## 2 Basic Theory
### Skeleton Simulation in Multi-Light-Source Scenarios
#### 2.1.1 3D Scenario Reproduction
To solve the unique calibration matrix of each sub-block, four pairs of marked point coordinates on ground surface plane and their corresponding pixel coordinates on image coordinate plane are required. However, manipulating massive markers to calculate parameter matrices of all sub-blocks will bring a heavy workload. In order to simplify the setup, the optimized block matrix based parameter calculation procedure is designed to be marker coordinates multiplexable and parallel computing friendly. From the top-view angle, the ground surface is divided into a matrix consisting of multiple intensive square sub-blocks as shown in Figure 1b. Each sub-block is a unit square area \(Sq_{sub}\) defined by the four corner markers, occupying one meter square area on the ground surface as shown in Figure 1b. The coordinate set of four markers on ground surface is defined as \(M_{sub}^{S}=\left\{\left(u_{i},v_{i}\right),i=1,2,3,4\right\}\), their corresponding image coordinate set is \(M_{sub}^{Im}=\left\{\left(x_{i},y_{i}\right),i=1,2,3,4\right\}\). For each sub-block \(Sq_{sub}\), a set of auxiliaries is introduced to simplify the calculation of parameter matrices \(A_{sub}\) based on Equation (10). The scale auxiliary parameters set includes \(\Delta x_{1}\), \(\Delta x_{2}\), \(\Delta y_{1}\), and \(\Delta y_{2}\). \[\left\{ \begin{array}{l}\Delta x_{1}=x_{2}-x_{3}\\ \Delta x_{2}=x_{4}-x_{3}\\ \Delta y_{1}=y_{2}-y_{3}\\ \Delta y_{2}=y_{4}-y_{3}\\ \Delta x_{3}=x_{1}-x_{2}+x_{3}-x_{4}\\ \Delta y_{3}=y_{1}-y_{2}+y_{3}-y_{4}\end{array} \right. \tag{10}\]  
Additionally, the parallel auxiliary parameters \(\Delta x_{3}\) and \(\Delta y_{3}\) are introduced as Equation (10) as well. If both auxiliary parameters \(\Delta x_{3}\) and \(\Delta y_{3}\) approach zero, the field of camera view is regarded as parallel with the sub-block. The translation parameter \(T_{sub}\), perspective parameter \(P_{sub}\) and linear parameter \(L_{sub}\) in each calibration matrix \(A_{sub}\) can be solved as:  
\[T_{sub}=\left[ \begin{array}{cc}a_{31}&a_{32}\end{array} \right]^{T}=\left[ \begin{array}{cc}x_{1}&y_{1}\end{array} \right]^{T} \tag{11a}\] \[P_{sub}=\left[ \begin{array}{cc}a_{13}&a_{23}\end{array} \right]^{T}=[\frac{ \Delta x_{3}\Delta y_{2}-\Delta x_{2}\Delta y_{1}}{\Delta x_{1}\Delta y_{2}- \Delta x_{2}\Delta y_{1}}\frac{\Delta x_{1}\Delta y_{3}-\Delta x_{1}\Delta y_{ 1}}{\Delta x_{1}\Delta y_{2}-\Delta x_{2}\Delta y_{1}}]^{T}\] (11b) \[L_{sub}=\left[ \begin{array}{cc}a_{11}&a_{12}\\ a_{21}&a_{22}\end{array} \right]=\left[ \begin{array}{cc}(x_{2}-x_{1}+a_{12}x _{2})&(y_{2}-y_{1}+a_{13}y_{2})\\ (x_{4}-x_{1}+a_{12}x_{3})&(y_{4}-y_{1}+a_{23}y_{4})\end{array} \right] \tag{11c}\]  
The extraction procedure of the block matrix based projection transformation calibration matrix can be simplified as:  
\[A_{sub}=P(M_{sub}^{S},M_{sub}^{Im})=\left[ \begin{array}{cc}L_{sub}&P_{sub}^ {T}\\ T_{sub}&1\end{array} \right] \tag{12}\]  
Block-Matrix Based Projection Transformation Deployment  
Based on Equation (9) and the calculated parameters in matrix \(A_{sub}\), real-world coordinates \(\left(u,v\right)\) of each point in one sub-block area can be calculated from the corresponding pixel coordinates \(\left(x,y\right)\). The presentation of extraction procedure can be simplified as:  
\[\left(u,v\right)^{T}=f(\left(x,y\right)^{T},A_{sub}) \tag{13}\]  
Noticeably, different from the original global calibration matrix \(A\), each sub-block calibration matrix \(A_{sub}\) is only deployed on the restricted regional transformation between the sub-block area on the ground and the corresponding pixel range in the image. Once all parameter matrices \(A_{sub}\) for different sub-blocks are extracted through the block matrix-based parameter calculation procedure, coordinates \(\left(x,y\right)\) of pixels belonging to different sub-blocks can be transformed to corresponding real-world coordinates \((u,v)\) inside the sub-block \(Sq_{sub}\).