## 2 Theory and Methods
### Model Improvements
#### 2.2.3 WIMIoU
The inner-box coordinates are obtained as follows:  
\[b_{1}^{\text{gt}}=x_{\text{c}}^{\text{gt}}-\frac{w_{\text{gt}}* \text{ratio}}{2},b_{\text{\it{r}}}^{\text{gt}}=x_{\text{\it{c}}}^{\text{gt}}+ \frac{w_{\text{gt}}*\text{ratio}}{2},b_{\text{\it{t}}}^{\text{gt}}=y_{\text{ \it{c}}}^{\text{gt}}+\frac{h_{\text{gt}}*\text{ratio}}{2},b_{\text{\it{b}}}^{ \text{gt}}=y_{\text{\it{c}}}^{\text{gt}}-\frac{h_{\text{gt}}*\text{ratio}}{2} \tag{6}\]  
\[b_{1}^{\text{prd}}=x_{\text{\it{c}}}^{\text{prd}}-\frac{w_{\text{prd}}*\text{ ratio}}{2},b_{\text{\it{r}}}^{\text{prd}}=x_{\text{\it{c}}}^{\text{prd}}+\frac{w_{\text{prd}}* \text{ratio}}{2},b_{\text{\it{t}}}^{\text{prd}}=y_{\text{\it{c}}}^{\text{prd}} +\frac{h_{\text{prd}}*\text{ratio}}{2},b_{\text{\it{b}}}^{\text{prd}}=y_{\text{ \it{c}}}^{\text{prd}}-\frac{h_{\text{prd}}*\text{ratio}}{2} \tag{7}\]  
where ratio is the ratio factor to find InnerIoU; \(b_{1}^{\text{gt}},b_{1}^{\text{gt}},b_{\text{\it{r}}}^{\text{gt}},b_{\text{ \it{t}}}^{\text{gt}},b_{1}^{\text{gt}},b_{\text{\it{r}}}^{\text{prd}},b_{\text{ \it{t}}}^{\text{prd}},b_{\text{\it{b}}}^{\text{prd}}\) represent the left horizontal coordinate, right horizontal coordinate, upper vertical coordinate, lower vertical coordinate of the target frame corrected by the ratio factor, and the left horizontal coordinate, right horizontal coordinate, upper vertical coordinate, and lower vertical coordinate of the prediction frame corrected by the ratio factor, respectively. The detection box range is reduced according to the ratio factor, forcing the model to focus on the target center area and reduce edge noise interference. Define InnerIoU and its loss function solved as follows:  
\[\text{InnerIoU}=\frac{\text{inter}}{\text{union}},L_{\text{Inner-IoU}}=1-\text{ InnerIoU}=1-\frac{\text{inter}}{\text{union}} \tag{8}\]  
where inter denotes the area of the intersection portion of the target frame and the prediction frame after the INNER operation, i.e., the numerator portion of InnerIoU; UNION denotes the area of the concatenation of the target frame and the prediction frame after the INNER operation, i.e., the denominator portion of InnerIoU; and L\({}_{\text{Inner-IoU}}\) denotes the loss function of InnerIoU. \[L_{\text{Inner-MPDIoU}}=L_{\text{MPDIoU}}+\text{IoU}-\text{InnerIoU} \tag{9}\]  
\[d_{1}^{2}=\left(x_{1}^{\text{gt}}-x_{1}^{\text{prd}}\right)^{2}+\left(y_{1}^{ \text{gt}}-y_{1}^{\text{prd}}\right)^{2},d_{2}^{2}=\left(x_{\text{r}}^{\text{ gt}}-x_{\text{r}}^{\text{prd}}\right)^{2}+\left(y_{\text{b}}^{\text{gt}}-y_{ \text{b}}^{\text{prd}}\right)^{2} \tag{2}\]0where L\({}_{\text{WIMIoU}}\) denotes the loss function of WIMIoU, L\({}_{\text{Inner-MPIoU}}\) denotes the loss function of Inner-MPIoU, and R\({}_{\text{WIMIoU}}\) denotes the compensation coefficients of the loss function WIMIoU. The loss weight is dynamically adjusted through the compensation coefficient. When the center of the predicted box deviates from the real box, the loss weight is increased to strengthen the center alignment constraint. The prediction bounding box of WIMIoU is plotted against the ground-truth bounding-box positional information in \[d_{1}^{2}=\left(x_{1}^{\text{gt}}-x_{1}^{\text{prd}}\right)^{2}+\left(y_{1}^{ \text{gt}}-y_{1}^{\text{prd}}\right)^{2},d_{2}^{2}=\left(x_{\text{r}}^{\text{ gt}}-x_{\text{r}}^{\text{prd}}\right)^{2}+\left(y_{\text{b}}^{\text{gt}}-y_{ \text{b}}^{\text{prd}}\right)^{2} \tag{2}\]1  
As illustrated in Figure 4, the red dashed lines depict MPDIoU's corner distance penalty mechanism, which quantifies positional deviations between predicted and ground-truth box corners. The two smaller inner boxes represent the contracted regions generated by InnerIoU through ratio-based scaling, emphasizing target-centric feature learning. Their overlapping area corresponds to WiseIoU's adaptive compensation zone, where dynamic weight adjustments prioritize center alignment accuracy based on offset severity.