## 2 Materials and Methods
### Fall Detection from Partially-Observed Depth-Map Video Sequences  
A partially-observed depth-map video sequence (\(V\)) that consists of \(s\) observed video subsequences can be represented as the union of the observed video subsequences. Specifically, \(V\) can be defined as follows: \begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt}} \hline \hline
**Angle** & **Description** & **Mathematical Formulation** \\ \hline \(\theta_{Lshank}\) & The angle between the left shank and a translated transverse plane (\(TP_{1}\)) that passes through the left ankle joint position, where the left shank is defined as a line in the space that passes through the left ankle and left knee joint positions. & (7) \\ \hline \(\theta_{Lhigh}\) & The angle between the left thigh and a translated transverse plane (\(TP_{2}\)) that passes through the left knee joint position, where the left thigh is defined as a line in the space that passes through the left hip and left knee joint positions. & (8) \\ \hline \(\theta_{L Knee}\) & The angle between the left thigh and left shank. & (9) \\ \hline \(\theta_{Rshank}\) & The angle between the right shank and a translated transverse plane (\(TP_{3}\)) that passes through the right ankle joint position, where the right shank is defined as the line in the space that passes through the right ankle and right knee joint positions. & (10) \\ \hline \(\theta_{Rhigh}\) & The angle between the right thigh and a translated transverse plane (\(TP_{4}\)) that passes through the right knee joint position, where the right thigh is defined as the line in the space that passes through the right hip and right knee joint positions. & (11) \\ \hline \(\theta_{Rknee}\) & The angle between the right thigh and a translated transverse plane (\(TP_{4}\)) that passes through the right knee joint position, where the right thigh is defined as the line in the space that passes through the right hip and right knee joint positions. & (12) \\ \hline \(\theta_{trunck}\) & The angle between the trunk and the transverse plane (\(TP\)), where the truck is defined as the line in the space that passes through the the hip center and shoulder center joint positions. & (13) \\ \hline \(\theta_{Lhip}\) & The angle between the trunk and left thigh. & (14) \\ \hline \(\theta_{Rhip}\) & The angle between the trunk and the right thigh. & (15) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The angle-based features employed in the pose profile. \[V\equiv\bigcup_{r=1}^{s}V_{r}[t_{r}(1):t_{r}(l)], \tag{17}\]  
where \(V_{r}[t_{r}(1):t_{r}(l)]\) is the \(r\)-th observed subsequence of frames and \(t_{r}(1)\) and \(t_{r}(l)\) are the indices of the first and last frames in the \(r\)-th observed video subsequence, respectively. In Figure 4, we illustrate the representation of partially-observed video sequences described in Equation (17). In particular, Figure 4 provides an example of a partially-observed depth-map video sequence that consists of three observed video subsequences. Such a partially-observed video can be represented as the union of the three observed video subsequences, namely \(V_{1}[t_{1}(1):t_{1}(l)]\), \(V_{2}[t_{2}(1):t_{2}(l)]\) and \(V_{3}[t_{3}(1):t_{3}(l)]\). In this work, we aim at classifying a partially-observed video sequence \(V\) into one of the four fall-related activities, namely walking (\(\omega_{1}\)), sitting (\(\omega_{2}\)), falling from standing (\(\omega_{3}\)) and falling from sitting (\(\omega_{4}\)). Let us denote the set of fall-related activities as \(\Omega=\{\omega_{c}\},c\in\{1,2,3,4\}\).