## 3 Methodology
### 2D Wavelet Decomposition  
Pavement textures refer to the surface characteristics of pavement, which are composed of micro-texture that is extremely intricate and difficult to analyze through naked eyes. To overcome this challenge, researchers have utilized various approaches, such as Fourier transform [19] and one-dimensional wavelet transform [42; 43; 44], to decompose the data for analysis. However, in this paper, a 2D wavelet transform algorithm was utilized for decomposition, which has the potential to provide more comprehensive and accurate results. We carried out the analysis using the powerful MATLAB Wavelet Toolbox, which provides numerous functionalities for signal processing and decomposition that are essential to our research.  
While working with different mother wavelets, it is crucial to choose the most suitable one for the analysis, as different wavelets can produce unique effects on the decomposition results. The different types of mother wavelets that can be selected include Haar, Morlet, Daubechies (db), and Symlets. Among these options, the db wavelet, which is composed of ten compactly supported orthonormal wavelet functions, has been proven to be the most-applied one for surface texture analysis. For instance, previous studies have used db3 to analyze surface texture due to its sharp spikes and close resemblance to a profile signal.  
However, other options, such as db4 to db10, exhibit smooth signal characteristics and are not suitable for texture analysis. As such, in this paper, we used db3 as the mother wavelet for conducting texture analysis. This is because it provides better resolution at high frequencies and enhances the accuracy of texture feature extraction. Overall, our approach to pavement texture analysis using the 2D wavelet transform algorithm and the appropriate mother wavelet selection was expected to deliver more detailed insights into the characteristics and properties of road surfaces.  
The data were decomposed into six levels, of which five were detail levels denoted as Level 1 (<0.3 mm), Level 2 (0.3-0.6 mm), Level 3 (0.6-1.2 mm), Level 4 (1.2-2.4 mm), Level 5 (2.4-4.8 mm), and Level 6 (>4.8 mm). According to the definition specification, the wavelengths range of micro-texture typically ranges from 1 \(\upmu\)m to 0.5 mm, whereas macro-texture typically ranges from 0.5 mm to 50 mm. In this regard, Levels 1 and 2 represent micro-texture, while Levels 3, 4, 5, and 6 represent macro-texture.  
In the x- or y-direction, each texture data were decomposed into six components using the 2D wavelet transform algorithm. Additionally, this algorithm decomposed the texture data into 6 \(\times\) 6 parts. Figure 9 shows the decomposition results after using the 2D wavelet transform algorithm.  
The energy (\(E\)) is an indicator that measures the overall condition of each part, and it can be calculated as:  
\[E=\sum_{x}\sum_{y}\abs{Z_{x,y}}^{2} \tag{1}\]  
Figure 8: The areal material ratio curve and functional and volume parameters.  
The energy characteristic represents the overall roughness, which exhibits significant variations among different specimens. Additionally, for the same specimen, the energy is sensitive to slight changes in the detection position. Therefore, to address this issue, the concept of relative energy (RE) was introduced. RE represents the proportion of energy at each scale in relation to the total energy and is defined as follows:  
\[RE_{ij}=\frac{E_{ij}}{\sum_{i=1}^{6}\sum_{i=1}^{6}E_{ij}},(1\leq i\leq 6,1\leq j \leq 6) \tag{2}\]  
RE was used in this paper to characterize pavement texture at multiple scales due to its comparability between different samples.