## 2 Morphological Characteristic Evaluation
### Angularity  
Aggregate angularity essentially describes the degree of sharpness of the angles. Corner roundness was proposed by [PERSON] [35] and others to describe the sharpness of the combination of corners and edges of a particle and was generally considered to be related to the radius of curvature or geometric fractal of the particle profile at the sharp end, while for quantitative evaluation, most researches have used the method of fitting similar polygons to obtain the angularity parameter of an aggregate. Fractals were first created by [PERSON] [36] and have been widely used in various disciplines through continuous development. Fractals are used as a theoretical tool to describe and evaluate some characteristics such as hierarchical continuity, irregularity or randomness [37], and asphalt mixtures exhibit statistical self-similarity in volume and properties, which is suitable for quantitative analysis with fractal theory. [PERSON] et al. [38] proposed a unified Fourier morphological analysis method to quantify the shape characteristics of aggregates. [PERSON] et al. [39] proposed an index to characterize the combined effect of rib angle and surface texture of coarse aggregates, i.e., AT, and investigated the statistical distribution pattern of the AT index for different grain sizes of basalt aggregates and their composite aggregates. [PERSON] et al. [40] studied the comparison of rib angles of different grain sizes of aggregates under different crusher production. The angularity index of aggregate with calculation equations and main characteristics is shown in Table 2. \begin{table}
\begin{tabular}{c c c} \hline
**Angularity Indexes** & **Equations \({}^{\star}\)** & **Characteristics** & **Literatures** \\ \hline Angularity & \(\frac{1}{2}\sum\limits_{m=m_{1}+1}^{m=m_{2}}\left[\left(\frac{a_{m}}{a_{0}} \right)^{2}+\left(\frac{b_{m}}{a_{0}}\right)^{2}\right]\) & Fourier series analysis was applied to measure the angularity of aggregate in PIAS. \\ Angularity factor (AF) & \(AF=\sum\limits_{p=1}^{N_{d}}\sum\limits_{q=1}^{N_{d}}\left[\left(\frac{a(p,q)}{ a_{0}}\right)^{2}+\left(\frac{b(p,q)}{a_{0}}\right)^{2}\right]\) & The fast Fourier transform method could be applied to calculate and analyze the regularity factor (AF) of aggregate in FTI. \\ Surface erosion-dilation & \(\frac{A_{1}-A_{2}}{A_{1}}\times 100\%\) & As a popular image processing technique, erosion dilation was utilized to analyze the angularity of aggregate particles in the UIAIA. \\ Gradient angularity (GA) & \(\frac{1}{3}-1\sum\limits_{i=1}^{n-3}|\theta_{i}-\theta_{i+3}|\) & Higher values of GA indicate more angular aggregate particles. \\ Angularity index (AI) (radius) & \(\sum\limits_{p=0}^{\theta-3600-A\theta}\left|\frac{\theta_{p,\theta-R_{EE}}}{R_ {EE}}\right|\theta\) & As the defining equation of angularity. \\
3D angularity (3 DA) & \(\frac{1}{3}\sum\limits_{\frac{1}{n}=1}^{\frac{n}{n}}\frac{P_{n,\theta}}{P_{n}}+ \frac{\sum\limits_{i=1}^{n}}{\sum\limits_{i=1}^{n}}\frac{P_{n,\theta}P_{n, \theta}}{P_{n}}+\frac{\sum\limits_{i=1}^{n}}{\sum\limits_{i=1}^{n}}\frac{P_{n, \theta}P_{n,\theta}}{P_{n}}\) & 3D parameters for more accurate characterization of aggregate morphology. \\ \hline \end{tabular}
\end{table} 
Table 2: Angularity index of aggregate with calculation equations and main characteristics. [PERSON] et al. [47] reconstructed the microstructure model of the asphalt mixes used for uniaxial compression tests based on X-ray CT scans, keeping the original morphology of the aggregates. The 3D finite element model with different aggregate angles was established and simulated for uniaxial compression tests by artificially reducing the aggregate angles by keeping the original morphology unchanged. The effect of the aggregate angle on the mechanical response of the asphalt mixture was quantitatively analyzed. In order to obtain the probability distribution of shape features, \(GA\) values were proposed in the previous study to quantify the 2D morphology of agglomerate particles by AIMS tests.