# A Normalization Strategy for Weakly Supervised 3D Hand Pose Estimation
## 3 Method
### Hand Pose Biomechanical Model  
To provide a clearer explanation, the following predefined notation and rules will be set:  
1. The kinematic chain starting from the root joint \(\mathbf{j}_{0}\) and extending to the fingertips. 2. The parent joint of a given joint \(\mathbf{j}_{i}\) is denoted as \(\mathbf{\dot{j}}_{p(i)}\). 3. A bone \(\mathbf{b}_{i}\) is defined as the vector pointing from the parent joint to its child joint, computed by \(\mathbf{b}_{i}=\mathbf{j}_{i+1}-\mathbf{j}_{p(i+1)}\). 4. The root bones vectors are represented as \(\mathbf{B_{root}}=\left\{\mathbf{b}_{i}\right\}_{i\in\{0,4,8,12,16\}}\), as indicated by the solid line in Figure 2a. 5. The angular distance between two vectors, \(v_{1}\) and \(v_{2}\), is denoted by \(\alpha(v_{1},v_{2})\) and is calculated using the arccosine function as \(\alpha(v_{1},v_{2})=\arccos\left(\frac{v_{1}^{T}v_{2}}{\|v_{1}\|_{2}\|v_{2}\|_ {2}}\right)\). 6. The normalized vector is defined as \(\mathrm{norm}(x)=\frac{x}{\|x\|_{2}}\). 7. The operator \(\mathbf{P_{xy}}(\mathbf{v})\) is used to project a vector \(\mathbf{v}\) orthogonally onto the \(\mathbf{x}-\mathbf{y}\) plane where \(\mathbf{x}\), \(\mathbf{y}\) are vectors. The construction of the HPBM was divided into two distinct components: palm modeling and finger modeling. The palm section encompasses joints \(\{\mathbf{j}_{0},\mathbf{j}_{1},\mathbf{j}_{5},\mathbf{j}_{9},\mathbf{j}_{13}, \mathbf{j}_{17}\}\), while the finger section comprises the remaining part of the hand. The approach defines the palm structure using biomechanical features like root bone length and angles, with \(\mathbf{j}_{0}\) as the reference position. Then the palm is utilized as the foundation to construct coordinate systems to determine finger flexion. The palmar structure of a particular individual was observed to remain relatively static in the spatio-temporal domain, representing a consistent biomechanical feature. To define the palmar structure, three parameters were used: the length of the root bones \(\ell_{root}\), the Angular Distances between the Root Bones (ARB) \(\phi_{\mathrm{ARB}}\), and the Curvatures of the Palmar Surface (CPS) \(\phi_{\mathrm{CPS}}\). \(\ell_{root}\) can be determined by calculating the Euclidean distance between the palmar joints, specifically,  
\(\ell_{root}=\left[\|\mathbf{b}_{0}\|_{2},\|\mathbf{b}_{4}\|_{2},\|\mathbf{b} _{8}\|_{2},\|\mathbf{b}_{12}\|_{2},\|\mathbf{b}_{16}\|_{2}\right]\). The ARB features are illustrated in Figure 2b, and are computed in the following equation:  
\[\phi_{\mathrm{ARB}i}=\alpha\Big{(}\mathbf{b}_{i+4},\mathbf{b}_{(i+1)\cdot 4} \Big{)},\quad\text{for }i\in\{0,1,2,3\}. \tag{1}\]  
Figure 2: Visualization of the biomechanical architecture of the hand. (**a**) Hand structure. (**b**) The angular distances between root bones. (**c**) The curvatures of the palmar surface. (**d**) The local coordinate system. The CPS features can represent the curvature characteristics of neighboring bone surfaces, as illustrated by the two yellow triangles highlighted by dashed lines in Figure 2c. Based on the physical structure of the hand, \(\phi_{\text{CPS}}\) can be obtained:  
\[\phi_{\text{CPS}i}=\arccos(e_{i}\cdot e_{i+1}),\quad\text{for}\;i\in\{0,1,2\}. \tag{2}\]  
where \(e_{i}\) is normal vector of the surface:  
\[e_{i}=\text{norm}\Big{(}\mathbf{b}_{i\cdot 4},\mathbf{b}_{(i+1)\cdot 4}\Big{)}, \quad\text{for}\;i\in\{0,1,2,3\}. \tag{3}\]  
During the transformation of the HPBM-encoded pose to a 3D-coordinate representation, the arccos function generates two-directional surfaces for each \(\phi_{\text{CPS}i}\), potentially resulting in multiple structural solutions for the palm. To address this, the relative surface direction \(d_{i}\) is introduced as a constraint to ensure that the unique pose is defined:  
\[d_{i}=\left\{ \begin{array}{ll}1,&\text{if}\;e_{i}\cdot\Big{(}\mathbf{b}_{(i+ 2)\cdot 4}-\mathbf{b}_{(i+1)\cdot 4}\Big{)}>0\\ -1,&\text{if}\;e_{i}\cdot\Big{(}\mathbf{b}_{(i+2)\cdot 4}-\mathbf{b}_{(i+1) \cdot 4}\Big{)}<0\end{array}