high reso- 
rban areas. 
he images 
v a 3D re- 
calibration 
itting, cor- 
lereo pairs. 
olar geom- 
chromatic 
the match- 
:s (Scholze 
btained by 
e segments 
illy incom- 
; are purged 
chromatic 
views. If a 
, its 3D po- 
to enhance 
D line seg- 
igure 5(c)) 
s described 
  
(c) 
air used for 
3D line seg- 
rements 
nts is a key 
different se- 
lished. The 
five possible labels form the set ©: 
1 
0 { w! ) = ridge, qO — gutter, 
3 
wi ) — gable, w® — convex, 
w® — concave (1) 
The names of the labels are chosen to be coherent, although 
they should not be taken literally. For example, a gutter 
segment just corresponds to the lower boundary of a patch, 
no matter if there is a gutter in the scene or not. Figure 
6 gives an overview. For convenience each semantic label 
(e.g. gutter) is represented by a variable w(*), where 
i encodes the actual label. In order to assign the seman- 
tic labels to the segments in a patch, geometric measure- 
ments are used. Measurements characterizing individual 
segments u; (unary attributes) and measurements between 
adjacent segments b;; (binary attributes) are distinguished. 
The actual measurements of 
1) 
ut length of segment 2 
ue slant angle of segment i relative to a 
horizontal plane 
po angle enclosed by adjacent (coplanar) 
segments ¢ and 7 
i : (signed) mid point height difference of 
adjacent segments ¢ and j (2) 
form the components of the attribute vectors u; = (ut? us”) 
1 2 
and bi; = (0,67). 
Ridge 
Gable 
  
Gutter 
Figure 6: Functional parts of a roof and their semantic la- 
bels. The label ridge is generally used for the upper bound- 
ary of a patch, gutter for the lower one. The boundaries of 
a patch are either labelled gable, convex connection or con- 
cave connection depending on the neighbourhood. 
3.3 Test Dataset 
A semi-automatically reconstructed test dataset has been 
used to learn the statistics of the geometric measurements. 
The dataset shows urban and sub-urban areas with four- 
fold overlap at an image scale of approximately 1:5000 
(IGP, 2001). Nearly 150 roofs, consisting of about 80 tri- 
angular and 270 quadrangular patches have been labelled 
manually using the semantic labels from Equation (1). Ad- 
ditionally to the semantic labels, the values of the geomet- 
ric measurements (unary and binary attributes) have been 
recorded. 
3.4 Initial Semantic Interpretation 
As will be described in the next Section, the patch recon- 
struction algorithm makes use of seed line segments for its 
search of patch hypotheses. Given the set of reconstructed 
3D line segments, a set of seed segments has to be derived 
which is used for the first pass of the reconstruction algo- 
rithm (cf. Figure 1(b)). In order to invoke the appropriate 
instantiation procedure, the label of the seed segment in a 
roof should be known in advance. Hence, the segments 
have to be attributed semantic labels from Equation (1). 
Of course, at this stage it is impossible to unambiguously 
assign a label to an arbitrary 3D line segment. Neverthe- 
less, class membership probabilities can be derived. Us- 
ing the distribution of only the unary attributes (line seg- 
ment length and slant) of the test dataset, the probability 
distributions for the respective labels are determined. For 
each class, the algorithm creates a list of seed line segments 
sorted by probability. 
4 PATCH RECONSTRUCTION 
Geometric patch reconstruction poses a twofold task. First, 
planar 3D line segment configuration corresponding to pla- 
nar roof patches are detected. Second, the outlines of patches, 
corresponding to the actual roof faces lying in these planes, 
have to be determined. 
4.1 Probabilistic Plane Detection 
For plane detection, the general approach is to rotate a half 
plane around a seed line segment, e. g. a tentative ridge line 
(cf. Figure 7). For each inclination of the half plane, the 
line segments in its neighbourhood which approximately 
lie in this plane are collected. To determine the number 
of planes and their positions, a Bayesian model selection 
procedure is applied. The plane model selection procedure 
maximizes the posterior probability 
p(m|x) — max (3) 
with x being the set of neighbouring 3D line segments 
an m the actual plane model. The plane model describes 
the number, positions and orientations of planes currently 
considered. Model selection is done taking into account 
model complexity. That is, a more complex model (con- 
taining more planes) may well describe the data better; but 
might also tend to over-fit the data. Consequently, a sim- 
pler model should be preferred if the more complex one 
does not describe the observations significantly better. This 
qualitative statement is quantified in form of an Ockham 
factor. A detailed explanation of the procedure is available 
in (Scholze, 2002). The result is a small number (usually 
1-4) of sets of plane hypotheses in the neighbourhood of 
a seed line together with their probabilities. The different 
sets of plane hypotheses correspond to different interpreta- 
tions of the scene. 
4.2 Maximum Expected Utility Patch Reconstruction 
During the Bayesian plane selection procedure, 3D line 
segments are associated with different plane hypotheses. 
—207— 
ec ES SES