## 3 Experimental Results and Discussion
### State Analysis  
First, we use the threshold value as the vertical velocity to detect the start point of the potential fall period. The threshold value is the same for all scenes. This means that the pixel difference between the top and bottom positions must be more than the predefined value, within the period spanning 12 frames, to begin the potential fall state. Figure 8 shows the vertical velocity feature used to detect a potential fall state. When the vertical velocity meets a predefined threshold, the normal state becomes a potential fall state. The end frame of the normal state is identified by detecting the start frame of the potential fall state. The start frame of the fall state can be detected by measuring the change in the height difference of the ellipse model. The plots of Figure 9 are an example of successful detection of the fall state near the 45 th frame. Note that the vertical axes of the two plots in Figure 9 are the height difference of the major axis of the ellipse model and the ratio of minor axis to major axis, respectively. The ratio of minor axis to major axis feature is also used to account for the changes in the length of the major axis due to the motions, such as bending, squatting, and sitting. In the case of bending, the length of the upper body reduces and, in the case of squatting, the length of the lower body reduces, consequently changing the length of the major axis when confounding motion occurs. However, in case of a general fall, the ratio of major axis to minor axis is almost always maintained. Therefore, we use these two features to detect the fall start point accurately. The no-motion phase is defined as no movement after the fall state. Therefore, in this state, the gradient values at the top and the bottom positions must be below the predefined threshold. Figure 10 shows an example where the gradient is lower than the threshold (see the red arrow between the 30 th and the 50 th frames). At the end of the no-motion phase, the most noticeable feature is the movement of a human's head or leg. Figure 8: Vertical velocity and the corresponding threshold used to detect the start of the potential fall state. In this study, the recovery state begins when the upper body is completely off the floor, which is the case of the change in the top position above the predefined threshold. However, in the case of a back fall, the position of the head is opposite to that in case of a front fall, so the change rate of the top position and bottom position is observed differently depending on the slope of the fall. As shown in Figure 11, the recovery state begins when the difference exceeds the threshold of the red arrow. If the height difference in the current frame is closer to the height difference in the normal state, the recovery state is changed to the normal state. The criterion used here is adaptively determined using the same value that was used to determine the fall. Every state and feature is described in Figure 12. For the evaluation experiments, we use optimized values for all of the thresholds (\(\delta_{1}=7\), \(\delta_{2}=0.25\), \(\delta_{3}=6\), \(\delta_{4}=4.5\), \(\delta_{5}=26\)) gathered by pilot experiments. Velocity threshold \(\delta_{1}\) refers to the commonly reported fall velocity. Moreover, \(\delta_{2}\) is the ratio of the minor axis to the major axis, which indicates how much the shape of a person changes. Symbols \(\delta_{3}\) and \(\delta_{4}\) represent the threshold for the top and bottom position gradient of the ellipse. These are the values that physically represent the movement of a person's head or leg in a gradient. \(\delta_{5}\) refers to the transition threshold from the no-motion phase to the recovery state, which indicates that there is a difference in head or leg height during the recovery. Figure 10: Fall detection feature that is applied to detect no-motion phase.