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ISPRS Commission III, Vol.34, Part 3A ,Photogrammetric Computer Vision“, Graz, 2002 
  
A useful choice for pu is the Lebesgue measure dzdy times 
the characteristic function xr of the region of interest 
(Fig. 1). If y 2 dzdyxr, we have 
d, (A, By = / SB OE NE 
r 
(8) 
We write dr (A, B) instead of d, (A, B). With c; :— a; — 
b;, equation (8) can be written as 
[1 [= [vy Co 
dr A, BY = (00,6 eode os dels dla 
Sy fut C2 
(9) 
This is a quadratic form, whose matrix depends on the do- 
main of integration I for the integrals (where we omitted 
the symbols dxdy for brevity). 
Another possibility is that j, equals the sum of several point 
masses at points (x;,y;). In this case we have 
dA BY = 2 /((ao — bo) + (a, — b1 )x; + (az — ba)y;)”. 
| (10) 
It can easily be shown (Pottmann Wallner, 2001), that the 
distance d,, defines a Euclidean metric in the set of planes 
of type (6), if and only if p is not concentrated in a straight 
line. 
In this way, approximation problems in the set of planes 
are transformed into approximation problems in the set of 
points in Euclidean 3-space, whose metric is based on d ,. 
The introduced metric depends on the choice of the refer- 
ence direction, which we identified with the z-axis of the 
underlying coordinate system. With decreasing angle be- 
tween the considered planes and the reference direction, 
the distance d,, is becoming geometrically meaningless. 
Therefore, it might be necessary to use different reference 
directions in order to fully cover the space of planes ap- 
propriately. This results in different local mappings of the 
space of planes to affine 3-space and in different Euclidean 
metrics. For the application we are dealing with in the next 
section, this strategy is sufficient. 
In a recent paper (Peternell Pottmann, 2001), the metric in 
the space of planes is investigated further and new appli- 
cations of this concept are presented. There, it is also dis- 
cussed how to measure distances for all planes; however, 
we do no longer obtain a Euclidean metric in the space of 
planes then. 
3.1 Application to the detection and reconstruction of 
planar faces in point clouds 
In the following we are interested in the detection and re- 
construction of planar faces in point clouds. A known so- 
lution to this problem uses the Gaussian sphere (Varady et 
al., 1998). For each data point, one locally fits a plane to 
the point and its nearest neighbors. The unit normal vec- 
tors of those planes describe points on the Gaussian unit 
  
Figure 2: Top: Image points of planes of regression. 
Bottom: Data points of roof points and the reconstructed 
building. 
sphere. Points from a planar face will give rise to nearly 
identical local regression planes and thus to a point clus- 
ter on the Gaussian sphere. The detection of planar faces 
is reduced to the detection of point clusters on the Gaus- 
sian sphere. An obvious disadvantage of this approach is 
that we lose information when going from the regression 
plane to the Gaussian sphere. Parallel planar faces cannot 
be separated directly on the Gaussian sphere. Moreover, 
the loss of information is critical in case of noisy data and 
complicated objects. 
These drawbacks can be overcome if we use approxima- 
tion in the space of planes: The local regression planes 
determine points in dual space. We use the Euclidean 
distance introduced there to detect point clusters. For 
each cluster, we then determine those original data points, 
which are close to the regression planes that determine the 
cluster. These data points belong to a planar region. Its 
plane can easily be computed as a regression plane. 
We provide an example from the reconstruction of 3D ur- 
ban models from airborne laser scanner data. This is cur- 
rently an important research topic in geodesy and pho- 
togrammetry. We do not review the literature on this prob- 
lem here, but just point to the very recent paper (Vossel- 
mann Dijkman, 2001). There, an extension of the Hough 
transform to 3 dimensions is used to recognize planar faces 
of the buildings' roofs. This approach has some similarity 
to ours, since the Hough transform is also a special duality. 
However, in our method the metric in the space of planes 
plays a crucial role and thus it can be expected that the 
present approach is more reliable. 
The example displayed in Fig. 2 shows both the point 
cluster in dual space and the reconstructed building. Our 
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