An image-based method to determine the particle size distribution (PSD) of fi ne-grained soil

This paper discusses the PSD eff ects on the behavior of soil and its properties. The coarse grain part of soil can be analyzed with sieve analysis. Evaluating the PSD of fi ne-grained soil with the traditional method (hydrometer analysis) is a time-consuming process, has laboratory complexities and the obtained results are not accurate. Due to these limitations, there is a demand for a method that can off er faster data with more accuracy for geotechnical engineering. In this paper, the digital image processing (DIP) method is proposed using microscopic images and MATLAB software to determine the PSD of fi ne-grained soil. The DIP method was compared with the hydrometer analysis. The analysis of the data obtained from these two methods indicated that the image based method is capable of providing faster and more comprehensive results of the PSD.


Introduction
One of the most useful pieces of data about soil is regarding the size of its particles. The PSD is an important part of soil mechanics, especially geotechnical engineering. The PSD of soil is utilized for soil classifi cation and consequently, it can be applied in the forecasting of every mechanical and hydrologic soil behavior containing the specifi c weight, cohesion, permeability, etc. The common method for determining PSD includes sieve analysis (ASTM D6913, 2017) and hydrometer analysis (ASTM D7928, 2017). Hydrometer analysis is based on the principle of sedimentation and Stoke's law so it has some limitations and it is a time-consuming process (Coates and Hulse, 2012). The fi eld of DIP is continuously developing and is used in many sciences. For example, digital image processing is used in criminology for fi ngerprint matching, automatically interpreting fi ngerprints for dermatoglyphic applications (Mardia et al., 1997). Image processing plays an important role in many medical treatments, for example, magnetic resonance imaging and digital mammography are an important sector of diagnoses (Pham et al., 2000). Based on computer vision, cell injection is experimentally done with 100% successful results (Wang et al., 2009 Classifi cation of soils through the DIP method has the benefi ts of reducing the number of operators, decreasing calculation errors, offering stable image-based data of a sample, and it is faster than the traditional method (hydrometer analysis). Many researchers have used image analysis to fi nd engineering properties.
Mahmoud et al., (2010) presented a new numerical method that mixes the discrete element method (DEM) and the DIP method for studying the mixed effects of grain properties on hot-mix asphalt (HMA). Tafesse et al., (2012) developed a novel image-based method for calculating the 3-D shape and size of soil particles. A Glow-In-the-Dark (GID) method was used to produce a bright background and images were analyzed with MATLAB software. Sochan et al., (2015) used images of sand particles to determine the shape parameters and compared the results with the Krumbein technique. Kuo et al., (1996) developed a 3-D image-based analysis of gravel that was carried out with adding gravel in trays that can keep the grain and rotate it ninety degrees for quality control of gravel utilized in HMA. Kuo, (2002) carried out a mathematical analysis for determining the relevance between the permanent deformation properties of HMA and its particle geometric irregularities. The image-based method has a signifi cant role in appraising HMA particles for engineering analysis. Rao and Tutumluer, (2000) used a novel imaging technique to determine the volume, shape, and angularity of coarse grain used in asphalt pavement. Sukumaran and Ashmawy, (2001) determined grain morphology by formulating two new parameters. Parameters can be used as quality control of material selection for civil engineering. Bowman et al., (2001) developed a quantitative method using Fourier shape descriptors and image analysis for sand particle shape characterization. A lot of studies on image-based particles size have been conducted as mentioned above. However, none of them have focused on fi ne-grained soil. The purpose of this paper is to describe a rapid technique called DIP to determine the fi ne-grained PSD of samples.

Material and methods
Image processing is a type of signal processing where the input is an image, such as a picture or a scene of a movie, and the output can either be an image, or a set of special characters or image variables. DIP is performed by computers and image processing software. DIP for analyzing the morphological characteristics of materials, especially materials used in civil engineering, includes the following steps: a) take images; b) image segmentation and thresholding; c) image processing and extraction of particle-specifi c characteristics; d) computing and expanding this information by parameters and functions.
In DIP, the last two steps are performed by software. There are currently various image processing programs, of which MATLAB software was selected for this paper. The input image to the software to extract the features must fi rst be calibrated. Calibrating the image means that the actual size of the image should be introduced to the computer. This is usually done by specifying a calibrated number based on the scale of the image and its magnifi cation. A calibrated number is a pixel size per unit micrometer, which plays a very important role in computing. In the following steps, with the help of the separation tool, the useful phases in the image are separated. There are several methods for separation in image processing science, but in this paper, the threshold method based on the histogram of the image was used. Finally, the features obtained from the image provide the required information with the help of the functions provided in the software. Depending on the size of the material, digital cameras can be used directly or connected to a microscope to provide the image. The computer makes calculations according to the pixels of the image, and, if using a high-quality image, the results will be precise.
In general, there is no defi nite defi nition for particle size. Particles of soil and rock are three-dimensional and have an irregular shape. Particle measurement varies depending on the shape of the particles. For spherical particles, the size is defi ned by diameter. For irregular particles, there are various defi nitions that particle size characteristics should include information on the type of measured diameter and also the shape of the particles. Some common defi nitions for determining the particle size used in microscopic observations are shown in Figure 1.
The microscopic images were prepared by an Olympus optical microscope BX53M attached to a computer system and a DP72 digital camera. The computer system is used to control the input for the MATLAB software and to show the output results. All these components are shown in Figure 2.

Calculation
In this study, after preparing fi ne-grained samples, the images were prepared for image analysis. An example of a microscopic image of soil particles is shown in Figure 3.
Using a simple analysis, the changes in particle size parameters were monitored by increasing the number of analyzed particles. The results of this analysis are shown in Figure 4. As shown in Figure 4, between 9000 and 12000 particles, the results are linear, and therefore, in this study, 12000 fi ne-grained particles were tested. To achieve more accurate results, soil particles should be well dispersed with appropriate methods. In this study, the sample was spread on a dry and fl at surface and a micro tweezer was used to disperse it. However, the sample preparation for fi ne-grained particles is diffi cult. After providing the images by the camera attached to the optical microscope, the fi nal step is to analyze the images with the software. The particle size in this method is expressed in terms of the diameter of the circle equivalent to the area of the two-dimensional image of the particle as shown in Figure 5.
Using microscopic image processing, particles can be measured in the range of one millimeter to one micrometer. It is notable that the hydrometer analysis is based on grain weight, and the raw data obtained from the image analysis is based on the number of grains. In order to compare the results of the two methods, the data obtained from the microscopic image analysis in terms of the number were converted to in terms of volume.
DIP makes a precise calculation of the area (A i ) possible by determining the number of pixels involved in the grains. This kind of investigation is an advancement over manual techniques. By the parameter t i , the A i is converted to volume (V i ) for grain i. The parameter t i is defi ned as Equation 1. (1) The parameter t i shows the thickness of grains that cannot be specifi ed by 2D microscopical calculations. In any case, it is accepted that t i will obey based on two assumptions.
Assumption one is that t i is constant and shows an identical third dimension for all grains. This situation shows the volume weighted distribution of fl at plates. In this situation, the V i distributions histogram of grains are independent of t i . The fraction of grains in the size range 11 ≤ S i ≤ 12 over n entire grains is defi ned as Equation 2: (2) In assumption one, the A i -weighted and V i -weighted distributions are the same.
Assumption two is that the t i increases at the same rate as the other dimensions. This situation shows the V iweighted distribution of globular grains. Here, t i is given by Equation 3. (3) Where K is a fi xed coeffi cient. Equation (1) and (3) are combined and yields Equation 4.

(4)
The K is a shape-based coeffi cient. For example, for globular grains, K is given by Equation 5.

(5)
The shape of a distributed histogram based on the V i of grains is independent of the K. The fraction of grains in the size range 11 ≤ S i ≤ 12, over n entire grains is given by Equation 6. (6)

Method validation
For validating the accuracy of the DIP method, a set of measurements of standard particles (Sigma-Aldrich) have been carried out. The average diameter of this product has been calibrated with the SIGMA-ALDRICH corporation. The average size and other properties of the certifi ed particles are given by the corporation and listed in Table 1. By utilizing the DIP method, the certifi ed   particles are measured to specify the PSD is shown in Figure 6. The result is compared to the data provided by the corporation. It obviously shows that the measured result is in good agreement with the data provided by the corporation so it can be concluded that the imaging instrument is accurate and reliable in sizing particles.

Soil sample
Tests were carried out on four various samples for comparing the DIP method and the hydrometer analysis. The soil properties were specifi ed and are mentioned in Table 2. Soil samples are shown in Figure 7. These samples are the most frequent soil type in Tabriz city. The (ASTM D2487, 2010) standard is used to determine the soil classifi cation. The samples microscopic images are shown in Figure 8.

Results and Discussions
Due to the type of application of PSD in civil engineering, the results should be in terms of the percentage of particle passing weight. In the image analysis method, a small amount of the sample is tested so that it cannot be easily weighed. Assuming that the particle density is constant, according to the equations described earlier, particle volumes are determined using their area. Curves based on the number of particles are converted into the curves based on the area and volume. Now it is possible to compare DIP and hydrometer analysis. The results of the DIP were compared to hydrometer analysis as shown in Figures 9, 10, 11, and 12. The hydrometer analysis was carried out in a ZAMIN-AZHAND-KAV laboratory.
The discrepancies in view of the two data categories are because of the reality that the hydrometer analysis creates only 11 data points compared to the many data points received through the DIP method. So, in spite of some differences, assent was noticeable between the DIP method and hydrometer analysis results. The difference between the graphs is due to the lack of data regarding the thickness of the particles. The results showed that particles larger than 40 microns have a more regular shape pattern than smaller particles and are closer to a full sphere shape.
According to Figure 11, excellent similarity was seen between the DIP method and hydrometer analysis results in silt soil because silt particles are bigger than clay particles and particles are spherical.
The results showed that the DIP method could be used for all type of fi ne-grained soil and it is more trustworthy than hydrometer analysis.

Conclusions
The PSD analysis using an image processing technique was studied in this paper. The particle size for