# A Normalization Strategy for Weakly Supervised 3D Hand Pose Estimation
## 4 Experiments
### Evaluation of Weakly Supervised Approach  
The evaluation results of the model trained on \(\mathcal{D}\) are presented in Table 3. The dataset \(\mathcal{D}\) is constructed solely based on prior knowledge, without any manual labeling. Notably, the model attained satisfactory performance despite the absence of manually annotated data. When the model was fed with GT and MediaPipe 2D keypoints, it both achieved higher accuracy compared to the model trained on non-normalized datasets. Here, the \(\mathcal{D}_{n}\) is used to denote the normalized HO-3D and FreiHAND datasets. Furthermore, weakly supervised training was conducted on the joint dataset \(\mathcal{D}\)+\(\mathcal{D}_{n}\), resulting in even more accurate estimation results, as shown in Table 3; the errors reduced from 7.19 to 4.57 and from 11.29 to 10.03 for GT and MediaPipe 2D keypoints, respectively.  
It should be noted that the generated weakly supervised data may not strictly adhere to physical plausibility. Despite the dynamic parameters \(\{\theta^{f},\theta^{a}\}\) being confined within the range \([\Theta_{\text{min}},\Theta_{\text{max}}]\), there exists interdependence among these parameters. For instance, when the index finger is angled to a certain degree, it consequently affects the permissible range of angles for the other fingers. Restricting \(\{\theta^{f},\theta^{a}\}\) to \([\Theta_{\text{min}},\Theta_{\text{max}}]\) does not necessarily guarantee physical plausibility. Currently, the method for quantitatively assessing the plausibility of poses has not yet been developed, and it will be addressed in future work.  
From Tables 1 and 3, it can be observed that despite the weakly supervised data not being strictly physically plausible, their incorporation with \(\mathcal{D}_{n}\) leads to further improvements in model performance compared to a model solely trained on \(\mathcal{D}_{n}\). For instance, with GT 2D keypoints as input, the **PJ** errors for individual RHD-A, RHD-B, RHD-C, and RHD-D exhibit reductions of 0.48, 0.31, 0.28, and 0.08, respectively. This improvement can be attributed to the broader range and diversity of poses included in the training data.  
To provide a more comprehensive understanding of the efficacy of the proposed method, a thorough comparison is conducted with numerous other state-of-the-art techniques. This comparison is presented in Table 3 and Figure 7. Compared to advanced methods [13; 31; 35; 44; 45; 46; 47], in terms of **PJ** and AUC metrics, the proposed method showed fewer errors. This achievement is accomplished without the need to incorporate training data from the RHD dataset. Solely leveraging the static biomechanical features of individuals enables the approach to achieve higher estimation accuracy.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{6}{c}{**Datasets**} \\ \cline{2-7}  & **RHD-A** & **RHD-B** & **RHD-C** & **RHD-D** & **RHD** \\ \hline \multirow{5}{*}{} & LDR [11] & - & - & - & - & 11.63 \\  & SS-GCN [17] & - & - & - & - & 13.29 \\  & RC2 CHP [44] & - & - & - & - & 13.14 \\ \hline \multirow{2}{*}{GT} & \(\mathcal{D}\) & 5.97 & 6.52 & 7.67 & 8.51 & 7.19 \\  & \(\mathcal{D}+\mathcal{D}_{n}\) & 4.73 & 4.93 & 4.42 & 4.25 & **4.57** \\ \cline{1-1} \cline{2-7}  & \(\mathcal{D}\) & 11.19 & 11.70 & 10.69 & 11.63 & 11.29 \\ \cline{1-1}  & \(\mathcal{D}+\mathcal{D}_{n}\) & 10.92 & 11.14 & 9.01 & 9.22 & **10.03** \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of the weakly supervised method with advanced techniques using the PA-MPJPE metric. \(\mathcal{D}_{n}\) denotes the normalized training dataset. “GT” and “MediaPipe” represent 2D keypoint sources, and VPose3D is employed as lifting model in this evaluation.