plans, all points from the point cloud in the segment are 
identified and tagged with the corresponding segment identifier 
(Figure 7). 
  
Figure 7. Point cloud classified by axis segment (limited by 
yellow plans). 
STAGE 4 
The fourth stage is to create two sections for each axis segment. 
For each axis segment a point is fixed at 25% of the segment 
length from the first point (called the main point), and a plan 
perpendicular to the axis (called the main plan) is calculated. 
The same is done for a point at 75% of the segment, as shown in 
figure 8: 
  
  
> ès $ Mee 
— 
Figure 8. The tunnel axis is cut at 2596 and 75% (blue plans). 
The aim of these two intermediate plans is to obtain the cross 
section of the tunnel. Each of these plans is used to define a 
box, with a thickness of 1 or 2 decimetres along the axis. Now, 
all the points from the point cloud, inside this box are extracted, 
projected in this plan and their polar coordinates (radius and 
angle) are calculated. The angle is calculated with origin at the 
upward direction (not exactly the vertical because of the axis 
slope). The selected points are grouped by angular classes, with 
a spacing defined by the user. For each angular interval the 
average and standard deviation of the radius are calculated, and 
those that differ from the mean more than two standard 
deviations, are excluded from the selection (only at this stage). 
With the others the centroid of the point set is calculated, and 
this mean point will be considered a point of the cross-section. 
The cross-section is composed by all the centroids of the 
angular intervals (Figure 9). This stage is repeated to calculate 
the second section of the tunnel segment, at 75% of the 
distance. At the final, two sections per segment are obtained. 
  
Figure 9. The two sections (at 25% (green) and 75% (orange) of 
the segment length) and the respective points. 
110 
STAGE 5 
The fifth stage is to project the points of the point cloud to a 
plan. For all the points of the point cloud the nearest cross- 
section is determined (the 25% or the 75% one). Then, the two 
nearest points of the cross-section are obtained, as well as the 
corresponding points on the other section, as shown in Figure 
10. 
erm 
np Ra EE 
  
  
Figure 10. The two closest points in the cross-section and its 
homologous in the other section (4 blue points) for the point 
under analysis (red point). 
The top of the tunnel is the origin for the determination of the 
distance along the cross-section. The point in analysis is 
projected under the line between the two points of the section 
and his length along the wall (along the section) until the top of 
the tunnel is measured (this step is repeated for the other section 
as well). Now, the distance between the point in analysis and 
the top of the tunnel is calculated through the weighted average 
of the two lengths, calculated before along the sections, 
assuming the weights are the distances between the point and 
the two sections (Figure 11). 
  
  
7 anne * 
  
Figure 11. The distance between the point in analysis and the 
top of the tunnel (blue line) along the two cross-sections (black 
arcs), calculated by a weighted average of the two distances. 
In Figure 11 the distance from the point in analysis (red point) 
to the top of the tunnel (blue line) is calculated with the average 
of the distances along the sections. 
p did.td,d.N . 
X= X + (Gens ) sin(6) ris 
= d,d,+dzd4 
Y=Y,+ cm ) cos(6) 
where — d;,d; = distance between point and cross-sections 
ds, d4 = length of the arcs along the cross-sections 
0 — axis segment's azimuth 
Xo, Yo = coord. of the point intersection at the axis 
All the points in the point cloud that are at a distance larger than 
a predefined tolerance (defined by the user) are considered not 
to be on the tunnel surface (eg. cables) and discarded from the 
projection process. Finally, the resulting points (2D points) have 
X coordinate as the distance along the tunnel axis and the Y 
coor 
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