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In order to register the boundary LiDAR points of buildings to 
the corresponding outline segments, a 2D similarity trans- 
formation is adopted: 
ehh m 
where (x,y) are the horizontal coordinates of a LIDAR point in 
a local coordinate system; (x',y') are the new coordinates in the 
map system after the transformation and (r,s) are the shifts of 
the origin. w = m cosa and u = m sina, where a is the rotation 
angle, and m is the scale factor. 
Assume that boundary LiDAR points with the new coordinates 
should fall exactly on an outline segment L: ax'tby'+c=0. 
Substitution of Eq. (5), followed by introducing measurement 
errors in the coordinates of the boundary points, leads to: 
[a(x +vy)+bly+v, w+ bx +v,)+ aly * v, ))u 
+ar+bs+c=0 , 
(5) 
where (a,b,c) can be calculated from the corresponding polygon 
data of an outline segment, and the residuals v, and v, represent 
two components of the distance vector v from a LiDAR point to 
the corresponding outline segment. 
The registration process is performed using the iterative RLS 
method (Klein and Foerstner 1984). The objective function in 
RLS consists of the sum of squares of the distances from 
boundary points to building outlines on a local xy-plane. In 
each iteration of the RLS adjustment, the corresponding outline 
segment for each boundary LiDAR point located now by new 
transformed coordinates must be re-determined. The procedure 
proceeds until the estimated standard deviation of the distances 
is convergent. 
3.2 Tensor analysis of residuals 
A resultant tensor T, is used to analyze the registration result of 
each building (You and Lin 201 1a). 
T, = ST = Se 
i=l , (6) 
where the residual tensor T; for each boundary point is obtained 
from the estimated residual vector v; =[v, V,, T. , and then the 
residual tensors of all boundary points of a building are added 
together to form a resultant tensor T, . To reduce the influence 
of the number of the points, the resultant tensor is normalized 
by dividing the number of the boundary points in this study. 
Based on the fact that the boundary points of a building 
surround the closed building polygon, it is evident that the 
resultant tensor T, is positive semi-definite and a rank-2 tensor. 
The eigenvalues (Amax, Amin) and eigenvectors (emax, Emin) Of the 
normalized resultant tensor can be derived by tensor 
decomposition as follows: 
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 
    
  
   
    
  
   
    
    
   
   
  
   
   
   
    
    
   
  
   
  
    
   
  
  
   
  
   
   
    
   
   
  
    
   
   
  
  
   
  
  
   
   
   
  
   
  
  
   
   
   
   
      
; (7) 
where Amax>Amin>0. This tensor can be geometrically visualized 
as an ellipse [22]. The eigenvectors represent the orientation of 
the ellipse, and the root square values of the eigenvalues 
represent the lengths of the principal axes. 
After registration, the tensor ellipse of each building polygon 
can be determined. If the tensor ellipse is beyond the tolerance 
circle, it means that the boundary LiDAR points are not 
sufficient to match the building outlines. This implies that these 
discrepancies may influence the results of the reconstruction. 
Therefore, it is possible to identify which building models are 
not reconstructed well using residual tensor analysis. 
3.3 Building model reconstruction 
After registration, an automatic reconstruction of 3D building 
models is applied. In this procedure, the height of each building 
outline node can be determined by the plane equation of a 
LiDAR surface segment, and then the structural lines derived 
from LiDAR data and the building outlines are automatically 
connected according to following rules (Lin et al. 2010): 
1. The 3D structural lines projected on the local xy- 
plane should be first extended to the boundary lines 
when they are shorter than they should be. 
2. If the intersection point of a structural line and a 
boundary line is near a node point within a small 
region, the structural line is directly connected to the 
node point (case A in Figure 3). 
3. If the height of a structural line at the intersection 
point is not different from the height of the boundary 
line, the structural line is directly connected to the 
boundary line and a new node of the boundary line is 
added (case B in Figure 3). 
4. If the height of a structural line at the intersection 
point is significantly different from the height of the 
boundary line, two new additional structural lines 
may be needed(case C in Figure 3). 
Then a 3D building model can be reconstructed. 
4. EXPERIMENT AND ANALYSIS 
An airborne LiDAR dataset for a 350 X 500 m? experimental 
area was acquired by an Optech ALTM 30/70. The flying 
height for the laser scanning was 500 m AGL. The average 
LiDAR point density was 6 pts/m?. The horizontal and vertical 
precision was about 25 cm and 15 cm, respectively. This dataset 
was referred to Taiwan geodetic datum 1997.0. The topographic 
map with a scale of 1:1000 for this area was produced from 
aerial images and is based on Taiwan geodetic datum 1967. 
After registration, a resultant tensor can be determined to 
analyze the registration result. If the tensor ellipse is beyond the 
tolerance circle, it means the discrepancy between the boundary 
points and the outlines of that building is obvious. In an actual 
experimental dataset, we illustrate some cases as mentioned 
above and present how the resultant tensor analysis can identify 
the incorrect reconstruction model. These cases can be divided