## 2 Proposed Method
### Residual Expanded Involution Module  
To combine the advantages of both operators, we propose a new residual involution module that is different from the Rednet module in [41]. Based on the design criterion set out [40], we see that when the input dimensions are smaller than the output dimensions, the low-rank space cannot represent the high-rank space, causing the characteristic bottleneck problem. The authors of [40,44] showed that the hard-swish activation function has features including smooth and non-monotonic; its nonlinear nature means that it retains more information. It has also been experimentally shown that the hard-swish activation function can boost the rank of the data so that the rank of the input is closer to the rank of the output, thus reducing the representation loss.  
Hence, the residual involution module in this paper adopts an expanded residual structure to retain as much information as possible. The difference between the REI module designed in this paper and other mainstream residual structures is shown in Figure 3.  
The REI module proposed in this paper involves dimensionally increasing convolution, dimensionally decreasing convolution, and involution. The operat  
Figure 3: Bottlenecks in (**a**) ResNet; (**b**) Mobilenet-v3; (**c**) Rednet-50 [41]; and (**d**) RIIAnet.  
Figure 2: Method of extracting feature maps in the involution operator.  
The feature map is expanded to the transition dimensions by convolution with a kernel size of \(1\times 1\) and a stride of one.
2. Based on the experiments presented in [41], after weighting the performance and the number of parameters, we apply an involution kernel with a size of \(7\times 7\), the number of groups is set to 16, and a stride of one is used to perform feature extraction on the feature maps.
3. The feature map extracted by involution is dimensionally reduced via conventional convolution with a kernel size of \(1\times 1\) and a stride of one.
4. The input feature maps are dimensionally expanded through conventional convolution with a kernel size of \(1\times 1\) and a stride of one, and are added to the feature map extracted.  
Here, we define \(C_{i}\) as the number of input channels, \(C_{m}\) as the number of transition channels, \(C_{j}\) as the number of output channels, \(H\) and \(W\) as the length and width of the feature map, respectively. The details of the steps of the REI module are shown in Figure 4.  
The advantages of this structure are as follows:  
* The extracted feature maps are dimensionally increased to enhance the nonlinear expression capability of the network, with a small increase in the computational effort.
* The involution operator is used to increase the perceptual field with fewer parameters to take advantage of the long-range dynamic modeling of involution and focus on the morphological information of the fracture.
* The dimensionality reduction process is used to decouple the spatial and cross-channel relationships of the feature maps [45] extracted by the involution module to achieve cross-channel information integration and enhance the feature expression capability of the network.
* We use the residual structure [46] to compensate for the loss of diversity information caused by the involution module, which makes the gradient flow propagate better and makes the network easier to train [47].  
When building lightweight, real-time segmentation neural networks, the REI module is not suitable for use in shallow feature extractors, for which we give a detailed analysis in Section 4.2.