## 2 Materials and Methods
### Theory of Filtering over Vertical Horizontal (HOVE) Angular Grid
#### 2.3.1 Horizontal and Vertical Angles within the HOVE Grid  
Figure 1A shows a horizontal-vertical angles (HOVE) gird, while Figure 2 represents a small sector of such a grid. Within the point grid, the closest points above and below as well as to the left and right of the point to be examined are determined (Figure 2, blue points). For this purpose, the corresponding search area must be defined. Within these search areas, the point with the smallest angular distance to the point to be examined is searched. Subsequently the angle from the examined point to the adjacent points is calculated using Equation (4):  
\[\alpha_{wi}=\tan^{-1}\left(\frac{\left(L_{pux}-L_{pual}\right)\times 2}{\sin \left(\beta_{pux}-\beta_{pual}\right)\times\left(L_{pux}+L_{pual}\right)}\right) \tag{4}\]  
where \(\alpha_{wi}\) is the angle from examined point to adjacent point; \(L_{pux}\) is the length from laser scanner to examined point; \(L_{pual}\) is the length from laser scanner to adjacent point; and \(\beta_{pux}\) is the angle from laser scanner to examined point; \(\beta_{pual}\): angle from laser scanner to adjacent point. In addition to the closest points, the points with the lowest \(\alpha_{uzi}\) below the point to be examined are determined (Figure 1, green points). From the angles found, the respective angle difference can be calculated. If no angle could be calculated for one side due to lack of points, the average of the negative angles of the next and after next point on the opposite side was taken for this side. If no closed point angle could be determined fir both sides, the difference of points with the lowest \(\alpha_{uzi}\) will be used for calculation. With the help of these angle differences, the following statements on the terrain model can be made. Find Distance Measurement Errors  
The dataset used includes some points that are obviously measurement errors. Even points which have a remarkably large length must be removed before the ending filtering. These points would otherwise be responsible for removing a lot of correct ground points. To identify these points, in a specific search range for each point, the horizontal as well as vertical back angle-difference to its neighbouring points is calculated. The left side of Figure 3 shows a schematic representation of the terrain cross-section, where the grey lines correspond with the vertical angles--to the adjacent points above and below the point to analyse--in Figure 2. The right side should give you an idea how these points could be situated in a cross-section of a measured point cloud. If this angle difference is greater than a previously defined threshold value, this point is prematurely removed as a measurement error point from the further calculation (Equation (5)):  
Figure 3: Terrain view from the side with measurement error. Figure 2: Horizontal-vertical (HOVE) grid and search area for horizontal and vertical angles. \[\theta_{\textit{drive}}\cap\theta_{\textit{dlibo}}>\lambda_{TLV} \tag{5}\]  
where \(\theta_{\textit{drive}}\) is the difference of the vertical back-angles; \(\theta_{\textit{dlibo}}\) is the difference of the horizontal back-angles; and \(\lambda_{TLV}\) is the threshold limit value. Reduction to a 2.5-Dimensional Terrain  
In this step, all points above a connecting line from other points below are removed from a three-dimensional point cloud (Figure 4). From the remaining points, a 2.5-dimensional terrain can be expected. These points would also be eliminated by Wedge Filtering. Since the filtering is only done iteratively via the wedge angle, the reduction with HOVE grid is more effective. As described in Section 2.3.1, beneath the angles to the adjacent points the angle to the point with the lowest \(\alpha\) is also calculated. If you take a small cross-section such as the red area in Figure 1A, points with a higher \(\theta\) and shorter length (Figure 1B) are with high probability vegetation points. The lowest \(\alpha_{\textit{wi}}\) of such points is less zero.