# A Normalization Strategy for Weakly Supervised 3D Hand Pose Estimation
## 3 Method
### Normalization Strategy  
The normalization strategy comprises two steps, namely relocating \(\mathbf{j}_{0}\) to the reference position and normalization of biomechanical features, as depicted in Figure 3. In a subsequent paper, each step will be discussed in detail. The current state of 3D pose annotation standardization is inadequate, as demonstrated in Figure 1c; the root joint \(\mathbf{j}_{0}\) can be marked either at the palmar center or the wrist, necessitating the initial normalization of \(\mathbf{j}_{0}\) position across all training sets. To accomplish this, a reference coordinate system, denoted as \(\mathbf{A}_{\text{root}}\), is established with \(\mathbf{j}_{0}\) as the origin. Specifically,  
\[\mathbf{A}_{\text{root}}^{x} =\text{norm}(\mathbf{j}_{13}-\mathbf{j}_{0})\] \[\mathbf{A}_{\text{root}}^{z} =\text{norm}((\mathbf{j}_{5}-\mathbf{j}_{0})\times\mathbf{A}_{ \text{root}}^{x}). \tag{9}\] \[\mathbf{A}_{\text{root}}^{y} =\mathbf{A}_{\text{root}}^{z}\times\mathbf{A}_{\text{root}}^{x}\]  
In the first step, depicted in Figure 3, the reference position of the root joint, denoted as \(\mathbf{j}_{0}^{r}\), was derived by averaging all target individual samples within the reference coordinate system \(\mathbf{A}_{\text{root}}\). Subsequently, all \(\mathbf{j}_{0}\) within the training samples were adjusted to align with the location of \(\mathbf{j}_{0}^{r}\). This procedure ensured that the training data adhered to a uniform annotation standard. Considering the challenge of neural networks acquiring accurate 2D-3D mapping knowledge from diverse training sets. An effort is made to preserve the dynamic biomechanical parameters while normalizing the static biomechanical features of training samples to the specific target individual. This process, outlined in Figure 3, involves mapping the 3D coordinate-encoded pose \(X\) to the HPBM-encoded pose \(\beta\). The parameters \(\left[\ell^{{}^{\prime}},\phi_{\text{ARB}}^{{}^{\prime}},\phi_{\text{CPS}}^{{} ^{\prime}},d^{{}^{\prime}}\right]\) are used to characterize the static parameters of the target individual. In \(\beta\), the dynamic parameters \(\theta^{{}^{\prime}}\) and \(\theta^{a}\) remain unchanged, while the static parameters are replaced with the corresponding features of the target individual. The resulting normalized pose is expressed as:  
\[\beta^{{}^{\prime}} =\text{norm}(\beta) \tag{10}\] \[=\left[\ell^{{}^{\prime}},\phi_{\text{ARB}}^{{}^{\prime}},\phi_{ \text{CPS}}^{{}^{\prime}},d^{{}^{\prime}},\theta^{{}^{\prime}},\theta^{a} \right].\]  
Next, the normalized pose \(\beta^{{}^{\prime}}\) is converted into a 3D coordinate-encoded pose \(X_{\text{norm}}\), ensuring consistency in biomechanical features across the training set. This procedure  
Figure 3: Flowchart of the normalization process. (**a**) Aligning the joint \(\mathbf{j}_{9}\) across various individual palm and superimposing the planes containing joints \(\mathbf{j}_{9}\), \(\mathbf{j}_{13}\), and \(\mathbf{j}_{0}\), the distribution of the \(\mathbf{j}_{0}\) is depicted by the red dots. (**b**) 3D Coordinate-Encoded Pose. (**c**) HPBM-Encoded Pose. Figures (**b**,**c**) depict the normalized process. can be regarded as the inverse operation of mapping \(X_{\text{norm}}\) to \(\beta^{{}^{\prime}}\), essentially solving the solution of the nonlinear function \(f:X_{\text{norm}}\mapsto\beta^{{}^{\prime}}\). In this study, the SLSQP algorithm is employed to solve it. It should be noted that BHPM only represents the biomechanical features and spatial structure of the hand but does not preserve the absolute orientation information. This is because 2D-3D pose pairs can be obtained by randomly rotating the 3D pose in the spatial domain and projecting it onto the 2D plane. Therefore, incorporation of absolute pose orientation information into the HPBM-encoded pose is deemed unnecessary for training.