## 2 Image Analysis Method  
### Summary of Previous Research  
Previous studies hypothesized the following. 1. All gravels used for high-speed railways are provided by the same supplier. 2. Mud pumping in the roadbed rarely occurs owing to reinforced roadbeds in the lower part of high-speed railways. If the relationships between the abrasion loss and shape index can be deduced for the same gravels as those used for high-speed railways under the aforementioned assumptions, the abrasion loss of gravels at actual high-speed railways can be traced back using the shape index based on such relation. The L.A. abrasion test [17,18] was conducted to measure the abrasion loss of gravels. The L.A. abrasion loss is defined as follow  
\[\text{Abrasion loss}\;(\%)=\frac{m_{1}-m_{2}}{m_{1}}\;\times\;100\]  
where \(m_{1}\) is the total mass of the gravels before L.A. abrasion test and \(m_{2}\) is the mass of the gravels larger than 1.7 mm after L.A. abrasion test. The abrasion loss was measured according to the number of revolutions in the L.A. abrasion test, and the shape indices of gravels were calculated using the image analysis method. Roundness, sphericity, aspect ratio, and angularity were employed for the shape indices of gravels. The same shape index may be calculated differently depending on the researcher [19,20]. In the present study, the shape indices were calculated as follows:1. Roundness was the ratio of the area of a circle, with the major axis of the gravel as the diameter, to the aggregate area, and is calculated using the equation proposed by [PERSON] et al. [21]. \[\text{Roundness}=\frac{A}{\pi L_{major}^{2}/4}\] where \(A\) is the area of a gravel aggregate, and \(L_{major}\) is the major axis length of the gravel. A small roundness indicates that particle is angular, while a large roundness indicates that particle is circular. 2. Sphericity was the ratio of the circumference of a circle, with the same area as the gravel aggregate, to the circumference of the gravel particle, and is calculated using the equation proposed by [PERSON] et al. [22]. \[\text{Sphericity}=\frac{L_{e}}{L_{a}}\] where \(L_{e}\) is circumference of a circle that has the same area as the gravel aggregate, and \(b\) is the circumference of the gravel particle. A small sphericity indicates that surface of particle is coarse, while a large sphericity indicates that surface of particle is smooth. 3. Aspect ratio was calculated as the ratio of the major axis to the minor axis of the gravel; thus, its value is always greater than 1.0, unlike other shape indices. 4. Angularity was calculated as proposed by [PERSON] et al. [23], in which the shape of gravel aggregates is represented using Fourier coefficients and then calculated as the sum of the squares of the coefficients of high-frequency bands. The relationships between abrasion loss according to the number of revolutions in the L.A. abrasion test and shape indices calculated using an image analysis method were derived as shown in Figure 1. The correlation among the abrasion loss, shape indices, and maintenance work was proved by applying these relationships to actual practice. As shown in Figure 2, the characteristics of gravel particles can be divided into three categories according to the scale [24]. Morphology (large scale) is an index that represents the overall shape  
Figure 1: Relationships between abrasion loss and the shape indices: (**a**) roundness; (**b**) sphericity; (**c**) aspect ratio; (**d**) angularity. of gravel particles; the roundness and aspect ratios used in this study, which demonstrate the closeness of gravel particles to a circle, are the shape indices belonging to this type. Roundness texture (intermediate scale) is an index that represents how angled the particle surface is, and sphericity belongs to this type. Surface texture (small scale) is an index that represents the detailed shape of the particle surface, and angularity belongs to this type. Table 1 presents the shape indices in this research according to scale.