Embedded Particle Size Measurement Method of Metal Mineral Polished Section Using Gaussian Mixture Model Based on Expectation Maximization Algorithm

Abstract

The study of process mineralogy plays a very important role in the field of mineral processing and metallurgy, in which the measurement of mineral-embedded particle size is one of the main research areas. The manual measurement method using a microscope has many problems, such as heavy workload and low measurement accuracy. In order to solve this problem, this paper proposes a Gaussian mixture model based on an expectation maximization (EM) algorithm to measure the embedded particle sizes of minerals of polished metal sections. Experiments are here performed on the polished section images of ilmenite and pyrite, and we compared the results with a microscope. The experimental results show that the proposed method has higher precision and accuracy in measuring the embedded particle sizes of metal minerals.

1. Introduction

Process mineralogy is a branch of mineralogy that serves the research and production practices of mineral processing [1]. It plays a very important role in the testing of mineral processing and metallurgy, in which the discrimination of mineral types and the statistics of embedded particle sizes of target minerals are the core areas of interest [2,3]. The sizes of mineral embedded particles directly affect the selection of beneficiation methods and technological processes. Through analyzing the embedded particle sizes of minerals, the degree of possible mineral liberation can be predicted at a certain grinding fineness, and the optimal grinding fineness required for the dissociation of useful minerals can be determined. Mineral separation is one of the necessary conditions of mineral beneficiation research and production practice. The degree of mineral separation directly affects the technical indexes of mineral beneficiation, such as concentrate grade and recovery rate. Therefore, it is very important to study the embedded particle sizes of ore in guiding the development of a beneficiation process [4].

At present, the most commonly used method for measuring mineral embedded particle size is based on industrial equipment. This method mainly uses scanning electron microscopy (SEM) to measure mineral embedded particle size, and then displays the analysis results on the computer with relevant software. There are a series of mature commercial products on the market. For example, the QEMSCAN system based on the Zeiss EVO50 scanning electron microscope developed by the Commonwealth Scientific and Industrial Research Organization of Australia [5], the MLA system developed by Dr. Gu Ying from Mineral Research Center of Queensland University in Australia [6], and the PTA system developed by the Norwegian Institute of Technology [7], etc.; however, these products have the problem of a high price.

At present, many countries still use microscopes to manually measure the embedded particles of metal minerals [8]. Technicians first observe the mineral polished section under the microscope to identify the main available minerals and gangue minerals contained in the polished section, and then measure the embedded particle size of the available minerals and gangue minerals. During this process, it is necessary to manually rotate or move the polished section continuously to measure the mineral particle size over as large an area as possible so as to improve the accuracy of embedded particle size statistics. However, this method mainly has the following problems: (1) the workload and labor intensity for technicians are large; (2) the measurement results are susceptible to human factors; (3) the measurement and statistical processes take a long time.

In order to solve the above problems effectively, a new approach to measuring mineral embedded particle size using digital image processing technology is proposed in this paper. By analyzing the image features of metal mineral polished section images, it can be found that most metal minerals have obvious color features, so we can identify and segment different minerals in the mineral polished section images according to the color features, and on this basis, measure and count the mineral distribution granularity.

Raw ore comprises a mixture with a large number of minerals; in order to segment useful minerals, it is necessary to adopt an effective image segmentation method because the accuracy of segmentation directly determines the effectiveness of a mineral embedded particle size measurement method based on digital image processing technology.

Among many image segmentation algorithms, the image segmentation algorithm based on clustering is a very practical algorithm, and it can gather similar pixels in the image space into one class according to certain requirements and laws so as to get different parts of the image. In recent years, a large number of clustering algorithms have been proposed, such as K-means [9,10], fuzzy C-mean clustering (FCM) [11,12], Spectral Clustering (SC) [13,14], Nonnegative Matrix Factorization (NMF) [15] and so on. These algorithms are simple, easy to understand and fast, so they have been widely used in many fields. The model-based clustering method makes use of the known mathematical model and achieves the best fitting between the given data set and the data model through the gradual approximation method.

A Gaussian mixture model is a commonly used mathematical model, offering a powerful framework for clustering, pattern recognition and multivariate density estimation [16,17,18,19]. Usually, the parameters of mixed models are estimated by maximizing the likelihood function, and the Expectation Maximization (EM) algorithm is a good tool to estimate the maximum likelihood of mixed models, as it can accurately estimate the parameters of mixed models with implicit variables [20,21,22]. Therefore, a Gaussian mixture model based on an EM algorithm for measuring the embedded particle sizes of metal mineral polished sections has been proposed in this paper. Firstly, mineral images are input, and then the parameters of the model are determined using the EM algorithm. Then, the Gaussian mixture model is used to cluster the images, and the segmented mineral images are obtained. Finally, Feret Diameter is used to measure and count the mineral embedded particles and their sizes.

The organization of this paper is as follows: Section 2 briefly introduces the Gaussian mixture model clustering algorithm based on the EM algorithm, Section 3 introduces the method for characterizing mineral embedded particle sizes, and Section 4 presents the experimental results and analysis. Finally, some conclusions and suggestions for future research are represented in Section 5.

2. Gaussian Mixture Model Clustering Based on EM Algorithm

The Expectation Maximization (EM) algorithm was originally proposed by Demaster, Laird and Rubin [23]. It is an iterative algorithm that estimates maximum likelihoods in the case of incomplete observation data, and converges the likelihood value towards the optimal value through multi-step iteration [24,25,26].

2.1. Gaussian Mixture Model (GMM)

Gaussian mixture models offer a probabilistic model that accurately quantifies things based on the Gaussian probability density function (normal distribution curve), and decomposes things into several models based this function (normal distribution curve), producing a model composed of multiple Gaussian distributions. Its overall density function is the weighted average of multiple Gaussian density functions. In this approach, each model has relatively independent parameters, and each Gaussian density function is called a sub-model. In addition, this hybrid model offers not only a probabilistic clustering model, but also a probabilistic graph model with hidden variables [27].

The specific Gaussian mixture model is defined as follows:

(1)

Assume that sample x is generated by one of the M Gaussian distributions, but it is impossible to observe which distribution is generated. A hidden variable $y\in \{1, 2\dots \dots ,M\}$ is introduced to indicate which Gaussian distribution the sample $x$ belongs to, and $y$ follows the multinomial distribution of Equation (1).

$$p\left(y=m;\pi \right)={\pi }_m,1\le m\le M$$

(1)

where $\pi =[{\pi }_1,{\pi }_2\dots \dots {,\pi }_m]$ is a parameter of the multinomial distribution, and ${\pi }_m$ represents the probability of sample $x$ generated by the $m$-th Gaussian distribution that satisfies ${\pi }_m\ge 0$, $\forall m$, $\sum _{m=1}^M{\pi }_m=1$.

(2)

When y = m, then the probability of sample $x$ takes the form shown in Equation (2) below.

$$p\left(\left.x\right|y=m{;\mu }_{m,}{\sigma }_m\right)=\frac{1}{\sqrt{2\pi {\sigma }_m^2}}{e}^{\left(-\frac{(x-{\mu }_m{)}^2}{2{\sigma }_m^2}\right)}$$

(2)

where ${\mu }_m$ and ${\sigma }_m^2$ represent the mean and variance of the $m$-th Gaussian distribution, respectively.

(3)

Calculated by Formulas (1) and (2), the probability density function and logarithmic marginal distribution of each sample $x^{\left(i\right)}$ are shown in Equation (3) below.

$$\begin{gathered} p\left(x^{\left(i\right)}\right)=\sum _{m=1}^M{\pi }_{m}p\left(x^{\left(i\right)}|y=m;{\mu }_m,{\sigma }_m\right)=\sum _{m=1}^M{\pi }_{m}N\left(x^{\left(i\right)};{\mu }_{m,}{\sigma }_m\right) \\ logp\left(x^{\left(i\right)}\right)=log{\sum }_{m=1}^M{\pi }_{m}N(x^{\left(i\right)};{\mu }_{m,}{\sigma }_m) \end{gathered}$$

(3)

It takes two steps to generate sample $x$ from the above model, including randomly generating a Gaussian distribution from $p\left(y;\pi \right)$ in the polynomial. We then assume that the m-th Gaussian distribution $(y=m)$ is selected, and select a sample $x$ from the Gaussian distribution.

2.2. EM Algorithm [23,28]

One of the most important problems in the Gaussian mixture model (GMM) is the parameter estimation of the model. The model may have observable variables or unobservable variables, whereby the unobservable variables are also called hidden variables. In the given data, if the given data are all observation variables, the maximum likelihood estimation can be used to solve the corresponding parameters of the model. However, the maximum likelihood estimation cannot be solved when there are hidden variables. Therefore, the EM algorithm is proposed to solve the parameter estimation problem of mixed models with hidden variables.

The basic idea of the EM algorithm is to estimate the values of model parameters according to the observed data obtained in advance; we then estimate the value of the missing data according to the parameter value of the previous step, and then estimate the parameter value again according to the estimated missing data plus the observed data. Finally, we iterate until final convergence.

The steps of the EM algorithm are as follows:

(1)

Step E: First, set the parameters and calculate the posterior probability of the sample $x^{\left(i\right)}$ belonging to the first Gaussian distribution, which is recorded as ${\gamma }_{im}$. The calculation formula of ${\gamma }_{im}$ is shown in Formula (4).

$${\gamma }_{im}=\frac{{\pi }_{m}N\left(x^{\left(i\right)};u_m,{\sigma }_m\right)}{\sum _{m=1}^M{\pi }_{m}N\left(x^{\left(i\right)};u_m,{\sigma }_m\right)}.$$

(4)

(2)

Step M: Maximize the likelihood function, convert the parameter estimation problem into an optimization problem, and obtain new parameter values. The specific calculations are shown in Formula (5).

$$\begin{gathered} u_m=\frac{1}{N_m}\sum _{i=1}^N{\gamma }_{im}{x}^{\left(i\right)} \\ {\pi }_n=\frac{Nm}{N} \\ {\sigma }_m^2=\frac{1}{N_m}{\sum }_{i=1}^N{\gamma }_{im}{\left(x^{\left(i\right)}-{\mu }_m\right)}^2 \end{gathered}$$

(5)

2.3. Gaussian Mixture Model Image Clustering Algorithm Based on EM Algorithm (EM-GMM Image Clustering Algorithm)

In this paper, the Gaussian mixture model image clustering algorithm based on the EM algorithm is proposed for use in mineral image segmentation. Firstly, mineral images are input, and the parameters of the model are determined using EM algorithm. Then, the Gaussian mixture model is used to cluster the images, and finally the segmented mineral images are obtained. The specific process is shown in Figure 1.

3. Method for Characterization of Embedded Particle Size

The measurement of mineral embedded particle size plays an important role in determining the grinding particle size in the mineral process. The current methods used for the measurement of embedded particle size in the mineral optical image include directional maximum intercept, directional random intercept, equivalent area diameter, etc. The specific method to be used as the measurement method is generally based on the measurement standards of relevant units’ instruments and the selection of manual measurement [1,29].

(1)

Directional maximum intercept

The so-called directional maximum intercept is used to measure the maximum diameter of the mineral particle in the image along a fixed direction for multiple particle sizes, as shown in Figure 2 (the particle sizes in the horizontal direction are expressed by d₁, d₂, d₃, d₄ and d₅) [29].

(2)

Directional random intercept

Directional random intercept is used to intercept the profiles of embedded particles in the image with equidistant directional measuring lines, and the length obtained is the directional random intercept. We then use these directional random intercepts to represent the particle size, as shown in Figure 3 the particle sizes of the same particle in the vertical direction in Figure 3 are represented by d₁, d₂, d₃, d₄ and d₅ [29].

(3)

Equivalent area diameter

Equivalent area diameter is used to replace the particle size of the target particle with the diameter of a circle with the same area as the contour of the target mineral particle in the image, as shown in Figure 4, where $\mathrm{d}$ represents the size of the target particle [30].

(4)

Feret Diameter

Ferret Diameter is also called caliper diameter, and its principle is that through the centers of irregularly shaped particles in an image, the particles are sandwiched between parallel lines running in any direction, that is, in a certain direction, the two straight lines are set tangential to the irregular pattern. Then, the distance between the parallel lines is taken as the Feret Diameter, as shown in Figure 5. The maximum diameter $d_{max}$ and minimum diameter $d_{min}$ of the Feret Diameter are shown in Figure 5 [31,32].

In order to accurately and intuitively understand the approximate shapes of mineral particles, the Feret Diameter maximum diameter method is adopted in this paper, and incorporated into the embedded particle size measurement system proposed for metal mineral polished sections of a Gaussian mixture model based on the EM algorithm.

4. Experimental Results and Analysis

In this section, the color moment is first used to judge whether the target mineral is present in the metal mineral polished section image, and at the same time to judge whether there are other minerals, and the K value is obtained, which is equal to the determined mineral type plus 2, and then the image is filtered spatially to improve the clustering effect. Finally, the Feret Diameter method is used to measure mineral embedded particle size. The K value is the number of cluster centers, which means that the pixels in the image need to be divided into K classes. In this paper, the number of mineral types plus 2 equals the K value of the clustering algorithm. The purpose of adding 2 is that in the backgrounds of metal mineral X-ray images, there are often two or more colors present according to the research context, rather than a single color. In order to reduce the influence of the background on the target mineral, it is necessary to classify it separately. The definition of color moment is shown in Formula (6). The detailed steps of the embedded particle size measurement method for a mineral metal polished section using a Gaussian mixture model based on the EM algorithm are shown in Figure 6.

$$\begin{gathered} u_i=\frac{1}{N}{\sum }_{j=1}^N{p}_{i,j} \\ {\sigma }_i={\left(\frac{1}{N}{\sum }_{j=1}^N{\left(p_{i,j}-u_i\right)}^2\right)}^{\frac{1}{2}} \\ {s}_i={\left(\frac{1}{N}\sum _{j=1}^N{\left(p_{i,j}-u_i\right)}^3\right)}^{\frac{1}{3}} \end{gathered}$$

(6)

where $u_i$ represents the first order moment of color, ${\sigma }_i$ is the second order moment,$s_i$ is the third order moment, N represents the number of pixels in the image, and $p_{i,j}$ is the probability of occurrence of pixels whose gray scale is j in the i color channel component of the color image.

In order to verify the performance of methods for measuring embedded particle sizes in metal mineral polished sections using the Gaussian mixture model based on the EM algorithm in this paper, we have taken ilmenite and pyrite as examples, and the mineral polished section images are shown in Figure 7, which were prepared by professional technicians from the Central Laboratory of Yunnan Geological and Mineral Exploration and Development Bureau; all images have a size of 512 × 512. The computer used in the experiment in this paper is Intel Core i7, with 16G memory. All algorithms were developed using MATLAB Release 2022 a.

Microscope measurements were completed in the central laboratory of the Bureau of Geology and Mineral Resources Exploration and Development in Yunnan Province. The polarizing microscope model was a LEICA DM4500 P LED, which was used to take pictures of and measure the embedded particle sizes in the mineral polished section images, and the operators conducted measurements according to relevant standards.

4.1. Experimental Result

4.1.1. Mineral Image Clustering Based on Gaussian Mixture Model of EM Algorithm

In order to ensure a better clustering effect, the metal mineral polished section images in Figure 7 were firstly filtered using mean filtering before clustering the image, and the filtering results are shown in Figure 8. Then, the EM-GMM image clustering algorithm was used to cluster the filtered image.

Before clustering, the mineral species in the polished section image were obtained through the color moment, and the K value of the clustering was obtained. The mineral species in Figure 7 is 1, so K = 2, the variance was set to 400, and the number of iterations was set to 100. The clustering results are shown in Figure 9.

4.1.2. Extraction of Target Minerals

Target mineral extraction is employed to extract target minerals from a clustering result graph. Before clustering, the location of the target mineral is identified using color moments. After this step, for Figure 10, the clustering effect was combined with the locations of the target minerals identified by color moments in order to segment and extract the target minerals. The extraction results are shown in Figure 10.

4.1.3. Target Detection by BoundingBox

How to detect each mineral particle accurately is very important, and the correctness of mineral particle detection directly affects the accuracy of measurement results. In order to accurately detect the particles of the target mineral in the polished section image, this section shows the use of BoundingBox with Regionprops in MATLAB to detect and identify the mineral particles of the target mineral. Each mineral particle of the target mineral was selected with a rectangular frame. The results of the frame selection are shown in Figure 11.

From Figure 6, Figure 8, Figure 9, Figure 10 and Figure 11, it can be seen that the EM algorithm-based mineral distribution particle size measurement method using a Gaussian mixture model proposed in this paper can accurately separate target minerals from the background, and achieve the effective extraction of target minerals. Finally, on this basis, measurements of the sizes of embedded particles of extracted target minerals are carried out in Section 4.1.4.

4.1.4. Mineral Embedded Particle Size Measurement

In this section, the Feret Diameter method is used to measure the sizes of embedded particles of the target minerals identified in Figure 12. The measurement results for ilmenite and pyrite are shown in

Table 1. Measurement results of embedded particle size using the method proposed in this paper.

Ilmenite (µm)
10.11	15.18	18.86	29.42	49.75	85.34	104.42	138.43	161.19	194.34
11.01	15.18	19.5	31.7	51.04	85.41	115.23	138.51	162.21	195.66
11.6	15.45	19.64	33.52	55.17	85.55	116.27	140.92	163.21	199.01
11.98	15.45	19.75	33.68	58.86	89.76	121.46	140.95	167.11	210.24
12.12	15.55	20.41	36.53	59.2	89.97	127.92	143.61	168.72	238.37
12.12	15.78	21.22	37.41	60.21	90.44	127.96	146.51	168.89	261.83
13.18	16.19	21.53	38.41	72.63	91.44	129.44	148.14	173.27	270.04
13.54	16.88	21.81	42.32	76.98	92.52	131.34	151.05	179.02	274.68
13.77	17.97	23.21	42.54	82.02	92.69	137.24	152.85	183.57	275.84
13.89	18.21	24.67	43.94	82.23	94.54	137.67	160.93	193.56
Pyrite (µm)
8.79	11.47	12.89	14.74	16.91	20.26	25.55	33	45.11	71.25
9.49	11.48	12.94	15.16	17.16	20.58	25.61	33.08	45.39	72.96
9.69	11.52	12.94	15.34	17.17	20.63	25.74	33.19	45.41	76.62
9.91	11.6	13	15.49	17.25	20.78	25.9	33.71	45.44	76.71
10.34	11.6	13.18	15.56	17.36	21.15	26.41	33.89	46.22	85
10.36	11.6	13.18	15.56	17.45	21.81	26.5	34.16	46.5	85.5
10.36	11.6	13.18	15.56	17.47	21.99	27.25	34.25	46.58	87.6
10.36	11.91	13.2	15.67	17.78	22	27.43	35.4	47.01	88.48
10.59	11.98	13.33	15.78	17.97	22.56	27.9	37.15	48.66	89.05
10.59	12.03	13.46	15.98	18.02	22.64	28.03	37.35	48.94	93
10.61	12.03	13.46	15.98	18.16	22.72	28.29	38.99	49.29	93.06
10.71	12.03	13.54	16.03	18.25	22.75	28.64	39.14	49.64	93.66
10.71	12.35	13.54	16.03	18.25	22.89	28.69	39.2	51.07	103.72
10.71	12.35	13.77	16.3	18.62	23.4	29.3	39.54	51.16	104.11
10.91	12.35	13.77	16.37	18.86	23.4	29.48	39.99	51.77	110.09
10.93	12.35	13.77	16.37	19.15	23.65	29.81	40.24	52.01	115.57
10.93	12.38	13.77	16.4	19.21	23.65	30	41.01	53.42	127.37
11.01	12.38	14.03	16.56	19.37	23.73	30.11	41.34	54.4	131.91
11.08	12.54	14.18	16.56	19.58	24.33	30.11	41.38	55.79	137.33
11.08	12.66	14.27	16.6	19.64	24.49	30.68	41.59	60.26	138.82
11.08	12.66	14.39	16.61	19.73	24.51	31.54	41.76	61.51	163.4
11.18	12.81	14.42	16.71	19.75	24.65	31.66	42.82	63.31	166.32
11.18	12.87	14.61	16.75	19.8	24.67	31.74	43.13	67.11	201.69
11.24	12.87	14.61	16.88	19.89	24.98	32.63	43.54	67.96	257.41
11.24	12.89	14.61	16.91	19.98	25.22	32.67	44.35	69.6	273.79
11.31	12.89	14.62	16.91	19.98	25.51	32.93	44.37	70.31

, and the unit is micrometers. In order to verify the performance of the metal mineral embedded particle size measurement method proposed in this paper, the embedded particle sizes of the target mineral in Figure 12 have also been measured by use of a polarizing microscope, as also shown in Figure 12. The measurement results are shown in

Table 2. Measurement results of embedded particle size using a microscope.

Ilmenite (µm)
10	15	15	20	30	40	60	85	90	120	140	150	160	180	210
10	15	20	20	30	40	60	85	90	130	140	150	160	180	240
10	15	20	20	35	45	75	90	95	130	140	150	165	190	260
10	15	20	20	35	50	75	90	105	130	140	150	170	190	270
10	15	20	25	35	50	80	90	115	130	140	160	170	195	275
15	15	20	25	40	55	80	90	115	135	145	160	170	200	275
15	15	20	30	40	60	85
Pyrite (µm)
10	10	10	10	15	15	20	20	20	25	30	40	45	50	90
10	10	10	10	15	15	20	20	20	25	30	40	45	55	90
10	10	10	10	15	15	20	20	20	25	30	40	45	60	90
10	10	10	10	15	15	20	20	20	25	30	40	45	60	105
10	10	10	10	15	15	20	20	25	30	30	40	45	60	105
10	10	10	10	15	15	20	20	25	30	30	40	45	65	110
10	10	10	10	15	15	20	20	25	30	30	40	45	65	115
10	10	10	10	15	15	20	20	25	30	30	40	45	65	130
10	10	10	10	15	15	20	20	25	30	30	40	50	70	130
10	10	10	15	15	15	20	20	25	30	30	40	50	70	140
10	10	10	15	15	15	20	20	25	30	35	40	50	70	140
10	10	10	15	15	15	20	20	25	30	35	40	50	70	160
10	10	10	15	15	15	20	20	25	30	35	40	50	75	165
10	10	10	15	15	15	20	20	25	30	35	45	50	75	200
10	10	10	15	15	15	20	20	25	30	35	45	50	85	275
10	10	10	15	15	20	20	20	25	30	35	45	50	85

, and the unit is micrometers.

From Table 1 and Table 2, we can see an abundance of large-particle minerals in ilmenite—97 ore particles were manually measured, while the method proposed in this article measured 99 ore particles, indicating little difference. However, there are many small and fine-grained minerals in pyrite, and some of them were overlooked during manual measurement. Therefore, 239 ore particles were measured manually, and the method proposed in this article measured 259 ore particles. In addition, the measurement results are all given in microns, but the results of the method proposed in this paper are accurate to two decimal places, meaning their accuracy is higher.

4.1.5. Performance Comparison

In order to further verify the performance of the method used for measuring the sizes of embedded particles size in a mineral polished section using a Gaussian mixture model based on the EM algorithm, the particle size measurement results of this method are compared with the results yielded by microscope measurement in this section. Firstly, the numbers of mineral particles, as well as the size ranges and proportions of mineral particles (Table 1 and Table 2), are counted, and the results are shown in

Table 3. Embedded particle size measurement results for ilmenite minerals.

Size Range/µm	The Number of Mineral Particles Measured by Embedded Particle Size Measurement Method	Proportion/%	The Number of Mineral Particles Measured by Microscope Method	Proportion/%
0–10	1	1	0	0
10–30	31	31	27	27.83
30–50	10	10	11	11.34
50–100	19	19	20	20.62
Above 100	39	39	39	40.21
Total	100	100	97	100

and

Table 4. Embedded particle size measurement results for pyrite minerals.

Size Range/µm	The Number of Mineral Particles Measured by Embedded Particle Size Measurement Method	Proportion/%	The Number of Mineral Particles Measured by Microscope Method	Proportion/%
0–10	5	1.92	0	0
10–30	168	64.62	148	61.92
30–50	48	18.46	52	21.76
50–100	26	10	27	11.30
Above 100	13	5	12	5.02
Total	260	100	239	100

. Then, the numbers of mineral particles in different size ranges measured by the proposed method and the microscope measurement method are compared, and the results are shown in Figure 12. Finally, the proportions of mineral particles in different size ranges are compared and the results are shown in Figure 13.

It can be seen from Table 3 and Table 4, and Figure 12 and Figure 13, that, firstly, in the whole particle size range, the total amounts of mineral particles measured by the method proposed in this paper are greater than those measured by microscope. When the mineral particle size was larger than 50 µm, the numbers of mineral particles measured by the two methods were not much different, and almost reached the same level, while we see a big difference when the mineral particle sizes were less than 50 µm. The reason is that, during manual measurement, there may be a leakage of ore particles with mineral sizes less than 50 µm, or some smaller particles might be artificially automatically ignored. Secondly, the measurement accuracy of the proposed method is higher than that of the microscope measurement method. Finally, the shapes of mineral proportion curves of the proposed method and the microscope measurement method are almost the same in different particle size ranges, which proves the accuracy of the proposed method of mineral embedded particle size measurement to some extent.

In order to further verify the effectiveness of the method proposed in this paper, the measurement times required by the proposed method and the microscope method are recorded in

Table 5. Measurement time comparison.

Method	Ilmenite/min	Pyrite/min
The method of this paper	2.05	2.18
Microscope method	6.24	7.36

, and the unit of time is minutes.

The measurement of mineral-embedded particle sizes using a microscope was carried out by professional technicians from the Central Laboratory of Yunnan Geological and Mineral Exploration and Development Bureau. The technology uses estimation methods to calculate mineral particle sizes, as shown in the data in Table 1 and Table 2. The embedded particle size data end with 0 or 5, so it takes about 2–10 s to calculate the embedded particle size of one mineral particle.

It can be seen from the data in Table 5 that the measurement times required by the method proposed in this paper are about one-third of the microscope measurement times, showing that the measurement method proposed in this paper can save a lot of time.

5. Conclusions

In view of the shortcomings of microscope measurement methods, such as high labor costs, high time consumption, high susceptibility to human factors, and the high prices of industrial equipment related to scanning electron microscopy, this paper proposes a method for measuring the embedded particle sizes of metal mineral polished sections using a Gaussian mixture model based on the EM algorithm via the cross-fusion of digital image processing technology and process mineralogy. The aim of this method is to effectively segment target minerals. Firstly, the Gaussian mixture model image clustering algorithm based on the EM algorithm is used to cluster and segment target minerals, and then the Feret Diameter method is used to measure mineral embedded particle sizes.

In order to verify the effectiveness of the proposed method, its performance has been assessed by taking the ilmenite and pyrite optical plate images as an example, and the measurement results of the method are compared with those of microscope measurement. The experimental results show that, compared with the microscope method, the proposed method achieves better performance in terms of measurement accuracy, measurement time, and the number of mineral particles measured. Therefore, the method for measuring embedded particle sizes in a metal mineral polished section using the Gaussian mixture model based on an EM algorithm represents a good approach. In future research, the embedded particle sizes of other metal minerals can be measured. In addition, methods for the embedded particle size measurement of non-metallic minerals can also be developed.