in segment S
1 i
, the second eigen vector in the remaining orthogo- nal direction of maximal variation, while the last points in the re-
maining orthogonal direction that has least variation, see also Fig- ure 1. As a consequence, the first two eigen vectors span in most
cases the least squares plane through the centralized points of segment S
1 i
. Therefore the desired projection is obtained by ex- pressing the centralized segment points of both segments with
respect to this eigenvector basis for the first, reference segemnt: ˜
S
1 i
:= E
1 i
T −1
· S
1 i
10 ˜
S
2 j
:= E
1 i
T −1
· S
2 j
11 In this new coordinate system, the first two coordinates express
the location of a point in the segment plane of S
1 i
, while the third coordinate gives the distance of the point to the plane.
In Figure 2 the results of such projection are illustrated.
2.6 2D Segment intersection
What remains, see also Figure 2, is to find the intersection of two planar segments s
1 i
and s
2 j
. To increase efficiency, this is done in two steps. First, for both segments s
1 i
and s
2 j
, the bounding Cartesian coordinates
x
m i
, x
M i
, y
m i
, y
M i
and x
m j
, x
M j
, y
m j
, y
M j
are determined, where e.g. x
m i
denotes the minimal x-coordinate of the points in segment s
1 i
and y
M j
the maximal y-coordinate of the points in segment s
2 j
. If the bounding rectangular boxes for both segments do not intersect, the segments themselves cer-
tainly do not intersect. If the bounding boxes do intersect, the second step only has to consider the points of both segments in
the intersection.
In the second step the points from each segment in the intersec- tion are organized in a raster, with a preset width, of, say, 5 cm.
If a raster cell contains a segment point, it is given the value 1,
otherwise it gets the value 0. The intersection of both segments
is now determined from overlaying the two rasters. Only if two corresponding raster cells both have the value
1, it belongs to the intersection of the two segments. This method is flexible in the
sense that it also deals with segments with holes, corresponding e.g. to a window in a door.
This method can be further refined by using quad-trees instead of simple rasters: the quad-tree structure can be used to more
precisely define the boundary of the segments. Each raster cell containing segment points, but 4-adjacent to an empty raster cell,
is subdivided into four smaller cells, until a minimal cell size is reached. The intersection of two such quad-trees, each corre-
sponding to a segment, basically works the same as intersecting two rasters.
2.7 Implementation
The described method is relatively easy to implement. Using least squares adjustment, Teunissen, 2000, the normal of each
segment can be derived. The same least squares adjustment can be used to project candidate corresponding segments to the least
squares plane of the first segment. What remains is construct- ing a raster or quad-tree for a relative small number of possibly
intersecting segment parts
3 RESULTS AND DISCUSSION
In this section, the potential of the method described in Section 2 is illustrated for a case study. It should be noted however that
the method used to obtain the results below was an initial method that differences at details from the method described in Section 2.
3.1 Data description
For the case study we consider two point clouds representing a part of the Rotterdam Central metro station. The selected area
covers an area of 8 × 3 m of the wall on the southern side of
the tunnel, see Figure 3. During the first epoch, acquired at April 24, 2006, approximately 350 thousand points of this specific part
of the tunnel were sampled from one standpoint using the Z+F Imager 5003 phase based scanner. During the third epoch, d.d.
November 20, 2007, 3.5 million points were obtained, from two different standpoints, compare also Figure 3, left, using the Faro
LS880 phase based scanner. From one of these standpoints, two scans were acquired, resulting in three available scans in this
epoch.
Registration of individual scans was performed using the soft- ware Cyclone from Leica. Both control points and object match-
ing were used. In Epoch 1, the reported accuracy was 3 mm, in Epoch 3, it was 6 mm. Control points were additionally measured
with a total station, and both point clouds were georeferenced to the same absolute coordinate system RDNAP. The two resulting
point clouds are shown in Figure 4. The effect of grafitti on the in- tensity measurements is clearly visible in both epochs. In Epoch
3, on the right, the footbridge is visible, indicated by a red ellipse, which was not yet present in Epoch 1.
3.2 Segmentation
Point clouds from both scans were segmented into planar patches using the region growing method described in Rabbani et al.,
2006. The method requires three parameters, a number, k, of neighbors for each point, an angular threshold, α
p
, to compare local surface normals, and a distance threshold, d
p
, that considers the distance between the point at hand and a growing segment.
Here, the parameter values k = 30, α
p
= 30
◦
and d
p
= .03m were used. These setting resulted in 20 segments for Epoch 1,
and 494 segments for Epoch 3. The increase in the number of segments from Epoch 1 to Epoch 3 already indicates that more
different objects are in the scene in Epoch 3.
3.3 Corresponding segments
