## 2 Theoretical Foundations
### DSCN Principle and Design  
The convolution operation in UNet is standardized, featuring a fixed receptive field that lacks adaptability to the shape variations of the target object. This inherent limitation hampers UNet's effectiveness in extracting features from complex object shapes. Deformable convolutional networks v3 (DCNv3) introduces deformable convolution [25] operations that enhance adaptability to the target object's shape. Moreover, DCNv3 employs a convolutional kernel with a global receptive field [26], enabling better capture of relationships between target objects. By learning sampling position offsets and modulation masks, the convolutional kernel can dynamically adjust to different shapes, thereby facili  
Figure 1: SWSDS-Net network architecture. tating more effective feature extraction from the target object. However, when employing large kernel convolutions in DCNv3, there is a significant increase in computational cost--particularly during training--as it necessitates storing numerous offsets and modulation masks, resulting in high memory consumption and slower training speeds. The paper thus proposes a simplification of DCNv3, resulting in the development of DSCN, a lightweight deformable convolution operation. The specific steps involved in this simplification process are as follows:  
The computational cost of DCNv3 increases quadratically as the kernel size grows due to its sampling in both the horizontal and the vertical directions. For instance, a \(3\times 3\) DCNv3 convolutional kernel necessitates 9 sampling points, while a \(5\times 5\) DCNv3 convolutional kernel requires 25 sampling points. Here, \(\triangle p_{k}\) represents the coordinate offset of the k-th sample point in the convolution kernel on the spatial domain, where \(k\) is equivalent to \((i,\,j)\) when expressed in spatial coordinates. \(G\) represents the quantity of deformable groups. Moreover, \(\Sigma_{g=1}^{G}\) represents the individual sampling founded on the learned offsets of each deformable group. \(\mathbf{y}^{1}(i_{0},\,j_{0})\) and \(\mathbf{y}^{2}(i_{0},\,j_{0})\), respectively, denote the values of the output feature map at position \((i_{0},\,j_{0})\) after conducting convolution operations along the \(x\)-axis and \(y\)-axis. \(m_{0,j}^{1}\) and \(m_{i,0}^{2}\) represent the weights of the modulation mask. \(\mathbf{w}^{1}\) and \(\mathbf{w}^{2}\) represent the weights of the linear mapping. \(\triangle p_{j}^{g}\) and \(\triangle p_{i}^{g}\), respectively, denote the offsets of the jth and ith sampling points of the jth deformable group. \[\mathbf{y}^{1}(i_{0},\,j_{0})=\sum_{g=1}^{G}\sum_{j=0}^{K_{w}-1}\mathbf{w}^{1}m_{0,j}^ {1}\mathbf{x}(i_{0},j_{0}+j+\triangle p_{j}^{g}) \tag{1}\]  
\[\mathbf{y}^{2}(i_{0},\,j_{0})=\sum_{g=1}^{G}\sum_{i=0}^{K_{h}-1}\mathbf{w}^{2}m_{i,0}^ {2}\mathbf{y}^{1}\Big{(}i_{0}+i+\triangle p_{i}^{g},j_{0}\,) \tag{2}\]  
The final output is obtained by substituting Equation (1) with Equation (2), and then sampling the superpixels along the \(x\)-axis and \(y\)-axis, respectively. The \(K_{w}\) and \(K_{h}\) represent the sampling size on the \(x\)-axis and \(y\)-axis. \[\mathbf{y}^{2}(i_{0},\,j_{0})=\sum_{g=1}^{G}\sum_{i=0}^{K_{h}-1}\sum_{j=0}^{K_{w} -1}\mathbf{w}^{\prime}m_{i,j}^{\prime}\mathbf{x}\Big{(}i_{0}+i+\triangle p_{i}^{g},j_ {0}\,+j+\triangle p_{i}^{g}) \tag{3}\]  
In Figure 2, The DSCN technique restricts the sampling to a single direction, such as the \(x\)-axis or \(y\)-axis, thereby enabling a \(3\times 3\) DSCN convolutional kernel to utilize only three sampling points and a \(5\times 5\) DSCN convolutional kernel to require just five sampling points. This approach of constraining the sampling direction effectively reduces the computational burden of DSCN, rendering it more suitable for lightweight CNN architectures.