## 4 Experimental Results and Analyses
### Ablation Experiment
#### 4.3.1 Ablation Experiments Under Different Multi-Level Supervision Strategies  
The multi-level supervision strategy is one of the core innovations of our method. In MS-WFFNet, considering that the MSE loss is based on the assumption of pixel independence, it is easy to cause blurring effects and a loss of local structural information. Therefore, we use it to supervise high-resolution density maps to ensure pixel-level accuracy. At the same time, due to the advantages of SSIM loss in preserving details and structural information, we apply SSIM loss to the four output regression results of the MSFA module for distributed supervision to strengthen the model's learning of structural information locally.  
To verify the effectiveness of this setting, we compared it with four other different supervision configurations. The results in Table 7 show that the performance of the ShanghaiTech PartA dataset is reduced when the multi-level MSE loss is introduced alone compared to the traditional \"MSE + SSIM\" combination (MAE increased from 64.8 to 69.0, and RMSE increased from 104.3 to 105.9). At the same time, a similar trend is also shown on the self-built dataset DTH. Compared with the \"MSE + SSIM\" combination (MAE is 4.1 and RMSE is 5.2), after the multi-level MSE loss is introduced alone, MAE increases to 4.7 and RMSE increases to 6.1. This shows that relying solely on MSE loss for distributed supervision is limited. In the \"Multi-level MSE + SSIM\" and \"Multi-level MSE + Multi-level SSIM\" settings, the performance is significantly improved by further introducing SSIM loss, which proves the effectiveness of SSIM loss in distributed supervision.  
Finally, our proposed \"MSE + Multi-level SSIM\" configuration achieves significantly better performance in both MAE and RMSE compared to other combinations, demonstrating its effectiveness in capturing both local details and global features. Furthermore, the experimental results of the \"Multi-level MSE + Multi-level SSIM\" setting reveal that the use of MSE loss for distributed supervision may interfere with feature learning across different paths, leading to reduced accuracy. It is important to note that we did not conduct experimental comparisons using only SSIM loss for distributed supervision. This decision stems from the fact that SSIM primarily measures image similarity and cannot accurately capture the discrepancies between density maps. Relying solely on SSIM loss might cause the model to overly focus on local similarities, potentially inducing negative learning effects and degrading overall performance.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Supervision Setting**} & \multicolumn{2}{c}{**ShanghaiTech PartA**} & \multicolumn{2}{c}{**DTH**} \\ \cline{2-5}  & **MAE** & **RMSE** & **MAE** & **RMSE** \\ \hline MSE + SSIM & 64.8 & 104.3 & 4.1 & 5.2 \\ Multi-level MSE & 69.0 & 105.9 & 4.7 & 6.1 \\ Multi-level MSE + SSIM & 65.8 & 105.1 & 4.2 & 5.4 \\ MSE + Multi-level SSIM (Ours) & **57.3** & **85.2** & **2.9** & **4.1** \\ Multi-level MSE + Multi-level SSIM & 63.6 & 101.1 & 3.9 & 5.0 \\ \hline \hline \end{tabular} Note: the best results are highlighted in **bold**.  
\end{table}
Table 7: Experimental results of applying different multi-level supervision strategies on the ShanghaiTech PartA dataset and the DTH dataset.