## 3 Proposed System Architecture
### Feature Extraction  
In this component, the features are extracted from the detected foreground object, including the centroid (C), area, height, width, aspect ratios (\(r\)), and the virtual grounding point (_VGP_). Specialized terminology and notations for feature extraction are provided in the following.  
With the use of _VGP_, we aim to define new parameters describing patterns of human action. Four steps are involved in constructing _VGP_, and the technical details are described as follows.  
Figure 2: Foreground refinement by using graph cut.  
1. Firstly, the position \(p\) at time \(t\) of the detected foreground object is defined as in Equation (2). \[p(t)=(x(t),y(t))\] (2) Then, the centroid of the object is obtained by Equation (3), as shown in Figure 3a. \[C(t)=(x_{c}(t),y_{c}(t))\] (3) Specifically, each \(x_{c}\) and \(y_{c}\) is simply formulated as in Equation (4). \[x_{c}=\sum_{i=1}^{N}\frac{x_{i}}{N},\ \ y_{c}=\sum_{i=1}^{N}\frac{y_{i}}{N}\] (4)
2. Secondly, a vertical line from the top-most row to the bottom-most row is created along the \(x\) axis of the centroid, as shown in Figure 3b. 3. Thirdly, a horizontal line from the left column to the right column is created along the \(y\) axis of the centroid at the bottom-most row, as shown in Figure 3c. 4. Finally, a point for \(VGP(t)\) is marked on the horizontal line of the bottom row along the \(y\) axis where the vertical line on the \(x\) axis extends from the centroid. Figure 3d describes the final result for \(VGP\), which can be formulated as in Equation (5). \[VGP(t)=(x_{VGP}(t),y_{VGP}(t))\] (5)  
We noticed that the virtual grounding point (\(VGP\)) can be simply obtained from the object centroid. In addition, the patterns of posture can be analyzed by observing pairs of changes in \(C\) and \(VGP\), as shown in Figure 4. The underlying pattern in Figure 4a indicates that the distance between \(C\) and \(VGP\) is initially quite short, and then lengthens as the pattern of the person's position changes from lying down to getting up. The distance between points \(C\) and \(VGP\) in the pattern for sitting down is quite short, and it shortens during the transition from standing to sitting, as shown in Figure 4b. Therefore, the differences between \(VGP\) and \(C\) are regarded as values for the supportive features for abnormal and normal patterns, as formulated in Equation (6). \[d(t)=y_{VGP}(t)-y_{c}(t)\] (6) where \(d\) is the distance of \(VGP\) from \(C\) along the \(y\) axis, \(y_{VGP}(t)\) means the virtual grounding point along the \(y\) axis at time \(t\), and \(y_{c}(t)\) represents the centroid along the \(y\) axis at time \(t\).  
Figure 3: Constructing a virtual grounding point by (**a**) centroid of object, (**b**) vertical line from top row to bottom row across the \(x\) axis, (**c**) horizontal line from the left to right columns across the \(y\) axis, (**d**) virtual grounding point.  
Moreover, information on the regularity of the object shape related to \(VGP\) is obtained by calculating the area. Finally, the aspect ratio (\(r\)) of the object is simply calculated to predict the posture as in Equation (7). The concepts for calculating area and aspect ratio are shown in Figure 5.  
\[r(t)=w(t)/h(t) \tag{7}\]  
where \(r(t)\) represents the aspect ratio of the object at time \(t\), and \(w\) and \(h\) refer to the width and height of the object, respectively.