# A New Approach to Fall Detection Based on the Human Torso Motion Model
## 3 Our Approach for Fall Detection
### Human Torso Motion Model  
Fall activity is the balance loss of a person. When a fall happens, it is always accompanied with sharp changes in the torso angle and the centroid height. In the proposed method, we take the torso angle and the centroid height as two key features. There are our thresholds in our fall detection model, where \(T_{\alpha}\) is the threshold for the start key frame detection, \(T_{\tau\omega}\) is the threshold of the changing rate of the torso angle, \(T_{\tau\omega}\) is the threshold of the velocity of the centroid height and \(T\) is the tracking time after the torso angle exceeded \(T_{\alpha}\). In HTMM. The changing rates of the torso angle and centroid height are calculated frame by frame after the torso angle exceeded \(T_{\alpha}\). The max values of these two features in the given period of time, \(T\), are compared to their thresholds, respectively. When both rates exceed their corresponding thresholds \(T_{\tau\omega}\) and \(T_{\tau\omega}\), a fall is detected. For the result shown in the limit of stability test (LOST), an adult person can keep his/her balance with the body leaning forward/backward no more than 12.5 degrees and leaning left/right no more than 16 degrees [35]. In LOST, the person is asked to keep the whole body in a line. However, there is always an angle between the lower body and the upper body in daily activities. However, the person's torso always keeps parallel with the gravity line. Therefore, we take the torso line rather than the body line to form the torso angle with the gravity line. In our experiments, the torso angle exceeding 13 degree is the start of our detection model. The usage of only the torso angle to detect falls is insufficient and will result in low accuracy and a high false alarm rate. For example, bending or controlled lying down will be judged as a fall. To address this issue, the centroid height is imported in our model as the second feature. Since a fall is an activity that usually happens in a short period of time (in our experiments, most of the fall samples last 1.1-1.6 s), for a video captured with 30 frames per second, there are 33-48 images during a fall. Thus, we built a motion model called HTMM for fall detection. In this model, we pay special attention to the changing rate of torso angle and centroid height in a given period of time. Assume that the given period of time is represented as \(\mathbb{T}\), and \(N\left(n\ \mid\ n\in\mathbb{Z}\wedge n\leq\mathbb{T}\ \times 30\right)\) denotes the frames' order in \(\mathbb{T}\). Then, the person's torso angle (\(\alpha\)), centroid height (\(H\)) and recorded time (\(T\)) in each frame can be represented as: \(\alpha=\left\{\alpha_{1},\,\alpha_{2},\,\ldots\,\,\alpha_{n}\right\}\), \(H=\left\{h_{1},\,h_{2},\,\ldots\,\,h_{n}\right\}\), \(T=\left\{t_{1},\,t_{2},\,\ldots\,\,t_{n}\right\}\). The changing rates of \(\alpha\) and \(H\) in \(\mathbb{T}\)  of each frame are calculated using Equations (7) and (8). \[V_{t}^{\alpha} =\bigg{\{}x|x_{i}\in\mathbb{R}\ \Lambda\ x_{i}=\frac{\alpha_{i}- \alpha_{1}}{t_{i}-t_{1}}\bigg{\}}, \tag{7}\] \[V_{t}^{h} =\bigg{\{}y|y_{i}\in\mathbb{R}\ \Lambda\ y_{i}=\frac{h_{i}-h_{1}}{t_{i}-t_{1}} \bigg{\}}, \tag{8}\]  
In HTMM, the max changing rates of torso angle and centroid height in the given period of time are used to compare with their thresholds, \(T_{vu}\) and \(T_{vh}\). When both of them exceed their thresholds, HTMM outputs lone to indicate that the activity is judged as a fall; else HTMM outputs zero to indicate that it is not. Our model can be represented as:  
\[\text{HTMM}(\alpha,h,t)\ =\ \left\{ \begin{array}{ll}1&\text{Max}(V_{t}^{ \alpha})>T_{vu}\ \Lambda\ \text{Max}>\ (\ V_{t}^{h})>T_{vh}\\ 0&\text{else}\end{array} \right., \tag{9}\]  
Figure 6 shows the general block diagram of our approach. Figure 5: The values of centroid height recorded in two daily activities videos: crouching and walking. (a) Centroid height in crouching activity video; (b) centroid height in walking activity video.