# High-Speed Dynamic Projection Mapping onto Human Arm with Realistic Skin Deformation +
## 4 Experiments
### Data Preparation  
We determined Equation (1) in advance using polynomial curve fitting of user-specific data. Six markers for data collection were placed along the forearm discretely, as shown in Figure 6. The 3D position of \(i\)-th marker \(M^{t}_{i}\) at time \(t\) was obtained by the motion tracking system. The data were processed as follows:  
\[ \begin{cases}o^{t}_{i}=|\mathbf{M}^{t}_{i}-\mathbf{I}^{t}_{\textit{wrist}}-\mathbf{M}^{t-1 }_{i}+\mathbf{I}^{t-1}_{\textit{wrist}}|+o^{t-1}_{i},\\ d^{t}_{i}=|\mathbf{M}^{t}_{i}-\mathbf{I}^{t}_{\textit{wrist}}|,\end{cases}  \tag{2}\]  
where \(o^{t}_{i}\) is the cumulative offset that is perpendicular to the axis of the forearm, of \(i\)-th marker at time \(t\), and \(d^{t}_{i}\) is the Euclidean distance between \(i\)-th marker and wrist at time \(t\)  
Figure 5: Default UV map of SMPL model for whole body [38] (**left**). UV map for the forearm after modification (**right**). Over 10,000 datasets of \((o_{i}^{t},d_{i}^{t})\) were collected for a user in the experiments, and the results of the following sixth-degree polynomial curve fitting were obtained:  
\[\mathbf{o}^{\prime}=\left[ \begin{array}{c}3E^{-12}\\ 2E^{-10}\\ -2E^{-8}\\ -1E^{-6}\\ 7E^{-5}\\ 0.0014\\ 0.0227\end{array} \right]^{T}\left[ \begin{array}{c}\gamma_{\textit{{\it wrist}}}^{ \mathbf{\chi}}^{\mathbf{\chi}}_{\textit{{\it wrist}}}^{\mathbf{\chi}}\\ \gamma_{\textit{{\it wrist}}}^{\mathbf{\chi}}_{\textit{{\it wrist}}}^{\mathbf{\chi}}\\ \gamma_{\textit{{\it wrist}}}^{\mathbf{\chi}}_{\textit{{\it wrist}}}^{\mathbf{\chi}}\\ \gamma_{\textit{{\it wrist}}}^{\mathbf{\chi}}_{\textit{{\it wrist}}}^{\mathbf{\chi}}\\ \gamma_{\textit{{\it wrist}}}^{\mathbf{\chi}}_{\textit{{\it wrist}}}^{\mathbf{\chi}}\\ 1\end{array} \right]\mathbf{d}+\mathbf{o}^{\bullet}, \tag{3}\]  
where the \(x\)-axis is parallel to the axis of the forearm in wrist coordinate; \(\mathbf{d}\in\mathbb{R}^{N}\) is a vector that contains the Euclidean distances in 3D world space from the arm-surface model's vertices to the wrist; \(\mathbf{o}^{\bullet}\in\mathbb{R}^{N}\) is a vector that contains the initial value before calculating cumulative offset of the arm-surface model's vertices; and \(\mathbf{o}^{\prime}\in\mathbb{R}^{N}\) is a vector that contains the cumulative offset of the arm-surface model's vertices according to corresponding \(\mathbf{\gamma}_{\textit{{\it wrist}}}^{x}\), \(\mathbf{d}\) and \(\mathbf{o}^{\bullet}\). Since \(\mathbf{o}^{\prime}\) and \(\mathbf{o}^{\bullet}\) are perpendicular to the axis of the forearm and \(\mathbf{d}\) is approximately parallel to the axis of the forearm, we accordingly obtained the conversion relationship as follows:  
\[\left\{ \begin{aligned} \mathbf{o}^{\prime}-\mathbf{o}^{\bullet}& =k(\mathbf{u}^{\prime}-\mathbf{u}^{\bullet}),\\ \mathbf{d}&=k(\mathbf{v}^{\bullet}-\mathbf{v}_{0}^{\bullet}), \end{aligned} \right. \tag{4}\]  
where \(k\) is the parameter that converts pixels to millimeters; and \(\mathbf{v}_{0}^{\bullet}\in\mathbb{R}^{N}\) is a vector that contains the \(v\)-axis value of the vertex \((u_{\textit{{\it wrist}}}^{\bullet},v_{\textit{{\it wrist}}}^{\bullet})\in( \mathbf{u}_{\textit{{\it wrist}}}^{\bullet},v_{\textit{{\it wrist}}}^{\bullet})\) for each vertex \((u^{\bullet},v^{\bullet})\in(\mathbf{u}^{\bullet},\mathbf{v}^{\bullet})\), where \((u_{\textit{{\it wrist}}}^{\bullet},v_{\textit{{\it wrist}}}^{\bullet})\) and \((u^{\bullet},v^{\bullet})\) have the minimum Euclidean distance in 2D texture coordinate space. Due to the modification of the UV map in Section 4.2, \(\mathbf{v}_{0}^{\bullet}\) is the zero vector and Equation (4) can be simplified as:  
\[\left\{ \begin{aligned} \mathbf{o}^{\prime}-\mathbf{o}^{\bullet}& =k(\mathbf{u}^{\prime}-\mathbf{u}^{\bullet}),\\ \mathbf{d}&=k\mathbf{v}^{\bullet}.\end{aligned} \right. \tag{5}\]  
Figure 6: Markers attached to the forearm for data collection. Hence, after combing Equation (3) and Equation (5), the regression model \(\mathbf{g}(\mathbf{v^{*}},\mathbf{\gamma}_{\textit{wrist}})\) was obtained:  
\[\mathbf{u^{\prime}}=\left[