# Wind Speed Measurement via Visual Recognition of Wind-Induced Waving Light Stick Target
## 2 Theoretical Modeling
### Imaging Formulation
#### 2.1.2 Side View  
Similar to the case of the bottom view, the theoretical model in side view is given in centered and deviated imaging. The projection length on the imaging plane is denoted as \(X_{\rm P}\), as in Figure 5. Other parameters are defined the same as in bottom view.  
The physics of the motion of the stick are similar to the preceding bottom view case where it has been adequately described.  
Figure 4: Imaging model in coupled view.  
By triangle similarity, we have:  
\[\frac{X_{\text{P}}}{f}=\frac{X}{D_{\text{O}}+L\text{-sin}\alpha+f}=\frac{L\text{ -cos}\alpha}{D_{\text{O}}+L\text{-sin}\alpha+f} \tag{13}\]  
The yaw angle \(\alpha\) can then be expressed as in Equation (14) below.  
\[\alpha=\tan^{-1}\left(\frac{-(D_{\text{O}}+f)\cdot\text{X}_{\text{P}}{}^{2}+f \cdot\sqrt{\left[L^{2}-(D_{\text{O}}+f)^{2}\right]\cdot\text{X}_{\text{P}}{}^{2 }+L^{2}\cdot f^{2}}}{f\cdot(D_{\text{O}}+f)\cdot\text{X}_{\text{P}}{}+X_{\text {P}}{}^{2}\cdot\sqrt{\left[L^{2}-(D_{\text{O}}+f)^{2}\right]\cdot\text{X}_{ \text{P}}{}^{2}+L^{2}\cdot f^{2}}}\right) \tag{14}\]  
The critical yaw angles \(\alpha_{\tau}\) and \(\alpha_{1}\) remain as previously defined, and are given in Equation (15) as:  
\[\alpha_{1}=\frac{\pi}{2} \tag{15}\]  
In the case where there is a deviation \(S\) between the camera optical center and the pure vertical state of the stick (Figure 6), the imaging model is extended to accommodate the range of view.  
Again, by triangle similarity, we have:  
\[\begin{array}{c}\frac{\left|\mathcal{O}^{\prime\prime}\,\text{P}^{\prime} \right|}{f}=\frac{(S+L\text{cos}\,\alpha)}{D_{\text{O}}+f+L\text{sin}\,\alpha} \\ \frac{\left|\mathcal{O}^{\prime\prime}\,\mathcal{O}^{\prime}\right|}{f}=\frac{ S}{D_{\text{O}}+f}\end{array} \tag{16}\]  
The imaging length variation \(X_{\text{P}}\) can then be formulated in Equation (17) as below.  
\[\frac{X_{P}}{f}=\frac{(S+L\text{cos}\,\alpha)}{D_{\text{O}}+f+L\text{sin}\, \alpha}-\frac{S}{D_{\text{O}}+f} \tag{17}\]  
Figure 5: Schematic of the side view model in centered imaging. (**a**) Schematic model; (**b**) geometry model.  
Figure 6: Schematic of the side view model in deviated imaging. (**a**) Schematic model; (**b**) geometry model.  
By trigonometric manipulations, the tangent of the yaw angle in Equation (18) is written as:  
\[\alpha=\tan^{-1}\left(\frac{-(D_{\text{O}}+f)^{3}\cdot{X_{p}}^{2}-S\cdot f\cdot(D _{\text{O}}+f)^{2}\cdot{X_{p}}+(D_{\text{O}}+f)\cdot f\cdot\sqrt{g(X_{p})}}{f \cdot(D_{\text{O}}+f)^{3}\cdot{X_{p}}+((D_{\text{O}}+f)\cdot{X_{p}}+S\cdot f) \cdot\sqrt{g(X_{p})}}\right) \tag{18}\]  
where the function g(\(X_{\text{P}}\)) under the square root is defined as:  
\[g(X_{\text{P}})=\left[L^{2}\cdot(D_{\text{O}}+f)^{2}-(D_{\text{O}}+f)^{4} \right]\cdot{X_{p}^{2}}+2f\cdot S\cdot L^{2}\cdot(D_{\text{O}}+f)\cdot{X_{p}}+ \left[S^{2}+(D_{\text{O}}+f)^{2}\right]\cdot f^{2}\cdot L^{2} \tag{19}\]  
To solve for the \(\alpha_{\text{r}}\) and \(\alpha_{\text{l}}\) we consider \(\Delta\text{OBO}_{\text{C}}\) and \(\Delta\text{OLO}_{\text{C}}\) where:  
\[\angle AOB=\tan^{-1}\left(\frac{D_{\text{O}}+f}{S}\right) \tag{20}\]  
\[\angle\text{BOO}_{\text{C}}=\tan^{-1}\left(\frac{\sqrt{(D_{\text{O}}+f)^{2}+S^ {2}-L^{2}}}{L}\right) \tag{21}\]  
The yaw angle range limit \(\alpha_{\text{r}}\) and \(\alpha_{\text{l}}\) can then be given according to Equations (22) and (23), respectively, below:  
\[\alpha_{\text{l}}=\frac{\pi}{2} \tag{22}\]  
\[\alpha_{\text{r}}=-\tan^{-1}\left(\frac{S\cdot\sqrt{(D_{\text{O}}+f)^{2}+S^{2 }-L^{2}}+L\cdot(D_{\text{O}}+f)}{S\cdot L-(D_{\text{O}}+f)\cdot\sqrt{(D_{\text {O}}+f)^{2}+S^{2}-L^{2}}}\right) \tag{23}\]