# A Depression Recognition Method Based on the Alteration of Video Temporal Angle Features
## 3 Proposed Framework
### Robustness Analysis of Angle Features
#### 3.3.1 Rotation Invariance  
The target undergoes rotation along the z-axis, as depicted in Figure 3. Patients frequently tilt their heads to the left and right during the recording of the interrogation video. Consequently, the tilting of the patient's head to the left and right can be considered facial rotation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sequence Number** & **Coordinate Definition** & **Meaning** \\ \hline A\({}_{1}\) & Ang(18,20,22) & Curvature of the left eyelash \\ A\({}_{2}\) & Ang(23,25,27) & Curvature of the right eyelash \\ A\({}_{3}\) & [PERSON](20,31,25) & Grain lift 1 \\ A\({}_{4}\) & [PERSON](20,28,25) & Grain lift 2 \\ A\({}_{5}\) & [PERSON](38,37,42) & Degree of left eye corner opening \\ A\({}_{6}\) & [PERSON](37,20,40) & Degree of left eye squinting \\ A\({}_{7}\) & [PERSON](47,46,45) & Degree of right eye corner opening \\ A\({}_{8}\) & [PERSON](43,25,46) & Degree of right eye squinting \\ A\({}_{9}\) & [PERSON](49,03,37) & Degree of left cheek contraction \\ A\({}_{10}\) & [PERSON](55,15,46) & Degree of right cheek contraction \\ A\({}_{11}\) & [PERSON](51,49,59) & Degree of left outer lip opening \\ A\({}_{12}\) & [PERSON](62,61,68) & Degree of left inner lip opening \\ A\({}_{13}\) & [PERSON](53,55,57) & Degree of right outer lip opening \\ A\({}_{14}\) & [PERSON](64,65,66) & Degree of right inner lip opening \\ A\({}_{15}\) & [PERSON](49,34,55) & Degree of mouth opening \\ A\({}_{16}\) & [PERSON](28,02,31) & Nose wrinkle 1 \\ A\({}_{17}\) & [PERSON](28,16,31) & Nose wrinkle 2 \\ A\({}_{18}\) & [PERSON](Le,31,Re) & Upside down \\ A\({}_{19}\) & [PERSON](01,30,03) & Right deflection \\ A\({}_{20}\) & [PERSON](17,30,15) & Left deflection \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Angle of facial landmarker. Figure 3: The rotational imaging model consists of two scenarios: (**a**) when the camera rotates around the optical axis during shooting, and (**b**) when the target rotates around the z-axis. **Theorem 1**. : _Let \(\Delta\)ABC be an image on the target surface \(\pi\) after camera imaging, and let \(\Delta A^{\prime}B^{\prime}C^{\prime}\) be its corresponding image. When the target rotates along the z-axis, the angles \(\angle A^{\prime}B^{\prime}C^{\prime}\), \(\angle B^{\prime}A^{\prime}C^{\prime}\), and \(\angle B^{\prime}C^{\prime}A^{\prime}\) remain unchanged. This property is referred to as the rotation invariance theorem._  
Proof of Theorem 1.: The rotation of the target around the z-axis is equivalent to the rotation of the camera around the optical axis. Figure 3a is the imaging model when the camera rotates around the optical axis, and Figure 3b is the imaging model when the target rotates around the z-axis. By establishing the camera space coordinate system \(OXYZ\), as shown in Figure 3a, let \(A_{1}B_{1}\) be the image of target \(AB\) in image plane \(\pi_{1}\). When the camera rotates to shoot, although the image \(A_{1}B_{1}\) of the target \(AB\) will change relative to the image coordinates, the length \(\,\mid A_{1}B_{1}\mid\) of \(A_{1}B_{1}\) will not change. This is shown in the vignette in Figure 3a. Thus, the following conclusions can be drawn: in the process of target rotation, the length of the image obtained by obliquely shooting the target at any angle will not change. Let the image of target \(ABC\) be \(A^{\prime}B^{\prime}C^{\prime}\); when the target point \(ABC\) rotates around the z-axis, a new target \(A_{1}B_{1}C_{1}\) is formed, and correspondingly, there is a new corresponding image \(A^{\prime}_{1}B^{\prime}_{1}C^{\prime}_{1}\) on the image plane. According to the theory of target rotation around the z-axis, there is the following Formula (1):  
\[ \begin{cases}|A_{1}B_{1}|=\left|A^{\prime}_{1}B^{\prime}_{1}\right|\\ |A_{1}C_{1}|=\left|A^{\prime}_{1}C^{\prime}_{1}\right|\\ |C_{1}B_{1}|=\left|C^{\prime}_{1}B^{\prime}_{1}\right|\end{cases}  \tag{1}\]  
That means \(\Delta A^{\prime}B^{\prime}C^{\prime}\cong\Delta A^{\prime}_{1}B^{\prime}_{1}C ^{\prime}_{1}\).