2010
In: Paparoditis N., Pierrot-Deseilligny M„ Mallet C., Tournaire O. (Eds), IAPRS, Vol. XXXVIII, Part 3A - Saint-Mandé, France, September 1-3, 2010
The cuboid is characterized as “broad and thin”. Parallelism of
the parameters and the coordinate axis is given by the model as
sumption of straight stairs that are perpendicular to the facade.
Furthermore, the centroid of the range query is located in the
mean of the two samples. Figure 4 illustrates the sampling for
the classification of stairs.
Figure 4: 2D projection: Two sampled points (dark green
spheres) and predicted points (dark purple) with centroid (light
green sphere) in between.
For windows the constraints of the pre-filtering are not sufficient
to ensure that most filtered samples are valid. Therefore, addi
tional verifications are applied to the sample before the predic
tion of supporters. The following feature was identified by its
information gain as the most evident one (out of 40 features) for
embrasures: The difference of y-coordinates of the median of two
point sets £ and C is greater than 31cm. If one of the sampled
points is on the left embrasure it defines the centroid of £. The
centroid of C is of the same y- and z-coordinate but translated
by —50cm in x-direction, i.e. to the left. £ and C are received
from the kd-tree by a range query with the centroid given by the
sampled point and the cuboid.
lx = Gsensor
l y = 2* max(depth W i n dow)
l z = 0.3 * median(height W indow)-
The factor 0.3 was determined empirically. Thus, £ contains
points on embrasures and C contains points on the facade next
to the embrasure. If the sample is accepted £ is the prediction
of the left embrasure. The same process is applied to the second
sample point, i.e. the right embrasure, analogously.
3.3 Estimation of boundaries
The most likely sample that has been selected during the pre
vious phase of classification only gives a precise description of
some parameters of the object. Therefore the boundaries of the
object are partly estimated. Windows, for instance, are so far
given by one point on each embrasure. Thus, their width is given
accurately, however, their height is unknown yet. Since their sur
face is given by vertical half-planes we have to estimate the top
and bottom of the objects. Analogously the width of stairs is un
known so far. Due to the regular shape of man-made objects the
boundaries are defined by one reference point and a set of shape
parameters. To estimate faces given by rectangular polygons we
apply clustering algorithms to the most evident features inferred
from the decision trees.
The two sample points of the window model and the prediction
of each embrasure, for example, suffice to define thresholds for
the x- and y-coordinates. On one hand, the x-coordinate of the
median point of the predictions defines the right or left border of
the window, whereas the front and the back boundary is approxi
mately given by the minimum and maximum y-coordinate of the
predictions. Since no points are observed on windowsills due to
occlusion, the estimation of the top and bottom boundary cannot
be done analogously to the estimation of left and right embrasure.
Therefore it is deduced from the clustering of another evident
feature, namely the standard deviation of y-coordinates <j y along
the embrasure in upwards and downwards direction. Similar to
the lines in sweep line algorithm (Shamos and Hoey (1976)) we
sweep the cuboids of spatial queries iteratively along the vertical
line through the points pj (J € {1,2}) of the sample. The starting
point and the end are given by
Zbottom, j — Zj 71 ledian ( flCig hi ■window )
ztop,j —— Zj ! median(height W i n dow)•
For each result set rij the standard deviation of the y-value
is calculated. The o y of cuboids that totally cover the embrasure
is much higher than the a y of cuboids that cover facades even if
they are structured with ornaments. Thus, clustering of cr y de
termines the top and bottom of windows. Figure 5 illustrates the
sweeping of range queries and the resulting standard deviation of
y-coordinates.
Figure 5: Clustering for the estimation of the boundaries: cr y of
vertically aligned spatial queries of left (purple) and right (green)
embrasure. The bright area in the centre indicates the estimated
z-range of the window.
4 RESULTS
We implemented the presented methods for straight stairs and
windows in MATLAB and applied them to nine 3D point clouds
of buildings of the district “Sudstadt” in Bonn, Germany. This
particular district has been chosen due to its challenging Wil-
helmian style buildings of which the facades show sophisticated
structures. The data sets that were used for testing were not part
of the ground truth database and were not used for the estimation
of probability density functions. However, the data sets of which
the ground truth database was constructed are also located in the
same district. The results are based on the independent estima
tion of single windows. The estimation was iterated until no more
windows were found, i.e. the remaining set of points was smaller
than the average number of points of the windows classified so far
or a maximum number of iterations was reached. Due to this iter
ative process the resulting windows have slightly different shape,
particularly for the depth.
To optimize the computing time and to show the robustness of our
method, we operated on subsets of 100.000 points of the original
point clouds. The accuracy of the classification and reconstruc
tion is given in table 1. Three of the 13 not detected windows
were subsumed by a too large reconstruction that covered two
small windows that are close together (“2inl”, cf. figure 6 top
left and top right windows). Therefore, one of them was counted
as not detected, and the other one had a large deviation of width.
The reason for the six erroneously detected windows was the re
construction of one ground truth window by two reconstructed
windows on top of each other (see figure 6 bottom right win
dow). This typically happened at thick window crosses that were
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