## 3 Methods
### RGCNN Model  
In this section, we will introduce the RGCNN and construct a new convolutional neural network based on g\({}^{\text{n}}\)Conv [22]. Before introducing g\({}^{\text{n}}\)Conv and its basic operation gConv, let us first recall the basic convolution operation. The traditional convolution operation for a given input image \(\text{X}\in\text{RH}\times\text{W}\times\text{C}\) and a convolutional kernel \(\text{K}\in\text{RKH}\times\text{KW}\times\text{C}\) is defined as follows:  
\[Y(t)=(X^{\bullet}K)(t)=\sum\
olimits_{i=1}^{K_{H}}\sum\
olimits_{j=1}^{K_{W }}X(t_{i},t_{j})\cdot K(i,j) \tag{8}\]  
where \({}^{\bullet}\) denotes the convolution operation, and \(Y\in\mathbb{R}^{(H-K_{H}+1)\times(W-K_{W}+1)\times C}\) represents the output feature map. Building upon this, g\({}^{\text{n}}\)Conv is a generalized convolutional operator capable of capturing higher-order feature interactions and modeling long-range dependencies and has shown effectiveness in downstream image tasks. This paper applies it to upstream image processing tasks. We introduce this module into the convolutional neural network to replace the convolutional layers of conventional convolutional neural networks. gConv is the basic operation of g\({}^{\text{n}}\)Conv and is considered a first-order interaction. Given the input features \(\text{X}_{\text{gin}}\in\mathbb{R}^{H\times W\times C}\), the resulting features \(\text{Y}_{\text{gout}}\) from gConv can be computed using Equation (5). \[ \begin{split}&[P_{0},Q_{0}]=F_{\text{Proj-in}}\big{(}X_{\text{gin}} \big{)}\\ & P_{1}=DWConv(Q_{0})\otimes P_{0}\\ & Y_{\text{gout}}=F_{\text{Proj-out}}(P_{1})\end{split}  \tag{9}\]  
In this context, \(F_{Proj-in}(\cdot)\) and \(F_{Proj-out}(\cdot)\) represent the input linear projection layer and the output linear projection layer of gConv, respectively; while \(\text{DWConv}(\cdot)\) denotes the depthwise convolution layer operation. In gConv, the input features \(X_{\text{gin}}\) are projected to \(\text{P}_{0},\text{Q}_{0}\big{(}\text{P}_{0},\text{Q}_{0}\in\mathbb{R}^{H \times W\times C}\big{)}\). The depthwise convolution is used to capture long-range dependencies and achieve single interactions between adjacent features through element-wise multiplication. The recursive gated convolution g\({}^{\text{n}}\)Conv was designed based on gConv to introduce higher-order interactions. To prevent excessive computational overhead from these higher order interactions, the number of channels involved in each order of computation is set to a lower count, with fewer channels used for lower-order interactions. Given the input features \(X_{\text{gmin}}\in\mathbb{R}^{\text{H}\times\text{W}\times\text{C}}\), the resulting features from g\({}^{\text{n}}\)Conv can be computed using Equation (7). \[ \begin{array}{l}[P_{0},Q_{0},Q_{1},\cdots,Q_{n-1}]=F_{\text{Proj-in}}(X_{g \text{min}})\\ P_{1}=DWConv(Q_{0})\otimes P_{0}/\alpha\\ P_{k+1}=DWConv(Q_{k})\otimes F_{\text{Proj},k}(P_{k})/\alpha\\ (Q_{k}\in Q_{0},Q_{1},\cdots,Q_{n-1})\\ Y_{g\text{nout}}=F_{\text{Proj-out}}(P_{n})\end{array}  \tag{10}\]  
In this context, \(F_{\text{Proj}-k}(\cdot)\) represents the linear projection layer used to adjust the number of P channels after the k-th (k = 1, 2, \(\ldots\), n \(-\) 1) order of interaction. The coefficient \(1/\alpha\) is used to scale the output. The network structure of g\({}^{4}\)Conv is outlined in the Figure 2. g\({}^{4}\)Conv is designed to realize arbitrary-order spatial interactions through a highly efficient implementation that leverages gated convolutions and recursive design principles. To enhance its performance, we construct the GNblock module to replace the convolution layers of convolutional neural networks, followed by normalization layers, and then output feature vectors through a multilayer perceptron (MLP), which is somewhat similar to the self-attention mechanism.