## 3 Methods
### Deep Dual-Branch Network  
We aim to achieve efficient feature extraction for specific scenes by leveraging multi-scale information and complementing multi-level information. To this end, we have redesigned the deep dual-branch backbone structure. Firstly, starting from the first downsampling, the network is divided into 7 stages (labeled as stages 0 to 6). After stage 2, an additional high-resolution branch is introduced. When the input image is 1024 \(\times\) 1024, the high-resolution branch is fixed at 128 \(\times\) 128, while the low-resolution branch extracts deep feature information through multiple downsamplings, ultimately resulting in an 8 \(\times\) 8 feature map. Secondly, to enhance the information exchange between the two branches, spatial and semantic information is shared once more among them through 4 Cross-stage Branches. The detailed structure is shown in Table 1.  
Regarding the design philosophy of the network architecture, as illustrated in Figure 1, the approach can be broken down into specific steps. Firstly, to reduce computational complexity, a Downsampling Block is constructed by stacking two consecutive 3 \(\times\) 3 convolutional layers, and a bottleneck block is added at the end of each branch to expand the output dimensions. Secondly, the cross-stage fusion of the Atrous branch includes bilateral Cross-stage Branches 1-2 (encompassing high-to-low and low-to-high fusion), the Cross-stage Branch 3 on the low-resolution branch (high-to-low fusion), and the Cross-stage Branch 4 on the high-resolution branch (channel fusion). For high-to-low fusion, there are two cases depending on the stage:  
Figure 1: The overall architecture of LDMNet, including the Mamba Fusion Block and the detailed structure of the cross-stage fusion branches. The notations such as % and 1/4 represent the downsampling multiples of the feature maps by the residual basic blocks. RAM denotes the Recombine Attention Module, FFM stands for Feature Fusion Mamba, and Seg. Head refers to the segmentation head. The detailed structure of the Cross-stage Branch is depicted within the dashed box. 1. Between stage 3 and stage 4, bilateral feature fusion is conducted using a convolutional block consisting of a \"3 \(\times\) 3 convolution + BN + ReLU\" sequence, followed by a downsampling operation with an Atrous convolution with a 3 \(\times\) 3 kernel and a dilation rate of 2. This is then followed by a 1 \(\times\) 1 convolution for channel reduction and finally fusion with the context branch. 2. Between stage 4 and stage 5, the convolutional block is doubled in sequence, with the rest of the settings remaining the same as in case a. For low-to-high fusion at the stage 4 position, the output features of the context branch are first compressed through 1 \(\times\) 1 convolution. They are then upscaled to 1/8 feature size using bilinear interpolation before being merged into the high-resolution branch. For channel fusion, only 1 \(\times\) 1 convolution is used to compress the feature channels. If \(X_{Hi}\) and \(X_{Li}\) represent the high-resolution and low-resolution feature maps of the \(i\)-th stage, respectively, then the cross-stage fusion branches of the 4 th and 5 th stages can be represented as follows:  
\[\left\{ \begin{array}{l}X_{H4}=T_{L-H}F_{L}X_{L4}+R(F_{H}X_{H3})\\ X_{H5}=T_{H-H}F_{H}X_{H3}+R(F_{H}X_{H4})\\ X_{L4}=T_{H-L}F_{H}X_{H3}+R(F_{L}X_{L3})\\ X_{L5}=T_{L-L}F_{L}X_{L3}+T_{H-L}F_{H}X_{H4}+R(F_{L}X_{L4})\end{array} \right. \tag{1}\]  
where \(F_{H}\) and \(F_{L}\) correspond to sequences of residual basic blocks with high and low resolutions, respectively, and \(T_{L-H}\), \(T_{H-L}\), \(T_{H-H}\), and \(T_{L-L}\) represent the transformation functions from low to high, from high to low, from high to high, and from low to low, respectively. \(R(\cdot)\) denotes the ReLU function. Finally, the context branch has one additional stage 6 compared to the high-resolution branch, which is used to downsample the feature map to 1/128 of the original feature map.