## 3 Method
### Gradient Density Loss Function
As a consequence, these highly challenging samples can potentially lead to a reduction in model accuracy. This is due to the fact that these challenging samples, acting as outliers, often possess gradient directions significantly distinct from those of other simple and moderately difficult samples. Forcing a nearly converged model to learn from these exceedingly challenging exceptional samples can result in inaccurate classification for a considerable number of other samples. After applying the gradient density loss function to weight the cross-entropy loss function, the relationship between the gradient magnitudes of samples and their corresponding quantities is illustrated in Figure 4.  
In Figure 4, the horizontal axis represents the initial gradient magnitudes, while the vertical axis illustrates the gradient magnitudes after employing distinct loss functions. The observation reveals that within the region associated with simple and moderately difficult samples, situated on the left side of Figure 3, the tendencies of the gradient density loss and focal loss align. This correspondence implies that these two loss functions demonstrate coherent behavior in discriminating between simple and moderately difficult samples. Conversely, for challenging samples situated on the right side of the graph, the gradient density loss effectively diminishes the gradient magnitudes. This adjustment serves to stabilize the model when confronted with difficult and exceptional samples, thereby amplifying training efficiency. After integrating the enhanced loss function, the comprehensive loss function for the image object detection method is formulated as  
\[Loss=\lambda_{1}Loss_{cls}+\lambda_{2}Loss_{L1}+\lambda_{3}Loss_{giou} \tag{14}\]  
where \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) denote the coefficients allocated to the classification loss, bounding box regression loss, and intersection over union (IoU) loss, correspondingly. The FL is a well-known loss function introduced to address the class imbalance problem in object detection. It assigns higher weights to hard-to-classify examples, reducing the contribution of easily classifiable examples to the loss. While FL effectively handles class imbalance, it does not explicitly consider the density of object instances within an image, potentially leading to suboptimal performance in scenarios where objects are densely packed. Figure 4: Gradient magnitudes with different weighted loss functions [20], where gradient density loss is represented by the red line and focal loss is represented by the blue line. Here, FL represents the focal loss, CE represents the cross-entropy, and GHM-C represents the gradient harmonizing mechanism classification. Figure 3: Distribution of sample gradient magnitudes and corresponding quantities [20]. Class-agnostic focal loss (CF) extends FL by making it class-agnostic, thus focusing on the overall detection quality without distinguishing between different object classes. This modification enhances the detection of objects irrespective of their classes, which is particularly useful when dealing with diverse or unknown objects. However, CF may not effectively address the issue of object density when objects of varying classes are densely distributed. GHM-C introduces gradient harmonization to address the imbalance between easy and hard examples during training. It effectively balances the learning process between different examples based on the gradient distribution, enhancing the network's ability to focus on challenging examples. However, similar to FL, GHM-C does not explicitly consider object density. In contrast, our proposed gradient density loss is specifically designed to address the challenge of object detection in scenarios with varying object densities. It combines the benefits of FL and GHM-C by incorporating density-based weighting into the loss calculation. Gradient density loss assigns higher weights to challenging examples, both in terms of classification difficulty and object density.