# Spatial Data Analysis for Deformation Monitoring of Bridge Structures
## 2 Deformation Analysis Using TLS
### Geometry-Based Approach for Deformation Analysis of Brgidge Structures  
In most cases, the structural elements of bridge structures consist of regular geometric shapes and their combinations. This fact enables us to model the chosen part of the monitored structure by geometric primitives and therefore leads to the utilization of a geometry-based deformation analysis approach to determine the displacements of the monitored object. To improve the quality (accuracy) of the deformation analysis, the monitored parts (points) can be modelled by regression planes fitted to segmented parts of the point cloud. The advantage of regression planes is improved precision due to the reduced measurement noise, which, in the end, results in an improved deformation analysis. In general, displacements of monitored points can be identified as coordinate differences between their coordinates in individual measurement epochs. For vertical displacements, this will be the difference between the heights (Z coordinates) of the monitored points. In the following section, an approach based on regression plane modelling is described. The heights of the monitored points are determined by modelling small planar surfaces using orthogonal regression. The monitored points are fixed in the horizontal plane (XY coordinates) in a local horizon, and only their heights are calculated (Z coordinates) (Figure 1). The segmentation of the point cloud is done by RANSAC [21] from a pre-segmented subset of points using fencing boxes. The fencing boxes are defined around the fixed position of the monitored points by their dimensions along the axis of a local coordinate system of the monitored structure. The parameters of the planes are calculated by orthogonal regression from the general equation of a plane:  
\[a\cdot X+b\cdot Y+c\cdot Z+d=0 \tag{1}\]  
where \(a\), \(b\), and \(c\) are the normal vector's parameters of the plane; \(X\), \(Y\), and \(Z\) are the coordinates of the point lying in the plane; and \(d\) is the scalar product of the normal vector of the plane and the position vector of any point of the plane. The parameters of normal vectors are calculated using singular value decomposition (SVD) [22]:  
\[\mathbf{A}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{\mathrm{T}} \tag{2}\]  
where the design matrix \(\mathbf{A}\) has the dimensions \(nx3\) where \(n\) is the number of points included in the plane's parameter estimation. The column vectors of \(\mathbf{U}^{\mathrm{n}\mathrm{n}\mathrm{n}}\) are normalized eigenvectors of matrix \(\mathbf{A}\mathbf{A}^{\mathrm{T}}\). The column vectors of  
The design matrix has the form:  
\[\mathbf{A}=\left( \begin{array}{ccc}(X_{1}-X_{0})&(Y_{1}-Y_{0})&(Z_{1}-Z_{0})\\ (X_{2}-X_{0})&(Y_{2}-Y_{0})&(Z_{2}-Z_{0})\\ \vdots&\vdots&\vdots\\ (X_{n}-X_{0})&(Y_{n}-Y_{0})&(Z_{n}-Z_{0})\end{array} \right) \tag{3}\]  
where \((X_{i}-X_{0})\), \((Y_{i}-Y_{0})\), and \((Z_{i}-Z_{0})\) are the reduced coordinates (reduced to the subset's centroid). The advantage of using SVD is that no initial parameters are needed for estimation. The above-mentioned procedure is performed by RANSAC, while the seed points are the points nearest to the defined positions of the monitored points. The heights (Z coordinates) of the monitored points are calculated by projecting the predefined position (XY) of these points onto the estimated regression planes (Figure 1) using the formula:  
\[Z_{P}=-\frac{a\cdot X+b\cdot Y+d}{c} \tag{4}\]  
The benefit of this approach is that the position of the monitored points does not change with the thermal expansion of the structure; the uncertainty caused by different scan densities (e.g., by different positions of the instrument) in the single epochs are also eliminated. Using uncertainty propagation law, the standard deviations of the results are calculated from the standard deviation of the vertical component of the registration error and the standard deviation of the regression plane.