# GNSS-Assisted Low-Cost Vision-Based Observation System for Deformation Monitoring
## 2 General Concept and Approach
### Design Concept of the VOS
As a result, the geometry of the initial square pixel will be distorted, or its size will change if the average value of the matrix resolution is applied. Consequently, this fact will affect the quality of digital image processing. Instead of the camera's field of view, the angle of the visible area along the image side \(\varphi\) is recommended:  
\[\varphi_{a}=arctg(a/S),\varphi_{b}=arctg(b/S) \tag{2}\]  
where \(a\) is the size of the visible image area along \(a\) axis, \(b\) is the size of the visible image area along the \(b\) axis, and \(S\) is the distance from the camera (sensor) to the target. This approach preserves the uniqueness of determining the resolution of the camera, which is denoted by the value \(c\):  
Figure 4.0  
where \(n\) and \(m\) are the sizes of the camera matrix. The camera's resolution was determined using expressions and (2). The necessary parameters were obtained from the typical camera specification and by calibration stand surveying from a fixed distance. The CMOS matrix size is 4320 \(\times\) 3240 px (6.17 mm \(\times\) 4.55 mm); the f-number equals 3.9. The results for the camera, a General Electric G100 with \(f\) = 72 mm, \(k\) = 5.62, and the surveying distance \(S\) = 1 m, are presented in Figure 4.1  
As expected, the resolution for the angle \(\gamma\) along each of the axes of the image takes different values, while the average value of the camera resolution is reduced. There is a serious flaw in the given concept. Such resolution determines the ideal case when the lens resolution is perfect. However, in real life the actual resolution is the sum of image resolution plus lens resolution. It is obvious that the lens also downgrades the final image resolution. The approximate resolution can be obtained under the premise that the standard lens provides the object discerned when the object has a size of at least 3 pixels. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & \begin{tabular}{c} **Distance \(S\)** \\ **(m)** \\ \end{tabular} & \
\begin{tabular}{c} **Size on the Ground** \\ **(mm)** \\ \end{tabular} & **Pixels** &
\begin{tabular}{c} **Resolution** \\ **(mm/px)** \\ \end{tabular} \\ \hline \(\varphi_{a}\) & 4\({}^{\circ}\)53\({}^{\prime}\)52.7\({}^{\prime\prime}\) & 1 & 85.694 & 4320 & 0.020 \\ \(\varphi_{b}\) & 3\({}^{\circ}\)36\({}^{\prime}\)57.5\({}^{\prime\prime}\) & 1 & 63.194 & 3240 & 0.020 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The results of resolution determination. Figure 7: Image acquisition via system camera CCD matrix. Thus, the resolution \(\varphi_{a,b}\) for the distance of 1 m equals to 0.058 mm. This result allows calculating the resolution error \(m_{\gamma}\) for different distances. \[m_{\gamma}=\frac{\varphi_{a,b}S}{f}. \tag{4}\]  
The results for expression (4) are given in Figure 8.  
It worth mentioning that these values can be decreased using sub-pixel processing algorithms. Thanks to the long focal length, the resolution error does not affect measurement accuracy significantly. The other factor that affects the accuracy, which should be accounted for to understand the real achievable accuracy that can be retrieved from the camera images, is the defocusing error. The defocusing error leads to image unsharpness that is described via the blur circle. The defocusing error is defined through the aperture that is presented as the relationship  
\[D=\frac{f}{N},\]  
where \(N\) is well known as an \(f\)-number, and for the selected camera \(N=\frac{f}{D}=3.9\). To calculate the defocusing error, let us use the main optics equation:  
\[\frac{1}{f}=\frac{1}{S}+\frac{1}{s}. \tag{5}\]  
The designations are clear from Figure 8. From Equation (4), we obtain the defocusing error \(\delta\):  
\[\delta=\frac{N\Delta f}{f},\;\Delta f=s-f,\;s=\frac{Sf}{S-f}. \tag{6}\]  
The actual error due to the defocusing error is determined as  
\[m_{\delta}=\frac{\delta S}{f}. \tag{7}\]  
Figure 8: The graph of the resolution error.