Some of the inferred 20 constraints are redundant. The
greedy algorithm identified a set of 15 independent con-
straints which are depicted together with the adjusted, i.e.,
constrained, boundary representation in Figure 6. Now the
ridge lines and the eaves are horizontal and the building’s
outline is rectangular. Of course, the detection and enforce-
ment of further constraints such as identical slopes for the
roof areas is conceivable.
Conclusions and Outlook
We derived analytical expressions for the uncertainty of
planes corresponding to the planar patches bounding a poly-
hedron. This paves the way to stochastic geometric reasoning
for generic city models, i.e., the detection and enforcement of
man-made structures and the corresponding constraints such
as orthogonality, parallelism, or identity, e.g., for groups of
buildings.
The estimation of a plane’s uncertainty based on a given
point cloud yields a normal equation matrix. Assuming a
continuous distribution function for points defining a planar
patch, we interpret the entries in the normal equation matrix
as moments for arbitrarily shaped
2
polygons. Furthermore,
by considering the signed areas of polygons, we are able to
cope with multiply-connected polygons, i.e., polygons with
interior boundaries defining holes.
As expected, the uncertainty of a plane corresponding to
a polygon depends on the polygon’s shape, on its area
A
, on
the sampling distance
D
of the equally distributed virtual
sampling points, and on the assumed uncertainty
σ
for the co-
ordinates of the sampling points. But only the product
σ
D
has
to be specified and the plane’s uncertainty scales with these
two factors. For the successful application of the approach, the
specification of an appropriate, plausible weight
w
= 1/
σ
2
for
the sampling point coordinates and the specification of a sam-
pling distance
D
is crucial. Thus, future investigations should
try to estimate unknown variance factors, too.
For the application at hand, we performed a stochastic
geometric reasoning followed by an adjustment of the build-
ing faces. By considering just the relations
orthogonality
and
identity
, already remarkable results are obtained for a real
data set: the building’s outline becomes rectangular and the
building’s eaves and ridge lines become horizontal.
The method for deriving the uncertainty of geometric
entities assuming pre-specified densities of observations can
easily be applied to other estimation problems. For standard
configurations, the resulting algebraic expressions not only
give insight into the structure of the estimation problem, but
can be used to derive otherwise unknown information about
the precision of geometric entities.
Figure 5. Two views of the boundary representation after merging adjacent faces with identical planes (13 faces). 20
orthogonality constraints have been detected.
Figure 6. Two views of the boundary representation obtained by adjustment of the planes with 15 enforced orthogonality
constraints. As a result the ridge lines and the eaves are horizontal and the building’s outline features right angles.
2. But not self-intersecting.
400
June 2018
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING