## 3 Methodology
### Normalized Wasserstein-Gaussian Distance  
The IoU metric, frequently employed in detecting general objects, demonstrates remarkable sensitivity in detecting small targets in infrared. Figure 5 demonstrates this,  
Figure 4: Schematic diagram of the Dynamic Head.  
showing a reduction in the IoU from 0.47 to 0.08, correlating with the shift in the small target's position from 1 to 3 pixels. Therefore, alternative evaluation metrics are needed to more accurately assess infrared small targets. The Normalized Wasserstein-Gaussian Distance is presented as an innovative metric. The Wasserstein distance is capable of gauging the likeness among distributions with little to no intersection and remains unaffected by objects of varying sizes. As a result, this approach addresses the similarity between positive and negative samples, as well as the rarity of positive samples in the training stage of infrared small targets.  
Infrared small targets' bounding box is depicted using a two-dimensional Gaussian model due to their unique intensity distribution characteristics.  
\[f(x|\mu,\Sigma)=\frac{\exp\Bigl{(}-\frac{1}{2}(x-\mu)^{T}\Sigma^{-1}(x-\mu) \Bigr{)}}{2\pi|\Sigma|^{\frac{1}{2}}} \tag{10}\]  
where \(x\), \(\mu\), and \(\Sigma\) symbolize the Gaussian distribution's two-dimensional coordinates, the average vector, and the covariance matrix, in that order. In the case of smaller targets, the two-dimensional spread of pixel values roughly meets  
\[\mu=\left[\begin{array}{c}c_{x}\\ c_{y}\end{array}\right]\Sigma=\left[\begin{array}{cc}\frac{\mu^{2}}{4}&0\\ 0&\frac{h^{2}}{4}\end{array}\right] \tag{11}\]  
The Wasserstein distance between two two-dimensional Gaussian distributions \(\mu_{1}=N(m_{1},\ \Sigma_{1})\) and \(\mu_{2}=N(m_{2},\ \Sigma_{2})\) can be defined as follows:  
\[W_{2}^{2}(\mu_{1},\ \mu_{2})=\|m_{1}-m_{2}\|_{2}^{2}+T\!r\!\left(\Sigma_{1}+ \Sigma_{2}-2\Bigl{(}\Sigma_{2}^{1/2}\Sigma_{1}\Sigma_{2}^{1/2}\Bigr{)}^{1/2}\right) \tag{12}\]  
This formula can be approximated as follows:  
\[W_{2}^{2}(\mu_{1},\ \mu_{2})=\|m_{1}-m_{2}\|_{2}^{2}+\left\|\Sigma_{1}^{1/2}- \Sigma_{2}^{1/2}\right\|_{2}^{2} \tag{13}\]  
Then, the distance between the Gaussian distributions \(N_{a}\) and \(N_{b}\) corresponding to the bounding boxes \(A=(cx_{a},\ cy_{a},\ w_{a},\ w_{b})\) and \(B=(cx_{b},\ cy_{b},\ w_{b},\ w_{b})\) can be expressed as follows:  
Figure 5: The motivation for introducing the Normalized Wasserstein–Gaussian Distance: (**a**) sensitivity analysis of the IoU on tiny-scale object. \(A\) represents the actual boundary box, while \(B\) and \(C\) represent predicted boundary boxes with deviations of 1 pixel and 3 pixels, respectively. (**b**) three-dimensional intensity distribution of infrared small targets.  
\[W_{2}^{2}(Na,Nb)=\left(\left[cx_{a},cy_{a},\frac{w_{a}}{2},\frac{h_{a}}{2}\right]^ {T},\left[cx_{b},cy_{b},\frac{w_{b}}{2},\frac{h_{b}}{2}\right]^{T}\right)_{2}^{2} \tag{14}\]  
Utilizing the exponential version of the aforementioned formula, the derived value is limited to a span of 0-1, with C representing a dataset-related constant [40].  
\[NWD(N_{a}\,\ N_{b})=\exp\left(-\frac{\sqrt{W_{2}^{2}(N_{a}\,\ N_{b})}}{C}\right) \tag{15}\]