z_p,r_p,n_p). (b) Example of trained data of δz = −0.2 mm. (c) Example of trained data of δz = 0 mm. (d) Example of trained data of δz = 0.2 mm. δz_max= 0.2 mm in all cases.

Download Full Size | PDF

3. Validation of estimation using three and four parameters with experimental results

If a well-characterized particle is used, the values of r_p and n_p can be validated by comparing the estimated parameters with the manufacturer's specifications. However, using experimental data for training is challenging because it is prohibitively difficult to obtain the true axial position z_p of moving particles. Therefore, after splitting the simulated data into 90% training data and 10% test data, the experimental data using 30 holograms of 30 frames is also used in the test dataset in Fig. 8. Here, to calculate the R² of the test data, only the test data after splitting the training data was used.

 

Fig. 8. Diagram of the estimation of experimental data.

Download Full Size | PDF

The estimated experimental parameters were validated as follows. We employed well-characterized particle polystyrene spheres (Thermo Fisher Scientific Inc. Catalog Number 4260A, Certified Batch 4260-018) to validate r_p and n_p. For the validation of z_p, we used a virtual hologram to mimic the shifted camera method, as in our previous study [18]. In this virtual hologram method, the estimated value z_p was extracted using the recorded hologram (Fig. 9). Next, the virtual hologram shift of Δz from the recorded hologram to a particle could be recorded using the reconstructed backpropagated wave from the recorded hologram via the angular spectrum method. Thus, the estimated value z_p′ was obtained using the virtual hologram; the difference ɛ_Δz = (z_p − z_p′)−Δz denotes a position error between two holograms. The RMSE of ɛ_Δz (RMSE_Δz) comprises two position errors, z_p and z_p′. These errors occurred independently, and the RMSE of one particle is RMSE_Δz /$\sqrt 2 $. For Δz = 5 mm, the region z for input curvature data is in the range 147.5 mm ≤ z ≤ 157.5 mm.

 

Fig. 9. Geometry of the virtual hologram method. (a) Recording hologram. (b) Virtual hologram recorded backpropagated object waves using the angular spectrum method. (c) Reconstruction of a virtual hologram.

Download Full Size | PDF

2.3 Estimated results using three parameters

In the estimation of parameters using experimental data, the estimated value was varied by a random state, which is a hyperparameter in RF and controls the sample randomness. Therefore, we used 50 random states derived random numbers for both splitting trained data and regressor and took an average of 50 times using different random states. We used training data (9,018 members) for three parameters, as mentioned in Section 2, and RF was performed. The setting parameters in RF were the same as in Section 2. We confirmed both the training and test R² to be 0.99, indicating the high accuracy of this method. While the specification of the mean of r_p was 29.3 µm, the MSD of r_p was −0.13 µm, which deviation is 0.2% of the particle’s diameter. As position error, the MSD was 18.1 µm, which is 31% of the particle’s diameter (Table 3).

Table 3. MSD and RMSE of the obtained parameters. The estimated parameters are radius, RI, and ɛ_△z, which is the deviation of (z_p – z_p′) from the correct distance of 5 mm. RMSE in ɛ_△z is RMSE of one particle.

View Table | View all tables in this article

2.4 Estimated using four parameters

The experimental results show that there is a small offset for the MSD of radii r_p and axial positions z_p. To obtain a more precise prediction, this offset can be decreased by assigning the fourth parameter δz. When the trained range δz_max in parameter δz is less than the expected range, the error of estimated parameters remains. If the range δz_max reaches the expected range by increasing δz_max, the error would be small. Hence, when capturing the smallest error in certain δz_max, we choose this δz_max as the expected range. Figure 10 shows the average of 50 times of the MSD and RMSE of parameters obtained from varying δz_max. Parameters $({z_p},{r_p},{n_p})$ for trained data are evenly distributed, spanned by the same value in Section 2, and δz is distributed spanned by −δz_max≤ δz ≤ δz_max at resolutions of 20 µm. We used 50 random states derived by random numbers, the same as in three parameters. We confirmed the R² of the trained and test data to be 0.99 for each δz_max. The case of δz_max= 0 corresponds to that of three parameters. We found that the MSD of both r_p and z_p successfully decrease to almost zero at a ratio δz_max/r_p ≈ 7, suggesting that δz_max = 200 µm is the expected range. In this range (δz_max/r_p= 6.8), the number of training data is 189,000, yielding a 1.1-GB model for each Δz. Table 3 shows the averaged estimated values of 50 times at δz_max/r_p= 6.8 for the four parameters. Despite that the RMSE (0.35 µm) of r_p does not decrease, this result is consistent because the values are comparable to the standard deviation (0.4 µm) of r_p stated by the manufacturer. Although the RMSE of the axial positional error was almost the same for each δz_max, the RMSE-to-diameter ratio was 0.85, which is smaller than the experimental value using the sign of curvature as a metric [17]. In fact, for a 10-µm diameter particle, the RMSE-to-diameter ratio was 1.28 in Ref. [17]. Although the RMSE of the axial positional error is smaller in our method, there is still a residual RMSE. The residual RMSE might stem from interfered noise by scattering in the finite water tank and mechanical and thermal stability in the experiment. Although the experimental RMSE of RI increases slightly, the deviation is as small as on the order of 10⁻³.

 

Fig. 10. MSD and RMSE of estimated parameters using 30 experimental data. The MSD and RMSE are the averages of 50 times using different random states in RF. (a)–(c) MSD and RMSE of radii, positional errors ɛ_Δz, and RIs, respectively.

Download Full Size | PDF

At δz_max/r_p= 6.8, one example of the histogram of ɛ_Δz for 30 hologram data using 30 frames is shown in Fig. 11. The histogram seemed to have a Gaussian profile, where the MSD of position errors was 0.6 µm, 1% of a particle’s diameter and the RMSE was 45.6 µm, 78% of a particle’s diameter. Note that the histogram in Fig. 11 contains two position errors, z_p and z_p.’ Meanwhile, in the nonlinear optimization method, the MSD of position errors was −26.1 µm, 45% of the particle’s diameter, and the RMSE was 92.0 µm, 157% of the particle’s diameter; the initial guess z_p in optimization was the same as the guess in ML. The reason for the worse result when using the conventional optimization method is probably the local minimum solution. For RFs, Table 3 shows that the particles were found to have a radius of 29.30 ± 0.36 µm (manufacturer’s specification: 29.3 ± 0.4 µm) and RI of 1.589 ± 0.002 (manufacturer’s bulk value specification: 1.59 for λ = 0.589 µm); thus, the values agree well with the manufacturer’s specifications.

 

Fig. 11. Standard deviation histogram of the experimental measurement between z_p and z_p′. The number of measurement data is 30. For estimated z_p using ML, the MSD is 0.6 µm, and the RMSE is 45.6 µm, which is 78% of the particle diameter.

Download Full Size | PDF

To more clearly verify the accuracy of the proposed method, we use other particles with different radii in the experiment and plan matching measurements of the known bimodal distribution. For this purpose, we use particles with another lot number whose radius is slightly larger (manufacturer’s specification: 30.2 ± 0.4 µm), and an experiment is performed using similar particles. As a result, the list in Table 4 shows that the particles are found to have a radius of 30.22 ± 0.56 µm and RI of 1.590 ± 0.002. Thus, the values agree well with the manufacturer’s specifications. Here, RMSE at a radius of 30.2 µm is slightly larger than the previous particle with a different lot number. This could have been caused by the signal noise due to mechanical unitability. The two different radius distributions shown in Fig. 12 indicate that particles with two different radii can be successfully distinguished owing to the fine resolution of the value of the trained radius. Moreover, the distributions with respect to z_p and n_p are nearly the same as those in the previous particle.

 

Fig. 12. Standard-deviation histogram of the experimental measurement using two differently sized particles. The number of each measurement data is 30. (a) Measurement of radius r_p. (b) Measurement between z_p and z_p′. (c) Measurement of refractive index n_p.

Download Full Size | PDF

Table 4. MSD and RMSE of the obtained parameters in the particles with different radii. The manufacturer’s specification is 30.2 ± 0.4 µm. The estimated parameters are radius, RI, and ɛ_△z, which is the deviation of (z_p – z_p′) from the correct distance of 5 mm. RMSE in ɛ_△z is RMSE of one particle.

View Table | View all tables in this article

It is not unusual for multiple particles to enter an observation FOV. Scattering waves from multiple particles interfere with each other and the phase curvature pattern of a particle fluctuates. We investigated the irregular patterns caused by 20 particles, and the curvature pattern was unaffected if the distance between two particles was longer than 0.23 mm. Therefore, we used the images satisfying this condition for confirmation in a sparse field.

In the ML calculations using Python, parallel computing using a multicore CPU desktop computer (AMD Ryzen Threadripper 3970X, 32 core, 64 GB RAM) was performed. In this experiment, although we estimated for two cases Δz = 0 and 5 mm for validating z_p, it is sufficient to perform only the case of Δz = 0 mm in the real estimation process. The processing time was 2.9 s for 30 holograms of Δz = 0 mm for three parameters and 80 s for four parameters at δz_max/r_p= 6.8. Therefore, compared with the optimization method, even the processing time for the four parameters in ML was reduced by a factor of 10,000, and once trained, an RF translated input experimental data into output parameter estimates extremely rapidly. The training time was about 3 days for each Δz. We could not use the graphics processing unit (GPU) computing in RF because it is not applicable in scikit-learn. If GPU calculations were used, the processing time would be much reduced. Moreover, if the system of interest is characterized by a more modest range of parameters, estimated results can be faster using a comparatively small set of training data spanning the relevant range. From our results, we conclude that our ML-based regression method has the advantage of fast calculation while simultaneously determining z_p, r_p, and n_p accurately, even for a particle with a 60-µm diameter.

4. Conclusions

We developed an ML-based regression method to determine the whole curvature-based axial position and characterize transparent particles. We employed a multioutput regressor using RF analysis. In this method, we devised a new technique using three and four parameters, adding the position error of a particle δz to three previously used parameters $({z_p},{r_p},{n_p})$. The technique could be applied for an optical setup using a collimated wave for illuminating particles and a bare image sensor with no lens. We verified the accuracy of our method both analytically and experimentally. Our simulation results showed that the MSD of axial position error was 4.8 µm with 4% noise. We verified the experimental results using the virtual hologram and as position error, the MSD and RMSE were 0.02% and 85% of the particle’s diameter, respectively. The size and RI of the measured particles had nearly the same values as the manufacturer’s specifications, demonstrating the accuracy of our method. The method was validated for characterizing the distribution of parameters $({z_p},{r_p},{n_p})$ of particles and the known bimodal distributions. This method can be used with factor of 10,000 faster calculations, instead of our previous nonlinear optimization method. In this study, we used 30 samples in this experiment and the estimated results were stochastic measurement. The increase in the number of samples would be expected to shrink the confidence interval on the estimates. A faster and higher precision artificial intelligence estimation method that uses multiple particles will be implemented in our future work.