# Automatic Bridge Crack Detection Using a Convolutional Neural Network
## 2 Materials and Methods
### Proposed Model
#### 2.2.1 Atrous Convolution  
Modern image classification networks integrate multi-scale context information through continuous pooling and down-sampling layers, resulting in a loss of detail information about the  
Figure 3: Description of the proposed convolutional neural network (CNN) structure.  
Figure 2: Examples of images with edges containing cracks. (**a**) a picture of the crack in the lower right corner, (**b**) a picture of the crack in the upper left corner.  
object edges and a degradation of the image resolution [25; 26]. To solve this problem, [PERSON] et al. [27] proposed a novel convolution method--atrous convolution (also called dilated convolution). Atrous convolution can exponentially expand the receptive field without a loss of resolution, resulting in a denser feature map [27].  
Let \(F:Z^{2}\to R\) denotes a discrete function. Let \(\Omega_{r}=\left[-r,r^{2}\right]\cap Z^{2}\) and \(k:\Omega_{r}\to R\) be a discrete filter of size \(\left(2r+1\right)^{2}\). Then, the discrete convolution operator \(\ast\) can be defined as:  
\[(F\ast k)(p)=\sum_{s+l=p}F(s)k(t), \tag{1}\]  
We now promote this operator. Let \(l\) be an atrous factor, and let \(\tau_{l}\) be defined as:  
\[(F\ast_{l}k)(p)=\sum_{s+ll=p}F(s)k(t), \tag{2}\]  
Here, \(\ast_{l}\) is the atrous convolution or l-atrous convolution. The discrete convolution \(\ast\) we are familiar with is an atrous convolution of \(l=1\).  
Atrous convolution enables an exponential expansion of the receptive field without a loss of resolution and coverage [27]. Let \(F_{0},F_{1},\ldots,F_{n-1}:Z^{2}\to R\) be discrete functions, and let \(k_{0},k_{1},\ldots,k_{n-2}:\Omega_{1}\to R\) be discrete \(3\times 3\) filters. Consider applying an atrous filter with an exponential increase:  
\[F_{i+1}=F_{i}\ast_{2^{i}}k_{i}\text{ for }1=0,\ 1,\ \ldots,\ \text{n}-2, \tag{3}\]  
The receptive field of the element \(p\) in \(F_{i+1}\) is defined as the set of elements in \(F_{0}\) that modify the value of \(F_{i+1}(p)\). Assume that the size of the receptive field of \(p\) in \(F_{i+1}\) is the number of these elements. Then, the receptive field size of each element in \(F_{i+1}\) is \(\left(2^{i+2}-1\right)\times\left(2^{i+2}-1\right)\). The receptive field is an exponentially increasing square, as shown in Figure 4.  
In our crack detection task, special attention was paid to the edge and texture information of the crack. However, continuous pooling and down-sampling layers will result in the loss of the object edge detail information and a reduced image resolution [25; 26]. For example, after four pooling layers, the information of any object smaller than 16 pixels would be lost. In the last few layers of the network, the resolution of the feature map is small, so using the atrous convolution instead of the maxpooling layer can effectively preserve the resolution of the feature map and provide more feature information  
Figure 4: Schematic diagram of the atrous convolution.  
for the final classification, thereby obtaining a higher accuracy. Therefore, atrous convolution with an atrous rate of 2 was introduced in the last three convolutional layers of the network. As a result, the loss of edge detail information or the reduction of the image resolution is avoided, while a larger receptive field is obtained.