## 2 Materials and Methods
### Dilated Self-Attention Network  
The deformable registration network in this paper is composed of two encoder-decoder networks with the same structure, which we call the dilated self-attention network (DSNet), as shown in Figure 2. In this network, we use dilated convolutions to increase the receptive field of the convolutional layer and thereby improve the performance of the network. In the deep neural network, downsampling is often used to increase the receptive field. However, due to the limited number of downsampling in the shallow network, the receptive field of the low-level features cannot be effectively expanded. It is difficult for the extracted features to fully reflect the differences between the input images. Therefore, some studies [25] have increased the receptive field of low-level features by adding Inception modules to the network. But for 3D convolution, especially in the shallow layer of the network (the size of the feature map is large at this time), the use of any large-scale convolution will bring huge memory and calculation consumption. But dilated convolution can bypass this problem. Dilated convolution can explicitly increase the receptive field of the feature map without adding additional parameters, enabling the network to capture more contextual information during the convolution process. At the same time, because there is no downsampling, the loss of detailed information in the image is avoided, and the ability of low-level features to perceive the difference in detail in the image is preserved. In computer vision, acquiring multi-scale features is crucial for object region detection. By adjusting the dilation rate of dilated convolutions, the size of the receptive field can be directly changed, and the network can capture features of different scales. Previous research [26] has elucidated that dilation convolution employs sparse sampling techniques, which, if a single dilation rate is used, can lead to a checkerboard pattern effect within the perceptual field of a pixel. In this case, voxels within the depth feature map can only perceive the underlying information in a checkerboard pattern, resulting in a large loss of local information. At the same time, the information correlation between voxels that are far away from each other is not strong, and the checkerboard effect reduces the coherence and consistency of the local information. For this reason, this study uses a cascade of three dilation convolutions with different dilation rates \(r\in\{1,2,3\}\) and integrates them into one module. This cascade of dilation convolutions with different dilation rates is able to expand the sensory field of the output of each layer without increasing the number of parameters, which makes the parameter utilization more efficient in capturing  
Figure 2: Structural diagram of the dilated self-attention network. the long-term dependence and global information in the input data, and thus simplifies the complexity of the model to a certain extent. In addition, the dilation convolution with different dilation rates is able to capture features at different scales, and by integrating these multi-scale features, information can be extracted more optimally, which in turn enhances the model's representation of the input data. As the sequence of dilation rates is redesigned, this will further optimize the feature extraction process and enhance the model's ability to recognize subtle features. This approach not only improves the model's sensitivity to local features but also enhances its understanding of global features, thus achieving better performance in various visual tasks. In this way, the model is able to achieve the in-depth mining of multi-dimensional features of the input data while keeping the number of parameters constant, which is important for improving the generalization ability and adaptability of the model.