# Improved Dab-Deformable Model for Runway Foreign Object Debris Detection in Airport Optical Images
## 3 Methodology
### Ldsf
#### 3.2.2 Space Mapping Enhancer  
In the spatial mapping enhancer, high-scoring key locations are extracted from the score map. The relative positions of key locations in the deep feature map are computed to facilitate the injection of key semantic information of deep features into the shallow feature map. Additionally, a \(3\times 3\) convolutional layer generates channel-sensitive enhancement coefficients, which are channel-wise modulated according to the spatial locations of the heatmap. The modulated heatmap is then fused with the shallow feature maps to produce a shallow high-resolution feature map containing key semantic information from deep features.  
\[\mathcal{K}_{s}^{(b)}=\{(x_{k},y_{k})\}_{k=1}^{K}=\text{argtopk}_{(i,j)\in \Omega_{s}}\,S_{s}^{(b)}(i,j) \tag{7}\]  
The above formulas complete the coordinate extraction of the top-\(K\) targets with the highest scores on the score map, where \(K\) is a hyperparameter, \(\Omega_{s}\) represents the spatial domain of the feature map \(\{1,\dots,W_{s}\}\times\{1,\dots,H_{s}\}\), and argtopk indicates the coordinate indices of the top-\(K\) maximum values. The extracted coordinates are stored in \(\mathcal{K}_{s}^{b}\).  
\[\begin{cases}x^{\prime}=\frac{W_{3}}{W_{s}}\cdot x_{s}\\ y^{\prime}=\frac{H_{3}}{H_{s}}\cdot y_{s}\end{cases} \tag{8}\]  
\(W_{3}\) and \(H_{3}\) are the dimensions of the shallow feature map; \(W_{s}\) and \(H_{s}\) are the dimensions of the deep feature map.  
\[M(x,y)=\sum_{s=4}^{5}\sum_{k=1}^{K}\sum_{k=-r}^{r}\sum_{k^{\prime}_{y}=-r}^{r }\mathbb{I}\big{(}\lfloor x^{\prime}\rfloor+\delta_{x},\lfloor y^{\prime} \rfloor+\delta_{y}\big{)} \tag{9}\]  
To generate the heatmap \(M(x,y)\), rectangular region accumulation is performed for coordinates near the target position. \(\delta_{x}\) and \(\delta_{y}\) are the feature values within the coordinate range \(r\), where \(r\) is a hyperparameter. \(\mathbb{I}(\cdot)\) is the indicator function that outputs \(1\) when the condition is satisfied; \(\lfloor\cdot\rfloor\) denotes the coordinate rounding operation.  
\[w(x,y)=\frac{1}{1+e^{-\frac{M(x,y)}{\mathbb{I}}}} \tag{10}\]  
Figure 3: The architecture of hotspot sensing network.  
The heatmap undergoes normalization and activation function compression to achieve unit area standardization \(w(x,y)\), thereby eliminating the impact of scale differences, where  
\[\Gamma=(2r+1)^{2}\]  
\(\Gamma\) represent sthe normalization factor, \(r\) is a hyperparameter defining the coordinate range, and \((2r+1)^{2}\) calculates the total number of pixels in the square neighborhood.  
\[\hat{f}_{3}(x,y)=f_{3}(x,y)\cdot[1+\sigma(\mathcal{W}_{e}*f_{3}(x,y))\cdot w(x,y)] \tag{11}\]  
Finally, joint spatial channel enhancement is applied to modulate the shallow feature map \(f_{3}(x,y)\). Here, \(\mathcal{W}_{e}\in\mathbb{R}^{\mathbb{C}_{\mathrm{in}}\times\mathbb{C}_{ \mathrm{in}}\times 3\times 3}\) is the \(3\times 3\) enhancement convolution kernel, which generates enhancement weights based on the input features to dynamically adjust the heatmap for local adaptability and global consistency.