## 2 Basic Theory
### Skeleton Simulation in Multi-Light-Source Scenarios
#### 2.1.1 3D Scenario Reproduction
Thus, the perspective transformation is introduced to recover the flat ground surface plane from the captured perspective view. The detailed perspective transformation parameters are given in parameter matrix \(P\). For linear transformation parameter matrix \(L\), the scale transformation parameters \(c_{x}\) and \(c_{y}\) and rotation angle \(\theta\) are included. \[L=\left[ \begin{array}{cc}c_{x}\cos\theta&c_{x}\sin\theta\\ -c_{y}sin\theta&c_{y}\cos\theta\end{array} \right] \tag{2}\]  
The translation transformation parameter matrix \(T\) is made up of translate values \(t_{x}\) and \(t_{y}\) in different axis directions. \[T=\left[ \begin{array}{cc}t_{x}&t_{y}\end{array} \right]^{T} \tag{3}\]  
The perspective transformation parameter matrix \(P\) is made up of perspective values \(p_{x}\) and \(p_{y}\) in different axis directions. \[P=\left[ \begin{array}{cc}p_{x}&p_{y}\end{array} \right]^{T} \tag{4}\]Based on the detailed transformation parameter matrices \(T\), \(L\) and \(P\), the projection transformation matrix \(A\) can be presented as:  
\[A=\left[ \begin{array}{cc}L&P\\ T^{T}&1\end{array} \right]=\left[ \begin{array}{ccc}c_{x}\cos\theta&c_{x}\sin \theta&p_{x}\\ -c_{y}sin\theta&c_{y}\cos\theta&p_{y}\\ t_{x}&t_{y}&1\end{array} \right]=\left[ \begin{array}{ccc}a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&1\end{array} \right] \tag{5}\]  
Based on the Equations (1) and (5), coordinates \(x^{\prime}\),\(y^{\prime}\) and \(w^{\prime}\) on the camera coordinate plane can be presented by coordinates \(u\), \(v\) on the ground surface plane and sub-parameters of matrix \(A\). \[x^{\prime} =a_{11}u+a_{21}v+a_{31} \tag{6a}\] \[y^{\prime} =a_{12}u+a_{22}v+a_{32}\] (6b) \[w^{\prime} =a_{13}u+a_{23}v+1 \tag{6c}\]  
Then, a linear transformation carried out to calculate pixel coordinates \(x\) and \(y\) in image coordinate plane. The transformation is controlled by the camera internal parameter \(w^{\prime}\). \[(x,y)^{T}=\left(\frac{x^{\prime}}{w^{\prime}},\frac{y^{\prime}}{w^{\prime}} \right)^{T} \tag{7}\]  
Eventually, the pixel coordinates \(x\) and \(y\) can be presented by ground surface plane coordinates \(u\), \(v\) and sub-parameters of projection transformation calibration matrix \(A\). \[x =\frac{a_{11}u+a_{21}v+a_{31}}{a_{13}u+a_{23}v+1} \tag{8a}\] \[y =\frac{a_{12}u+a_{22}v+a_{32}}{a_{13}u+a_{23}v+1} \tag{8b}\]  
Additionally, if the human shadow pixel coordinates \(x\) and \(y\) and projection calibration matrix \(A\) is acknowledged, the real-world coordinates \(u\) and \(v\) of the human shadow can be extracted based on solving the Equations (8a) and (8b). The procedure of solving real-world coordinates \(u\) and \(v\) is simplified in Equation (9). \[(u,v)^{T}=f((x,y)^{T},A) \tag{9}\]  
Block Matrix-Based Projection Transformation Parameter Calculation  
The traditional plane-to-plane projection transformation is designed for ideal scenarios with continuous flat ground surface. Nevertheless, the realistic scenarios contain uneven ground surfaces with discontinuous pavement levels. Thus, the single projection transformation calibration matrix \(A\) is not capable of ensuring precise projection transformation for all sub-blocks of the uneven ground surface. In order to deploy the projection transformation on realistic scenarios with high precision, a block matrix-based projection transformation is proposed in this part. Instead of deploying imprecise plane-to-plane global transformation, the proposed method launches a set of precise sub-transformations. Each single sub-transformation covers only one partially flat sub-block on the ground surface, ensuring the precise projection transformation between a surface sub-block and the corresponding image subset. For each sub-block, the unique projection transformation calibration matrix \(A_{sub}\) is non identical with the parameter matrices belonging to other sub-blocks. The parameter matrices \(A_{sub}\) of different sub-blocks are calculated separately based on Equations (8a) and (8b).