## 3 Proposed Method
### Multi-Scale Nonlinear Fusion Module  
There exists a large number of small objects and many other objects with multi-scales in the UAV dataset, resulting in the poor detection performance of YOLOv8 [25]. At the same time, the C2f module is not capable of learning global salient features because the conventional convolutions only aggregate features in the neighborhood. Therefore, we introduce the Multi-scale Nonlinear Fusion Module (MNFM), as shown in Figure 6, which combines multi-scale fusion with nonlinear mapping, which helps to obtain a more salient feature representation of the target. The MNFM employs a cascaded and cross-layer connection approach to achieve the fusion of feature maps at different scales while utilizing the Nonlinear Fusion Module for nonlinear mapping to obtain more prominent target features. Specifically, the MNFM first divides the input feature map into two parts: lower-level detail features and higher-level semantic features. Then, it performs a concatenation operation between the lower-level feature map and the higher-level feature map, merging them into a more comprehensive feature representation. This approach retains detailed information from the lower-level while introducing higher-level semantic and contextual information. At the same time, the NFM Bottleneck enhances the saliency of target features through variable convolutions and nonlinear mapping operations.  
The specific implementation formula of the Multi-scale Nonlinear Fusion Module is as follows:  
\[y_{\text{MNFM}}=\text{Conv}(\text{Concat}\{\text{Split}(x),\text{NFM}(x),\text {NFM}(\text{NFM}(x)),\ldots\}) \tag{7}\]  
The NFM achieves the nonlinear mapping of features through multiple branches and skip connections. Specifically, the process involves passing the feature map through different mapping branch units, where each mapping branch unit consists of two cascaded deformable convolutions [22; 23]. And then, the results from different mapping branch units are integrated together. The specific implementation formula of the Nonlinear Fusion Module is as follows:  
\[y_{\text{NFM}}=x+\text{dconv}(\text{dconv}(x))+\text{dconv}(\text{dconv}(x ))+\text{dconv}(\text{dconv}(x))+\ldots \tag{8}\]Due to the following characteristics of deformable convolutions, it is highly suitable to utilize them as nonlinear mapping units. Deformable convolutions possess the capability of long-range modeling and can dynamically modify the shape of convolution kernels while interacting with local or global features through the adjustment of offset parameters. In addition, it also has the ability of adaptive spatial aggregation. When aggregating different features from convolutions, the offsets and weights are learnable and vary according to the input. For a given input \(x\in R^{\mathrm{C}\times H\times W}\) and a pixel point \(p_{0}\), the implementation process of deformable convolution is as follows:  
\[y(p_{0})=\sum_{k=1}^{K}w_{k}m_{k}x(p_{0}+p_{k}+\Delta p_{k}) \tag{9}\]  
where \(K\) represents the number of sampling points, \(w_{k}\) is the projection weight of the corresponding sampling points, \(m_{k}\) represents the modulation scalar of the corresponding sampling points, and \(\triangle p_{k}\) represents the offset of the corresponding sampling points.  
To a certain extent, the NFM bottleneck can compensate for the shortcomings of the backbone network in extracting features from small objects and objects with multiple scales. Its mapping branch units incorporate multiple sets of mechanisms, which have a more significant global modeling ability and are more suitable for retaining the diversity of features during feature fusion.  
In a word, the MNFM integrates nonlinear transformations and multi-scale feature fusion into the backbone to enhance the salient features of small targets, thereby improving the detection performance on small targets.