## 3 Methods
### Linear Correlation Attention Module
This mechanism significantly improves segmentation performance in underwater scenarios, particularly under conditions of drastic lighting changes. Through ablation experiments, we find that applying the LCA module to the P4 layer of the backbone network GELAN achieves the best results. The LCA module, by leveraging Pearson correlation, enhances the model's understanding of complex relationships between features, particularly in maintaining semantic consistency when handling variations in the color tones of similar objects. This leads to a significant improvement in the accuracy and stability of instance segmentation. The specific structure is illustrated in Figure 3, and the computational process is as follows. Firstly, the input feature map \(X\) undergoes feature transformation. Specifically, \(X\) is transformed into lower-dimensional feature representations through two \(1\times 1\) convolution layers, generating the feature representations \(\theta(X)\) and \(\varnothing(X)\), which are defined as follows:  
\[\theta(X)=W_{\theta}X \tag{1}\]  
\[\varnothing(X)=W_{\varnothing}X \tag{2}\]  
Here, \(W_{\theta}\) and \(W_{\varnothing}\) are \(|x|\) convolution kernels used to reduce the dimensionality of the input feature map and extract semantic feature representations. At the same time, the input feature map undergoes a linear transformation to generate the feature map \(g(X)\):  
\[g(X)=W_{g}X \tag{3}\]  
where \(W_{g}\) is also a \(|x|\) convolution kernel used for further processing of the input feature map. Figure 3: The architecture of Linear Correlation Attention Module, the \({}^{*}\) denotes multiplication. Next, we compute the similarity matrix. First, we calculate the average values of the feature maps \(\theta(X)\) and \(\varnothing(X)\) as follows:  
\[\theta_{avg}=\frac{1}{N}{\sum\limits_{i=1}^{N}{\theta(X_{i})}} \tag{4}\]  
\[\varnothing_{avg}=\frac{1}{N}{\sum\limits_{i=1}^{N}{\varnothing(X_{i})}} \tag{5}\]  
Here, \(N\) represents the total number of pixels in the feature map, and \(X_{i}\) represents the \(i\)-th pixel in the input feature map \(X\). Then, we compute the centered representations of the feature maps by subtracting their respective mean values:  
\[\theta^{\prime}(X)=\theta(X)-\theta_{avg} \tag{6}\]  
\[\varnothing^{\prime}(X)=\varnothing(X)-\varnothing_{avg} \tag{7}\]  
Here, \(\theta^{\prime}(X)\) and \(\varnothing^{\prime}(X)\) represent the centered feature maps, which are normalized by removing the mean to prepare for subsequent correlation calculations. The Pearson correlation coefficient matrix \(f(i,j)\) is then computed as follows:  
Figure 2:0  
where \(i\) and \(j\) represent the indices of pixels or positions in the feature map, and \(X_{i}\) and \(X_{j}\) represent the pixel values at the \(i\)-th and \(j\)-th positions in the input feature map \(X\). This equation measures the Pearson correlation between the feature vectors at positions \(i\) and \(j\). Next, we normalize the Pearson correlation coefficients to obtain the normalized similarity matrix:  
Figure 2:1  
Here, the constant 0.5 is used to linearly scale the Pearson correlation coefficients from the range of \([-1,1]\) to \([0,1]\). This transformation ensures that all similarity values are non-negative, making them suitable for subsequent processing steps. The use of 0.5 effectively maps the negative correlations to a positive scale, thus preventing potential issues that could arise from negative similarity values and ensuring the stability of the subsequent operations. Then, the normalized similarity matrix \(f_{norm}(i,j)\) is used to compute a weighted sum over the feature map \(g(X)\), yielding the augmented feature \(Y_{i}\):  
Figure 2:2  
Here, the symbol \"\(\otimes\)\" represents matrix multiplication, which is used to multiply the normalized similarity matrix \(f_{norm}(i,j)\) with the feature map \(g(X_{j})\); then, a weighted summation is performed over all positions to generate the enhanced feature \(Y_{i}\).