## 2 Basic Theory
### Skeleton Simulation in Multi-Light-Source Scenarios
#### 2.1.3 3D Joint Position Estimation and Skeleton Synthesis  
In a multiple light source scenario, more than one human shadow is projected on the ground surface at the same time. In order to identify shadow areas generated by different light sources, 2D human shadow contour \(Sh\) and joint position region \(Sp^{joint}\) are footnoted with corresponding light source identifier \(i\) as shown in Equation (21). Additionally, the point coordinates \((u,v)\) inside the each region \(Sp^{joint}_{i}\) are footnoted as \((u^{joint}_{i},v^{joint}_{i})\). \[Sp^{joint}_{i}\subset Sh_{i} \tag{21}\]  
In order to estimate the 3D joint position \(Mp^{joint}\) of each major joint, the light beams \(L^{joint}_{i}\) from different light sources \(S_{i}\) blocked by \(Mp^{joint}\) are reconstructed first. Then, the 3D position of \(Mp^{joint}\) is calculated based on allocating the shared voxel area between multiple reconstructed light beams \(L^{joint}_{i}\). Finally, the human skeleton is synthesized based on the calculated 3D joint position set \(\{Mp^{joint}\}\). For the first step, each light beam \(L^{joint}_{i}\) is generated as a 3D cone with its vertex on the light source position \(S_{i}=(u_{i},v_{i},h_{i})\). The underside of each cone is the joint area \(Sp^{joint}_{i}\) in the shadow. While any light beam \(L^{joint}_{i}\) from light source \(S_{i}\) is blocked by the certain joint part \(Mp^{joint}\) of human body \(M\), the joint shadow area \(Sp^{joint}_{i}=\{(u^{joint}_{i},v^{joint}_{i})\}\) is produced on the ground. Thus, the direction of blocked light beam \(L^{joint}_{i}\) leads to shadow area \(Sp^{joint}_{i}\subset Sh_{i}\), going through human body part \(Mp^{joint}\). If \(w\in[0,h_{i}]\) is introduced as the height component in the cone expression of \(L^{joint}_{i}\), the 3D space caused by \(L^{joint}_{i}\) can be presented as Equation (22). \[L^{joint}_{i}=(\frac{(h_{i}-w)^{joint}_{i}}{h_{i}},\frac{(h_{i}-w)^{joint}_{ i}}{h_{i}},w) \tag{22}\]  
The 3D light beam shape extraction procedure presented by Section 2.1.3 is simplified in the stage Equation (23). For the simplicity of the subsequent presentations, function \(Occ\) is introduced to cover the 3D light beam shape extraction procedure for the occupied 3D cone shape \(L^{joint}_{i}\) based on the corresponding joint position \(Sp^{joint}_{i}\) and the light source position \(S_{i}\). \[L^{joint}_{i}=Occ(Sp^{joint}_{i},S_{i}) \tag{23}\]  
Then, for multiple light beams \(L^{joint}_{i}\) generated by different light sources \(S_{i}\), \(Mp^{joint}\) is the shared subset for all reconstructed \(L^{joint}_{i}\). Thus, the 3D position \(Mp^{joint}\) can be generated based on the intersection of all reconstructed \(L^{joint}_{i}\) as shown in Figure 2(a). \(sum_{l}\) is the sum of light sources in the scenario. \[Mp^{joint}=\underset{i=1}{\overset{sum_{S}}{\cap}}L^{joint}_{i} \tag{24}\]  
Figure 2(a) demonstrates the recovery of neck joint area \(Mp^{neck}\) based on two related shadow areas \(Sp^{neck}_{a}\) and \(Sp^{neck}_{b}\) generated by light sources \(S_{a}\) and \(S_{b}\). Finally, 3D human skeleton \(Sk\) with multiple joint positions is synthesized by calculating \(Mp^{joint}\) joint by joint. Figure 2(b) presents the skeleton synthesis procedure of a human being based on human shadow information under a multi light source scenario. The illustrated human skeleton synthesis procedure is presented in Algorithm 2 and simplified in Equation (25). \[Sk=Syn(\{Sh_{i}\},\{S_{i}\}) \tag{25}\]  
However, there are two restrictions for the deployment of the basic theory:  
* **Condition (1)** Two or more light sources are required in the scene. * **Condition (2)** Relative angular positions between human body and different light sources should be different. ```
Input: \(\{Sh_{i}\}\): 2D human shadow contour on the ground surface. \(\{S_{i}\}\): 3D positions of multiple light sources \(\{(u_{i},v_{i},h_{i})\}\). Output: \(Sk\): 3D human skeleton synthesis based on seven major joint positions.