# Wind Speed Measurement via Visual Recognition of Wind-Induced Waving Light Stick Target
## 2 Theoretical Modeling
### Imaging Formulation
#### 2.1.1 Bottom View  
The theoretical model at the bottom view is given in centered and deviated imaging, where in centered imaging, the initial stick line passes through the optical center, whereas in deviated imaging, there is a distance \(S\) between the stick line and optical center. In centered imaging at the bottom view, the fixed end of the stick is denoted as O, and the moving end rotates in a circular trajectory. The stick yaw angle is defined with respect to its initial vertical position as \(\alpha\). The imaging length on the projection plane is denoted as \(X_{\text{P}}\). Parameters including the stick length \(L\), camera focal length \(f\), and object distance \(D_{\text{O}}\) of the stick fixed end are all measured and calibrated before detection as in Figure 2.  
When the stick is propelled by horizontal wind and rotates in different yaw angles from the beginning, the imaging length \(X_{\text{P}}\) on the imaging plane experiences a rise until the link between the optical center and stick moving end becomes normal to the stick itself. The corresponding yaw angle is described as \(\alpha_{1}\). After the perpendicular point L, the imaging length \(X_{\text{P}}\) drops as the yaw angle still climbs, even if the stick is lifted up to the horizontal limit. Accordingly, the symmetrical perpendicular point at the right side is denoted as R when the detected yaw angle reaches its limit of \(\alpha_{\text{r}}\) in Figure 3b. When the stick rotates in yaw angle \(\alpha\), the imaging length on imaging plane is defined as \(X_{\text{P}}=\text{I}\,\text{A}^{\prime}\text{B}^{\prime}\,\text{I}\,\). The distance between moving end B and optical centerline, denoted as \(X\), follows a linear relation with the imaging length \(X_{\text{P}}\) considering triangle similarity in Equation (1). \[\frac{X_{\text{P}}}{f}=\frac{X}{D_{\text{O}}-L\cdot\text{cos}\alpha+f}=\frac{ L\cdot\text{sin}\alpha}{D_{\text{O}}-L\cdot\text{cos}\alpha+f} \tag{1}\]  
Figure 2: Schematic of the bottom view model in centered imaging. (**a**) Schematic model; (**b**) geometry model. The yaw angle \(\alpha\) can then be expressed in Equation (2) as below. \[\alpha=\tan^{-1}\left(\frac{(D_{\text{O}}+f)\cdot f\cdot X_{\text{P}}-X_{\text{P} }\cdot\sqrt{\left[L^{2}-(D_{\text{O}}+f)^{2}\right]\cdot X_{\text{P}}{}^{2}+L^{2 }\cdot f^{2}}}{(D_{\text{O}}+f)\cdot X_{\text{P}}{}^{2}+f\cdot\sqrt{\left[L^{2} -(D_{\text{O}}+f)^{2}\right]\cdot X_{\text{P}}{}^{2}+L^{2}\cdot f^{2}}}\right) \tag{2}\]  
The critical yaw angle is defined as \(\alpha_{1}\) at left and \(\alpha_{\text{r}}\) at right when the maximum imaging length occurs at the tangential location. In the right triangle OOLO, the yaw angle range can be written as in Equation (3) below. \[\alpha_{1}=-\alpha_{\text{r}}=\cos^{-1}\left(\frac{L}{D_{\text{O}}+f}\right) \tag{3}\]  
where the positive angle is defined as clockwise, and negative is defined as anti-clockwise rotation. Following the centered imaging case, similar formulations are hereby presented for the deviated imaging case where there is a deviation \(S\) between the camera optical center and the pure vertical state of the stick (Figure 3). According to triangle similarity, we have:  
\[\frac{\left|\frac{\left|\text{O}^{\prime\prime}\text{O}^{\prime}\right|}{f} \right|=\frac{S}{D_{\text{O}}+f}}{f} \tag{4}\]  
The imaging length variation \(X_{\text{P}}\) can be formulated in Equations (3)-(6) as below.