## 2 Materials and Methods
### Human Activity Representation
#### 2.2.1 Motion Profile  
The movements of different body parts during various human activities can be described with respect to three anatomical planes, namely the transverse plane (TP), coronal plane (CP) and sagittal plane (SP), intersecting at a specific point on the human body [31]. Inspired by the concept of the anatomical planes, we propose a view-invariant motion profile that can capture the movements of different body parts during fall-related activities. Specifically, to construct a view-invariant motion profile that is independent of the location of the Kinect sensor, we build a body-attached coordinate system that is centered at a specific joint position. Then, the 3D positions of the skeletal joints, which are acquired using the Kinect sensor, are recomputed with respect to the origin of the body-attached coordinate system. Using the recomputed skeletal joint positions, we construct the three anatomical planes, such that the three planes are intersecting at the origin of the body-attached coordinate system. Finally, the movement of a specific body part can be described in terms of the position of the displacement vector, which extends between the initial and final positions of the movement with respect to the three anatomical planes. For example, the movement of the left hand in the upward direction can be represented in terms of the position of the displacement vector constructed between the initial and final positions of the hand movement with respect to the three anatomical planes. The construction procedure of the motion profile for each frame in the depth-map video sequence is as follows:  
1. In order to analyze the depth-map video sequence, we utilize a sliding window of size \(W\) frames and overlap of size \(O\) frames between any two consecutive positions of the sliding window. Moreover, for a given position of the sliding window, the frames in the window are numbered \begin{table}
\begin{tabular}{l l} \hline \hline
**Activity** & **Sub-Activities** \\ \hline Walking & Walking forward (i.e., walking towards the Kinect sensor) and walking backward (i.e., walking away from the Kinect sensor). \\ Sitting & Stand still pose, bending down for sitting and sitting pose. \\ Falling from standing & Stand still pose, falling from standing pose and fall pose. \\ Falling from sitting & Sitting pose, falling from sitting pose and fall pose. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The sub-activities encapsulated within each activity. sequentially between 1 and \(W\). For each window position, we build a motion profile for the human activities incorporated within the frames of the window as follows:  
1.1 We create a body-attached coordinate system that is centered at the hip center (\(hc\)) joint. Then, we recalculate the positions of all of the other skeletal joints with respect to the \(hc\) joint. Figure 3 illustrates the constructed body-attached coordinate system. 1.2 Using the recalculated joint positions, we define the three anatomical planes, i.e., the TP, CP and SP, such that the three planes intersect at the \(hc\) joint (see Figure 3). Each plane is defined using three non-collinear skeletal joint positions as described below: \[TP\equiv<hc,l\bar{hp},r\bar{hp}>.\] (1) \[CP\equiv<hc,lsh,rsh>.\] (2) \[SP\equiv<hc,sc,sp>.\] (3) In Equation (1), \(l\bar{hp}\) and \(r\bar{hp}\) are the 3D positions of the left and right hip joints after applying a translation transformation along the Y-axis of the body-attached coordinate system to align the two joints with the \(hc\) joint. 1.3 We compute the displacement vectors for a subset of the skeletal joints, denoted as \(s_{dv}\), that are related to the fall-event. In this study, the subset \(s_{dv}\) is composed of the following joints: sp, hd, rhd, lhd, rft and lft.