# An Improved Crack Identification Method for Asphalt Concrete Pavement
## 2 Methodology
### Image Feature Enhancement Method of Asphalt Concrete Pavement Cracks
#### 2.1.1 Pavement Crack Image Denoising Preprocessing
Each square contains basic image information, and blocks with similar information in the image are combined to form a new image block. During the merging process, calculate the variance of the original image block after merging with each adjacent basic image block, and select the image block with the smallest variance after merging as the new image block, that is, the original image block that is most similar to the basic image block structure information, and then continue to search for a new basic image block for merging. Set a threshold value for the image block. When the variance of the image block exceeds this threshold value, the merging of the current image block will be stopped and the merging of the next image block will start. Because the variance of the merged image blocks needs to be calculated each time when searching for the most similar basic image blocks, the search speed is slow. In response to this issue, this study simplifies the search process by following Theorem 1, which states that the variance between the basic image block and the original image block is the minimum value. For samples \(Y\) and \(X_{s}(s=1,2,\cdots,n)\), \(k\), \(\mu_{0}\) and \(Y_{0}^{2}\) are sample size, mean and variance of \(Y\), respectively, and \(R\), \(\mu_{s}\) and \(Y_{s}^{2}\) are sample size, mean and variance of \(X_{s}(s=1,2,\cdots,n)\), respectively. From sample \(X_{s}(s=1,2,\cdots,n)\), select a group, and from sample \(Y\), form a new sample \(Z\); the sample size, mean and variance of new sample \(Z\) are \(k+R\), \(\mu_{0}^{\prime}\) and \({Y_{0}^{\prime}}^{2}\), respectively. Among \(k\), \(m\) and \(Y_{0}^{2}\), \(\mu_{0}\) are constants, and \(\mu_{s}\) and \(Y_{s}^{2}\) are variables. If in sample \(X_{s}(s=1,2,\cdots,n)\), variables \(\mu_{s}\) and \(Y_{s}^{2}\) satisfy the value of the expression \((m-1)Y_{s}^{2}+\frac{mk}{m+k}(\mu_{s}-\mu_{0})^{2}\) as the smallest, then \(Y\) and \(X_{s}(s=1,2,\cdots,n)\) form the new sample \(Z\) in \(R+k\), and the sample variance \(Y_{0}^{\prime}\) is minimum. The variance of sample \(Y\) is calculated as follows:  
\[Y_{0}^{2}=\frac{\sum\limits_{i=1}^{k}{(y_{i}-\mu_{0})}^{2}}{k-1} \tag{4}\]  
The variance of sample \(X_{s}(s=1,2,\cdots,n)\) is calculated as follows:  
\[Y_{s}^{2}=\frac{\sum\limits_{i=1}  
Due to the fact that the new sample \(Z\) is created by \(Y\) and \(X_{s}(s=1,2,\cdot\cdot\cdot,n)\), it is composed of:  
\[\sum\limits_{i=1}^{k+R}{(z_{i}-\mu_{0}^{\prime})}^{2}=\left[\sum\limits_{i=1}^{k} {(y_{i}-\mu_{0}^{\prime})}+\sum\limits_{i=k+1}^{k+R}{(x_{i}-\mu_{0}^{\prime})} \right]^{2} \tag{7}\]  
\[(k+R)\mu_{0}^{\prime}=k\mu_{0}+R\mu_{s} \tag{8}\]  
Equation (7) can be converted to:  
\[{\rm{Y^{\prime}}}_{0}^{2}=\frac{\sum\limits_{i=1}^{k}{(y_{i}-\mu_{0}^{\prime} )}^{2}+\sum\limits_{i=k+1}^{k+R}{(x_{i}-\mu_{0}^{\prime})}^{2}}{k+R-1} \tag{9}\]  
Set \(\lambda=\frac{R}{R+k}\), \(\Delta\mu=\mu_{s}-\mu_{0}\), we can obtain from Equation (8):  
\[\mu_{0}^{\prime}=\mu_{0}+\lambda\Delta\mu \tag{10}\]  
According to the above equations, it can be deduced that:  
\[\left\{{ \begin{array}{*{20}{c}}{(y_{i}-\mu_{0}^{\prime})}^{2}=(y_{i}-\mu_{ 0}+\lambda\Delta\mu)\\ {(x_{i}-\mu_{0}^{\prime})}^{2}=(x_{i}-\mu_{s}+(1-\lambda)\Delta\mu)\\ \end{array} }\right. \tag{11}\]  
According to Equation (11):  
\[\left\{{ \begin{array}{*{20}{c}}{\sum\limits_{i=1}^{k}{(y_{i}-\mu_{0}^{\prime })}^{2}=\sum\limits_{i=1}^{k}{(y_{i}-\mu_{0})}^{2}+k{(\lambda\Delta\mu)}^{2}} \\ {\sum\limits_{i=k+1}^{k+R}{(x_{i}-\mu_{0}^{\prime})}^{2}=\sum\limits_{i=k+1}^{ k+R}{(x_{i}-\mu_{s}+R{((1-\lambda)\Delta\mu)})}^{2}}\\ \end{array} }\right. \tag{12}\]  
Substitute Equation (12) into (9) to obtain:  
\[{\rm{Y^{\prime}}}_{0}^{2}=\frac{(k-1){\rm{Y}}_{0}^{2}+(R-1){\rm{Y}}_{s}^{2}} {k+R-1}+\frac{Rk{(\mu_{s}-\mu_{0})}^{2}}{R+k} \tag{13}\]  
The image block corresponding to the minimum value of Equation (11) is taken as a new image to complete the block processing of an asphalt concrete pavement crack image. Step 2: Conduct fractional differential image enhancement as follows.