into two categories: (1) There are not enough boundary points 
from the LiDAR data to depict the complete building outlines, 
as shown in Cases I. (2) The building outlines are not detailed 
enough, as shown in Cases II. These two categories indicate 
that boundary LiDAR points do not sufficiently match the 
building outlines. 
Case I: There are not enough boundary points from the LiDAR 
data to depict the complete building outlines. 
(1) Building with a horizontal or slanted roof plane. In these 
buildings, the roof faces are partly or fully covered by 
neighboring building or trees. 
  
  
  
  
Figure 3. Building with a horizontal or slanted roof plane. 
(2) Building with a ridge or hips. In these buildings, the 
important structural lines, like hips, are partly or fully covered 
by trees etc. 
  
  
  
  
Figure 4. Building with a ridge or hips. 
Case II: Building outlines are not detailed enough. 
(1) Two adjoining buildings with flat or slanted roof planes or 
roofs with two different ridges. In these buildings, the building 
outlines of these two buildings are drawn together with one 
outline. The border line between these two buildings is not 
drawn in the topographic map. 
  
emm 
&ssáesehpeig 
  
  
Figure 5. Two adjoining buildings with flat or slanted roof 
planes or roofs with two different ridges. 
  
(2) Some small buildings on top of a big building. In this case, 
the outlines of the small buildings are not drawn on the 
topographic map. 
  
    
  
  
  
Figure 6. Some small buildings on top of a big building. 
According to these case analyses, the tensor ellipse is available 
for checking the incorrectly reconstructed models when the 
automatic reconstruction is completed. In our experience, it is 
good enough that the radius of the tolerance circle is set to 2 
times the estimated standard deviation of the discrepancy. 
After reconstruction, some of the building models may be 
incorrect when the discrepancy between the tensor ellipse and 
the tolerance circle at the buildings is significant. In Case I-1, a 
building with a single roof plane, the building models can be 
still reconstructed correctly, since the roof planes and the 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
    
heights of the nodes can be determined by the LiDAR surface 
segments. In Case I-2, a building with multiple intersecting roof 
planes, the building model can be also reconstructed correctly 
as long as all of the structural lines of the building are extracted 
exactly or inferred correctly from the LiDAR data. The 
structural line can be still inferred correctly when a part of it is 
covered. However, if the structural line is completely covered, 
the reconstructed building model will be incorrect. A building 
with multiple non-intersecting roof planes, such as buildings 
with different heights of roof planes that are close to one 
another (Case II-1), or small structures on the roofs (Case II-2). 
The reconstructed building models are always wrong in these 
cases, since these small structures are usually not shown on the 
topographic map or the closely adjacent buildings are drawn 
with a single outline. All these incorrect cases can be observed 
in the tensor analysis stage. 
Incorrect building models may occur in the results of the 
automatic reconstruction due to either insufficient LIDAR data 
or topographic map information. Hence manual refinement of 
the reconstructed building models may be necessary after the 
automatic procedure. Since the incorrectly reconstructed 
building models can be detected according to the tensor 
analysis developed in this study, it reduce both the time and 
cost of building reconstruction. 
The quality of reconstructed building models is evaluated by 
tensor analysis. In Figure 5, a closed polygon represents a 
building. The results show that a total of 33 of 108 buildings are 
incorrect building models as shown with gray polygons in 
Figure 5. 
  
  
  
  
2 Tolerance 
Figure 7. Incorrect building models (gray polygons) in the test 
area. 
In Figure 7, a closed polygon represents a building. Our results 
have shown that in total, 41 tensor ellipses are beyond the 
tolerance circle. There are 6, 5 and 30 buildings with single roof 
planes, multiple intersecting roof planes and multiple non- 
intersecting roof planes respectively. These building models 
should be manually checked after the automatic reconstruction 
procedure. The 6 buildings with single roof planes are 
automatically reconstructed very well, as mentioned before, and