## 3 Proposed Methods
### Highway Crack Detection Methods
#### 3.1.2 MHDC Module  
Considering the complexity, connectivity, and narrowness of cracks, it is also important to increase the receptive field of feature points in the central part of the network while preserving detailed information. Dilated convolution is an ideal alternative to pooling in which the receptive field can expand and multi-scale contextual information captured without changing the feature map size.  
Dilated convolution has an additional dilation rate parameter \(r\) to indicate the size of dilation compared with normal convolution. As shown in Figure 3, the receptive field size is 7 after successive stacks of three ordinary convolutions of size \(3\times 3\) and stride 1, while the receptive field can reach 15 after using three convolutions with dilation rates of 1, 2, and 4, respectively. Thus, it is easy to see that the dilated convolution can significantly expand the receptive field and cause the receptive field grow exponentially. However, when successively stacking the dilated convolutions, the dilation rate of each dilated convolution should be set according to the standardized design strategy, hybrid dilated convolution (HDC), to avoid the \"gridding effect\" problem. The HDC design strategy can be described as follows: (1) The dilation rate of each convolution of the stacked layers cannot have a convention greater than 1, such as \(r=[2,4,6]\), there is a maximum convention of 2, and there will be a gridding effect. (2) Design the dilation rate as a sawtooth structure, such as \([1,2,3,1,2,3]\). (3) Satisfy the following expression:  
\[M_{i}=\max[M_{i+1}-2r_{i},M_{i+1}-2(M_{i+1}-r_{i}),r_{i}] \tag{1}\]  
where \(r_{i}\) is the dilation rate of layer \(i\), and \(M_{i}\) means the maximum distance between two non-zero values in layer \(i\). Then, assuming a total of \(n\) layers, the default \(M_{n}=r_{n}\). The design goal is to make \(M_{2}<=k\). For example, for kernel size \(k=3\), an \(r=[1,2,5]\) pattern works as \(M_{2}=2\); however, an \(r=[1,2,9]\) pattern does not work as \(M_{2}=5\).  
Figure 4: The MHDC module implementation details. DCGE is a combination of dilated convolution, GN, and ELU, and \(r\) represents the dilation rate of the dilated convolution.  
The usage modes of dilated convolution can be broadly classified into two, cascaded-mode-like [32] and parallel-mode-like [33], both of which have shown a powerful ability to improve segmentation accuracy. The MHDC module proposed in this paper then takes full advantage of both and combines the two modes by means of short connections.  
As shown in Figure 4, if the expansion rate of the stacked dilated convolution is 1, 2, 4, 8, and 16, the receptive field of each layer is 3, 7, 15, 31, and 63, respectively. When the dilation rate is 1, the dilated convolution is the normal convolution. The encoder part of CrackNet network has four downsampling layers, and the image size for this experiment is \(512\times 512\), after four times of max-pooling. The uppermost HDC layer of the MHDC module designed in this paper uses dilated convolutional layers with dilation rates of 1, 2, 4, and 8, and its receptive field is \(31\times 31\), which covers most of the pixel points of the underlying feature map and can capture more contextual information. The other layers of HDC are 3, 7, and 15 in order of receptive field from bottom to top. By stitching the four layers of the multi-scale feature map together and reducing the dimensions through \(1\times 1\) convolution with 512 channels, multi-scale feature information is obtained, and deeper features are extracted without changing the size of the feature map.