## 4 Results
### Effective Distance  
The error introduced by the pixel shift grows with the tangent function in the proposed model. For example, in our case a 416 \(\times\) 416 image with a horizontal FOV of 63.46\({}^{\circ}\) and the camera height \(h=1.2\) m. Assuming the detection shifts 1 pixel at \(D=2.4\) m (\(y=0\) in image coordinate),  
\[y_{err}=1.2\times\tan(63.46+\frac{2\times 26.54\times 1}{416})-1.2\times\tan(63.46 )=0.013\;\text{m} \tag{14}\]  
\[y_{err}=y_{156}-y_{155}=0.196\;\text{m} \tag{15}\]  
In Equation (14), the error is 0.013 m, which is 0.56% of the distance. However, if 1-pixel shifting occurs at \(D=10\) m (\(y=155\) in image coordinate), the error becomes 0.196 m, which is 1.94% of the distance. Figure 7 illustrates the relationship between the object distance and the error introduced by the pixel shifting. For the object at 10 m, 2 to 3 pixels shifting made by the detection will introduce a 3.96% to 6.05% error. It is unignorable and it is necessary to define the effective distance for the proposed model.  
According to [PERSON] [39] and [PERSON] et al. [40], the mean walking speed of pedestrians is 1.69 m/s, 1.43 m/s, and 0.792 to 1.53 m/s for younger individuals, older individuals, and visually impaired people, respectively. Therefore, assuming there is an object located 10 m away, it takes 6.5 s (1.53 m/s) for the user to reach it. Assuming the system needs 0.5 s to manipulate, there are 6 seconds left for the users to respond and adjust their route. On the other hand, if a user is wearing a camera at 1.2 m height (average chest height of a human), the blind area is around 2.4 m. It means that the object closer than 2.4 m could not be or could only be partially detected. Since the position-based model is only effective for the object of which the bottom is completely captured, the partial detection will introduce error to the system (Figure 8). Hence, the effective distance of NFD is assumed to be 2.4 m to 10 m. For the estimation of the distance of objects shorter than the effective distance, it is suggested to use the previous frames of images for the determination.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline  & **Camera Height (m)** & **Distance (m)** & **Height (Pixel)** & **Focal Length (Pixel/m)** \\ \hline (a) & 0.9 & 3.0 & 121 & 418.60 \\ (b) & 0.9 & 3.5 & 103 & 419.19 \\ (c) & 0.9 & 4.0 & 89 & 413.95 \\ (d) & 1.1 & 3.0 & 123 & 429.07 \\ (e) & 1.1 & 3.5 & 105 & 427.33 \\ (f) & 1.1 & 4.0 & 92 & 427.91 \\ (g) & 1.3 & 3.0 & 122 & 425.58 \\ (h) & 1.3 & 3.5 & 103 & 419.19 \\ (i) & 1.3 & 4.0 & 92 & 427.91 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Readings of focal length experiment on long fence height.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline  & **Camera Height (m)** & **Distance (m)** & **Height (Pixel)** & **Focal Length (Pixel/m)** \\ \hline (a) & 0.9 & 3.0 & 121 & 422.09 \\ (b) & 0.9 & 3.5 & 104 & 423.26 \\ (c) & 0.9 & 4.0 & 90 & 418.6 \\ (d) & 1.1 & 3.0 & 125 & 436.05 \\ (e) & 1.1 & 3.5 & 103 & 419.19 \\ (f) & 1.1 & 4.0 & 92 & 427.91 \\ (g) & 1.3 & 3.0 & 122 & 425.58 \\ (h) & 1.3 & 3.5 & 105 & 427.33 \\ (i) & 1.3 & 4.0 & 93 & 432.56 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Readings of focal length experiment on short fence height.