# A Normalization Strategy for Weakly Supervised 3D Hand Pose Estimation
## 4 Experiments
### Evaluation of Normalization Strategy
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**2D Pose**} & \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Datasets**} & \multicolumn{2}{c}{**w/o norm**} & \multicolumn{2}{c}{**w/ norm**} \\ \cline{3-6}  & & & **PJ**\(\downarrow\) & **AUC**\(\uparrow\) & **PJ**\(\downarrow\) & **AUC**\(\uparrow\) \\ \hline \multirow{8}{*}{GT} & \multirow{8}{*}{VPose3D} & STB & 12.37 & 0.960 & 6.78 & 0.995 \\  & & RHD-A & 9.91 & 0.984 & 5.21 & 0.996 \\  & & RHD-B & 9.03 & 0.988 & 5.24 & 0.996 \\  & & RHD-C & 7.87 & 0.992 & 4.70 & 0.997 \\  & & RHD-D & 7.62 & 0.993 & 4.33 & 0.998 \\ \cline{2-6}  & & STB & 13.52 & 0.958 & 6.77 & 0.995 \\  & & RHD-A & 11.18 & 0.970 & 4.48 & 0.995 \\  & & RT-B & 10.35 & 0.977 & 4.76 & 0.994 \\  & & RHD-C & 7.89 & 0.988 & 4.07 & 0.996 \\  & & RHD-D & 7.47 & 0.990 & 3.71 & 0.998 \\ \hline \multirow{8}{*}{MediaPipe} & \multirow{8}{*}{VPose3D} & STB & 11.39 & 0.964 & 10.10 & 0.989 \\  & & RHD-A & 13.53 & 0.945 & 11.35 & 0.949 \\ \cline{1-1}  & & RHD-B & 13.07 & 0.947 & 11.43 & 0.946 \\ \cline{1-1}  & & RHD-C & 10.83 & 0.971 & 9.18 & 0.971 \\ \cline{1-1}  & & RHD-D & 10.92 & 0.968 & 9.45 & 0.966 \\ \cline{1-1} \cline{2-6}  & & STB & 12.42 & 0.963 & 12.33 & 0.951 \\ \cline{1-1}  & & RHD-A & 14.70 & 0.929 & 12.23 & 0.931 \\ \cline{1-1}  & & RT-GCN & RHD-B & 14.29 & 0.930 & 12.11 & 0.930 \\ \cline{1-1}  & & RHD-C & 11.58 & 0.958 & 9.50 & 0.962 \\ \cline{1-1}  & & RHD-D & 11.54 & 0.957 & 9.66 & 0.959 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The effect of normalization strategy. At the inference process, both ground truth (GT) 2D keypoints and those generated by the MediaPipe framework were fed to the lifting model. “\(w/\) norm” denotes the training dataset with normalization, while “w/o norm” signifies the dataset without normalization. “PJ” denotes PA-MPJPE, the “\(\
icefrac{{1}}{{2}}\)”/\(\
icefrac{{1}}{{2}}\)” denote the lower/higher, the better. The AUC has a threshold ranging from 20 to 50 mm. The comparison in Figure 4 also reveals that static biomechanical features within the same individual exhibit minimal variation, with fluctuations largely attributed to the annotation process. Notably, the disparities between non-normalized training data and STB datasets exhibit a greater margin compared to those of RHD datasets. This observation substantiates why models trained on non-normalized datasets exhibit better performance on the RHD dataset than on the STB dataset. Table 2 compares the errors in static biomechanical features between models trained with and without normalization on different datasets. Across all datasets (STB, RHD-A, RHD-B, RHD-C, RHD-D), the model trained with normalization (w/ norm) consistently demonstrates lower errors in biomechanical features compared to the model trained without normalization (w/o norm). For instance, in the STB dataset, the error in \(\ell\) is reduced from 71.05 to 8.43, showcasing a substantial improvement. Similar trends are observed in other biomechanical features, such as \(\phi_{\text{ARB}}\) and \(\phi_{\text{CPS}}\). These results emphasize the effectiveness of the normalization method in enhancing the accuracy. The consistency in the palmar structure of the training data, constrained by \([\ell,\phi_{\text{ARB}},\phi_{\text{CPS}},d]\), leads to significantly improved accuracy in estimating the palmar joints \([\mathbf{j}_{0},\mathbf{j}_{1},\mathbf{j}_{5},\mathbf{j}_{9},\mathbf{j}_{13}, \mathbf{j}_{17}]\), as evident in the heatmap in Figure 5. This improvement is particularly pronounced for \(\mathbf{j}_{0}\). Similarly, when trained on a dataset constrained by \(\left[\ell,\theta^{\ell},\theta^{\ell}\right]\), the prediction accuracy of finger joints is also improved. For instance, the position of \(\mathbf{j}_{5}\) is constrained by the predefined palmar structure, and the position distribution of \([\mathbf{j}_{6},\mathbf{j}_{7},\mathbf{j}_{8}]\) is closer to that of the target individual due to the constraints \(|b_{5}|,|b_{6}|,|b_{7}|\).