ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
component analysis is applied to the resulting image in order to 
create additional candidates for planar patches. Figure 6d shows 
the final segment label image created for one of the building 
regions from Figure 5e. The r.m.s. errors of planar adjustment 
varies between X5 cm and +15 cm for the segments 
corresponding to the "homogeneous" points. The segments 
having a r.m.s. error larger than +10 cm possibly still 
correspond to more than one roof plane. In the planar regions 
created by the analysis of the originally inconsistent points, the 
r.m.s. errors vary between 425 cm and +5 m. Some of these 
regions correspond to trees, and other regions still correspond 
to more than one roof plane. In the future, a further analysis will 
split these regions into smaller ones corresponding to even 
smaller planes in object space. This can be accomplished, e.g., 
by a height segmentation of the DSM in these regions. Table 1 
shows the distribution of the r.m.s. errors of the planar fit. 
  
  
r.m.s. error [m] Regions Pixels [96] 
0.00 - 0.05 241 30.6 
0.05 - 0.10 333 44.4 
0.10 - 0.15 96 9.3 
0.15 - 0.20 133 8.8 
0.20 - 0.50 26 0.9 
0.50 - 1.00 10 0.2 
1.00 - 2.00 14 0.5 
2.00 - 3.00 42 2.6 
3.00 — 4.00 36 1.8 
>= 4.00 15 0.9 
  
  
  
  
  
Table 1. Distribution of the r.m.s. errors of the planar fit. 
Regions: number of planar regions in the respective range of 
r.m.s. errors. Pixels: percentage of pixels in these regions 
compared to the number of all pixels in all planar regions. 
68% of the pixels in the building candidate regions are 
classified as belonging to a planar region. 
5.2 Grouping planar segments to create polyhedral models 
To derive the neighborhood relations of the planar segments, a 
Voronoi diagram based on a distance transformation of the 
segment label image has to be created (Ameri, 2000): each pixel 
inside the region of interest not yet assigned to a planar segment 
is assigned to the nearest segment. The distances of pixels from 
the nearest segment are computed by using a 3-4 chamfer mask. 
Figure 7 shows a Voronoi diagram of the segment label image 
from Figure 6d. From the Voronoi diagram, the neighborhood 
relations of the planar segments are derived, and the borders of 
the Voronoi regions can be extracted as the first estimates for 
the border polygons of the planar segments (Figure 8). 
  
Figure 7. A Voronoi diagram of the label image in Figure 6d. 
After deriving the neighborhood relations, neighboring planar 
segments have to be grouped. There are three possibilities for 
the relations of two neighboring planes (Baillard et al., 1999). 
First, they might be co-planar, which is found out by a 
statistical test applied to the plane parameters. In this case, they 
have to be merged. Second, two neighboring planes might 
intersect consistently, which is the case if the intersection line is 
close to the initial boundary. In this case, the intersection line 
has to be computed, and both region boundaries have to be 
updated to contain the intersection line. Third, if the planes do 
not intersect in a consistent way, there is a step edge, and a 
vertical wall has to be inserted at the border of these segments. 
After grouping neighboring planes, the bounding polygons of 
all enhanced planar regions have to be completed. (Moons et 
al., 1998) give a method for doing so and for regularizing the 
shape of these polygons at building corners. Finally, the planar 
polygons have to be combined to form a polyhedral model, and 
vertical walls as well as a floor have to be added to the model. 
   
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Figure 8. The roof polygons of the b 
back-projected to an aerial image. 
The tools for grouping planes and for computing intersections 
and the positions of step edges have not yet been implemented. 
Figure 9 shows a VRML visualization of a 3D model created 
from intersecting vertical prisms bounded by the borders of the 
Voronoi regions with the respective 3D roof planes. The 
structure of the roofs is correctly resembled, but the intersection 
lines of neighboring roof planes are not yet computed correctly. 
However, the visualization shows the high potential of the 
method for generating roof planes from LIDAR data. 
  
Figure 9. VRML visualization of a model created from the 
boundary polygons of the Voronoi diagram in Figure 5e. 
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