## 3 Methodology
### Camouflaged Object Detection Based on Attention Enhancement
#### 3.2.2 Joint Dense Skip Attention Module  
Our main purpose in establishing the joint dense jump attention module is to obtain richer semantic expressions from the contextual channels and spaces, that is, to aggregate them at multiple levels. In the following equations, \(f(\cdot)\) represents convolutional operation, and \(cat(\cdot)\) represents channel-wise feature concatenation. The output result of each encoder is denoted as \(E_{2}\), \(E_{3}\), and \(E_{4}\), corresponding to the number of channels of sizes 32, 64, and 128, expressed as in Equations (2)-(4):  
\[E_{2}^{\prime} =E_{2}\otimes f(Up(E_{3}))\otimes f(Up(Up(E_{4}))) \tag{2}\] \[E_{3}^{\prime\prime} =f(\text{cat}[E_{3}\otimes f(Up(E_{4})),f(Up(E_{4}))])\] (3) \[E_{2}^{\prime\prime} =f\big{(}Up\big{(}E_{3}^{\prime\prime}\big{)}\big{)} \tag{4}\]  
We obtain an aggregated representation, which is the scale of \(E_{2}\), through \(A\), \(A=f(cat[E_{2}^{\prime},E_{2}^{\prime\prime}])\in\mathbb{R}^{(32+64+128)\times H _{2}\times W_{2}}\). \(UP\) refers to upsampling; for instance, \(\text{Up}(E_{3})\) signifies upsampling the features of the \(E_{3}\) layer, typically resizing them to match the dimensions of the \(E_{2}\) layer. After the aggregation, it still represents the contextual information. The current research on COD has separated the channel and spatial attention modules for decoding and expanding the receptive field. So, we first distinguish the relatively important channel contextual information:  
\[\alpha_{c}=\sigma\Bigg{(}\frac{\text{exp}\Big{(}\mathcal{P}_{q}(\widetilde{A })\Big{(}\mathcal{P}_{k}(\widetilde{A})\Big{)}^{\top}\Big{)}}{\sum\text{exp} \Big{(}\mathcal{P}_{q}(\widetilde{A})\Big{(}\mathcal{P}_{k}(\widetilde{A}) \Big{)}^{\top}\Big{)}}\mathcal{P}_{v}(\widetilde{A})\Bigg{)} \tag{5}\]  
where \(\tilde{A}\in\mathbb{R}^{C\times 1\times 1}\) represents channel level pooling representation, and \(\mathcal{P}(\cdot)\) represents the use of \(1\times 1\) convolutional kernel operation, utilizing the softmax function and self-attention, combined with the sigmoid function to distinguish important channels and obtain deep semantic information, as shown in Equation (5). In order to obtain more detailed deep semantic information, we perform aggregation transformation on it and set the confidence channel weights as follows: \(A_{c}=(A\otimes\alpha_{c})+A\). Subsequently, we rely on \(\alpha_{c}\) and the confidence ratio \(\gamma\). The distribution retains the confidence channels, as shown in Equation (6):  
\[\tilde{A}_{c}=A_{c}\otimes\text{ mask }\Bigg{\{}\begin{array}{ll}\text{ mask }&=1,&\text{if }\alpha_{c}>P^{-1}(\gamma)\\ \text{mask }&=0,&\text{otherwise}\end{array} \tag{6}\]We use \(P^{-1}(\gamma)\) to denote the \(\gamma\) quantile of \(\alpha_{c}\). Then, the refined \(\tilde{A}_{c}\) is computed spatially to discriminate the camouflaged object detection and generate the first decoder representation \(D_{0}\in\mathbb{R}^{1\times H_{2}\times W_{2}}\), and the announcement is as follows Equation (7):  
\[D_{0}=\frac{\exp\left(\mathcal{Z}_{q}\left(\widetilde{A}_{c}\right)\left( \mathcal{Z}_{k}\left(\widetilde{A}_{c}\right)\right)^{\top}\right)}{\sum\exp \left(\mathcal{Z}_{q}\left(\widetilde{A}_{c}\right)\left(\mathcal{Z}_{k}\left( \widetilde{A}_{c}\right)\right)^{\top}\right)}\mathcal{Z}_{v}\left( \widetilde{A}_{c}\right)+\mathcal{Z}_{v}\left(\widetilde{A}_{c}\right) \tag{7}\]  
We use a convolution operation to map the input features to a new space \(A_{c}\). We generate a decoder \(D_{0}\) using \(A_{c}\), which is of the same size as \(A_{c}\). The features projected by \(\mathcal{Z}(\cdot)\) through convolution operation onto \(A_{c}\). We upsample \(D_{0}\) to the size of the original input to obtain \(DS_{0}\), which is a graph used to provide deep supervision.