## 2 Proposed Method
### Involution Operator and Optimization Measures  
#### 2.1.1 Feature Extraction Method Based on Involution  
Pavement cracks often have long and narrow shapes and diverse branches, but the small-sized convolution kernels commonly used in mainstream networks have difficulty in capturing the global features at one time, which limits the acquisition of context information by the convolution layer and reduces the accuracy of segmentation by the algorithm. To improve the accuracy of network identification of cracks, this paper introduces the involution [41] operator, which reduces the redundant information and strengthens the spatial modeling by sharing the weights of the channel dimensions. The channel information is then cleverly decoupled from the spatial information by the REI module, which was designed to achieve higher performance with fewer parameters.  
Involution is an operator built on ordinary convolution that has the opposite properties of regular two-dimensional convolution. Based on the research of [PERSON] et al. [41], in the spatial dimension, the same convolution kernel weights are shared on the same feature map called spatial invariance; in the channel dimension, each column of channels enjoys different groups of convolution weights called channel specificity. Convolution has properties such  
Figure 1: Overall model framework of RIIAnet.  
as being spatially agnostic and channel specific, while involution has the opposite properties because of the different methods of feature extraction.  
The involution kernel is denoted as \(H\times W\times K\times K\times G\), where \(H\) denotes the length of the input feature map, \(W\) denotes its width, \(K\) denotes the size of the involution kernel, and \(G\) denotes the number of groupings. All channels share the G-group convolution kernel. The steps in the involution operation are as follows:  
1. The dimensions of the input feature maps are changed by averaging pooling.
2. The channel dimensions are changed to \(G\times K\times K\) by two convolution layers of size \(1\times 1\) with weight \(W\).
3. A grid of size \(K\) is used to spread the input feature maps and divide them into \(G\) groups.
4. The obtained weights are multiplied and added to the spread feature maps. The feature mapping of each involution is obtained by multiplying and adding the values of the input features.  
The mapping is defined as:  
\[Y_{i,j,k}=\sum_{(u,v)\in\Delta x}\mathcal{H}_{i,j,\mu+[\frac{K}{2}],\
u+[ \frac{K}{2}],[\frac{K\bar{G}}{c}]}X_{i+u,j+v,k} \tag{1}\]  
where \(\mathcal{H}\in\mathbb{R}^{H\times W\times K\times K\times G}\) denotes the involution kernel. The generic form of the involution kernel generation is as follows:  
\[\mathcal{H}_{i,j}=\phi\Big{(}X_{\Psi_{i,j}}\Big{)} \tag{2}\]  
where \(\Psi_{i,j}\) is an index set of \((i,j)\) coordinate neighborhoods, and \(X_{\Psi_{i,j}}\) represents the feature maps of the slices after being passed through an unfolding operation. The function \(\phi\) denotes the generated form of the weights \(W\), and its expression is:  
\[\phi\big{(}X_{i,j}\big{)}=W_{1}\sigma\big{(}W_{0}X_{i,j}\big{)} \tag{3}\]  
where \(W_{0}\in\mathbb{R}^{\frac{C}{\gamma}\times C}\) and \(W_{1}\in\mathbb{R}^{(K\times K\times G)\times\frac{C}{\gamma}}\) are the linear transformation matrices, \(r\) denotes the channel reduction rate, and \(\sigma\) denotes the batch normalization and ReLU activation functions.  
The involution operator is defined here with \(N=1\), \(G=1\), \(K=2\), and its features are extracted in the manner shown in Figure 2.  
To visually show the difference between the convolution and involution feature extraction methods, we put the analysis and visualization results of them in Section 3.3.