um
lelled as a
geometry,
r, 3D line
reliminary
ep can be
verall roof
s). For an
tainty and
nstruction
g (Cord et
ar, 2001).
tup
netric build-
onent is a
ty, various
Yodel. The
) space to-
econstruct
rect topol-
d method-
etailed de-
ed in Sec-
steps from
nents. Ad-
e used test
steps lead-
described.
f this geo-
and refine
sults of the
y of build-
t of planar
es and the
ight lines,
oof as en-
suffices to
ar patches,
es) are ob-
would suf-
E^ ]
3D line segments
-
|
| |
| (a) ^
po EE Y seeds from
|
+ seme e)
geometric ES
reconstruction |
a
[ | seeds from
In un ) semantics (d)
r
| semantic
labelling
sr
semantic model
refinement
Lie en
Figure 1: Structure of geometric and semantic modelling
system. From the input data (a), consisting of 3D line seg-
ments, a set of seed segments (b) is chosen by virtue of
some geometric properties. In the neighbourhood of these,
preliminary geometric models consisting of planar faces in
3D space are reconstructed (c). Firstly, the semantic la-
belling performed in (e) allows to detect missing parts of
the model and thus to determine additional seed segments
(d) which in turn are used for geometric model instantia-
tion. Secondly, the semantic labels are used to improve the
overall model topology (f).
fice to use triangular patches only. However, since quad-
rangular patches are very common in building roofs, in-
cluding them is advantageous for the practical application.
The 3D line segments delineating the border of a patch will
be referred to as patch segments. The term edge is delib-
erately not used because of its widespread use in computer
vision. Besides the collection of patches constituting the
roof, the relations between the patches are of importance
and will be integrated in the model as constraints.
2.1 Patches
A roof patch is a triangular or quadrangular face in 3D
space. It can be accessed through two types of representa-
tions, which are kept in parallel. On the one hand, a para-
metric representation is provided, allowing direct inference
of quantities such as angles or lengths. Three coordinate
systems are involved in the parametrization: a world co-
ordinate system O, a patch centered coordinate system 0!
and a 2D coordinate system in the patch plane O". The re-
lation between O and O' is illustrated in Figure 2. Figures
3(a,b) show the parametrization of the triangular and quad-
rangular patch models in the patch plane coordinate system
O"'. The advantages of the chosen parametric representa-
tion are that the quantities involved have an obvious mean-
ing. Probability distributions have been obtained from a
test dataset. Additionally, this representation allows to in-
corporate symmetries between different patches of a roof
model.
In parallel to the parametric representation, a representa-
tion based on the 3D world coordinates of the corner points
Py,..., Pa of a patch is kept as well. Although this dual
representation is redundant, its importance transpires when
modelling the relations between the patches of a roof. Pos-
sible relations are given by coincidence or equality con-
straints and can hold either between coordinates or param-
eters of different patches. Consequently, the dual repre-
sentation allows to integrate diverse constraints such as
geometric symmetry or topological connectivity simulta-
neously and at the same level of complexity.
Figure 2: Patch model; global and patch centered coor-
dinate systems. T is the transformation from world co-
ordinates to the local patch coordinate system including a
translation t and a rotation around the z-axis by $1. $1
is the angle between the x’, y'-plane and the x, y-plane. z
and z' are parallel; the z^, y'-plane is horizontal. The slant
of the patch plane is given by $», which corresponds to the
rotation performed to get the 2D patch coordinate system
O". (P,) denote the corner points of the patch ordered in
mathematical positive sense.
(a) (b)
Figure 3: Patch models; patch plane coordinate system.
Q'' is the origin of the 2D coordinate system in the patch
plane. (p) denote the projections of the corner points into
the patch plane. /;; denotes the line segments given by pi
and p;. /;? and [3o are parallel to z". The inclination of the
bordering segments is denoted by o and o» respectively.
The full parametrization of a patch is given by specifying
the height ^, and the width w in case of a quadrangular
patch.
2.20 Constraints
A roof is described by its constituting patches and the re-
lations between them. These relations are modelled as
constraints between parameters or coordinates of different
patches in a roof similar to what has been done to describe
dependencies within CAD-models (Seybold et al., 1997,
Ault, 1999). Once a relation between patches has been es-
—205—
