=> 
/D9 ‚Max 
Figure 7: Illustration of rotating a half-plane around a ref- 
erence line for plane instantiation. $» is the slant angle of 
the plane, measured relative to a horizontal plane. 
The next step now is to determine the outlines of pos- 
sible patches, containing the segments in the individual 
plane hypotheses. The outlines are found by computing 
the extremal points of the convex hull of the segments in 
a tentative plane. The extremal points correspond to the 
end points of the associated line segments, projected into 
the patch plane, with minimal/maximal coordinates. Then, 
the patch parameters are adjusted to contain the extremal 
points. 
At this stage of the reconstruction, different patch and plane 
configurations exist in parallel. To select the optimal con- 
figuration Utility Theory is applied. The Utility Function 
used consists of two parts. One part quantifies the reliabil- 
ity of the patch hypothesis (Scholze et al., 2001), whereas 
the second part takes into account the compatibility be- 
tween different patches. Finally, the patch and plane con- 
figuration with Maximum Expected Utility is entered into 
a preliminary roof model. (A detailed description of plane 
and patch instantiation will be published elsewhere.) 
5 ROOF RECONSTRUCTION 
The patch hypothesis generation procedure described in 
Section 4 is sufficient to reconstruct roofs whose patches 
lie on planes which contain the seed line. To complete the 
reconstruction of more complicated roofs, a semantic in- 
terpretation of already reconstructed patches is performed. 
5.1 Semantic Model Completion 
Geometry based reconstruction results in a preliminary roof 
model where patches might be missing. To locate the miss- 
ing patches the outline of the reconstruction is scrutinized. 
The key idea is now to identify patch segments on the 
outline which actually should correspond to an internal 
boundary of the roof — that is, a concave or convex joint 
of roof patches. If such segments could be identified (if 
present at all), these in turn form a set of seed segments 
which are fed into the reconstruction algorithm again (cf. Fig- 
ure 1(d)). 
More precisely, in order to assign the semantic labels from 
Equation (1) to the patch segments, the unary and binary 
attribute vectors are computed for each patch individually 
in a first step. This involves a classification of the mea- 
surements according to the attribute distributions of the 
test dataset. For the unary attributes, the classes corre- 
spond to the semantic labels given in Equation (1). Since 
the binary attributes describe relations between pairs of 
line segments, their classes are given by compatible la- 
bel combinations (e.g. (ridge;gable)). A nonparamet- 
ric classification technique, namely Linear Discriminant 
Analysis (LDA) is used to classify the unknown observa- 
tions (Fisher, 1936). During LDA, the input data (unary 
and binary attributes respectively) are transformed as to 
obtain maximal separation between the classes. Optimal 
classification results would be achieved, when the deci- 
sion boundaries perfectly separate the classes. However, 
the clusters corresponding to the different semantic labels 
in this space are overlapping for both unary and binary at- 
tributes. This makes an unambiguous label assignment im- 
possible. This is overcome by an iterative procedure which 
determines the labels for all segments in a patch in such a 
way, that the entire label assignment (per patch) has max- 
imal probability, exploiting unary and binary attributes si- 
multaneously (Christmas et al., 1995). 
5.2 Semantic Model Refinement 
It is desirable to have a topologically correct, that is, a 
point-wise connected reconstruction of the roof. There- 
fore, corner points of patches with compatible semantic la- 
bels are forced to coincide in a final gluing step. Thus, in 
this step, topological correctness is preferred over geomet- 
ric precision. Due to coincidence constraints introduced 
into the model, the gluing step does not open gaps at other 
locations in the model. 
6 RESULTS AND CONCLUSION 
The presented results are obtained using a state-of-the-art 
dataset, produced by Eurosense Belfotop n.v. The image 
characteristics are: 1:4000 image scale and geometrically 
accurate film scanning with 20 x 20 um? pixel size corre- 
sponding to 8x8 cm? on the ground. The 3D line segments 
are obtained using three overlapping views. The precise 
sensor orientation is known. To emphasise the quality of 
the reconstructed roof geometry no texture mapping is ap- 
plied. 
Figure 8(a) shows a reconstruction result for the upper left 
building in Figure 5(c), which was completely reconstructed 
from its seed (here: ridge) line by one pass of the recon- 
struction algorithm. The reconstruction results are detailed 
and topologically correct. For instance the small differ- 
ence in the slope of the two patches on the right side of 
the roof has been correctly detected. Figure 8(b) shows 
the reconstruction of the roof of the corner building in Fig- 
ure 5(c). The triangular patches on the front side do not 
lie in a plane given by the seed line. However, driven by 
the semantic labels attributed to the partially reconstructed 
roof, the missing patches could be successfully found. 
Concluding, in this paper we have presented a novel ap- 
proach for the probabilistic modelling of building roofs. 
The proposed theoretical foundation and technical imple- 
mentation leads to stable and reliable results. Future work 
will focus on exploiting the knowledge available in the 
form of test datasets to a broader extent, with the goal to 
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