# GNSS-Assisted Low-Cost Vision-Based Observation System for Deformation Monitoring
## 2 General Concept and Approach
### Displacement Determination between Sensors  
Under the effect of the structure's bending \(R\) or torsion \(\theta\), each VOS sensor will be subject to vertical and horizontal displacement (Figure 14). The VOS chain \(I\) in the vertical plane determines two displacement components depending on the chain orientation \(\delta x_{iI}(y_{iI}),\ \delta z_{iI}\), whereas in the horizontal plane it is \(\delta x_{iI},\ \delta y_{iI}\). The final deformation relative to the ground floor is the sum of partial ones:  
\[\delta x_{I}=\sum\delta x_{iI},\ \delta y_{I}=\sum\delta y_{iI},\delta z_{I}= \sum\delta z_{iI}. \tag{10}\]  
Figure 14: Displacements along vertical chains and monitoring parameters (top view) in the horizontal plane. Figure 13: The QR target size at the different distances. These displacements are relative. The total displacement of the structure \(\Delta\), including vertical displacement \(\Delta z\) and its roll \(\Delta L(H)\), are absolute quantities. Their values are yielded by referencing the VOS to the external targets (Figure 15). Therefore, the displacements that VOS measurements can establish have the following form:  
\[\Delta x_{I}=\Delta x+\Delta L_{x}+\sum\delta x_{i1},\ \Delta y_{I}=\Delta y+ \Delta L_{y}+\sum\delta y_{i1},\Delta z_{I}=\Delta z+\sum\delta z_{i1}. \tag{11}\]  
According to Figure 14, displacement in the horizontal plane for a chain sensor to sensor will be determined as shown in Figure 16.  
It is proposed to determine the VOS sensor displacement in the horizontal plane using an image comparison algorithm based on the phase correlation method [95]. Considering a pair of images, we will take one of them as the reference, denoted as \(A\), and the second as the target, denoted as \(B\). Let \(f_{A}(x,y)\) and \(f_{B}(x,y)\) be images, one of which is shifted by \((x_{0},y_{0})\) relative to the other, and \(F_{A}(u,v)\) and \(F_{B}(u,v)\) are their Fourier transforms, then:  
\[f_{A}(x,y)=f_{B}(x-x_{0},y-y_{0}), \tag{12}\]  
Figure 16: Displacement geometry in the horizontal plane. Figure 15: Total displacements of the structure and its roll. \[F_{A}(u,v)=e^{-2\pi(ux_{0}+vy_{0})}F_{B}(u,v), \tag{13}\] \[R=\frac{F_{B}(u,v)F_{A}^{*}(u,v)}{|F_{B}(u,v)F_{A}(u,v)|}, \tag{14}\]  
where \(R\) is the cross-spectrum, and \(F_{A}^{*}\) is a complex conjugate of \(F\). Calculating the inverse Fourier transformation of the cross-spectrum, we obtain the impulse function:  
Figure 14:0  
Having found the maximum of this function, we determine the required displacement. Now let us find the rotation angle \(\theta_{0}\) under the premise of displacement \((x_{0},y_{0})\) using polar coordinates:  
Figure 14:1  
Figure 14:2  
Figure 14:3  
It is possible to determine the target and camera displacements using this algorithm. To do this, the first image from the camera is taken as the reference (\(A\)), all subsequent ones are considered as input ones (\(B\)), and then the image center is searched \((n_{0},m_{0})\). The shift of each new image relative to the reference \((\delta_{n},\delta_{m})\) will describe the camera movement (Figure 17a). To find the target displacement, each new image from the camera is taken as the reference image (A), and the target template is used as the input one (B). Therefore, by finding the target center \((n_{0},m_{0})\) in the first and subsequent images, it is possible to determine the displacement of the target in the image coordinate system (Figure 17b). To obtain the data independent of the camera displacement, the displacement has to be transformed regarding the center of each image in the image set. In both cases, knowing the image resolution, the displacements in pixels \((\delta_{n},\delta_{m})\) can be transformed into displacements in millimeters \((\delta_{a},\delta_{b})\).