Particle-size distribution determination using optical sensing and neural networks

Previously, methods such as smoothing and statistical and Backus–Gilbert inversion techniques were used to find profiles of particle distributions.[1]–[4] The smoothing technique requires a judicious choice of two parameters that control the smoothness of the solution. Statistical inversion techniques require prior knowledge of the statistical properties of the unknown function and the measurement errors. The Backus–Gilbert technique requires a good compromise between the spread and the variance.

In this Letter we present an alternative method based on an artificial neural network technique. The neural network technique offers a different approach in that the network memorizes the experience gained by the training; even though the training may take a long time, once the training is done the inversion can be performed instantly. As an example, we use a simple inverse problem to find particle distribution by using single scattering. We use this example to demonstrate that this technique has a potential for more general multiple scattering problems.

We consider the inverse problem of finding the particle-size distribution from the measurements of backscattered light on an optically thin medium containing particles. The first-order scattering approximation is used. The measured quantity is the backscattered intensity β(λ_i) at three different wavelengths λ_i(i = 1, 2, 3), and it is related to the size-distribution function n(r) by a Fredholm integral equation of the first kind as

Here we utilize a layered perceptron neural network to determine particle-size distributions.[6] The size-distribution function is assumed to be a log-normal distribution function so that it is characterized by the mean radius r_m and the standard deviation σ. The inverse problem is to obtain r_m and σ for given β(λ_i). Although β(λ_i) is linearly related to the distribution n(r), the relationship between input β(λ_i) and r_m and σ is nonlinear. Earlier techniques such as the Backus–Gilbert technique can handle only linear inversion. It is also our purpose here to demonstrate that the neural network can perform a nonlinear inversion problem. The inverse problem by Kitamura,[6] however, is to obtain n(r) at 31 points, and therefore the relationship between input and output is linear. We also found that β(λ_i)'s can become close together for some r_m and σ, resulting in a nonunique inverse solution. An algorithm is presented to find the ranges from r_m and σ so that unique solutions can be obtained. Finally we show that increasing the number of iterations in training the neural network causes the inverse solutions to tend to converge to the real value.

An artificial neural network can be defined as a highly connected array of elementary processors called neurons. Here we consider the multilayer perceptron (MLP) type of artificial neural network.[7]–[11]

As shown in Fig. 1, the MLP-type neural network consists of one input layer, one or more hidden layers, and one output layer. Each layer employs several neurons, and each neuron in the same layer is connected to the neurons in the adjacent layer with different weights. A schematic diagram of this model is depicted in Fig. 1. We use three inputs [β(λ₁), β(λ₂), β(λ₃)] and two output neurons (r_m, σ). Signals pass from the input layer, through the hidden layers, to the output layer. Except in the input layer, each neuron receives a signal that is a linearly weighted sum of all the outputs from the neurons of the former layer. The neuron then produces its output signal by passing the summed signal through the sigmoid function 1/(1 + e^−x).

The backpropagation learning algorithm is employed for training the neural network. Basically this algorithm uses the gradient descent algorithm to get the best estimates of the interconnected weights, and the weights are adjusted after every iteration. The iteration process stops when a minimum of the difference between the desired and actual output is searched by the gradient descent algorithm.[10],[11]

We consider the backscattering of light from a volume distribution of spherical particles with 31 radii ranging from 0.01 to 40 μm. We assume that the size-distribution function n(r) is governed by the log-normal function so that it is characterized by two quantities: the mean radius r_m and the standard deviation σ. Therefore it is given by

First we conduct a study of the forward problem of finding β(λ_i) for various r_m and σ. Since the radius of particles varies from 0.01 to 40 μm, i.e., −2 ≤ log(r) ≤ 1.66, the ranges for r_m and σ are chosen so that −1 ≤ log(r_m) ≤ 0.44 and 0.03 ≤ log(σ) ≤ 1. Thus the actual size of particles ranging from (r_m/σ) to r_mσ will be within the range for r. The inverse problem is generally nonunique in getting r_m and σ for given β(λ_i) if r_m and σ are allowed as in the above ranges. In what follows, we restrict the ranges for r_m and σ such that unique solution can be obtained. The algorithm in finding such ranges is also discussed.

Both log(r_m) and log(σ) are divided into 10 intervals for generating the training and testing data. We chose the refractive index of the particle to be m = 1.53 − j0.008 and the wavelengths to be λ₁ = 0.53 μm, λ₂ = 1.06 μm, and λ₃ = 2.12 μm. The study of β(λ_i) reveals that for some values of log(r_m) and log(σ), the inputs are close to one another, resulting in nonunique solution for r_m and σ with such β(λ_i). For a unique solution of r_m and σ for a given β(λ_i), the change in β(λ_i) for a given change of r_m and σ must be sufficiently large. Therefore we define the distance D, a measure of separation of β(λ_i), as

In order to ensure that the β(λ_i)'s are sufficiently separated, we require that D exceed a minimum distance D_m. To find D_m, we first notice that there is a large difference in magnitude between β(λ_i, σ_j, r_ml) and β(λ_i, σ_k, r_ml) for k > j. For instance, we have β(λ_i, σ₁, r_m1) ∼ 10⁻¹⁵ and β(λ_i, σ_M, r_m1) ∼ 10⁻⁶. Thus D_m cannot be fixed for all σ_j but should vary according to σ_j. In addition, for the same σ_j, the value for β(λ_i, σ_j, r_ml) increases from l = 1 to l = N. The lowest value occurs when l = 1, Hence the minimum distance D_m is chosen proportionally to β(λ_i) obtained from the first mean radius r_m1. Specifically,

The constant D₁ in Eq. (4) controls the size of the allowable region for log(r_m) and log(σ). A large value of D₁ will generally create a small allowable region, but the values of β(λ_i) are reasonably separated, and therefore unique sets of β(λ_i) can be obtained. On the other hand, a small value of D₁ will create a large allowable region, but the sets of β(λ_i) are close to one another. Unique sets of β(λ_i) are thus difficult to obtain, resulting in a large percentage of error in obtaining the unknown size distribution. A value of D₁ ranging from 0.1 to 50 has been tested for finding the suitable D₁. It was found that a value of 10 for D₁ is a good compromise between the percentage error and the size of the allowable region for log(r_m) and log(σ). With such a value, the allowable region is found to be −0.328 ≤ log(r_m) ≤ 0.44 and 0.03 ≤ log(σ) ≤ 0.5, as shown in Fig. 2.

Based on the allowable region discussed above, a group of 480 sets of data was generated from Eq. (1). In order to maximize the computing accuracy of the neural network, we first normalize all the data from zero to unity. We use 462 data sets to train the neural network. The remaining 18 sets are used to test the system. Finally, as shown in Fig. 3, the results are converted back to the original values.

Figure 3 shows the performance of the neural network in terms of absolute percentage error for log(r_m) and log(σ). It is clear from the figure that an increasing number of iterations tends to cause the outputs to converge to the real value and hence lowers the absolute percentage of errors. Except when the desired outputs log(r_m) and log(σ) are small, the neural network yields good results for most of the testing data, with an absolute percentage of error of less than 10%.

In summary, we have presented an inverse technique of finding particle-size distribution by using optical sensing and a neural network. The size-distribution function is assumed to be a log-normal function, so that it is characterized by a mean radius r_m and a standard deviation σ. It was shown that the neural network yields good results for the testing data, with an absolute percentage of errors of less than 10% for most of the testing input β(λ_i). A major advantage of this technique is that, once the neural network is trained, the inverse problem of obtaining the size distributions can be solved speedily and efficiently (in a small fraction of a second) by a microcomputer workstation.

This research is partially supported by a grant-in-aid for scientific research from the Ministry of Education, Science, and Culture of Japan (Monbusho), the National Science Foundation, and NASA.

Figures

 

Fig. 1 Structure of a multilayered perceptron-type artificial neural network.

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Fig. 2 Region of the allowable log(r_m) and log(σ) for D₁ = 10.

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Fig. 3 Performance of the neural network in generating (a) the standard deviation of particle size, log(σ), and (b) the mean particle size, log(r_m), from the given backscattered intensities, β(λ_i), in terms of absolute percentage error. Increasing the number of iterations tends to converge to the true value and hence lowers the absolute percentage error.

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